CINXE.COM

<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Welfare Approximation in Additively Separable Hedonic Games</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.06017v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S1" title="In Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S2" title="In Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Related work</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S3" title="In Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Preliminaries</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4" title="In Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Deterministic Games</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.SS1" title="In 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Welfare Inapproximability for Restricted Valuations</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.SS2" title="In 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Logarithmic Approximation for Nonnegative Total Value</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5" title="In Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Beyond Worst-Case Analysis</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.SS1" title="In 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1 </span>Erdős-Rényi Graphs</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.SS2" title="In 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2 </span>Random Multipartite Graphs</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.SS2.SSS1" title="In 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2.1 </span>Low Perturbation Regime for Random Turán Graphs</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.SS2.SSS2" title="In 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2.2 </span>High Perturbation Regime for Random Turán Graphs</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.SS3" title="In 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.3 </span>Balanced Multipartite Graphs</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.SS3.SSS1" title="In 5.3 Balanced Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.3.1 </span>Low Perturbation Regime for Random Balanced Multipartite Graphs</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.SS3.SSS2" title="In 5.3 Balanced Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.3.2 </span>High Perturbation Regime for Random Balanced Multipartite Graphs</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S6" title="In Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Conclusion</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Welfare Approximation in Additively Separable Hedonic Games</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Martin Bullinger </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">University of Oxford </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Vaggos Chatziafratis </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">University of California, Santa Cruz </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Parnian Shahkar </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">University of California, Irvine </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id3.3">Partitioning a set of <math alttext="n" class="ltx_Math" display="inline" id="id1.1.m1.1"><semantics id="id1.1.m1.1a"><mi id="id1.1.m1.1.1" xref="id1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="id1.1.m1.1b"><ci id="id1.1.m1.1.1.cmml" xref="id1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="id1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="id1.1.m1.1d">italic_n</annotation></semantics></math> items or agents while maximizing the value of the partition is a fundamental algorithmic task. We study this problem in the specific setting of maximizing social welfare in additively separable hedonic games. Unfortunately, this task faces strong computational boundaries: Extending previous results, we show that approximating welfare by a factor of <math alttext="n^{1-\varepsilon}" class="ltx_Math" display="inline" id="id2.2.m2.1"><semantics id="id2.2.m2.1a"><msup id="id2.2.m2.1.1" xref="id2.2.m2.1.1.cmml"><mi id="id2.2.m2.1.1.2" xref="id2.2.m2.1.1.2.cmml">n</mi><mrow id="id2.2.m2.1.1.3" xref="id2.2.m2.1.1.3.cmml"><mn id="id2.2.m2.1.1.3.2" xref="id2.2.m2.1.1.3.2.cmml">1</mn><mo id="id2.2.m2.1.1.3.1" xref="id2.2.m2.1.1.3.1.cmml">−</mo><mi id="id2.2.m2.1.1.3.3" xref="id2.2.m2.1.1.3.3.cmml">ε</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="id2.2.m2.1b"><apply id="id2.2.m2.1.1.cmml" xref="id2.2.m2.1.1"><csymbol cd="ambiguous" id="id2.2.m2.1.1.1.cmml" xref="id2.2.m2.1.1">superscript</csymbol><ci id="id2.2.m2.1.1.2.cmml" xref="id2.2.m2.1.1.2">𝑛</ci><apply id="id2.2.m2.1.1.3.cmml" xref="id2.2.m2.1.1.3"><minus id="id2.2.m2.1.1.3.1.cmml" xref="id2.2.m2.1.1.3.1"></minus><cn id="id2.2.m2.1.1.3.2.cmml" type="integer" xref="id2.2.m2.1.1.3.2">1</cn><ci id="id2.2.m2.1.1.3.3.cmml" xref="id2.2.m2.1.1.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id2.2.m2.1c">n^{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="id2.2.m2.1d">italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT</annotation></semantics></math> is <span class="ltx_ERROR undefined" id="id3.3.1">\NP</span>-hard, even for severely restricted weights. However, we can obtain a randomized <math alttext="\mathcal{O}(\log n)" class="ltx_Math" display="inline" id="id3.3.m3.1"><semantics id="id3.3.m3.1a"><mrow id="id3.3.m3.1.1" xref="id3.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="id3.3.m3.1.1.3" xref="id3.3.m3.1.1.3.cmml">𝒪</mi><mo id="id3.3.m3.1.1.2" xref="id3.3.m3.1.1.2.cmml">⁢</mo><mrow id="id3.3.m3.1.1.1.1" xref="id3.3.m3.1.1.1.1.1.cmml"><mo id="id3.3.m3.1.1.1.1.2" stretchy="false" xref="id3.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="id3.3.m3.1.1.1.1.1" xref="id3.3.m3.1.1.1.1.1.cmml"><mi id="id3.3.m3.1.1.1.1.1.1" xref="id3.3.m3.1.1.1.1.1.1.cmml">log</mi><mo id="id3.3.m3.1.1.1.1.1a" lspace="0.167em" xref="id3.3.m3.1.1.1.1.1.cmml">⁡</mo><mi id="id3.3.m3.1.1.1.1.1.2" xref="id3.3.m3.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="id3.3.m3.1.1.1.1.3" stretchy="false" xref="id3.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id3.3.m3.1b"><apply id="id3.3.m3.1.1.cmml" xref="id3.3.m3.1.1"><times id="id3.3.m3.1.1.2.cmml" xref="id3.3.m3.1.1.2"></times><ci id="id3.3.m3.1.1.3.cmml" xref="id3.3.m3.1.1.3">𝒪</ci><apply id="id3.3.m3.1.1.1.1.1.cmml" xref="id3.3.m3.1.1.1.1"><log id="id3.3.m3.1.1.1.1.1.1.cmml" xref="id3.3.m3.1.1.1.1.1.1"></log><ci id="id3.3.m3.1.1.1.1.1.2.cmml" xref="id3.3.m3.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id3.3.m3.1c">\mathcal{O}(\log n)</annotation><annotation encoding="application/x-llamapun" id="id3.3.m3.1d">caligraphic_O ( roman_log italic_n )</annotation></semantics></math>-approximation on instances for which the sum of input valuations is nonnegative. Finally, we study two stochastic models of aversion-to-enemies games, where the weights are derived from Erdős-Rényi or multipartite graphs. We obtain constant-factor and logarithmic-factor approximations with high probability.</p> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Partitioning a set of items or agents, say humans or machines, is a fundamental problem that has been studied across many disciplines such as computer science, economics, or mathematics. For instance, it is relevant in the context of clustering, an important task in machine learning with far-reaching applications like image segmentation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx22" title="">DMC15</a>]</cite>, or for community detection, which helps in understanding networks, e.g., of societies or physical systems <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx32" title="">New04</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Our paper takes a game-theoretic perspective and considers the prominent model of <em class="ltx_emph ltx_font_italic" id="S1.p2.1.1">additively separable hedonic games</em> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx10" title="">BJ02</a>]</cite>. We assume that there is a set of agents that has to be partitioned into coalitions and agents have preferences over the coalitions that they are part of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx21" title="">DG80</a>]</cite>. Preferences are given by a weighted graph, where the agents are the vertices and the edge weights encode the valuation between agents. The utility of an agent for a coalition is the sum of weights of edges towards members of this coalition. This class of games is quite expressive and contains more structured subclasses of games. For instance, an agent might divide the other agents into friends and enemies and could simply try to maximize the number of friends within their coalition while minimizing the number of enemies. A priority between these two objectives can be captured by the exact edge weights: for example, if there is a large negative weight for enemies and a small positive weight for friends, then minimizing enemies is much more important than maximizing friends, as conceptualized in so-called <em class="ltx_emph ltx_font_italic" id="S1.p2.1.2">aversion-to-enemies games</em> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx19" title="">DBHS06</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.2">A fundamental quantity for evaluating a possible output is its <em class="ltx_emph ltx_font_italic" id="S1.p3.2.1">social welfare</em> (also called utilitarian welfare) which is the sum of all agents utilities. Unfortunately, maximizing this quantity faces significant computational boundaries. Aziz et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx1" title="">ABS13</a>]</cite> show that it is <span class="ltx_ERROR undefined" id="S1.p3.2.2">\NP</span>-hard to maximize, and, even worse, approximating maximum welfare by a factor of at least <math alttext="n^{1-\varepsilon}" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><msup id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml"><mi id="S1.p3.1.m1.1.1.2" xref="S1.p3.1.m1.1.1.2.cmml">n</mi><mrow id="S1.p3.1.m1.1.1.3" xref="S1.p3.1.m1.1.1.3.cmml"><mn id="S1.p3.1.m1.1.1.3.2" xref="S1.p3.1.m1.1.1.3.2.cmml">1</mn><mo id="S1.p3.1.m1.1.1.3.1" xref="S1.p3.1.m1.1.1.3.1.cmml">−</mo><mi id="S1.p3.1.m1.1.1.3.3" xref="S1.p3.1.m1.1.1.3.3.cmml">ε</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><apply id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p3.1.m1.1.1.1.cmml" xref="S1.p3.1.m1.1.1">superscript</csymbol><ci id="S1.p3.1.m1.1.1.2.cmml" xref="S1.p3.1.m1.1.1.2">𝑛</ci><apply id="S1.p3.1.m1.1.1.3.cmml" xref="S1.p3.1.m1.1.1.3"><minus id="S1.p3.1.m1.1.1.3.1.cmml" xref="S1.p3.1.m1.1.1.3.1"></minus><cn id="S1.p3.1.m1.1.1.3.2.cmml" type="integer" xref="S1.p3.1.m1.1.1.3.2">1</cn><ci id="S1.p3.1.m1.1.1.3.3.cmml" xref="S1.p3.1.m1.1.1.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">n^{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT</annotation></semantics></math> is <span class="ltx_ERROR undefined" id="S1.p3.2.3">\NP</span>-hard for any <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mrow id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml"><mi id="S1.p3.2.m2.1.1.2" xref="S1.p3.2.m2.1.1.2.cmml">ε</mi><mo id="S1.p3.2.m2.1.1.1" xref="S1.p3.2.m2.1.1.1.cmml">></mo><mn id="S1.p3.2.m2.1.1.3" xref="S1.p3.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><apply id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1"><gt id="S1.p3.2.m2.1.1.1.cmml" xref="S1.p3.2.m2.1.1.1"></gt><ci id="S1.p3.2.m2.1.1.2.cmml" xref="S1.p3.2.m2.1.1.2">𝜀</ci><cn id="S1.p3.2.m2.1.1.3.cmml" type="integer" xref="S1.p3.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">italic_ε > 0</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx26" title="">FKV22</a>]</cite>. Our paper aims at circumventing this computational boundary.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.6">First, we investigate the inapproximability of maximum welfare. Notably, the result of Flammini et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx26" title="">FKV22</a>]</cite> is for aversion-to-enemies games, which use valuations <math alttext="-n" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mrow id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml"><mo id="S1.p4.1.m1.1.1a" xref="S1.p4.1.m1.1.1.cmml">−</mo><mi id="S1.p4.1.m1.1.1.2" xref="S1.p4.1.m1.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><apply id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1"><minus id="S1.p4.1.m1.1.1.1.cmml" xref="S1.p4.1.m1.1.1"></minus><ci id="S1.p4.1.m1.1.1.2.cmml" xref="S1.p4.1.m1.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">- italic_n</annotation></semantics></math> and <math alttext="1" class="ltx_Math" display="inline" id="S1.p4.2.m2.1"><semantics id="S1.p4.2.m2.1a"><mn id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><cn id="S1.p4.2.m2.1.1.cmml" type="integer" xref="S1.p4.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.1d">1</annotation></semantics></math>, i.e., the negative valuation is dependent on the number of agents <math alttext="n" class="ltx_Math" display="inline" id="S1.p4.3.m3.1"><semantics id="S1.p4.3.m3.1a"><mi id="S1.p4.3.m3.1.1" xref="S1.p4.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.p4.3.m3.1b"><ci id="S1.p4.3.m3.1.1.cmml" xref="S1.p4.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.p4.3.m3.1d">italic_n</annotation></semantics></math>. We complement this by showing an <math alttext="n^{1-\varepsilon}" class="ltx_Math" display="inline" id="S1.p4.4.m4.1"><semantics id="S1.p4.4.m4.1a"><msup id="S1.p4.4.m4.1.1" xref="S1.p4.4.m4.1.1.cmml"><mi id="S1.p4.4.m4.1.1.2" xref="S1.p4.4.m4.1.1.2.cmml">n</mi><mrow id="S1.p4.4.m4.1.1.3" xref="S1.p4.4.m4.1.1.3.cmml"><mn id="S1.p4.4.m4.1.1.3.2" xref="S1.p4.4.m4.1.1.3.2.cmml">1</mn><mo id="S1.p4.4.m4.1.1.3.1" xref="S1.p4.4.m4.1.1.3.1.cmml">−</mo><mi id="S1.p4.4.m4.1.1.3.3" xref="S1.p4.4.m4.1.1.3.3.cmml">ε</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.p4.4.m4.1b"><apply id="S1.p4.4.m4.1.1.cmml" xref="S1.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S1.p4.4.m4.1.1.1.cmml" xref="S1.p4.4.m4.1.1">superscript</csymbol><ci id="S1.p4.4.m4.1.1.2.cmml" xref="S1.p4.4.m4.1.1.2">𝑛</ci><apply id="S1.p4.4.m4.1.1.3.cmml" xref="S1.p4.4.m4.1.1.3"><minus id="S1.p4.4.m4.1.1.3.1.cmml" xref="S1.p4.4.m4.1.1.3.1"></minus><cn id="S1.p4.4.m4.1.1.3.2.cmml" type="integer" xref="S1.p4.4.m4.1.1.3.2">1</cn><ci id="S1.p4.4.m4.1.1.3.3.cmml" xref="S1.p4.4.m4.1.1.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.4.m4.1c">n^{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.4.m4.1d">italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT</annotation></semantics></math>-inapproximability result on instances in which the valuations are restricted to <math alttext="\{-v^{-},0,1\}" class="ltx_Math" display="inline" id="S1.p4.5.m5.3"><semantics id="S1.p4.5.m5.3a"><mrow id="S1.p4.5.m5.3.3.1" xref="S1.p4.5.m5.3.3.2.cmml"><mo id="S1.p4.5.m5.3.3.1.2" stretchy="false" xref="S1.p4.5.m5.3.3.2.cmml">{</mo><mrow id="S1.p4.5.m5.3.3.1.1" xref="S1.p4.5.m5.3.3.1.1.cmml"><mo id="S1.p4.5.m5.3.3.1.1a" xref="S1.p4.5.m5.3.3.1.1.cmml">−</mo><msup id="S1.p4.5.m5.3.3.1.1.2" xref="S1.p4.5.m5.3.3.1.1.2.cmml"><mi id="S1.p4.5.m5.3.3.1.1.2.2" xref="S1.p4.5.m5.3.3.1.1.2.2.cmml">v</mi><mo id="S1.p4.5.m5.3.3.1.1.2.3" xref="S1.p4.5.m5.3.3.1.1.2.3.cmml">−</mo></msup></mrow><mo id="S1.p4.5.m5.3.3.1.3" xref="S1.p4.5.m5.3.3.2.cmml">,</mo><mn id="S1.p4.5.m5.1.1" xref="S1.p4.5.m5.1.1.cmml">0</mn><mo id="S1.p4.5.m5.3.3.1.4" xref="S1.p4.5.m5.3.3.2.cmml">,</mo><mn id="S1.p4.5.m5.2.2" xref="S1.p4.5.m5.2.2.cmml">1</mn><mo id="S1.p4.5.m5.3.3.1.5" stretchy="false" xref="S1.p4.5.m5.3.3.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.5.m5.3b"><set id="S1.p4.5.m5.3.3.2.cmml" xref="S1.p4.5.m5.3.3.1"><apply id="S1.p4.5.m5.3.3.1.1.cmml" xref="S1.p4.5.m5.3.3.1.1"><minus id="S1.p4.5.m5.3.3.1.1.1.cmml" xref="S1.p4.5.m5.3.3.1.1"></minus><apply id="S1.p4.5.m5.3.3.1.1.2.cmml" xref="S1.p4.5.m5.3.3.1.1.2"><csymbol cd="ambiguous" id="S1.p4.5.m5.3.3.1.1.2.1.cmml" xref="S1.p4.5.m5.3.3.1.1.2">superscript</csymbol><ci id="S1.p4.5.m5.3.3.1.1.2.2.cmml" xref="S1.p4.5.m5.3.3.1.1.2.2">𝑣</ci><minus id="S1.p4.5.m5.3.3.1.1.2.3.cmml" xref="S1.p4.5.m5.3.3.1.1.2.3"></minus></apply></apply><cn id="S1.p4.5.m5.1.1.cmml" type="integer" xref="S1.p4.5.m5.1.1">0</cn><cn id="S1.p4.5.m5.2.2.cmml" type="integer" xref="S1.p4.5.m5.2.2">1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.5.m5.3c">\{-v^{-},0,1\}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.5.m5.3d">{ - italic_v start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT , 0 , 1 }</annotation></semantics></math>, where <math alttext="v^{-}\geq 1" class="ltx_Math" display="inline" id="S1.p4.6.m6.1"><semantics id="S1.p4.6.m6.1a"><mrow id="S1.p4.6.m6.1.1" xref="S1.p4.6.m6.1.1.cmml"><msup id="S1.p4.6.m6.1.1.2" xref="S1.p4.6.m6.1.1.2.cmml"><mi id="S1.p4.6.m6.1.1.2.2" xref="S1.p4.6.m6.1.1.2.2.cmml">v</mi><mo id="S1.p4.6.m6.1.1.2.3" xref="S1.p4.6.m6.1.1.2.3.cmml">−</mo></msup><mo id="S1.p4.6.m6.1.1.1" xref="S1.p4.6.m6.1.1.1.cmml">≥</mo><mn id="S1.p4.6.m6.1.1.3" xref="S1.p4.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.6.m6.1b"><apply id="S1.p4.6.m6.1.1.cmml" xref="S1.p4.6.m6.1.1"><geq id="S1.p4.6.m6.1.1.1.cmml" xref="S1.p4.6.m6.1.1.1"></geq><apply id="S1.p4.6.m6.1.1.2.cmml" xref="S1.p4.6.m6.1.1.2"><csymbol cd="ambiguous" id="S1.p4.6.m6.1.1.2.1.cmml" xref="S1.p4.6.m6.1.1.2">superscript</csymbol><ci id="S1.p4.6.m6.1.1.2.2.cmml" xref="S1.p4.6.m6.1.1.2.2">𝑣</ci><minus id="S1.p4.6.m6.1.1.2.3.cmml" xref="S1.p4.6.m6.1.1.2.3"></minus></apply><cn id="S1.p4.6.m6.1.1.3.cmml" type="integer" xref="S1.p4.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.6.m6.1c">v^{-}\geq 1</annotation><annotation encoding="application/x-llamapun" id="S1.p4.6.m6.1d">italic_v start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ≥ 1</annotation></semantics></math> is an arbitrary but fixed (and, therefore, globally bounded) number.<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>By rescaling valuations, this is equivalent to assuming that, in addition to a neutral valuation of <math alttext="0" class="ltx_Math" display="inline" id="footnote1.m1.1"><semantics id="footnote1.m1.1b"><mn id="footnote1.m1.1.1" xref="footnote1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="footnote1.m1.1c"><cn id="footnote1.m1.1.1.cmml" type="integer" xref="footnote1.m1.1.1">0</cn></annotation-xml></semantics></math>, there is a single positive and negative valuation, where the former is bounded by the absolute value of the latter.</span></span></span> This sounds discouraging but it strengthens the impression that negative valuations seem to be the reason for computational boundaries.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">In the remainder of the paper, we provide several possibilities to achieve better approximation guarantees. First, we consider the restricted domain of games in which the sum of all valuations is nonnegative. This assumption still allows for the existence of rather negative valuations, however, it disallows an overall bias towards negative valuations. We make use of a result from the correlation clustering literature <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx18" title="">CW04</a>]</cite> to prove the existence of a randomized algorithm that approximates social welfare by a factor of <math alttext="\mathcal{O}(\log n)" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><mrow id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p5.1.m1.1.1.3" xref="S1.p5.1.m1.1.1.3.cmml">𝒪</mi><mo id="S1.p5.1.m1.1.1.2" xref="S1.p5.1.m1.1.1.2.cmml">⁢</mo><mrow id="S1.p5.1.m1.1.1.1.1" xref="S1.p5.1.m1.1.1.1.1.1.cmml"><mo id="S1.p5.1.m1.1.1.1.1.2" stretchy="false" xref="S1.p5.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p5.1.m1.1.1.1.1.1" xref="S1.p5.1.m1.1.1.1.1.1.cmml"><mi id="S1.p5.1.m1.1.1.1.1.1.1" xref="S1.p5.1.m1.1.1.1.1.1.1.cmml">log</mi><mo id="S1.p5.1.m1.1.1.1.1.1a" lspace="0.167em" xref="S1.p5.1.m1.1.1.1.1.1.cmml">⁡</mo><mi id="S1.p5.1.m1.1.1.1.1.1.2" xref="S1.p5.1.m1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S1.p5.1.m1.1.1.1.1.3" stretchy="false" xref="S1.p5.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><apply id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1"><times id="S1.p5.1.m1.1.1.2.cmml" xref="S1.p5.1.m1.1.1.2"></times><ci id="S1.p5.1.m1.1.1.3.cmml" xref="S1.p5.1.m1.1.1.3">𝒪</ci><apply id="S1.p5.1.m1.1.1.1.1.1.cmml" xref="S1.p5.1.m1.1.1.1.1"><log id="S1.p5.1.m1.1.1.1.1.1.1.cmml" xref="S1.p5.1.m1.1.1.1.1.1.1"></log><ci id="S1.p5.1.m1.1.1.1.1.1.2.cmml" xref="S1.p5.1.m1.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">\mathcal{O}(\log n)</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">caligraphic_O ( roman_log italic_n )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.3">Second, we consider two stochastic models of aversion-to-enemies games in which we achieve approximation guarantees with high probability. We start by assuming a basic model where valuations originate from an Erdős-Rényi graph. We show that a constant approximation of maximum welfare is possible. Subsequently, we define a stochastic model inspired by team management where every agent has a role, such as project manager, software engineer, UX designer, or marketing specialist. Coalitions represent teams and each role should be present in a team at most once. This scenario can be conceptualized by making agents with the same role mutually incompatible by introducing large negative valuations. In other words, the compatibility of agents is captured by a multipartite graph where the roles induce a partition of the vertices. However, in reality, even agents of different roles might be incompatible for various reasons. We model this by introducing a parameter <math alttext="p" class="ltx_Math" display="inline" id="S1.p6.1.m1.1"><semantics id="S1.p6.1.m1.1a"><mi id="S1.p6.1.m1.1.1" xref="S1.p6.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.p6.1.m1.1b"><ci id="S1.p6.1.m1.1.1.cmml" xref="S1.p6.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.1.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.p6.1.m1.1d">italic_p</annotation></semantics></math> that captures the probability of agents being incompatible. In our stochastic model, every pair of agents admitting different roles are incompatible independently. Based on the magnitude of <math alttext="p" class="ltx_Math" display="inline" id="S1.p6.2.m2.1"><semantics id="S1.p6.2.m2.1a"><mi id="S1.p6.2.m2.1.1" xref="S1.p6.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.p6.2.m2.1b"><ci id="S1.p6.2.m2.1.1.cmml" xref="S1.p6.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.p6.2.m2.1d">italic_p</annotation></semantics></math> we obtain perturbation regimes that lead to different approximation guarantees. In the low perturbation regime, we can approximate maximum welfare by a constant factor, whereas a high perturbation regime allows for a <math alttext="\log n" class="ltx_Math" display="inline" id="S1.p6.3.m3.1"><semantics id="S1.p6.3.m3.1a"><mrow id="S1.p6.3.m3.1.1" xref="S1.p6.3.m3.1.1.cmml"><mi id="S1.p6.3.m3.1.1.1" xref="S1.p6.3.m3.1.1.1.cmml">log</mi><mo id="S1.p6.3.m3.1.1a" lspace="0.167em" xref="S1.p6.3.m3.1.1.cmml">⁡</mo><mi id="S1.p6.3.m3.1.1.2" xref="S1.p6.3.m3.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.3.m3.1b"><apply id="S1.p6.3.m3.1.1.cmml" xref="S1.p6.3.m3.1.1"><log id="S1.p6.3.m3.1.1.1.cmml" xref="S1.p6.3.m3.1.1.1"></log><ci id="S1.p6.3.m3.1.1.2.cmml" xref="S1.p6.3.m3.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.3.m3.1c">\log n</annotation><annotation encoding="application/x-llamapun" id="S1.p6.3.m3.1d">roman_log italic_n</annotation></semantics></math>-approximation.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Related work</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">Hedonic games were introduced by Drèze and Greenberg <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx21" title="">DG80</a>]</cite> as an ordinal model of coalition formation, in which agents state their preferences as rankings over coalitions. Their broad consideration started, however, only <math alttext="20" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><mn id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">20</mn><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><cn id="S2.p1.1.m1.1.1.cmml" type="integer" xref="S2.p1.1.m1.1.1">20</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">20</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">20</annotation></semantics></math> years later <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx17" title="">CRM01</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx12" title="">BKS01</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx10" title="">BJ02</a>]</cite>. Much of their popularity today is due to the introduction of additively separable hedonic games by Bogomolnaia and Jackson <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx10" title="">BJ02</a>]</cite> in this era. An introduction to hedonic games is provided in the book chapters by Aziz and Savani <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx2" title="">AS16</a>]</cite> and Bullinger et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx9" title="">BER24</a>]</cite>.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.3">While these first papers were in the realm of economic theory, they soon sparked a broader consideration of hedonic games in computer science. This led to increased attention of algorithmic properties of solution concepts, including their computational complexity <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx16" title="">CH02</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx3" title="">Bal04</a>]</cite>. Social welfare was first realized to be a demanding objective by Aziz et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx1" title="">ABS13</a>]</cite> who showed that it is <span class="ltx_ERROR undefined" id="S2.p2.3.1">\NP</span>-hard to compute even if valuations are restricted to be only <math alttext="-1" class="ltx_Math" display="inline" id="S2.p2.1.m1.1"><semantics id="S2.p2.1.m1.1a"><mrow id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml"><mo id="S2.p2.1.m1.1.1a" xref="S2.p2.1.m1.1.1.cmml">−</mo><mn id="S2.p2.1.m1.1.1.2" xref="S2.p2.1.m1.1.1.2.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.1b"><apply id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1"><minus id="S2.p2.1.m1.1.1.1.cmml" xref="S2.p2.1.m1.1.1"></minus><cn id="S2.p2.1.m1.1.1.2.cmml" type="integer" xref="S2.p2.1.m1.1.1.2">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.1c">-1</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.1d">- 1</annotation></semantics></math> or <math alttext="1" class="ltx_Math" display="inline" id="S2.p2.2.m2.1"><semantics id="S2.p2.2.m2.1a"><mn id="S2.p2.2.m2.1.1" xref="S2.p2.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.p2.2.m2.1b"><cn id="S2.p2.2.m2.1.1.cmml" type="integer" xref="S2.p2.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.p2.2.m2.1d">1</annotation></semantics></math>. Subsequently, Flammini et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx26" title="">FKV22</a>]</cite> significantly strengthened this to the <math alttext="n^{1-\varepsilon}" class="ltx_Math" display="inline" id="S2.p2.3.m3.1"><semantics id="S2.p2.3.m3.1a"><msup id="S2.p2.3.m3.1.1" xref="S2.p2.3.m3.1.1.cmml"><mi id="S2.p2.3.m3.1.1.2" xref="S2.p2.3.m3.1.1.2.cmml">n</mi><mrow id="S2.p2.3.m3.1.1.3" xref="S2.p2.3.m3.1.1.3.cmml"><mn id="S2.p2.3.m3.1.1.3.2" xref="S2.p2.3.m3.1.1.3.2.cmml">1</mn><mo id="S2.p2.3.m3.1.1.3.1" xref="S2.p2.3.m3.1.1.3.1.cmml">−</mo><mi id="S2.p2.3.m3.1.1.3.3" xref="S2.p2.3.m3.1.1.3.3.cmml">ε</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.p2.3.m3.1b"><apply id="S2.p2.3.m3.1.1.cmml" xref="S2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p2.3.m3.1.1.1.cmml" xref="S2.p2.3.m3.1.1">superscript</csymbol><ci id="S2.p2.3.m3.1.1.2.cmml" xref="S2.p2.3.m3.1.1.2">𝑛</ci><apply id="S2.p2.3.m3.1.1.3.cmml" xref="S2.p2.3.m3.1.1.3"><minus id="S2.p2.3.m3.1.1.3.1.cmml" xref="S2.p2.3.m3.1.1.3.1"></minus><cn id="S2.p2.3.m3.1.1.3.2.cmml" type="integer" xref="S2.p2.3.m3.1.1.3.2">1</cn><ci id="S2.p2.3.m3.1.1.3.3.cmml" xref="S2.p2.3.m3.1.1.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.3.m3.1c">n^{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.3.m3.1d">italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT</annotation></semantics></math>-inapproximability result for aversion-to-enemies games mentioned in the introduction.</p> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.1">Beyond social welfare, other welfare objectives have been explored. Some early papers on hedonic games already studied Pareto optimality, a less demanding notion of welfare studied throughout economics <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx21" title="">DG80</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx10" title="">BJ02</a>]</cite>. Pareto-optimal coalition structures can be computed in polynomial time under fairly general assumptions including symmetric valuations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx1" title="">ABS13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx14" title="">Bul20</a>]</cite>. However, this yields no approximation of social welfare because Pareto-optimal outcomes may have negative social welfare <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx23" title="">EFF20</a>]</cite>. Moreover, Aziz et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx1" title="">ABS13</a>]</cite> also considered egalitarian welfare, which aims at maximizing the utility of the worst-off agent.</p> </div> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.3">Despite its challenges for the offline model, welfare approximation has also been studied in an online variant of additively separable hedonic games <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx27" title="">FMM<sup class="ltx_sup"><span class="ltx_text ltx_font_italic">+</span></sup>21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx13" title="">BR23</a>]</cite>. Flammini et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx27" title="">FMM<sup class="ltx_sup"><span class="ltx_text ltx_font_italic">+</span></sup>21</a>]</cite> consider a general model where no finite competitive ratio is possible if the utility range is unbounded. Moreover, Bullinger and Romen <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx13" title="">BR23</a>]</cite> study a model where the algorithm is allowed to dissolve coalitions into singleton coalitions, which allows to achieve a coalition structure with a social welfare that is at most a factor of <math alttext="\Theta(n)" class="ltx_Math" display="inline" id="S2.p4.1.m1.1"><semantics id="S2.p4.1.m1.1a"><mrow id="S2.p4.1.m1.1.2" xref="S2.p4.1.m1.1.2.cmml"><mi id="S2.p4.1.m1.1.2.2" mathvariant="normal" xref="S2.p4.1.m1.1.2.2.cmml">Θ</mi><mo id="S2.p4.1.m1.1.2.1" xref="S2.p4.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.p4.1.m1.1.2.3.2" xref="S2.p4.1.m1.1.2.cmml"><mo id="S2.p4.1.m1.1.2.3.2.1" stretchy="false" xref="S2.p4.1.m1.1.2.cmml">(</mo><mi id="S2.p4.1.m1.1.1" xref="S2.p4.1.m1.1.1.cmml">n</mi><mo id="S2.p4.1.m1.1.2.3.2.2" stretchy="false" xref="S2.p4.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.1.m1.1b"><apply id="S2.p4.1.m1.1.2.cmml" xref="S2.p4.1.m1.1.2"><times id="S2.p4.1.m1.1.2.1.cmml" xref="S2.p4.1.m1.1.2.1"></times><ci id="S2.p4.1.m1.1.2.2.cmml" xref="S2.p4.1.m1.1.2.2">Θ</ci><ci id="S2.p4.1.m1.1.1.cmml" xref="S2.p4.1.m1.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.1.m1.1c">\Theta(n)</annotation><annotation encoding="application/x-llamapun" id="S2.p4.1.m1.1d">roman_Θ ( italic_n )</annotation></semantics></math> worse than the maximum possible welfare. In particular, they show that maximum weight matchings achieve an <math alttext="n" class="ltx_Math" display="inline" id="S2.p4.2.m2.1"><semantics id="S2.p4.2.m2.1a"><mi id="S2.p4.2.m2.1.1" xref="S2.p4.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.p4.2.m2.1b"><ci id="S2.p4.2.m2.1.1.cmml" xref="S2.p4.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.p4.2.m2.1d">italic_n</annotation></semantics></math>-approximation of social welfare <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx13" title="">BR23</a>]</cite>. This essentially matches the aforementioned inapproximability by a factor of <math alttext="n^{1-\varepsilon}" class="ltx_Math" display="inline" id="S2.p4.3.m3.1"><semantics id="S2.p4.3.m3.1a"><msup id="S2.p4.3.m3.1.1" xref="S2.p4.3.m3.1.1.cmml"><mi id="S2.p4.3.m3.1.1.2" xref="S2.p4.3.m3.1.1.2.cmml">n</mi><mrow id="S2.p4.3.m3.1.1.3" xref="S2.p4.3.m3.1.1.3.cmml"><mn id="S2.p4.3.m3.1.1.3.2" xref="S2.p4.3.m3.1.1.3.2.cmml">1</mn><mo id="S2.p4.3.m3.1.1.3.1" xref="S2.p4.3.m3.1.1.3.1.cmml">−</mo><mi id="S2.p4.3.m3.1.1.3.3" xref="S2.p4.3.m3.1.1.3.3.cmml">ε</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.p4.3.m3.1b"><apply id="S2.p4.3.m3.1.1.cmml" xref="S2.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p4.3.m3.1.1.1.cmml" xref="S2.p4.3.m3.1.1">superscript</csymbol><ci id="S2.p4.3.m3.1.1.2.cmml" xref="S2.p4.3.m3.1.1.2">𝑛</ci><apply id="S2.p4.3.m3.1.1.3.cmml" xref="S2.p4.3.m3.1.1.3"><minus id="S2.p4.3.m3.1.1.3.1.cmml" xref="S2.p4.3.m3.1.1.3.1"></minus><cn id="S2.p4.3.m3.1.1.3.2.cmml" type="integer" xref="S2.p4.3.m3.1.1.3.2">1</cn><ci id="S2.p4.3.m3.1.1.3.3.cmml" xref="S2.p4.3.m3.1.1.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.3.m3.1c">n^{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.3.m3.1d">italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx26" title="">FKV22</a>]</cite>. Finally, social welfare has been considered in a mechanism design perspective aiming at strategyproof preference elicitation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx25" title="">FKMZ21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx26" title="">FKV22</a>]</cite>.</p> </div> <div class="ltx_para" id="S2.p5"> <p class="ltx_p" id="S2.p5.1">Beyond welfare, the most common objectives in hedonic games are notions of stability <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx10" title="">BJ02</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx33" title="">SD10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx1" title="">ABS13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx35" title="">Woe13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx29" title="">GS19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx6" title="">BBT24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx7" title="">BBW23</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx4" title="">BB22</a>]</cite>. Rather than the global guarantees provided by welfare notions, stability assumes a more strategic perspective in that it requires the absence of beneficial deviations by single agents or groups of agents. Single-deviation stability often leads to <span class="ltx_ERROR undefined" id="S2.p5.1.1">\NP</span>-completeness <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx33" title="">SD10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx6" title="">BBT24</a>]</cite>, whereas group stability can even be <math alttext="\Sigma_{2}^{p}" class="ltx_Math" display="inline" id="S2.p5.1.m1.1"><semantics id="S2.p5.1.m1.1a"><msubsup id="S2.p5.1.m1.1.1" xref="S2.p5.1.m1.1.1.cmml"><mi id="S2.p5.1.m1.1.1.2.2" mathvariant="normal" xref="S2.p5.1.m1.1.1.2.2.cmml">Σ</mi><mn id="S2.p5.1.m1.1.1.2.3" xref="S2.p5.1.m1.1.1.2.3.cmml">2</mn><mi id="S2.p5.1.m1.1.1.3" xref="S2.p5.1.m1.1.1.3.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.p5.1.m1.1b"><apply id="S2.p5.1.m1.1.1.cmml" xref="S2.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p5.1.m1.1.1.1.cmml" xref="S2.p5.1.m1.1.1">superscript</csymbol><apply id="S2.p5.1.m1.1.1.2.cmml" xref="S2.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p5.1.m1.1.1.2.1.cmml" xref="S2.p5.1.m1.1.1">subscript</csymbol><ci id="S2.p5.1.m1.1.1.2.2.cmml" xref="S2.p5.1.m1.1.1.2.2">Σ</ci><cn id="S2.p5.1.m1.1.1.2.3.cmml" type="integer" xref="S2.p5.1.m1.1.1.2.3">2</cn></apply><ci id="S2.p5.1.m1.1.1.3.cmml" xref="S2.p5.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.1.m1.1c">\Sigma_{2}^{p}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.1.m1.1d">roman_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT</annotation></semantics></math>-complete <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx35" title="">Woe13</a>]</cite>. Interestingly, symmetric valuations lead to the existence of stable outcomes based on single-agent deviations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx10" title="">BJ02</a>]</cite>, but their computation is still infeasible. It is <span class="ltx_ERROR undefined" id="S2.p5.1.2">\PLS</span>-complete, i.e., complete for the complexity class capturing problems that guarantee solutions based on local search algorithms <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx29" title="">GS19</a>]</cite>. An interesting objective that combines ideas of stability and global guarantees is popularity <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx1" title="">ABS13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx4" title="">BB22</a>]</cite>, which is akin to weak Condorcet winners as studied in social choice theory <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx8" title="">BCE<sup class="ltx_sup"><span class="ltx_text ltx_font_italic">+</span></sup>16</a>]</cite>.</p> </div> <div class="ltx_para" id="S2.p6"> <p class="ltx_p" id="S2.p6.1">While all the literature discussed so far considers a deterministic model, stochastic models have been studied to some extent <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx24" title="">FFKV23</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx11" title="">BK24</a>]</cite>. In particular, Bullinger and Kraiczy <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx11" title="">BK24</a>]</cite> show how to obtain stable outcomes if valuations are drawn uniformly at random. Their algorithm runs in three stages, the first of which will turn out to be useful in obtaining welfare guarantees as well, see <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.SS1" title="5.1 Erdős-Rényi Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">5.1</span></a>. By contrast, Fioravanti et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx24" title="">FFKV23</a>]</cite> consider a deterministic game model and aim at computing outcomes that are stable with high probability.</p> </div> <div class="ltx_para" id="S2.p7"> <p class="ltx_p" id="S2.p7.6">Finally, hedonic games are also related to other graph partitioning problems such as correlation clustering. The input typically consists of a complete graph with edges labeled as “<math alttext="+" class="ltx_Math" display="inline" id="S2.p7.1.m1.1"><semantics id="S2.p7.1.m1.1a"><mo id="S2.p7.1.m1.1.1" xref="S2.p7.1.m1.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S2.p7.1.m1.1b"><plus id="S2.p7.1.m1.1.1.cmml" xref="S2.p7.1.m1.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.1.m1.1c">+</annotation><annotation encoding="application/x-llamapun" id="S2.p7.1.m1.1d">+</annotation></semantics></math>” or “<math alttext="-" class="ltx_Math" display="inline" id="S2.p7.2.m2.1"><semantics id="S2.p7.2.m2.1a"><mo id="S2.p7.2.m2.1.1" xref="S2.p7.2.m2.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S2.p7.2.m2.1b"><minus id="S2.p7.2.m2.1.1.cmml" xref="S2.p7.2.m2.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.2.m2.1c">-</annotation><annotation encoding="application/x-llamapun" id="S2.p7.2.m2.1d">-</annotation></semantics></math>” to indicate similarity or dissimilarity, respectively <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx5" title="">BBC04</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx34" title="">Swa04</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx20" title="">DEFI06</a>]</cite>. The goal is to find a partition that maximizes agreements as measured by the sum of “<math alttext="+" class="ltx_Math" display="inline" id="S2.p7.3.m3.1"><semantics id="S2.p7.3.m3.1a"><mo id="S2.p7.3.m3.1.1" xref="S2.p7.3.m3.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S2.p7.3.m3.1b"><plus id="S2.p7.3.m3.1.1.cmml" xref="S2.p7.3.m3.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.3.m3.1c">+</annotation><annotation encoding="application/x-llamapun" id="S2.p7.3.m3.1d">+</annotation></semantics></math>” edges inside clusters plus “<math alttext="-" class="ltx_Math" display="inline" id="S2.p7.4.m4.1"><semantics id="S2.p7.4.m4.1a"><mo id="S2.p7.4.m4.1.1" xref="S2.p7.4.m4.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S2.p7.4.m4.1b"><minus id="S2.p7.4.m4.1.1.cmml" xref="S2.p7.4.m4.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.4.m4.1c">-</annotation><annotation encoding="application/x-llamapun" id="S2.p7.4.m4.1d">-</annotation></semantics></math>” edges across different clusters. Other objectives where the goal is to minimize errors of the partition, measured by “<math alttext="-" class="ltx_Math" display="inline" id="S2.p7.5.m5.1"><semantics id="S2.p7.5.m5.1a"><mo id="S2.p7.5.m5.1.1" xref="S2.p7.5.m5.1.1.cmml">−</mo><annotation-xml encoding="MathML-Content" id="S2.p7.5.m5.1b"><minus id="S2.p7.5.m5.1.1.cmml" xref="S2.p7.5.m5.1.1"></minus></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.5.m5.1c">-</annotation><annotation encoding="application/x-llamapun" id="S2.p7.5.m5.1d">-</annotation></semantics></math>” edges within clusters plus “<math alttext="+" class="ltx_Math" display="inline" id="S2.p7.6.m6.1"><semantics id="S2.p7.6.m6.1a"><mo id="S2.p7.6.m6.1.1" xref="S2.p7.6.m6.1.1.cmml">+</mo><annotation-xml encoding="MathML-Content" id="S2.p7.6.m6.1b"><plus id="S2.p7.6.m6.1.1.cmml" xref="S2.p7.6.m6.1.1"></plus></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.6.m6.1c">+</annotation><annotation encoding="application/x-llamapun" id="S2.p7.6.m6.1d">+</annotation></semantics></math>” edges across clusters have also been extensively studied <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx15" title="">CGW05</a>]</cite>. By contrast, our social welfare objective in hedonic games is different in that it accounts only for the edges within the coalitions (in particular, it ignores the edges across different coalitions).</p> </div> <div class="ltx_para" id="S2.p8"> <p class="ltx_p" id="S2.p8.3">Going beyond worst-case analysis, the two stochastic models we study for hedonic games in the second part of our paper relate to various stochastic models with random (or semi-random) edges that have been proposed for correlation clustering. For example, Mathieu and Schudy <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx31" title="">MS10</a>]</cite> investigates a noisy model on complete graphs, where they start from an arbitrary partition of the vertices into clusters and for each pair of vertices, the edge information (either <math alttext="1" class="ltx_Math" display="inline" id="S2.p8.1.m1.1"><semantics id="S2.p8.1.m1.1a"><mn id="S2.p8.1.m1.1.1" xref="S2.p8.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.p8.1.m1.1b"><cn id="S2.p8.1.m1.1.1.cmml" type="integer" xref="S2.p8.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.p8.1.m1.1d">1</annotation></semantics></math> or <math alttext="-1" class="ltx_Math" display="inline" id="S2.p8.2.m2.1"><semantics id="S2.p8.2.m2.1a"><mrow id="S2.p8.2.m2.1.1" xref="S2.p8.2.m2.1.1.cmml"><mo id="S2.p8.2.m2.1.1a" xref="S2.p8.2.m2.1.1.cmml">−</mo><mn id="S2.p8.2.m2.1.1.2" xref="S2.p8.2.m2.1.1.2.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p8.2.m2.1b"><apply id="S2.p8.2.m2.1.1.cmml" xref="S2.p8.2.m2.1.1"><minus id="S2.p8.2.m2.1.1.1.cmml" xref="S2.p8.2.m2.1.1"></minus><cn id="S2.p8.2.m2.1.1.2.cmml" type="integer" xref="S2.p8.2.m2.1.1.2">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.2.m2.1c">-1</annotation><annotation encoding="application/x-llamapun" id="S2.p8.2.m2.1d">- 1</annotation></semantics></math>) is corrupted independently with probability <math alttext="p" class="ltx_Math" display="inline" id="S2.p8.3.m3.1"><semantics id="S2.p8.3.m3.1a"><mi id="S2.p8.3.m3.1.1" xref="S2.p8.3.m3.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.p8.3.m3.1b"><ci id="S2.p8.3.m3.1.1.cmml" xref="S2.p8.3.m3.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.3.m3.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.p8.3.m3.1d">italic_p</annotation></semantics></math>. Other average-case models and extensions to arbitrary graphs (not necessarily complete) have been studied by Makarychev et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx30" title="">MMV15</a>]</cite>, with the goal of designing provably good approximation algorithms.</p> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Preliminaries</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.19">Consider a finite set <math alttext="N" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mi id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><ci id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">N</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">italic_N</annotation></semantics></math> of <math alttext="n:=|N|" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><mrow id="S3.p1.2.m2.1.2" xref="S3.p1.2.m2.1.2.cmml"><mi id="S3.p1.2.m2.1.2.2" xref="S3.p1.2.m2.1.2.2.cmml">n</mi><mo id="S3.p1.2.m2.1.2.1" lspace="0.278em" rspace="0.278em" xref="S3.p1.2.m2.1.2.1.cmml">:=</mo><mrow id="S3.p1.2.m2.1.2.3.2" xref="S3.p1.2.m2.1.2.3.1.cmml"><mo id="S3.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S3.p1.2.m2.1.2.3.1.1.cmml">|</mo><mi id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml">N</mi><mo id="S3.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S3.p1.2.m2.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><apply id="S3.p1.2.m2.1.2.cmml" xref="S3.p1.2.m2.1.2"><csymbol cd="latexml" id="S3.p1.2.m2.1.2.1.cmml" xref="S3.p1.2.m2.1.2.1">assign</csymbol><ci id="S3.p1.2.m2.1.2.2.cmml" xref="S3.p1.2.m2.1.2.2">𝑛</ci><apply id="S3.p1.2.m2.1.2.3.1.cmml" xref="S3.p1.2.m2.1.2.3.2"><abs id="S3.p1.2.m2.1.2.3.1.1.cmml" xref="S3.p1.2.m2.1.2.3.2.1"></abs><ci id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">n:=|N|</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">italic_n := | italic_N |</annotation></semantics></math> agents. A <em class="ltx_emph ltx_font_italic" id="S3.p1.19.1">coalition</em> is a nonempty subset of <math alttext="N" class="ltx_Math" display="inline" id="S3.p1.3.m3.1"><semantics id="S3.p1.3.m3.1a"><mi id="S3.p1.3.m3.1.1" xref="S3.p1.3.m3.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.1b"><ci id="S3.p1.3.m3.1.1.cmml" xref="S3.p1.3.m3.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.1c">N</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.1d">italic_N</annotation></semantics></math>. We denote by <math alttext="\mathcal{N}_{i}:=\{S\subseteq N\colon i\in S\}" class="ltx_Math" display="inline" id="S3.p1.4.m4.2"><semantics id="S3.p1.4.m4.2a"><mrow id="S3.p1.4.m4.2.2" xref="S3.p1.4.m4.2.2.cmml"><msub id="S3.p1.4.m4.2.2.4" xref="S3.p1.4.m4.2.2.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p1.4.m4.2.2.4.2" xref="S3.p1.4.m4.2.2.4.2.cmml">𝒩</mi><mi id="S3.p1.4.m4.2.2.4.3" xref="S3.p1.4.m4.2.2.4.3.cmml">i</mi></msub><mo id="S3.p1.4.m4.2.2.3" lspace="0.278em" rspace="0.278em" xref="S3.p1.4.m4.2.2.3.cmml">:=</mo><mrow id="S3.p1.4.m4.2.2.2.2" xref="S3.p1.4.m4.2.2.2.3.cmml"><mo id="S3.p1.4.m4.2.2.2.2.3" stretchy="false" xref="S3.p1.4.m4.2.2.2.3.1.cmml">{</mo><mrow id="S3.p1.4.m4.1.1.1.1.1" xref="S3.p1.4.m4.1.1.1.1.1.cmml"><mi id="S3.p1.4.m4.1.1.1.1.1.2" xref="S3.p1.4.m4.1.1.1.1.1.2.cmml">S</mi><mo id="S3.p1.4.m4.1.1.1.1.1.1" xref="S3.p1.4.m4.1.1.1.1.1.1.cmml">⊆</mo><mi id="S3.p1.4.m4.1.1.1.1.1.3" xref="S3.p1.4.m4.1.1.1.1.1.3.cmml">N</mi></mrow><mo id="S3.p1.4.m4.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.p1.4.m4.2.2.2.3.1.cmml">:</mo><mrow id="S3.p1.4.m4.2.2.2.2.2" xref="S3.p1.4.m4.2.2.2.2.2.cmml"><mi id="S3.p1.4.m4.2.2.2.2.2.2" xref="S3.p1.4.m4.2.2.2.2.2.2.cmml">i</mi><mo id="S3.p1.4.m4.2.2.2.2.2.1" xref="S3.p1.4.m4.2.2.2.2.2.1.cmml">∈</mo><mi id="S3.p1.4.m4.2.2.2.2.2.3" xref="S3.p1.4.m4.2.2.2.2.2.3.cmml">S</mi></mrow><mo id="S3.p1.4.m4.2.2.2.2.5" stretchy="false" xref="S3.p1.4.m4.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.2b"><apply id="S3.p1.4.m4.2.2.cmml" xref="S3.p1.4.m4.2.2"><csymbol cd="latexml" id="S3.p1.4.m4.2.2.3.cmml" xref="S3.p1.4.m4.2.2.3">assign</csymbol><apply id="S3.p1.4.m4.2.2.4.cmml" xref="S3.p1.4.m4.2.2.4"><csymbol cd="ambiguous" id="S3.p1.4.m4.2.2.4.1.cmml" xref="S3.p1.4.m4.2.2.4">subscript</csymbol><ci id="S3.p1.4.m4.2.2.4.2.cmml" xref="S3.p1.4.m4.2.2.4.2">𝒩</ci><ci id="S3.p1.4.m4.2.2.4.3.cmml" xref="S3.p1.4.m4.2.2.4.3">𝑖</ci></apply><apply id="S3.p1.4.m4.2.2.2.3.cmml" xref="S3.p1.4.m4.2.2.2.2"><csymbol cd="latexml" id="S3.p1.4.m4.2.2.2.3.1.cmml" xref="S3.p1.4.m4.2.2.2.2.3">conditional-set</csymbol><apply id="S3.p1.4.m4.1.1.1.1.1.cmml" xref="S3.p1.4.m4.1.1.1.1.1"><subset id="S3.p1.4.m4.1.1.1.1.1.1.cmml" xref="S3.p1.4.m4.1.1.1.1.1.1"></subset><ci id="S3.p1.4.m4.1.1.1.1.1.2.cmml" xref="S3.p1.4.m4.1.1.1.1.1.2">𝑆</ci><ci id="S3.p1.4.m4.1.1.1.1.1.3.cmml" xref="S3.p1.4.m4.1.1.1.1.1.3">𝑁</ci></apply><apply id="S3.p1.4.m4.2.2.2.2.2.cmml" xref="S3.p1.4.m4.2.2.2.2.2"><in id="S3.p1.4.m4.2.2.2.2.2.1.cmml" xref="S3.p1.4.m4.2.2.2.2.2.1"></in><ci id="S3.p1.4.m4.2.2.2.2.2.2.cmml" xref="S3.p1.4.m4.2.2.2.2.2.2">𝑖</ci><ci id="S3.p1.4.m4.2.2.2.2.2.3.cmml" xref="S3.p1.4.m4.2.2.2.2.2.3">𝑆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.2c">\mathcal{N}_{i}:=\{S\subseteq N\colon i\in S\}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.4.m4.2d">caligraphic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := { italic_S ⊆ italic_N : italic_i ∈ italic_S }</annotation></semantics></math> the set of all coalitions that agent <math alttext="i" class="ltx_Math" display="inline" id="S3.p1.5.m5.1"><semantics id="S3.p1.5.m5.1a"><mi id="S3.p1.5.m5.1.1" xref="S3.p1.5.m5.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p1.5.m5.1b"><ci id="S3.p1.5.m5.1.1.cmml" xref="S3.p1.5.m5.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m5.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m5.1d">italic_i</annotation></semantics></math> belongs to. A <em class="ltx_emph ltx_font_italic" id="S3.p1.19.2">coalition structure</em> (or <em class="ltx_emph ltx_font_italic" id="S3.p1.19.3">partition</em>) is a partition <math alttext="\pi" class="ltx_Math" display="inline" id="S3.p1.6.m6.1"><semantics id="S3.p1.6.m6.1a"><mi id="S3.p1.6.m6.1.1" xref="S3.p1.6.m6.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S3.p1.6.m6.1b"><ci id="S3.p1.6.m6.1.1.cmml" xref="S3.p1.6.m6.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.6.m6.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S3.p1.6.m6.1d">italic_π</annotation></semantics></math> of <math alttext="N" class="ltx_Math" display="inline" id="S3.p1.7.m7.1"><semantics id="S3.p1.7.m7.1a"><mi id="S3.p1.7.m7.1.1" xref="S3.p1.7.m7.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S3.p1.7.m7.1b"><ci id="S3.p1.7.m7.1.1.cmml" xref="S3.p1.7.m7.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.7.m7.1c">N</annotation><annotation encoding="application/x-llamapun" id="S3.p1.7.m7.1d">italic_N</annotation></semantics></math> into coalitions, i.e., <math alttext="\bigcup_{C\in\pi}C=N" class="ltx_Math" display="inline" id="S3.p1.8.m8.1"><semantics id="S3.p1.8.m8.1a"><mrow id="S3.p1.8.m8.1.1" xref="S3.p1.8.m8.1.1.cmml"><mrow id="S3.p1.8.m8.1.1.2" xref="S3.p1.8.m8.1.1.2.cmml"><msub id="S3.p1.8.m8.1.1.2.1" xref="S3.p1.8.m8.1.1.2.1.cmml"><mo id="S3.p1.8.m8.1.1.2.1.2" xref="S3.p1.8.m8.1.1.2.1.2.cmml">⋃</mo><mrow id="S3.p1.8.m8.1.1.2.1.3" xref="S3.p1.8.m8.1.1.2.1.3.cmml"><mi id="S3.p1.8.m8.1.1.2.1.3.2" xref="S3.p1.8.m8.1.1.2.1.3.2.cmml">C</mi><mo id="S3.p1.8.m8.1.1.2.1.3.1" xref="S3.p1.8.m8.1.1.2.1.3.1.cmml">∈</mo><mi id="S3.p1.8.m8.1.1.2.1.3.3" xref="S3.p1.8.m8.1.1.2.1.3.3.cmml">π</mi></mrow></msub><mi id="S3.p1.8.m8.1.1.2.2" xref="S3.p1.8.m8.1.1.2.2.cmml">C</mi></mrow><mo id="S3.p1.8.m8.1.1.1" xref="S3.p1.8.m8.1.1.1.cmml">=</mo><mi id="S3.p1.8.m8.1.1.3" xref="S3.p1.8.m8.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.8.m8.1b"><apply id="S3.p1.8.m8.1.1.cmml" xref="S3.p1.8.m8.1.1"><eq id="S3.p1.8.m8.1.1.1.cmml" xref="S3.p1.8.m8.1.1.1"></eq><apply id="S3.p1.8.m8.1.1.2.cmml" xref="S3.p1.8.m8.1.1.2"><apply id="S3.p1.8.m8.1.1.2.1.cmml" xref="S3.p1.8.m8.1.1.2.1"><csymbol cd="ambiguous" id="S3.p1.8.m8.1.1.2.1.1.cmml" xref="S3.p1.8.m8.1.1.2.1">subscript</csymbol><union id="S3.p1.8.m8.1.1.2.1.2.cmml" xref="S3.p1.8.m8.1.1.2.1.2"></union><apply id="S3.p1.8.m8.1.1.2.1.3.cmml" xref="S3.p1.8.m8.1.1.2.1.3"><in id="S3.p1.8.m8.1.1.2.1.3.1.cmml" xref="S3.p1.8.m8.1.1.2.1.3.1"></in><ci id="S3.p1.8.m8.1.1.2.1.3.2.cmml" xref="S3.p1.8.m8.1.1.2.1.3.2">𝐶</ci><ci id="S3.p1.8.m8.1.1.2.1.3.3.cmml" xref="S3.p1.8.m8.1.1.2.1.3.3">𝜋</ci></apply></apply><ci id="S3.p1.8.m8.1.1.2.2.cmml" xref="S3.p1.8.m8.1.1.2.2">𝐶</ci></apply><ci id="S3.p1.8.m8.1.1.3.cmml" xref="S3.p1.8.m8.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.8.m8.1c">\bigcup_{C\in\pi}C=N</annotation><annotation encoding="application/x-llamapun" id="S3.p1.8.m8.1d">⋃ start_POSTSUBSCRIPT italic_C ∈ italic_π end_POSTSUBSCRIPT italic_C = italic_N</annotation></semantics></math> and for each pair of coalitions <math alttext="C,C^{\prime}\in\pi" class="ltx_Math" display="inline" id="S3.p1.9.m9.2"><semantics id="S3.p1.9.m9.2a"><mrow id="S3.p1.9.m9.2.2" xref="S3.p1.9.m9.2.2.cmml"><mrow id="S3.p1.9.m9.2.2.1.1" xref="S3.p1.9.m9.2.2.1.2.cmml"><mi id="S3.p1.9.m9.1.1" xref="S3.p1.9.m9.1.1.cmml">C</mi><mo id="S3.p1.9.m9.2.2.1.1.2" xref="S3.p1.9.m9.2.2.1.2.cmml">,</mo><msup id="S3.p1.9.m9.2.2.1.1.1" xref="S3.p1.9.m9.2.2.1.1.1.cmml"><mi id="S3.p1.9.m9.2.2.1.1.1.2" xref="S3.p1.9.m9.2.2.1.1.1.2.cmml">C</mi><mo id="S3.p1.9.m9.2.2.1.1.1.3" xref="S3.p1.9.m9.2.2.1.1.1.3.cmml">′</mo></msup></mrow><mo id="S3.p1.9.m9.2.2.2" xref="S3.p1.9.m9.2.2.2.cmml">∈</mo><mi id="S3.p1.9.m9.2.2.3" xref="S3.p1.9.m9.2.2.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.9.m9.2b"><apply id="S3.p1.9.m9.2.2.cmml" xref="S3.p1.9.m9.2.2"><in id="S3.p1.9.m9.2.2.2.cmml" xref="S3.p1.9.m9.2.2.2"></in><list id="S3.p1.9.m9.2.2.1.2.cmml" xref="S3.p1.9.m9.2.2.1.1"><ci id="S3.p1.9.m9.1.1.cmml" xref="S3.p1.9.m9.1.1">𝐶</ci><apply id="S3.p1.9.m9.2.2.1.1.1.cmml" xref="S3.p1.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.p1.9.m9.2.2.1.1.1.1.cmml" xref="S3.p1.9.m9.2.2.1.1.1">superscript</csymbol><ci id="S3.p1.9.m9.2.2.1.1.1.2.cmml" xref="S3.p1.9.m9.2.2.1.1.1.2">𝐶</ci><ci id="S3.p1.9.m9.2.2.1.1.1.3.cmml" xref="S3.p1.9.m9.2.2.1.1.1.3">′</ci></apply></list><ci id="S3.p1.9.m9.2.2.3.cmml" xref="S3.p1.9.m9.2.2.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.9.m9.2c">C,C^{\prime}\in\pi</annotation><annotation encoding="application/x-llamapun" id="S3.p1.9.m9.2d">italic_C , italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_π</annotation></semantics></math> with <math alttext="C\neq C^{\prime}" class="ltx_Math" display="inline" id="S3.p1.10.m10.1"><semantics id="S3.p1.10.m10.1a"><mrow id="S3.p1.10.m10.1.1" xref="S3.p1.10.m10.1.1.cmml"><mi id="S3.p1.10.m10.1.1.2" xref="S3.p1.10.m10.1.1.2.cmml">C</mi><mo id="S3.p1.10.m10.1.1.1" xref="S3.p1.10.m10.1.1.1.cmml">≠</mo><msup id="S3.p1.10.m10.1.1.3" xref="S3.p1.10.m10.1.1.3.cmml"><mi id="S3.p1.10.m10.1.1.3.2" xref="S3.p1.10.m10.1.1.3.2.cmml">C</mi><mo id="S3.p1.10.m10.1.1.3.3" xref="S3.p1.10.m10.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.10.m10.1b"><apply id="S3.p1.10.m10.1.1.cmml" xref="S3.p1.10.m10.1.1"><neq id="S3.p1.10.m10.1.1.1.cmml" xref="S3.p1.10.m10.1.1.1"></neq><ci id="S3.p1.10.m10.1.1.2.cmml" xref="S3.p1.10.m10.1.1.2">𝐶</ci><apply id="S3.p1.10.m10.1.1.3.cmml" xref="S3.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S3.p1.10.m10.1.1.3.1.cmml" xref="S3.p1.10.m10.1.1.3">superscript</csymbol><ci id="S3.p1.10.m10.1.1.3.2.cmml" xref="S3.p1.10.m10.1.1.3.2">𝐶</ci><ci id="S3.p1.10.m10.1.1.3.3.cmml" xref="S3.p1.10.m10.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.10.m10.1c">C\neq C^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.10.m10.1d">italic_C ≠ italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> it holds that <math alttext="C\cap C^{\prime}=\emptyset" class="ltx_Math" display="inline" id="S3.p1.11.m11.1"><semantics id="S3.p1.11.m11.1a"><mrow id="S3.p1.11.m11.1.1" xref="S3.p1.11.m11.1.1.cmml"><mrow id="S3.p1.11.m11.1.1.2" xref="S3.p1.11.m11.1.1.2.cmml"><mi id="S3.p1.11.m11.1.1.2.2" xref="S3.p1.11.m11.1.1.2.2.cmml">C</mi><mo id="S3.p1.11.m11.1.1.2.1" xref="S3.p1.11.m11.1.1.2.1.cmml">∩</mo><msup id="S3.p1.11.m11.1.1.2.3" xref="S3.p1.11.m11.1.1.2.3.cmml"><mi id="S3.p1.11.m11.1.1.2.3.2" xref="S3.p1.11.m11.1.1.2.3.2.cmml">C</mi><mo id="S3.p1.11.m11.1.1.2.3.3" xref="S3.p1.11.m11.1.1.2.3.3.cmml">′</mo></msup></mrow><mo id="S3.p1.11.m11.1.1.1" xref="S3.p1.11.m11.1.1.1.cmml">=</mo><mi id="S3.p1.11.m11.1.1.3" mathvariant="normal" xref="S3.p1.11.m11.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.11.m11.1b"><apply id="S3.p1.11.m11.1.1.cmml" xref="S3.p1.11.m11.1.1"><eq id="S3.p1.11.m11.1.1.1.cmml" xref="S3.p1.11.m11.1.1.1"></eq><apply id="S3.p1.11.m11.1.1.2.cmml" xref="S3.p1.11.m11.1.1.2"><intersect id="S3.p1.11.m11.1.1.2.1.cmml" xref="S3.p1.11.m11.1.1.2.1"></intersect><ci id="S3.p1.11.m11.1.1.2.2.cmml" xref="S3.p1.11.m11.1.1.2.2">𝐶</ci><apply id="S3.p1.11.m11.1.1.2.3.cmml" xref="S3.p1.11.m11.1.1.2.3"><csymbol cd="ambiguous" id="S3.p1.11.m11.1.1.2.3.1.cmml" xref="S3.p1.11.m11.1.1.2.3">superscript</csymbol><ci id="S3.p1.11.m11.1.1.2.3.2.cmml" xref="S3.p1.11.m11.1.1.2.3.2">𝐶</ci><ci id="S3.p1.11.m11.1.1.2.3.3.cmml" xref="S3.p1.11.m11.1.1.2.3.3">′</ci></apply></apply><emptyset id="S3.p1.11.m11.1.1.3.cmml" xref="S3.p1.11.m11.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.11.m11.1c">C\cap C^{\prime}=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.p1.11.m11.1d">italic_C ∩ italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ∅</annotation></semantics></math>. Note that there is no bound on the number or size of coalitions. For an agent <math alttext="i\in N" class="ltx_Math" display="inline" id="S3.p1.12.m12.1"><semantics id="S3.p1.12.m12.1a"><mrow id="S3.p1.12.m12.1.1" xref="S3.p1.12.m12.1.1.cmml"><mi id="S3.p1.12.m12.1.1.2" xref="S3.p1.12.m12.1.1.2.cmml">i</mi><mo id="S3.p1.12.m12.1.1.1" xref="S3.p1.12.m12.1.1.1.cmml">∈</mo><mi id="S3.p1.12.m12.1.1.3" xref="S3.p1.12.m12.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.12.m12.1b"><apply id="S3.p1.12.m12.1.1.cmml" xref="S3.p1.12.m12.1.1"><in id="S3.p1.12.m12.1.1.1.cmml" xref="S3.p1.12.m12.1.1.1"></in><ci id="S3.p1.12.m12.1.1.2.cmml" xref="S3.p1.12.m12.1.1.2">𝑖</ci><ci id="S3.p1.12.m12.1.1.3.cmml" xref="S3.p1.12.m12.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.12.m12.1c">i\in N</annotation><annotation encoding="application/x-llamapun" id="S3.p1.12.m12.1d">italic_i ∈ italic_N</annotation></semantics></math>, we denote by <math alttext="\pi(i)" class="ltx_Math" display="inline" id="S3.p1.13.m13.1"><semantics id="S3.p1.13.m13.1a"><mrow id="S3.p1.13.m13.1.2" xref="S3.p1.13.m13.1.2.cmml"><mi id="S3.p1.13.m13.1.2.2" xref="S3.p1.13.m13.1.2.2.cmml">π</mi><mo id="S3.p1.13.m13.1.2.1" xref="S3.p1.13.m13.1.2.1.cmml">⁢</mo><mrow id="S3.p1.13.m13.1.2.3.2" xref="S3.p1.13.m13.1.2.cmml"><mo id="S3.p1.13.m13.1.2.3.2.1" stretchy="false" xref="S3.p1.13.m13.1.2.cmml">(</mo><mi id="S3.p1.13.m13.1.1" xref="S3.p1.13.m13.1.1.cmml">i</mi><mo id="S3.p1.13.m13.1.2.3.2.2" stretchy="false" xref="S3.p1.13.m13.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.13.m13.1b"><apply id="S3.p1.13.m13.1.2.cmml" xref="S3.p1.13.m13.1.2"><times id="S3.p1.13.m13.1.2.1.cmml" xref="S3.p1.13.m13.1.2.1"></times><ci id="S3.p1.13.m13.1.2.2.cmml" xref="S3.p1.13.m13.1.2.2">𝜋</ci><ci id="S3.p1.13.m13.1.1.cmml" xref="S3.p1.13.m13.1.1">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.13.m13.1c">\pi(i)</annotation><annotation encoding="application/x-llamapun" id="S3.p1.13.m13.1d">italic_π ( italic_i )</annotation></semantics></math> the coalition that <math alttext="i" class="ltx_Math" display="inline" id="S3.p1.14.m14.1"><semantics id="S3.p1.14.m14.1a"><mi id="S3.p1.14.m14.1.1" xref="S3.p1.14.m14.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p1.14.m14.1b"><ci id="S3.p1.14.m14.1.1.cmml" xref="S3.p1.14.m14.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.14.m14.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p1.14.m14.1d">italic_i</annotation></semantics></math> belongs to in <math alttext="\pi" class="ltx_Math" display="inline" id="S3.p1.15.m15.1"><semantics id="S3.p1.15.m15.1a"><mi id="S3.p1.15.m15.1.1" xref="S3.p1.15.m15.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S3.p1.15.m15.1b"><ci id="S3.p1.15.m15.1.1.cmml" xref="S3.p1.15.m15.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.15.m15.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S3.p1.15.m15.1d">italic_π</annotation></semantics></math>. We denote the set of all partitions of <math alttext="N" class="ltx_Math" display="inline" id="S3.p1.16.m16.1"><semantics id="S3.p1.16.m16.1a"><mi id="S3.p1.16.m16.1.1" xref="S3.p1.16.m16.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S3.p1.16.m16.1b"><ci id="S3.p1.16.m16.1.1.cmml" xref="S3.p1.16.m16.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.16.m16.1c">N</annotation><annotation encoding="application/x-llamapun" id="S3.p1.16.m16.1d">italic_N</annotation></semantics></math> by <math alttext="\Pi_{N}" class="ltx_Math" display="inline" id="S3.p1.17.m17.1"><semantics id="S3.p1.17.m17.1a"><msub id="S3.p1.17.m17.1.1" xref="S3.p1.17.m17.1.1.cmml"><mi id="S3.p1.17.m17.1.1.2" mathvariant="normal" xref="S3.p1.17.m17.1.1.2.cmml">Π</mi><mi id="S3.p1.17.m17.1.1.3" xref="S3.p1.17.m17.1.1.3.cmml">N</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p1.17.m17.1b"><apply id="S3.p1.17.m17.1.1.cmml" xref="S3.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S3.p1.17.m17.1.1.1.cmml" xref="S3.p1.17.m17.1.1">subscript</csymbol><ci id="S3.p1.17.m17.1.1.2.cmml" xref="S3.p1.17.m17.1.1.2">Π</ci><ci id="S3.p1.17.m17.1.1.3.cmml" xref="S3.p1.17.m17.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.17.m17.1c">\Pi_{N}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.17.m17.1d">roman_Π start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT</annotation></semantics></math>, and the set of all partitions containing exactly two coalitions as <math alttext="\Pi_{N}^{(2)}" class="ltx_Math" display="inline" id="S3.p1.18.m18.1"><semantics id="S3.p1.18.m18.1a"><msubsup id="S3.p1.18.m18.1.2" xref="S3.p1.18.m18.1.2.cmml"><mi id="S3.p1.18.m18.1.2.2.2" mathvariant="normal" xref="S3.p1.18.m18.1.2.2.2.cmml">Π</mi><mi id="S3.p1.18.m18.1.2.2.3" xref="S3.p1.18.m18.1.2.2.3.cmml">N</mi><mrow id="S3.p1.18.m18.1.1.1.3" xref="S3.p1.18.m18.1.2.cmml"><mo id="S3.p1.18.m18.1.1.1.3.1" stretchy="false" xref="S3.p1.18.m18.1.2.cmml">(</mo><mn id="S3.p1.18.m18.1.1.1.1" xref="S3.p1.18.m18.1.1.1.1.cmml">2</mn><mo id="S3.p1.18.m18.1.1.1.3.2" stretchy="false" xref="S3.p1.18.m18.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p1.18.m18.1b"><apply id="S3.p1.18.m18.1.2.cmml" xref="S3.p1.18.m18.1.2"><csymbol cd="ambiguous" id="S3.p1.18.m18.1.2.1.cmml" xref="S3.p1.18.m18.1.2">superscript</csymbol><apply id="S3.p1.18.m18.1.2.2.cmml" xref="S3.p1.18.m18.1.2"><csymbol cd="ambiguous" id="S3.p1.18.m18.1.2.2.1.cmml" xref="S3.p1.18.m18.1.2">subscript</csymbol><ci id="S3.p1.18.m18.1.2.2.2.cmml" xref="S3.p1.18.m18.1.2.2.2">Π</ci><ci id="S3.p1.18.m18.1.2.2.3.cmml" xref="S3.p1.18.m18.1.2.2.3">𝑁</ci></apply><cn id="S3.p1.18.m18.1.1.1.1.cmml" type="integer" xref="S3.p1.18.m18.1.1.1.1">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.18.m18.1c">\Pi_{N}^{(2)}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.18.m18.1d">roman_Π start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT</annotation></semantics></math>, i.e., <math alttext="\Pi_{N}^{(2)}:=\{\pi\in\Pi_{N}\colon|\pi|=2\}" class="ltx_Math" display="inline" id="S3.p1.19.m19.4"><semantics id="S3.p1.19.m19.4a"><mrow id="S3.p1.19.m19.4.4" xref="S3.p1.19.m19.4.4.cmml"><msubsup id="S3.p1.19.m19.4.4.4" xref="S3.p1.19.m19.4.4.4.cmml"><mi id="S3.p1.19.m19.4.4.4.2.2" mathvariant="normal" xref="S3.p1.19.m19.4.4.4.2.2.cmml">Π</mi><mi id="S3.p1.19.m19.4.4.4.2.3" xref="S3.p1.19.m19.4.4.4.2.3.cmml">N</mi><mrow id="S3.p1.19.m19.1.1.1.3" xref="S3.p1.19.m19.4.4.4.cmml"><mo id="S3.p1.19.m19.1.1.1.3.1" stretchy="false" xref="S3.p1.19.m19.4.4.4.cmml">(</mo><mn id="S3.p1.19.m19.1.1.1.1" xref="S3.p1.19.m19.1.1.1.1.cmml">2</mn><mo id="S3.p1.19.m19.1.1.1.3.2" stretchy="false" xref="S3.p1.19.m19.4.4.4.cmml">)</mo></mrow></msubsup><mo id="S3.p1.19.m19.4.4.3" lspace="0.278em" rspace="0.278em" xref="S3.p1.19.m19.4.4.3.cmml">:=</mo><mrow id="S3.p1.19.m19.4.4.2.2" xref="S3.p1.19.m19.4.4.2.3.cmml"><mo id="S3.p1.19.m19.4.4.2.2.3" stretchy="false" xref="S3.p1.19.m19.4.4.2.3.1.cmml">{</mo><mrow id="S3.p1.19.m19.3.3.1.1.1" xref="S3.p1.19.m19.3.3.1.1.1.cmml"><mi id="S3.p1.19.m19.3.3.1.1.1.2" xref="S3.p1.19.m19.3.3.1.1.1.2.cmml">π</mi><mo id="S3.p1.19.m19.3.3.1.1.1.1" xref="S3.p1.19.m19.3.3.1.1.1.1.cmml">∈</mo><msub id="S3.p1.19.m19.3.3.1.1.1.3" xref="S3.p1.19.m19.3.3.1.1.1.3.cmml"><mi id="S3.p1.19.m19.3.3.1.1.1.3.2" mathvariant="normal" xref="S3.p1.19.m19.3.3.1.1.1.3.2.cmml">Π</mi><mi id="S3.p1.19.m19.3.3.1.1.1.3.3" xref="S3.p1.19.m19.3.3.1.1.1.3.3.cmml">N</mi></msub></mrow><mo id="S3.p1.19.m19.4.4.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.p1.19.m19.4.4.2.3.1.cmml">:</mo><mrow id="S3.p1.19.m19.4.4.2.2.2" xref="S3.p1.19.m19.4.4.2.2.2.cmml"><mrow id="S3.p1.19.m19.4.4.2.2.2.2.2" xref="S3.p1.19.m19.4.4.2.2.2.2.1.cmml"><mo id="S3.p1.19.m19.4.4.2.2.2.2.2.1" stretchy="false" xref="S3.p1.19.m19.4.4.2.2.2.2.1.1.cmml">|</mo><mi id="S3.p1.19.m19.2.2" xref="S3.p1.19.m19.2.2.cmml">π</mi><mo id="S3.p1.19.m19.4.4.2.2.2.2.2.2" stretchy="false" xref="S3.p1.19.m19.4.4.2.2.2.2.1.1.cmml">|</mo></mrow><mo id="S3.p1.19.m19.4.4.2.2.2.1" xref="S3.p1.19.m19.4.4.2.2.2.1.cmml">=</mo><mn id="S3.p1.19.m19.4.4.2.2.2.3" xref="S3.p1.19.m19.4.4.2.2.2.3.cmml">2</mn></mrow><mo id="S3.p1.19.m19.4.4.2.2.5" stretchy="false" xref="S3.p1.19.m19.4.4.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.19.m19.4b"><apply id="S3.p1.19.m19.4.4.cmml" xref="S3.p1.19.m19.4.4"><csymbol cd="latexml" id="S3.p1.19.m19.4.4.3.cmml" xref="S3.p1.19.m19.4.4.3">assign</csymbol><apply id="S3.p1.19.m19.4.4.4.cmml" xref="S3.p1.19.m19.4.4.4"><csymbol cd="ambiguous" id="S3.p1.19.m19.4.4.4.1.cmml" xref="S3.p1.19.m19.4.4.4">superscript</csymbol><apply id="S3.p1.19.m19.4.4.4.2.cmml" xref="S3.p1.19.m19.4.4.4"><csymbol cd="ambiguous" id="S3.p1.19.m19.4.4.4.2.1.cmml" xref="S3.p1.19.m19.4.4.4">subscript</csymbol><ci id="S3.p1.19.m19.4.4.4.2.2.cmml" xref="S3.p1.19.m19.4.4.4.2.2">Π</ci><ci id="S3.p1.19.m19.4.4.4.2.3.cmml" xref="S3.p1.19.m19.4.4.4.2.3">𝑁</ci></apply><cn id="S3.p1.19.m19.1.1.1.1.cmml" type="integer" xref="S3.p1.19.m19.1.1.1.1">2</cn></apply><apply id="S3.p1.19.m19.4.4.2.3.cmml" xref="S3.p1.19.m19.4.4.2.2"><csymbol cd="latexml" id="S3.p1.19.m19.4.4.2.3.1.cmml" xref="S3.p1.19.m19.4.4.2.2.3">conditional-set</csymbol><apply id="S3.p1.19.m19.3.3.1.1.1.cmml" xref="S3.p1.19.m19.3.3.1.1.1"><in id="S3.p1.19.m19.3.3.1.1.1.1.cmml" xref="S3.p1.19.m19.3.3.1.1.1.1"></in><ci id="S3.p1.19.m19.3.3.1.1.1.2.cmml" xref="S3.p1.19.m19.3.3.1.1.1.2">𝜋</ci><apply id="S3.p1.19.m19.3.3.1.1.1.3.cmml" xref="S3.p1.19.m19.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S3.p1.19.m19.3.3.1.1.1.3.1.cmml" xref="S3.p1.19.m19.3.3.1.1.1.3">subscript</csymbol><ci id="S3.p1.19.m19.3.3.1.1.1.3.2.cmml" xref="S3.p1.19.m19.3.3.1.1.1.3.2">Π</ci><ci id="S3.p1.19.m19.3.3.1.1.1.3.3.cmml" xref="S3.p1.19.m19.3.3.1.1.1.3.3">𝑁</ci></apply></apply><apply id="S3.p1.19.m19.4.4.2.2.2.cmml" xref="S3.p1.19.m19.4.4.2.2.2"><eq id="S3.p1.19.m19.4.4.2.2.2.1.cmml" xref="S3.p1.19.m19.4.4.2.2.2.1"></eq><apply id="S3.p1.19.m19.4.4.2.2.2.2.1.cmml" xref="S3.p1.19.m19.4.4.2.2.2.2.2"><abs id="S3.p1.19.m19.4.4.2.2.2.2.1.1.cmml" xref="S3.p1.19.m19.4.4.2.2.2.2.2.1"></abs><ci id="S3.p1.19.m19.2.2.cmml" xref="S3.p1.19.m19.2.2">𝜋</ci></apply><cn id="S3.p1.19.m19.4.4.2.2.2.3.cmml" type="integer" xref="S3.p1.19.m19.4.4.2.2.2.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.19.m19.4c">\Pi_{N}^{(2)}:=\{\pi\in\Pi_{N}\colon|\pi|=2\}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.19.m19.4d">roman_Π start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT := { italic_π ∈ roman_Π start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT : | italic_π | = 2 }</annotation></semantics></math>. A coalition is called a <em class="ltx_emph ltx_font_italic" id="S3.p1.19.4">singleton coalition</em> if it contains exactly one agent. The partition where every agent is in a singleton coalition is called the <em class="ltx_emph ltx_font_italic" id="S3.p1.19.5">singleton partition</em>.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.7">In a hedonic game, every agent possesses preferences over the coalitions in <math alttext="\mathcal{N}_{i}" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><msub id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.1.m1.1.1.2" xref="S3.p2.1.m1.1.1.2.cmml">𝒩</mi><mi id="S3.p2.1.m1.1.1.3" xref="S3.p2.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><apply id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p2.1.m1.1.1.1.cmml" xref="S3.p2.1.m1.1.1">subscript</csymbol><ci id="S3.p2.1.m1.1.1.2.cmml" xref="S3.p2.1.m1.1.1.2">𝒩</ci><ci id="S3.p2.1.m1.1.1.3.cmml" xref="S3.p2.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">\mathcal{N}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">caligraphic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. We use the model of additively separable hedonic games by Bogomolnaia and Jackson <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx10" title="">BJ02</a>]</cite> in which these preferences are obtained from cardinal valuations that can be encoded by a complete and directed weighted graph. Formally, a <em class="ltx_emph ltx_font_italic" id="S3.p2.7.1">cardinal hedonic game</em> is a pair <math alttext="(N,u)" class="ltx_Math" display="inline" id="S3.p2.2.m2.2"><semantics id="S3.p2.2.m2.2a"><mrow id="S3.p2.2.m2.2.3.2" xref="S3.p2.2.m2.2.3.1.cmml"><mo id="S3.p2.2.m2.2.3.2.1" stretchy="false" xref="S3.p2.2.m2.2.3.1.cmml">(</mo><mi id="S3.p2.2.m2.1.1" xref="S3.p2.2.m2.1.1.cmml">N</mi><mo id="S3.p2.2.m2.2.3.2.2" xref="S3.p2.2.m2.2.3.1.cmml">,</mo><mi id="S3.p2.2.m2.2.2" xref="S3.p2.2.m2.2.2.cmml">u</mi><mo id="S3.p2.2.m2.2.3.2.3" stretchy="false" xref="S3.p2.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.2.m2.2b"><interval closure="open" id="S3.p2.2.m2.2.3.1.cmml" xref="S3.p2.2.m2.2.3.2"><ci id="S3.p2.2.m2.1.1.cmml" xref="S3.p2.2.m2.1.1">𝑁</ci><ci id="S3.p2.2.m2.2.2.cmml" xref="S3.p2.2.m2.2.2">𝑢</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.m2.2c">(N,u)</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.m2.2d">( italic_N , italic_u )</annotation></semantics></math> where <math alttext="u=(u_{i}\colon\mathcal{N}_{i}\to\mathbb{R})_{i\in N}" class="ltx_math_unparsed" display="inline" id="S3.p2.3.m3.1"><semantics id="S3.p2.3.m3.1a"><mrow id="S3.p2.3.m3.1b"><mi id="S3.p2.3.m3.1.1">u</mi><mo id="S3.p2.3.m3.1.2">=</mo><msub id="S3.p2.3.m3.1.3"><mrow id="S3.p2.3.m3.1.3.2"><mo id="S3.p2.3.m3.1.3.2.1" stretchy="false">(</mo><msub id="S3.p2.3.m3.1.3.2.2"><mi id="S3.p2.3.m3.1.3.2.2.2">u</mi><mi id="S3.p2.3.m3.1.3.2.2.3">i</mi></msub><mo id="S3.p2.3.m3.1.3.2.3" lspace="0.278em" rspace="0.278em">:</mo><msub id="S3.p2.3.m3.1.3.2.4"><mi class="ltx_font_mathcaligraphic" id="S3.p2.3.m3.1.3.2.4.2">𝒩</mi><mi id="S3.p2.3.m3.1.3.2.4.3">i</mi></msub><mo id="S3.p2.3.m3.1.3.2.5" stretchy="false">→</mo><mi id="S3.p2.3.m3.1.3.2.6">ℝ</mi><mo id="S3.p2.3.m3.1.3.2.7" stretchy="false">)</mo></mrow><mrow id="S3.p2.3.m3.1.3.3"><mi id="S3.p2.3.m3.1.3.3.2">i</mi><mo id="S3.p2.3.m3.1.3.3.1">∈</mo><mi id="S3.p2.3.m3.1.3.3.3">N</mi></mrow></msub></mrow><annotation encoding="application/x-tex" id="S3.p2.3.m3.1c">u=(u_{i}\colon\mathcal{N}_{i}\to\mathbb{R})_{i\in N}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.3.m3.1d">italic_u = ( italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : caligraphic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → blackboard_R ) start_POSTSUBSCRIPT italic_i ∈ italic_N end_POSTSUBSCRIPT</annotation></semantics></math> is a vector of <em class="ltx_emph ltx_font_italic" id="S3.p2.7.2">utility functions</em>. An <em class="ltx_emph ltx_font_italic" id="S3.p2.7.3">additively separable hedonic game</em> (ASHG) is specified by a vector <math alttext="v=(v_{i}\colon N\to\mathbb{R})_{i\in N}" class="ltx_math_unparsed" display="inline" id="S3.p2.4.m4.1"><semantics id="S3.p2.4.m4.1a"><mrow id="S3.p2.4.m4.1b"><mi id="S3.p2.4.m4.1.1">v</mi><mo id="S3.p2.4.m4.1.2">=</mo><msub id="S3.p2.4.m4.1.3"><mrow id="S3.p2.4.m4.1.3.2"><mo id="S3.p2.4.m4.1.3.2.1" stretchy="false">(</mo><msub id="S3.p2.4.m4.1.3.2.2"><mi id="S3.p2.4.m4.1.3.2.2.2">v</mi><mi id="S3.p2.4.m4.1.3.2.2.3">i</mi></msub><mo id="S3.p2.4.m4.1.3.2.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S3.p2.4.m4.1.3.2.4">N</mi><mo id="S3.p2.4.m4.1.3.2.5" stretchy="false">→</mo><mi id="S3.p2.4.m4.1.3.2.6">ℝ</mi><mo id="S3.p2.4.m4.1.3.2.7" stretchy="false">)</mo></mrow><mrow id="S3.p2.4.m4.1.3.3"><mi id="S3.p2.4.m4.1.3.3.2">i</mi><mo id="S3.p2.4.m4.1.3.3.1">∈</mo><mi id="S3.p2.4.m4.1.3.3.3">N</mi></mrow></msub></mrow><annotation encoding="application/x-tex" id="S3.p2.4.m4.1c">v=(v_{i}\colon N\to\mathbb{R})_{i\in N}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.4.m4.1d">italic_v = ( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_N → blackboard_R ) start_POSTSUBSCRIPT italic_i ∈ italic_N end_POSTSUBSCRIPT</annotation></semantics></math> of (single-agent) <em class="ltx_emph ltx_font_italic" id="S3.p2.7.4">valuation functions</em>. It is then defined as the cardinal hedonic game <math alttext="(N,u)" class="ltx_Math" display="inline" id="S3.p2.5.m5.2"><semantics id="S3.p2.5.m5.2a"><mrow id="S3.p2.5.m5.2.3.2" xref="S3.p2.5.m5.2.3.1.cmml"><mo id="S3.p2.5.m5.2.3.2.1" stretchy="false" xref="S3.p2.5.m5.2.3.1.cmml">(</mo><mi id="S3.p2.5.m5.1.1" xref="S3.p2.5.m5.1.1.cmml">N</mi><mo id="S3.p2.5.m5.2.3.2.2" xref="S3.p2.5.m5.2.3.1.cmml">,</mo><mi id="S3.p2.5.m5.2.2" xref="S3.p2.5.m5.2.2.cmml">u</mi><mo id="S3.p2.5.m5.2.3.2.3" stretchy="false" xref="S3.p2.5.m5.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.5.m5.2b"><interval closure="open" id="S3.p2.5.m5.2.3.1.cmml" xref="S3.p2.5.m5.2.3.2"><ci id="S3.p2.5.m5.1.1.cmml" xref="S3.p2.5.m5.1.1">𝑁</ci><ci id="S3.p2.5.m5.2.2.cmml" xref="S3.p2.5.m5.2.2">𝑢</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.5.m5.2c">(N,u)</annotation><annotation encoding="application/x-llamapun" id="S3.p2.5.m5.2d">( italic_N , italic_u )</annotation></semantics></math>, where for any agent <math alttext="i\in N" class="ltx_Math" display="inline" id="S3.p2.6.m6.1"><semantics id="S3.p2.6.m6.1a"><mrow id="S3.p2.6.m6.1.1" xref="S3.p2.6.m6.1.1.cmml"><mi id="S3.p2.6.m6.1.1.2" xref="S3.p2.6.m6.1.1.2.cmml">i</mi><mo id="S3.p2.6.m6.1.1.1" xref="S3.p2.6.m6.1.1.1.cmml">∈</mo><mi id="S3.p2.6.m6.1.1.3" xref="S3.p2.6.m6.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.6.m6.1b"><apply id="S3.p2.6.m6.1.1.cmml" xref="S3.p2.6.m6.1.1"><in id="S3.p2.6.m6.1.1.1.cmml" xref="S3.p2.6.m6.1.1.1"></in><ci id="S3.p2.6.m6.1.1.2.cmml" xref="S3.p2.6.m6.1.1.2">𝑖</ci><ci id="S3.p2.6.m6.1.1.3.cmml" xref="S3.p2.6.m6.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.6.m6.1c">i\in N</annotation><annotation encoding="application/x-llamapun" id="S3.p2.6.m6.1d">italic_i ∈ italic_N</annotation></semantics></math> and coalition <math alttext="C\in\mathcal{N}_{i}" class="ltx_Math" display="inline" id="S3.p2.7.m7.1"><semantics id="S3.p2.7.m7.1a"><mrow id="S3.p2.7.m7.1.1" xref="S3.p2.7.m7.1.1.cmml"><mi id="S3.p2.7.m7.1.1.2" xref="S3.p2.7.m7.1.1.2.cmml">C</mi><mo id="S3.p2.7.m7.1.1.1" xref="S3.p2.7.m7.1.1.1.cmml">∈</mo><msub id="S3.p2.7.m7.1.1.3" xref="S3.p2.7.m7.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.7.m7.1.1.3.2" xref="S3.p2.7.m7.1.1.3.2.cmml">𝒩</mi><mi id="S3.p2.7.m7.1.1.3.3" xref="S3.p2.7.m7.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.7.m7.1b"><apply id="S3.p2.7.m7.1.1.cmml" xref="S3.p2.7.m7.1.1"><in id="S3.p2.7.m7.1.1.1.cmml" xref="S3.p2.7.m7.1.1.1"></in><ci id="S3.p2.7.m7.1.1.2.cmml" xref="S3.p2.7.m7.1.1.2">𝐶</ci><apply id="S3.p2.7.m7.1.1.3.cmml" xref="S3.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.p2.7.m7.1.1.3.1.cmml" xref="S3.p2.7.m7.1.1.3">subscript</csymbol><ci id="S3.p2.7.m7.1.1.3.2.cmml" xref="S3.p2.7.m7.1.1.3.2">𝒩</ci><ci id="S3.p2.7.m7.1.1.3.3.cmml" xref="S3.p2.7.m7.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.7.m7.1c">C\in\mathcal{N}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.7.m7.1d">italic_C ∈ caligraphic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, it holds that</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="u_{i}(C):=\sum_{j\in C}v_{i}(j)\text{.}" class="ltx_Math" display="block" id="S3.Ex1.m1.2"><semantics id="S3.Ex1.m1.2a"><mrow id="S3.Ex1.m1.2.3" xref="S3.Ex1.m1.2.3.cmml"><mrow id="S3.Ex1.m1.2.3.2" xref="S3.Ex1.m1.2.3.2.cmml"><msub id="S3.Ex1.m1.2.3.2.2" xref="S3.Ex1.m1.2.3.2.2.cmml"><mi id="S3.Ex1.m1.2.3.2.2.2" xref="S3.Ex1.m1.2.3.2.2.2.cmml">u</mi><mi id="S3.Ex1.m1.2.3.2.2.3" xref="S3.Ex1.m1.2.3.2.2.3.cmml">i</mi></msub><mo id="S3.Ex1.m1.2.3.2.1" xref="S3.Ex1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S3.Ex1.m1.2.3.2.3.2" xref="S3.Ex1.m1.2.3.2.cmml"><mo id="S3.Ex1.m1.2.3.2.3.2.1" stretchy="false" xref="S3.Ex1.m1.2.3.2.cmml">(</mo><mi id="S3.Ex1.m1.1.1" xref="S3.Ex1.m1.1.1.cmml">C</mi><mo id="S3.Ex1.m1.2.3.2.3.2.2" rspace="0.278em" stretchy="false" xref="S3.Ex1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex1.m1.2.3.1" rspace="0.111em" xref="S3.Ex1.m1.2.3.1.cmml">:=</mo><mrow id="S3.Ex1.m1.2.3.3" xref="S3.Ex1.m1.2.3.3.cmml"><munder id="S3.Ex1.m1.2.3.3.1" xref="S3.Ex1.m1.2.3.3.1.cmml"><mo id="S3.Ex1.m1.2.3.3.1.2" movablelimits="false" xref="S3.Ex1.m1.2.3.3.1.2.cmml">∑</mo><mrow id="S3.Ex1.m1.2.3.3.1.3" xref="S3.Ex1.m1.2.3.3.1.3.cmml"><mi id="S3.Ex1.m1.2.3.3.1.3.2" xref="S3.Ex1.m1.2.3.3.1.3.2.cmml">j</mi><mo id="S3.Ex1.m1.2.3.3.1.3.1" xref="S3.Ex1.m1.2.3.3.1.3.1.cmml">∈</mo><mi id="S3.Ex1.m1.2.3.3.1.3.3" xref="S3.Ex1.m1.2.3.3.1.3.3.cmml">C</mi></mrow></munder><mrow id="S3.Ex1.m1.2.3.3.2" xref="S3.Ex1.m1.2.3.3.2.cmml"><msub id="S3.Ex1.m1.2.3.3.2.2" xref="S3.Ex1.m1.2.3.3.2.2.cmml"><mi id="S3.Ex1.m1.2.3.3.2.2.2" xref="S3.Ex1.m1.2.3.3.2.2.2.cmml">v</mi><mi id="S3.Ex1.m1.2.3.3.2.2.3" xref="S3.Ex1.m1.2.3.3.2.2.3.cmml">i</mi></msub><mo id="S3.Ex1.m1.2.3.3.2.1" xref="S3.Ex1.m1.2.3.3.2.1.cmml">⁢</mo><mrow id="S3.Ex1.m1.2.3.3.2.3.2" xref="S3.Ex1.m1.2.3.3.2.cmml"><mo id="S3.Ex1.m1.2.3.3.2.3.2.1" stretchy="false" xref="S3.Ex1.m1.2.3.3.2.cmml">(</mo><mi id="S3.Ex1.m1.2.2" xref="S3.Ex1.m1.2.2.cmml">j</mi><mo id="S3.Ex1.m1.2.3.3.2.3.2.2" stretchy="false" xref="S3.Ex1.m1.2.3.3.2.cmml">)</mo></mrow><mo id="S3.Ex1.m1.2.3.3.2.1a" xref="S3.Ex1.m1.2.3.3.2.1.cmml">⁢</mo><mtext id="S3.Ex1.m1.2.3.3.2.4" xref="S3.Ex1.m1.2.3.3.2.4a.cmml">.</mtext></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex1.m1.2b"><apply id="S3.Ex1.m1.2.3.cmml" xref="S3.Ex1.m1.2.3"><csymbol cd="latexml" id="S3.Ex1.m1.2.3.1.cmml" xref="S3.Ex1.m1.2.3.1">assign</csymbol><apply id="S3.Ex1.m1.2.3.2.cmml" xref="S3.Ex1.m1.2.3.2"><times id="S3.Ex1.m1.2.3.2.1.cmml" xref="S3.Ex1.m1.2.3.2.1"></times><apply id="S3.Ex1.m1.2.3.2.2.cmml" xref="S3.Ex1.m1.2.3.2.2"><csymbol cd="ambiguous" id="S3.Ex1.m1.2.3.2.2.1.cmml" xref="S3.Ex1.m1.2.3.2.2">subscript</csymbol><ci id="S3.Ex1.m1.2.3.2.2.2.cmml" xref="S3.Ex1.m1.2.3.2.2.2">𝑢</ci><ci id="S3.Ex1.m1.2.3.2.2.3.cmml" xref="S3.Ex1.m1.2.3.2.2.3">𝑖</ci></apply><ci id="S3.Ex1.m1.1.1.cmml" xref="S3.Ex1.m1.1.1">𝐶</ci></apply><apply id="S3.Ex1.m1.2.3.3.cmml" xref="S3.Ex1.m1.2.3.3"><apply id="S3.Ex1.m1.2.3.3.1.cmml" xref="S3.Ex1.m1.2.3.3.1"><csymbol cd="ambiguous" id="S3.Ex1.m1.2.3.3.1.1.cmml" xref="S3.Ex1.m1.2.3.3.1">subscript</csymbol><sum id="S3.Ex1.m1.2.3.3.1.2.cmml" xref="S3.Ex1.m1.2.3.3.1.2"></sum><apply id="S3.Ex1.m1.2.3.3.1.3.cmml" xref="S3.Ex1.m1.2.3.3.1.3"><in id="S3.Ex1.m1.2.3.3.1.3.1.cmml" xref="S3.Ex1.m1.2.3.3.1.3.1"></in><ci id="S3.Ex1.m1.2.3.3.1.3.2.cmml" xref="S3.Ex1.m1.2.3.3.1.3.2">𝑗</ci><ci id="S3.Ex1.m1.2.3.3.1.3.3.cmml" xref="S3.Ex1.m1.2.3.3.1.3.3">𝐶</ci></apply></apply><apply id="S3.Ex1.m1.2.3.3.2.cmml" xref="S3.Ex1.m1.2.3.3.2"><times id="S3.Ex1.m1.2.3.3.2.1.cmml" xref="S3.Ex1.m1.2.3.3.2.1"></times><apply id="S3.Ex1.m1.2.3.3.2.2.cmml" xref="S3.Ex1.m1.2.3.3.2.2"><csymbol cd="ambiguous" id="S3.Ex1.m1.2.3.3.2.2.1.cmml" xref="S3.Ex1.m1.2.3.3.2.2">subscript</csymbol><ci id="S3.Ex1.m1.2.3.3.2.2.2.cmml" xref="S3.Ex1.m1.2.3.3.2.2.2">𝑣</ci><ci id="S3.Ex1.m1.2.3.3.2.2.3.cmml" xref="S3.Ex1.m1.2.3.3.2.2.3">𝑖</ci></apply><ci id="S3.Ex1.m1.2.2.cmml" xref="S3.Ex1.m1.2.2">𝑗</ci><ci id="S3.Ex1.m1.2.3.3.2.4a.cmml" xref="S3.Ex1.m1.2.3.3.2.4"><mtext id="S3.Ex1.m1.2.3.3.2.4.cmml" xref="S3.Ex1.m1.2.3.3.2.4">.</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex1.m1.2c">u_{i}(C):=\sum_{j\in C}v_{i}(j)\text{.}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m1.2d">italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_C ) := ∑ start_POSTSUBSCRIPT italic_j ∈ italic_C end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.1">In words, the utility of a coalition is derived from single-agent values, which are aggregated by summing the values of the agents in this coalition. Since valuation functions fully specify an ASHG, we also speak of the ASHG <math alttext="(N,v)" class="ltx_Math" display="inline" id="S3.p3.1.m1.2"><semantics id="S3.p3.1.m1.2a"><mrow id="S3.p3.1.m1.2.3.2" xref="S3.p3.1.m1.2.3.1.cmml"><mo id="S3.p3.1.m1.2.3.2.1" stretchy="false" xref="S3.p3.1.m1.2.3.1.cmml">(</mo><mi id="S3.p3.1.m1.1.1" xref="S3.p3.1.m1.1.1.cmml">N</mi><mo id="S3.p3.1.m1.2.3.2.2" xref="S3.p3.1.m1.2.3.1.cmml">,</mo><mi id="S3.p3.1.m1.2.2" xref="S3.p3.1.m1.2.2.cmml">v</mi><mo id="S3.p3.1.m1.2.3.2.3" stretchy="false" xref="S3.p3.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.1.m1.2b"><interval closure="open" id="S3.p3.1.m1.2.3.1.cmml" xref="S3.p3.1.m1.2.3.2"><ci id="S3.p3.1.m1.1.1.cmml" xref="S3.p3.1.m1.1.1">𝑁</ci><ci id="S3.p3.1.m1.2.2.cmml" xref="S3.p3.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.2d">( italic_N , italic_v )</annotation></semantics></math>. Note that ASHGs can be encoded by a weighted graph where agents are vertices and edge weights are given by the valuations.</p> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.12">We extend utilities over coalitions to utilities over partitions by defining <math alttext="u_{i}(\pi):=u_{i}(\pi(i))" class="ltx_Math" display="inline" id="S3.p4.1.m1.3"><semantics id="S3.p4.1.m1.3a"><mrow id="S3.p4.1.m1.3.3" xref="S3.p4.1.m1.3.3.cmml"><mrow id="S3.p4.1.m1.3.3.3" xref="S3.p4.1.m1.3.3.3.cmml"><msub id="S3.p4.1.m1.3.3.3.2" xref="S3.p4.1.m1.3.3.3.2.cmml"><mi id="S3.p4.1.m1.3.3.3.2.2" xref="S3.p4.1.m1.3.3.3.2.2.cmml">u</mi><mi id="S3.p4.1.m1.3.3.3.2.3" xref="S3.p4.1.m1.3.3.3.2.3.cmml">i</mi></msub><mo id="S3.p4.1.m1.3.3.3.1" xref="S3.p4.1.m1.3.3.3.1.cmml">⁢</mo><mrow id="S3.p4.1.m1.3.3.3.3.2" xref="S3.p4.1.m1.3.3.3.cmml"><mo id="S3.p4.1.m1.3.3.3.3.2.1" stretchy="false" xref="S3.p4.1.m1.3.3.3.cmml">(</mo><mi id="S3.p4.1.m1.1.1" xref="S3.p4.1.m1.1.1.cmml">π</mi><mo id="S3.p4.1.m1.3.3.3.3.2.2" rspace="0.278em" stretchy="false" xref="S3.p4.1.m1.3.3.3.cmml">)</mo></mrow></mrow><mo id="S3.p4.1.m1.3.3.2" rspace="0.278em" xref="S3.p4.1.m1.3.3.2.cmml">:=</mo><mrow id="S3.p4.1.m1.3.3.1" xref="S3.p4.1.m1.3.3.1.cmml"><msub id="S3.p4.1.m1.3.3.1.3" xref="S3.p4.1.m1.3.3.1.3.cmml"><mi id="S3.p4.1.m1.3.3.1.3.2" xref="S3.p4.1.m1.3.3.1.3.2.cmml">u</mi><mi id="S3.p4.1.m1.3.3.1.3.3" xref="S3.p4.1.m1.3.3.1.3.3.cmml">i</mi></msub><mo id="S3.p4.1.m1.3.3.1.2" xref="S3.p4.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S3.p4.1.m1.3.3.1.1.1" xref="S3.p4.1.m1.3.3.1.1.1.1.cmml"><mo id="S3.p4.1.m1.3.3.1.1.1.2" stretchy="false" xref="S3.p4.1.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.p4.1.m1.3.3.1.1.1.1" xref="S3.p4.1.m1.3.3.1.1.1.1.cmml"><mi id="S3.p4.1.m1.3.3.1.1.1.1.2" xref="S3.p4.1.m1.3.3.1.1.1.1.2.cmml">π</mi><mo id="S3.p4.1.m1.3.3.1.1.1.1.1" xref="S3.p4.1.m1.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.p4.1.m1.3.3.1.1.1.1.3.2" xref="S3.p4.1.m1.3.3.1.1.1.1.cmml"><mo id="S3.p4.1.m1.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S3.p4.1.m1.3.3.1.1.1.1.cmml">(</mo><mi id="S3.p4.1.m1.2.2" xref="S3.p4.1.m1.2.2.cmml">i</mi><mo id="S3.p4.1.m1.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S3.p4.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.p4.1.m1.3.3.1.1.1.3" stretchy="false" xref="S3.p4.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.3b"><apply id="S3.p4.1.m1.3.3.cmml" xref="S3.p4.1.m1.3.3"><csymbol cd="latexml" id="S3.p4.1.m1.3.3.2.cmml" xref="S3.p4.1.m1.3.3.2">assign</csymbol><apply id="S3.p4.1.m1.3.3.3.cmml" xref="S3.p4.1.m1.3.3.3"><times id="S3.p4.1.m1.3.3.3.1.cmml" xref="S3.p4.1.m1.3.3.3.1"></times><apply id="S3.p4.1.m1.3.3.3.2.cmml" xref="S3.p4.1.m1.3.3.3.2"><csymbol cd="ambiguous" id="S3.p4.1.m1.3.3.3.2.1.cmml" xref="S3.p4.1.m1.3.3.3.2">subscript</csymbol><ci id="S3.p4.1.m1.3.3.3.2.2.cmml" xref="S3.p4.1.m1.3.3.3.2.2">𝑢</ci><ci id="S3.p4.1.m1.3.3.3.2.3.cmml" xref="S3.p4.1.m1.3.3.3.2.3">𝑖</ci></apply><ci id="S3.p4.1.m1.1.1.cmml" xref="S3.p4.1.m1.1.1">𝜋</ci></apply><apply id="S3.p4.1.m1.3.3.1.cmml" xref="S3.p4.1.m1.3.3.1"><times id="S3.p4.1.m1.3.3.1.2.cmml" xref="S3.p4.1.m1.3.3.1.2"></times><apply id="S3.p4.1.m1.3.3.1.3.cmml" xref="S3.p4.1.m1.3.3.1.3"><csymbol cd="ambiguous" id="S3.p4.1.m1.3.3.1.3.1.cmml" xref="S3.p4.1.m1.3.3.1.3">subscript</csymbol><ci id="S3.p4.1.m1.3.3.1.3.2.cmml" xref="S3.p4.1.m1.3.3.1.3.2">𝑢</ci><ci id="S3.p4.1.m1.3.3.1.3.3.cmml" xref="S3.p4.1.m1.3.3.1.3.3">𝑖</ci></apply><apply id="S3.p4.1.m1.3.3.1.1.1.1.cmml" xref="S3.p4.1.m1.3.3.1.1.1"><times id="S3.p4.1.m1.3.3.1.1.1.1.1.cmml" xref="S3.p4.1.m1.3.3.1.1.1.1.1"></times><ci id="S3.p4.1.m1.3.3.1.1.1.1.2.cmml" xref="S3.p4.1.m1.3.3.1.1.1.1.2">𝜋</ci><ci id="S3.p4.1.m1.2.2.cmml" xref="S3.p4.1.m1.2.2">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.3c">u_{i}(\pi):=u_{i}(\pi(i))</annotation><annotation encoding="application/x-llamapun" id="S3.p4.1.m1.3d">italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_π ) := italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_π ( italic_i ) )</annotation></semantics></math>. Given coalitions <math alttext="C,C^{\prime}\in\mathcal{N}_{i}" class="ltx_Math" display="inline" id="S3.p4.2.m2.2"><semantics id="S3.p4.2.m2.2a"><mrow id="S3.p4.2.m2.2.2" xref="S3.p4.2.m2.2.2.cmml"><mrow id="S3.p4.2.m2.2.2.1.1" xref="S3.p4.2.m2.2.2.1.2.cmml"><mi id="S3.p4.2.m2.1.1" xref="S3.p4.2.m2.1.1.cmml">C</mi><mo id="S3.p4.2.m2.2.2.1.1.2" xref="S3.p4.2.m2.2.2.1.2.cmml">,</mo><msup id="S3.p4.2.m2.2.2.1.1.1" xref="S3.p4.2.m2.2.2.1.1.1.cmml"><mi id="S3.p4.2.m2.2.2.1.1.1.2" xref="S3.p4.2.m2.2.2.1.1.1.2.cmml">C</mi><mo id="S3.p4.2.m2.2.2.1.1.1.3" xref="S3.p4.2.m2.2.2.1.1.1.3.cmml">′</mo></msup></mrow><mo id="S3.p4.2.m2.2.2.2" xref="S3.p4.2.m2.2.2.2.cmml">∈</mo><msub id="S3.p4.2.m2.2.2.3" xref="S3.p4.2.m2.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p4.2.m2.2.2.3.2" xref="S3.p4.2.m2.2.2.3.2.cmml">𝒩</mi><mi id="S3.p4.2.m2.2.2.3.3" xref="S3.p4.2.m2.2.2.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.2.m2.2b"><apply id="S3.p4.2.m2.2.2.cmml" xref="S3.p4.2.m2.2.2"><in id="S3.p4.2.m2.2.2.2.cmml" xref="S3.p4.2.m2.2.2.2"></in><list id="S3.p4.2.m2.2.2.1.2.cmml" xref="S3.p4.2.m2.2.2.1.1"><ci id="S3.p4.2.m2.1.1.cmml" xref="S3.p4.2.m2.1.1">𝐶</ci><apply id="S3.p4.2.m2.2.2.1.1.1.cmml" xref="S3.p4.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.p4.2.m2.2.2.1.1.1.1.cmml" xref="S3.p4.2.m2.2.2.1.1.1">superscript</csymbol><ci id="S3.p4.2.m2.2.2.1.1.1.2.cmml" xref="S3.p4.2.m2.2.2.1.1.1.2">𝐶</ci><ci id="S3.p4.2.m2.2.2.1.1.1.3.cmml" xref="S3.p4.2.m2.2.2.1.1.1.3">′</ci></apply></list><apply id="S3.p4.2.m2.2.2.3.cmml" xref="S3.p4.2.m2.2.2.3"><csymbol cd="ambiguous" id="S3.p4.2.m2.2.2.3.1.cmml" xref="S3.p4.2.m2.2.2.3">subscript</csymbol><ci id="S3.p4.2.m2.2.2.3.2.cmml" xref="S3.p4.2.m2.2.2.3.2">𝒩</ci><ci id="S3.p4.2.m2.2.2.3.3.cmml" xref="S3.p4.2.m2.2.2.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.2.m2.2c">C,C^{\prime}\in\mathcal{N}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.2.m2.2d">italic_C , italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, we say that agent <math alttext="i" class="ltx_Math" display="inline" id="S3.p4.3.m3.1"><semantics id="S3.p4.3.m3.1a"><mi id="S3.p4.3.m3.1.1" xref="S3.p4.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p4.3.m3.1b"><ci id="S3.p4.3.m3.1.1.cmml" xref="S3.p4.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p4.3.m3.1d">italic_i</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S3.p4.12.1">prefers</em> <math alttext="C" class="ltx_Math" display="inline" id="S3.p4.4.m4.1"><semantics id="S3.p4.4.m4.1a"><mi id="S3.p4.4.m4.1.1" xref="S3.p4.4.m4.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.p4.4.m4.1b"><ci id="S3.p4.4.m4.1.1.cmml" xref="S3.p4.4.m4.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.4.m4.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.p4.4.m4.1d">italic_C</annotation></semantics></math> over <math alttext="C^{\prime}" class="ltx_Math" display="inline" id="S3.p4.5.m5.1"><semantics id="S3.p4.5.m5.1a"><msup id="S3.p4.5.m5.1.1" xref="S3.p4.5.m5.1.1.cmml"><mi id="S3.p4.5.m5.1.1.2" xref="S3.p4.5.m5.1.1.2.cmml">C</mi><mo id="S3.p4.5.m5.1.1.3" xref="S3.p4.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.p4.5.m5.1b"><apply id="S3.p4.5.m5.1.1.cmml" xref="S3.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S3.p4.5.m5.1.1.1.cmml" xref="S3.p4.5.m5.1.1">superscript</csymbol><ci id="S3.p4.5.m5.1.1.2.cmml" xref="S3.p4.5.m5.1.1.2">𝐶</ci><ci id="S3.p4.5.m5.1.1.3.cmml" xref="S3.p4.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.5.m5.1c">C^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.5.m5.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> if <math alttext="u_{i}(C)\geq u_{i}(C^{\prime})" class="ltx_Math" display="inline" id="S3.p4.6.m6.2"><semantics id="S3.p4.6.m6.2a"><mrow id="S3.p4.6.m6.2.2" xref="S3.p4.6.m6.2.2.cmml"><mrow id="S3.p4.6.m6.2.2.3" xref="S3.p4.6.m6.2.2.3.cmml"><msub id="S3.p4.6.m6.2.2.3.2" xref="S3.p4.6.m6.2.2.3.2.cmml"><mi id="S3.p4.6.m6.2.2.3.2.2" xref="S3.p4.6.m6.2.2.3.2.2.cmml">u</mi><mi id="S3.p4.6.m6.2.2.3.2.3" xref="S3.p4.6.m6.2.2.3.2.3.cmml">i</mi></msub><mo id="S3.p4.6.m6.2.2.3.1" xref="S3.p4.6.m6.2.2.3.1.cmml">⁢</mo><mrow id="S3.p4.6.m6.2.2.3.3.2" xref="S3.p4.6.m6.2.2.3.cmml"><mo id="S3.p4.6.m6.2.2.3.3.2.1" stretchy="false" xref="S3.p4.6.m6.2.2.3.cmml">(</mo><mi id="S3.p4.6.m6.1.1" xref="S3.p4.6.m6.1.1.cmml">C</mi><mo id="S3.p4.6.m6.2.2.3.3.2.2" stretchy="false" xref="S3.p4.6.m6.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.p4.6.m6.2.2.2" xref="S3.p4.6.m6.2.2.2.cmml">≥</mo><mrow id="S3.p4.6.m6.2.2.1" xref="S3.p4.6.m6.2.2.1.cmml"><msub id="S3.p4.6.m6.2.2.1.3" xref="S3.p4.6.m6.2.2.1.3.cmml"><mi id="S3.p4.6.m6.2.2.1.3.2" xref="S3.p4.6.m6.2.2.1.3.2.cmml">u</mi><mi id="S3.p4.6.m6.2.2.1.3.3" xref="S3.p4.6.m6.2.2.1.3.3.cmml">i</mi></msub><mo id="S3.p4.6.m6.2.2.1.2" xref="S3.p4.6.m6.2.2.1.2.cmml">⁢</mo><mrow id="S3.p4.6.m6.2.2.1.1.1" xref="S3.p4.6.m6.2.2.1.1.1.1.cmml"><mo id="S3.p4.6.m6.2.2.1.1.1.2" stretchy="false" xref="S3.p4.6.m6.2.2.1.1.1.1.cmml">(</mo><msup id="S3.p4.6.m6.2.2.1.1.1.1" xref="S3.p4.6.m6.2.2.1.1.1.1.cmml"><mi id="S3.p4.6.m6.2.2.1.1.1.1.2" xref="S3.p4.6.m6.2.2.1.1.1.1.2.cmml">C</mi><mo id="S3.p4.6.m6.2.2.1.1.1.1.3" xref="S3.p4.6.m6.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.p4.6.m6.2.2.1.1.1.3" stretchy="false" xref="S3.p4.6.m6.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.6.m6.2b"><apply id="S3.p4.6.m6.2.2.cmml" xref="S3.p4.6.m6.2.2"><geq id="S3.p4.6.m6.2.2.2.cmml" xref="S3.p4.6.m6.2.2.2"></geq><apply id="S3.p4.6.m6.2.2.3.cmml" xref="S3.p4.6.m6.2.2.3"><times id="S3.p4.6.m6.2.2.3.1.cmml" xref="S3.p4.6.m6.2.2.3.1"></times><apply id="S3.p4.6.m6.2.2.3.2.cmml" xref="S3.p4.6.m6.2.2.3.2"><csymbol cd="ambiguous" id="S3.p4.6.m6.2.2.3.2.1.cmml" xref="S3.p4.6.m6.2.2.3.2">subscript</csymbol><ci id="S3.p4.6.m6.2.2.3.2.2.cmml" xref="S3.p4.6.m6.2.2.3.2.2">𝑢</ci><ci id="S3.p4.6.m6.2.2.3.2.3.cmml" xref="S3.p4.6.m6.2.2.3.2.3">𝑖</ci></apply><ci id="S3.p4.6.m6.1.1.cmml" xref="S3.p4.6.m6.1.1">𝐶</ci></apply><apply id="S3.p4.6.m6.2.2.1.cmml" xref="S3.p4.6.m6.2.2.1"><times id="S3.p4.6.m6.2.2.1.2.cmml" xref="S3.p4.6.m6.2.2.1.2"></times><apply id="S3.p4.6.m6.2.2.1.3.cmml" xref="S3.p4.6.m6.2.2.1.3"><csymbol cd="ambiguous" id="S3.p4.6.m6.2.2.1.3.1.cmml" xref="S3.p4.6.m6.2.2.1.3">subscript</csymbol><ci id="S3.p4.6.m6.2.2.1.3.2.cmml" xref="S3.p4.6.m6.2.2.1.3.2">𝑢</ci><ci id="S3.p4.6.m6.2.2.1.3.3.cmml" xref="S3.p4.6.m6.2.2.1.3.3">𝑖</ci></apply><apply id="S3.p4.6.m6.2.2.1.1.1.1.cmml" xref="S3.p4.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.p4.6.m6.2.2.1.1.1.1.1.cmml" xref="S3.p4.6.m6.2.2.1.1.1">superscript</csymbol><ci id="S3.p4.6.m6.2.2.1.1.1.1.2.cmml" xref="S3.p4.6.m6.2.2.1.1.1.1.2">𝐶</ci><ci id="S3.p4.6.m6.2.2.1.1.1.1.3.cmml" xref="S3.p4.6.m6.2.2.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.6.m6.2c">u_{i}(C)\geq u_{i}(C^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.p4.6.m6.2d">italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_C ) ≥ italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>. Moreover, we say that <math alttext="i" class="ltx_Math" display="inline" id="S3.p4.7.m7.1"><semantics id="S3.p4.7.m7.1a"><mi id="S3.p4.7.m7.1.1" xref="S3.p4.7.m7.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p4.7.m7.1b"><ci id="S3.p4.7.m7.1.1.cmml" xref="S3.p4.7.m7.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.7.m7.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p4.7.m7.1d">italic_i</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S3.p4.12.2">strictly prefers</em> <math alttext="C" class="ltx_Math" display="inline" id="S3.p4.8.m8.1"><semantics id="S3.p4.8.m8.1a"><mi id="S3.p4.8.m8.1.1" xref="S3.p4.8.m8.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.p4.8.m8.1b"><ci id="S3.p4.8.m8.1.1.cmml" xref="S3.p4.8.m8.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.8.m8.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.p4.8.m8.1d">italic_C</annotation></semantics></math> over <math alttext="C^{\prime}" class="ltx_Math" display="inline" id="S3.p4.9.m9.1"><semantics id="S3.p4.9.m9.1a"><msup id="S3.p4.9.m9.1.1" xref="S3.p4.9.m9.1.1.cmml"><mi id="S3.p4.9.m9.1.1.2" xref="S3.p4.9.m9.1.1.2.cmml">C</mi><mo id="S3.p4.9.m9.1.1.3" xref="S3.p4.9.m9.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.p4.9.m9.1b"><apply id="S3.p4.9.m9.1.1.cmml" xref="S3.p4.9.m9.1.1"><csymbol cd="ambiguous" id="S3.p4.9.m9.1.1.1.cmml" xref="S3.p4.9.m9.1.1">superscript</csymbol><ci id="S3.p4.9.m9.1.1.2.cmml" xref="S3.p4.9.m9.1.1.2">𝐶</ci><ci id="S3.p4.9.m9.1.1.3.cmml" xref="S3.p4.9.m9.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.9.m9.1c">C^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.9.m9.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> if <math alttext="u_{i}(C)>u_{i}(C^{\prime})" class="ltx_Math" display="inline" id="S3.p4.10.m10.2"><semantics id="S3.p4.10.m10.2a"><mrow id="S3.p4.10.m10.2.2" xref="S3.p4.10.m10.2.2.cmml"><mrow id="S3.p4.10.m10.2.2.3" xref="S3.p4.10.m10.2.2.3.cmml"><msub id="S3.p4.10.m10.2.2.3.2" xref="S3.p4.10.m10.2.2.3.2.cmml"><mi id="S3.p4.10.m10.2.2.3.2.2" xref="S3.p4.10.m10.2.2.3.2.2.cmml">u</mi><mi id="S3.p4.10.m10.2.2.3.2.3" xref="S3.p4.10.m10.2.2.3.2.3.cmml">i</mi></msub><mo id="S3.p4.10.m10.2.2.3.1" xref="S3.p4.10.m10.2.2.3.1.cmml">⁢</mo><mrow id="S3.p4.10.m10.2.2.3.3.2" xref="S3.p4.10.m10.2.2.3.cmml"><mo id="S3.p4.10.m10.2.2.3.3.2.1" stretchy="false" xref="S3.p4.10.m10.2.2.3.cmml">(</mo><mi id="S3.p4.10.m10.1.1" xref="S3.p4.10.m10.1.1.cmml">C</mi><mo id="S3.p4.10.m10.2.2.3.3.2.2" stretchy="false" xref="S3.p4.10.m10.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.p4.10.m10.2.2.2" xref="S3.p4.10.m10.2.2.2.cmml">></mo><mrow id="S3.p4.10.m10.2.2.1" xref="S3.p4.10.m10.2.2.1.cmml"><msub id="S3.p4.10.m10.2.2.1.3" xref="S3.p4.10.m10.2.2.1.3.cmml"><mi id="S3.p4.10.m10.2.2.1.3.2" xref="S3.p4.10.m10.2.2.1.3.2.cmml">u</mi><mi id="S3.p4.10.m10.2.2.1.3.3" xref="S3.p4.10.m10.2.2.1.3.3.cmml">i</mi></msub><mo id="S3.p4.10.m10.2.2.1.2" xref="S3.p4.10.m10.2.2.1.2.cmml">⁢</mo><mrow id="S3.p4.10.m10.2.2.1.1.1" xref="S3.p4.10.m10.2.2.1.1.1.1.cmml"><mo id="S3.p4.10.m10.2.2.1.1.1.2" stretchy="false" xref="S3.p4.10.m10.2.2.1.1.1.1.cmml">(</mo><msup id="S3.p4.10.m10.2.2.1.1.1.1" xref="S3.p4.10.m10.2.2.1.1.1.1.cmml"><mi id="S3.p4.10.m10.2.2.1.1.1.1.2" xref="S3.p4.10.m10.2.2.1.1.1.1.2.cmml">C</mi><mo id="S3.p4.10.m10.2.2.1.1.1.1.3" xref="S3.p4.10.m10.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.p4.10.m10.2.2.1.1.1.3" stretchy="false" xref="S3.p4.10.m10.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.10.m10.2b"><apply id="S3.p4.10.m10.2.2.cmml" xref="S3.p4.10.m10.2.2"><gt id="S3.p4.10.m10.2.2.2.cmml" xref="S3.p4.10.m10.2.2.2"></gt><apply id="S3.p4.10.m10.2.2.3.cmml" xref="S3.p4.10.m10.2.2.3"><times id="S3.p4.10.m10.2.2.3.1.cmml" xref="S3.p4.10.m10.2.2.3.1"></times><apply id="S3.p4.10.m10.2.2.3.2.cmml" xref="S3.p4.10.m10.2.2.3.2"><csymbol cd="ambiguous" id="S3.p4.10.m10.2.2.3.2.1.cmml" xref="S3.p4.10.m10.2.2.3.2">subscript</csymbol><ci id="S3.p4.10.m10.2.2.3.2.2.cmml" xref="S3.p4.10.m10.2.2.3.2.2">𝑢</ci><ci id="S3.p4.10.m10.2.2.3.2.3.cmml" xref="S3.p4.10.m10.2.2.3.2.3">𝑖</ci></apply><ci id="S3.p4.10.m10.1.1.cmml" xref="S3.p4.10.m10.1.1">𝐶</ci></apply><apply id="S3.p4.10.m10.2.2.1.cmml" xref="S3.p4.10.m10.2.2.1"><times id="S3.p4.10.m10.2.2.1.2.cmml" xref="S3.p4.10.m10.2.2.1.2"></times><apply id="S3.p4.10.m10.2.2.1.3.cmml" xref="S3.p4.10.m10.2.2.1.3"><csymbol cd="ambiguous" id="S3.p4.10.m10.2.2.1.3.1.cmml" xref="S3.p4.10.m10.2.2.1.3">subscript</csymbol><ci id="S3.p4.10.m10.2.2.1.3.2.cmml" xref="S3.p4.10.m10.2.2.1.3.2">𝑢</ci><ci id="S3.p4.10.m10.2.2.1.3.3.cmml" xref="S3.p4.10.m10.2.2.1.3.3">𝑖</ci></apply><apply id="S3.p4.10.m10.2.2.1.1.1.1.cmml" xref="S3.p4.10.m10.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.p4.10.m10.2.2.1.1.1.1.1.cmml" xref="S3.p4.10.m10.2.2.1.1.1">superscript</csymbol><ci id="S3.p4.10.m10.2.2.1.1.1.1.2.cmml" xref="S3.p4.10.m10.2.2.1.1.1.1.2">𝐶</ci><ci id="S3.p4.10.m10.2.2.1.1.1.1.3.cmml" xref="S3.p4.10.m10.2.2.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.10.m10.2c">u_{i}(C)>u_{i}(C^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.p4.10.m10.2d">italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_C ) > italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>. We use the same terminology for partitions. Given an ASHG <math alttext="(N,v)" class="ltx_Math" display="inline" id="S3.p4.11.m11.2"><semantics id="S3.p4.11.m11.2a"><mrow id="S3.p4.11.m11.2.3.2" xref="S3.p4.11.m11.2.3.1.cmml"><mo id="S3.p4.11.m11.2.3.2.1" stretchy="false" xref="S3.p4.11.m11.2.3.1.cmml">(</mo><mi id="S3.p4.11.m11.1.1" xref="S3.p4.11.m11.1.1.cmml">N</mi><mo id="S3.p4.11.m11.2.3.2.2" xref="S3.p4.11.m11.2.3.1.cmml">,</mo><mi id="S3.p4.11.m11.2.2" xref="S3.p4.11.m11.2.2.cmml">v</mi><mo id="S3.p4.11.m11.2.3.2.3" stretchy="false" xref="S3.p4.11.m11.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.11.m11.2b"><interval closure="open" id="S3.p4.11.m11.2.3.1.cmml" xref="S3.p4.11.m11.2.3.2"><ci id="S3.p4.11.m11.1.1.cmml" xref="S3.p4.11.m11.1.1">𝑁</ci><ci id="S3.p4.11.m11.2.2.cmml" xref="S3.p4.11.m11.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.11.m11.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S3.p4.11.m11.2d">( italic_N , italic_v )</annotation></semantics></math> and a partition <math alttext="\pi" class="ltx_Math" display="inline" id="S3.p4.12.m12.1"><semantics id="S3.p4.12.m12.1a"><mi id="S3.p4.12.m12.1.1" xref="S3.p4.12.m12.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S3.p4.12.m12.1b"><ci id="S3.p4.12.m12.1.1.cmml" xref="S3.p4.12.m12.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.12.m12.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S3.p4.12.m12.1d">italic_π</annotation></semantics></math>, we define its <em class="ltx_emph ltx_font_italic" id="S3.p4.12.3">social welfare</em> as</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{SW}(\pi):=\sum_{i\in N}u_{i}(\pi)=\sum_{C\in\pi\colon i,j\in C}v_{i}(% j)\text{.}" class="ltx_Math" display="block" id="S3.Ex2.m1.5"><semantics id="S3.Ex2.m1.5a"><mrow id="S3.Ex2.m1.5.6" xref="S3.Ex2.m1.5.6.cmml"><mrow id="S3.Ex2.m1.5.6.2" xref="S3.Ex2.m1.5.6.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex2.m1.5.6.2.2" xref="S3.Ex2.m1.5.6.2.2.cmml">𝒮</mi><mo id="S3.Ex2.m1.5.6.2.1" xref="S3.Ex2.m1.5.6.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex2.m1.5.6.2.3" xref="S3.Ex2.m1.5.6.2.3.cmml">𝒲</mi><mo id="S3.Ex2.m1.5.6.2.1a" xref="S3.Ex2.m1.5.6.2.1.cmml">⁢</mo><mrow id="S3.Ex2.m1.5.6.2.4.2" xref="S3.Ex2.m1.5.6.2.cmml"><mo id="S3.Ex2.m1.5.6.2.4.2.1" stretchy="false" xref="S3.Ex2.m1.5.6.2.cmml">(</mo><mi id="S3.Ex2.m1.3.3" xref="S3.Ex2.m1.3.3.cmml">π</mi><mo id="S3.Ex2.m1.5.6.2.4.2.2" rspace="0.278em" stretchy="false" xref="S3.Ex2.m1.5.6.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex2.m1.5.6.3" rspace="0.111em" xref="S3.Ex2.m1.5.6.3.cmml">:=</mo><mrow id="S3.Ex2.m1.5.6.4" xref="S3.Ex2.m1.5.6.4.cmml"><munder id="S3.Ex2.m1.5.6.4.1" xref="S3.Ex2.m1.5.6.4.1.cmml"><mo id="S3.Ex2.m1.5.6.4.1.2" movablelimits="false" xref="S3.Ex2.m1.5.6.4.1.2.cmml">∑</mo><mrow id="S3.Ex2.m1.5.6.4.1.3" xref="S3.Ex2.m1.5.6.4.1.3.cmml"><mi id="S3.Ex2.m1.5.6.4.1.3.2" xref="S3.Ex2.m1.5.6.4.1.3.2.cmml">i</mi><mo id="S3.Ex2.m1.5.6.4.1.3.1" xref="S3.Ex2.m1.5.6.4.1.3.1.cmml">∈</mo><mi id="S3.Ex2.m1.5.6.4.1.3.3" xref="S3.Ex2.m1.5.6.4.1.3.3.cmml">N</mi></mrow></munder><mrow id="S3.Ex2.m1.5.6.4.2" xref="S3.Ex2.m1.5.6.4.2.cmml"><msub id="S3.Ex2.m1.5.6.4.2.2" xref="S3.Ex2.m1.5.6.4.2.2.cmml"><mi id="S3.Ex2.m1.5.6.4.2.2.2" xref="S3.Ex2.m1.5.6.4.2.2.2.cmml">u</mi><mi id="S3.Ex2.m1.5.6.4.2.2.3" xref="S3.Ex2.m1.5.6.4.2.2.3.cmml">i</mi></msub><mo id="S3.Ex2.m1.5.6.4.2.1" xref="S3.Ex2.m1.5.6.4.2.1.cmml">⁢</mo><mrow id="S3.Ex2.m1.5.6.4.2.3.2" xref="S3.Ex2.m1.5.6.4.2.cmml"><mo id="S3.Ex2.m1.5.6.4.2.3.2.1" stretchy="false" xref="S3.Ex2.m1.5.6.4.2.cmml">(</mo><mi id="S3.Ex2.m1.4.4" xref="S3.Ex2.m1.4.4.cmml">π</mi><mo id="S3.Ex2.m1.5.6.4.2.3.2.2" stretchy="false" xref="S3.Ex2.m1.5.6.4.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex2.m1.5.6.5" rspace="0.111em" xref="S3.Ex2.m1.5.6.5.cmml">=</mo><mrow id="S3.Ex2.m1.5.6.6" xref="S3.Ex2.m1.5.6.6.cmml"><munder id="S3.Ex2.m1.5.6.6.1" xref="S3.Ex2.m1.5.6.6.1.cmml"><mo id="S3.Ex2.m1.5.6.6.1.2" movablelimits="false" xref="S3.Ex2.m1.5.6.6.1.2.cmml">∑</mo><mrow id="S3.Ex2.m1.2.2.2" xref="S3.Ex2.m1.2.2.2.cmml"><mrow id="S3.Ex2.m1.2.2.2.4" xref="S3.Ex2.m1.2.2.2.4.cmml"><mi id="S3.Ex2.m1.2.2.2.4.2" xref="S3.Ex2.m1.2.2.2.4.2.cmml">C</mi><mo id="S3.Ex2.m1.2.2.2.4.1" xref="S3.Ex2.m1.2.2.2.4.1.cmml">∈</mo><mi id="S3.Ex2.m1.2.2.2.4.3" xref="S3.Ex2.m1.2.2.2.4.3.cmml">π</mi></mrow><mo id="S3.Ex2.m1.2.2.2.3" lspace="0.278em" rspace="0.278em" xref="S3.Ex2.m1.2.2.2.3.cmml">:</mo><mrow id="S3.Ex2.m1.2.2.2.5" xref="S3.Ex2.m1.2.2.2.5.cmml"><mrow id="S3.Ex2.m1.2.2.2.5.2.2" xref="S3.Ex2.m1.2.2.2.5.2.1.cmml"><mi id="S3.Ex2.m1.1.1.1.1" xref="S3.Ex2.m1.1.1.1.1.cmml">i</mi><mo id="S3.Ex2.m1.2.2.2.5.2.2.1" xref="S3.Ex2.m1.2.2.2.5.2.1.cmml">,</mo><mi id="S3.Ex2.m1.2.2.2.2" xref="S3.Ex2.m1.2.2.2.2.cmml">j</mi></mrow><mo id="S3.Ex2.m1.2.2.2.5.1" xref="S3.Ex2.m1.2.2.2.5.1.cmml">∈</mo><mi id="S3.Ex2.m1.2.2.2.5.3" xref="S3.Ex2.m1.2.2.2.5.3.cmml">C</mi></mrow></mrow></munder><mrow id="S3.Ex2.m1.5.6.6.2" xref="S3.Ex2.m1.5.6.6.2.cmml"><msub id="S3.Ex2.m1.5.6.6.2.2" xref="S3.Ex2.m1.5.6.6.2.2.cmml"><mi id="S3.Ex2.m1.5.6.6.2.2.2" xref="S3.Ex2.m1.5.6.6.2.2.2.cmml">v</mi><mi id="S3.Ex2.m1.5.6.6.2.2.3" xref="S3.Ex2.m1.5.6.6.2.2.3.cmml">i</mi></msub><mo id="S3.Ex2.m1.5.6.6.2.1" xref="S3.Ex2.m1.5.6.6.2.1.cmml">⁢</mo><mrow id="S3.Ex2.m1.5.6.6.2.3.2" xref="S3.Ex2.m1.5.6.6.2.cmml"><mo id="S3.Ex2.m1.5.6.6.2.3.2.1" stretchy="false" xref="S3.Ex2.m1.5.6.6.2.cmml">(</mo><mi id="S3.Ex2.m1.5.5" xref="S3.Ex2.m1.5.5.cmml">j</mi><mo id="S3.Ex2.m1.5.6.6.2.3.2.2" stretchy="false" xref="S3.Ex2.m1.5.6.6.2.cmml">)</mo></mrow><mo id="S3.Ex2.m1.5.6.6.2.1a" xref="S3.Ex2.m1.5.6.6.2.1.cmml">⁢</mo><mtext id="S3.Ex2.m1.5.6.6.2.4" xref="S3.Ex2.m1.5.6.6.2.4a.cmml">.</mtext></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex2.m1.5b"><apply id="S3.Ex2.m1.5.6.cmml" xref="S3.Ex2.m1.5.6"><and id="S3.Ex2.m1.5.6a.cmml" xref="S3.Ex2.m1.5.6"></and><apply id="S3.Ex2.m1.5.6b.cmml" xref="S3.Ex2.m1.5.6"><csymbol cd="latexml" id="S3.Ex2.m1.5.6.3.cmml" xref="S3.Ex2.m1.5.6.3">assign</csymbol><apply id="S3.Ex2.m1.5.6.2.cmml" xref="S3.Ex2.m1.5.6.2"><times id="S3.Ex2.m1.5.6.2.1.cmml" xref="S3.Ex2.m1.5.6.2.1"></times><ci id="S3.Ex2.m1.5.6.2.2.cmml" xref="S3.Ex2.m1.5.6.2.2">𝒮</ci><ci id="S3.Ex2.m1.5.6.2.3.cmml" xref="S3.Ex2.m1.5.6.2.3">𝒲</ci><ci id="S3.Ex2.m1.3.3.cmml" xref="S3.Ex2.m1.3.3">𝜋</ci></apply><apply id="S3.Ex2.m1.5.6.4.cmml" xref="S3.Ex2.m1.5.6.4"><apply id="S3.Ex2.m1.5.6.4.1.cmml" xref="S3.Ex2.m1.5.6.4.1"><csymbol cd="ambiguous" id="S3.Ex2.m1.5.6.4.1.1.cmml" xref="S3.Ex2.m1.5.6.4.1">subscript</csymbol><sum id="S3.Ex2.m1.5.6.4.1.2.cmml" xref="S3.Ex2.m1.5.6.4.1.2"></sum><apply id="S3.Ex2.m1.5.6.4.1.3.cmml" xref="S3.Ex2.m1.5.6.4.1.3"><in id="S3.Ex2.m1.5.6.4.1.3.1.cmml" xref="S3.Ex2.m1.5.6.4.1.3.1"></in><ci id="S3.Ex2.m1.5.6.4.1.3.2.cmml" xref="S3.Ex2.m1.5.6.4.1.3.2">𝑖</ci><ci id="S3.Ex2.m1.5.6.4.1.3.3.cmml" xref="S3.Ex2.m1.5.6.4.1.3.3">𝑁</ci></apply></apply><apply id="S3.Ex2.m1.5.6.4.2.cmml" xref="S3.Ex2.m1.5.6.4.2"><times id="S3.Ex2.m1.5.6.4.2.1.cmml" xref="S3.Ex2.m1.5.6.4.2.1"></times><apply id="S3.Ex2.m1.5.6.4.2.2.cmml" xref="S3.Ex2.m1.5.6.4.2.2"><csymbol cd="ambiguous" id="S3.Ex2.m1.5.6.4.2.2.1.cmml" xref="S3.Ex2.m1.5.6.4.2.2">subscript</csymbol><ci id="S3.Ex2.m1.5.6.4.2.2.2.cmml" xref="S3.Ex2.m1.5.6.4.2.2.2">𝑢</ci><ci id="S3.Ex2.m1.5.6.4.2.2.3.cmml" xref="S3.Ex2.m1.5.6.4.2.2.3">𝑖</ci></apply><ci id="S3.Ex2.m1.4.4.cmml" xref="S3.Ex2.m1.4.4">𝜋</ci></apply></apply></apply><apply id="S3.Ex2.m1.5.6c.cmml" xref="S3.Ex2.m1.5.6"><eq id="S3.Ex2.m1.5.6.5.cmml" xref="S3.Ex2.m1.5.6.5"></eq><share href="https://arxiv.org/html/2503.06017v1#S3.Ex2.m1.5.6.4.cmml" id="S3.Ex2.m1.5.6d.cmml" xref="S3.Ex2.m1.5.6"></share><apply id="S3.Ex2.m1.5.6.6.cmml" xref="S3.Ex2.m1.5.6.6"><apply id="S3.Ex2.m1.5.6.6.1.cmml" xref="S3.Ex2.m1.5.6.6.1"><csymbol cd="ambiguous" id="S3.Ex2.m1.5.6.6.1.1.cmml" xref="S3.Ex2.m1.5.6.6.1">subscript</csymbol><sum id="S3.Ex2.m1.5.6.6.1.2.cmml" xref="S3.Ex2.m1.5.6.6.1.2"></sum><apply id="S3.Ex2.m1.2.2.2.cmml" xref="S3.Ex2.m1.2.2.2"><ci id="S3.Ex2.m1.2.2.2.3.cmml" xref="S3.Ex2.m1.2.2.2.3">:</ci><apply id="S3.Ex2.m1.2.2.2.4.cmml" xref="S3.Ex2.m1.2.2.2.4"><in id="S3.Ex2.m1.2.2.2.4.1.cmml" xref="S3.Ex2.m1.2.2.2.4.1"></in><ci id="S3.Ex2.m1.2.2.2.4.2.cmml" xref="S3.Ex2.m1.2.2.2.4.2">𝐶</ci><ci id="S3.Ex2.m1.2.2.2.4.3.cmml" xref="S3.Ex2.m1.2.2.2.4.3">𝜋</ci></apply><apply id="S3.Ex2.m1.2.2.2.5.cmml" xref="S3.Ex2.m1.2.2.2.5"><in id="S3.Ex2.m1.2.2.2.5.1.cmml" xref="S3.Ex2.m1.2.2.2.5.1"></in><list id="S3.Ex2.m1.2.2.2.5.2.1.cmml" xref="S3.Ex2.m1.2.2.2.5.2.2"><ci id="S3.Ex2.m1.1.1.1.1.cmml" xref="S3.Ex2.m1.1.1.1.1">𝑖</ci><ci id="S3.Ex2.m1.2.2.2.2.cmml" xref="S3.Ex2.m1.2.2.2.2">𝑗</ci></list><ci id="S3.Ex2.m1.2.2.2.5.3.cmml" xref="S3.Ex2.m1.2.2.2.5.3">𝐶</ci></apply></apply></apply><apply id="S3.Ex2.m1.5.6.6.2.cmml" xref="S3.Ex2.m1.5.6.6.2"><times id="S3.Ex2.m1.5.6.6.2.1.cmml" xref="S3.Ex2.m1.5.6.6.2.1"></times><apply id="S3.Ex2.m1.5.6.6.2.2.cmml" xref="S3.Ex2.m1.5.6.6.2.2"><csymbol cd="ambiguous" id="S3.Ex2.m1.5.6.6.2.2.1.cmml" xref="S3.Ex2.m1.5.6.6.2.2">subscript</csymbol><ci id="S3.Ex2.m1.5.6.6.2.2.2.cmml" xref="S3.Ex2.m1.5.6.6.2.2.2">𝑣</ci><ci id="S3.Ex2.m1.5.6.6.2.2.3.cmml" xref="S3.Ex2.m1.5.6.6.2.2.3">𝑖</ci></apply><ci id="S3.Ex2.m1.5.5.cmml" xref="S3.Ex2.m1.5.5">𝑗</ci><ci id="S3.Ex2.m1.5.6.6.2.4a.cmml" xref="S3.Ex2.m1.5.6.6.2.4"><mtext id="S3.Ex2.m1.5.6.6.2.4.cmml" xref="S3.Ex2.m1.5.6.6.2.4">.</mtext></ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex2.m1.5c">\mathcal{SW}(\pi):=\sum_{i\in N}u_{i}(\pi)=\sum_{C\in\pi\colon i,j\in C}v_{i}(% j)\text{.}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex2.m1.5d">caligraphic_S caligraphic_W ( italic_π ) := ∑ start_POSTSUBSCRIPT italic_i ∈ italic_N end_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_π ) = ∑ start_POSTSUBSCRIPT italic_C ∈ italic_π : italic_i , italic_j ∈ italic_C end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.p5"> <p class="ltx_p" id="S3.p5.2">Hence, the social welfare is the sum of the utilities which, in an ASHG, is equivalent to the sum of all valuations between agents in the same coalition. We denote by <math alttext="\pi^{*}" class="ltx_Math" display="inline" id="S3.p5.1.m1.1"><semantics id="S3.p5.1.m1.1a"><msup id="S3.p5.1.m1.1.1" xref="S3.p5.1.m1.1.1.cmml"><mi id="S3.p5.1.m1.1.1.2" xref="S3.p5.1.m1.1.1.2.cmml">π</mi><mo id="S3.p5.1.m1.1.1.3" xref="S3.p5.1.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S3.p5.1.m1.1b"><apply id="S3.p5.1.m1.1.1.cmml" xref="S3.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p5.1.m1.1.1.1.cmml" xref="S3.p5.1.m1.1.1">superscript</csymbol><ci id="S3.p5.1.m1.1.1.2.cmml" xref="S3.p5.1.m1.1.1.2">𝜋</ci><times id="S3.p5.1.m1.1.1.3.cmml" xref="S3.p5.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.1.m1.1c">\pi^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.1.m1.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> a partition that maximizes <math alttext="\mathcal{SW}" class="ltx_Math" display="inline" id="S3.p5.2.m2.1"><semantics id="S3.p5.2.m2.1a"><mrow id="S3.p5.2.m2.1.1" xref="S3.p5.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p5.2.m2.1.1.2" xref="S3.p5.2.m2.1.1.2.cmml">𝒮</mi><mo id="S3.p5.2.m2.1.1.1" xref="S3.p5.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.p5.2.m2.1.1.3" xref="S3.p5.2.m2.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.2.m2.1b"><apply id="S3.p5.2.m2.1.1.cmml" xref="S3.p5.2.m2.1.1"><times id="S3.p5.2.m2.1.1.1.cmml" xref="S3.p5.2.m2.1.1.1"></times><ci id="S3.p5.2.m2.1.1.2.cmml" xref="S3.p5.2.m2.1.1.2">𝒮</ci><ci id="S3.p5.2.m2.1.1.3.cmml" xref="S3.p5.2.m2.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.2.m2.1c">\mathcal{SW}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.2.m2.1d">caligraphic_S caligraphic_W</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p6"> <p class="ltx_p" id="S3.p6.9">ASHGs admit various interesting subclasses when restricting valuations. Following Dimitrov et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx19" title="">DBHS06</a>]</cite>, an ASHG <math alttext="(N,v)" class="ltx_Math" display="inline" id="S3.p6.1.m1.2"><semantics id="S3.p6.1.m1.2a"><mrow id="S3.p6.1.m1.2.3.2" xref="S3.p6.1.m1.2.3.1.cmml"><mo id="S3.p6.1.m1.2.3.2.1" stretchy="false" xref="S3.p6.1.m1.2.3.1.cmml">(</mo><mi id="S3.p6.1.m1.1.1" xref="S3.p6.1.m1.1.1.cmml">N</mi><mo id="S3.p6.1.m1.2.3.2.2" xref="S3.p6.1.m1.2.3.1.cmml">,</mo><mi id="S3.p6.1.m1.2.2" xref="S3.p6.1.m1.2.2.cmml">v</mi><mo id="S3.p6.1.m1.2.3.2.3" stretchy="false" xref="S3.p6.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.1.m1.2b"><interval closure="open" id="S3.p6.1.m1.2.3.1.cmml" xref="S3.p6.1.m1.2.3.2"><ci id="S3.p6.1.m1.1.1.cmml" xref="S3.p6.1.m1.1.1">𝑁</ci><ci id="S3.p6.1.m1.2.2.cmml" xref="S3.p6.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.1.m1.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S3.p6.1.m1.2d">( italic_N , italic_v )</annotation></semantics></math> is called an <em class="ltx_emph ltx_font_italic" id="S3.p6.9.1">aversion-to-enemies game</em> if <math alttext="v_{i}(j)\in\{-n,1\}" class="ltx_Math" display="inline" id="S3.p6.2.m2.3"><semantics id="S3.p6.2.m2.3a"><mrow id="S3.p6.2.m2.3.3" xref="S3.p6.2.m2.3.3.cmml"><mrow id="S3.p6.2.m2.3.3.3" xref="S3.p6.2.m2.3.3.3.cmml"><msub id="S3.p6.2.m2.3.3.3.2" xref="S3.p6.2.m2.3.3.3.2.cmml"><mi id="S3.p6.2.m2.3.3.3.2.2" xref="S3.p6.2.m2.3.3.3.2.2.cmml">v</mi><mi id="S3.p6.2.m2.3.3.3.2.3" xref="S3.p6.2.m2.3.3.3.2.3.cmml">i</mi></msub><mo id="S3.p6.2.m2.3.3.3.1" xref="S3.p6.2.m2.3.3.3.1.cmml">⁢</mo><mrow id="S3.p6.2.m2.3.3.3.3.2" xref="S3.p6.2.m2.3.3.3.cmml"><mo id="S3.p6.2.m2.3.3.3.3.2.1" stretchy="false" xref="S3.p6.2.m2.3.3.3.cmml">(</mo><mi id="S3.p6.2.m2.1.1" xref="S3.p6.2.m2.1.1.cmml">j</mi><mo id="S3.p6.2.m2.3.3.3.3.2.2" stretchy="false" xref="S3.p6.2.m2.3.3.3.cmml">)</mo></mrow></mrow><mo id="S3.p6.2.m2.3.3.2" xref="S3.p6.2.m2.3.3.2.cmml">∈</mo><mrow id="S3.p6.2.m2.3.3.1.1" xref="S3.p6.2.m2.3.3.1.2.cmml"><mo id="S3.p6.2.m2.3.3.1.1.2" stretchy="false" xref="S3.p6.2.m2.3.3.1.2.cmml">{</mo><mrow id="S3.p6.2.m2.3.3.1.1.1" xref="S3.p6.2.m2.3.3.1.1.1.cmml"><mo id="S3.p6.2.m2.3.3.1.1.1a" xref="S3.p6.2.m2.3.3.1.1.1.cmml">−</mo><mi id="S3.p6.2.m2.3.3.1.1.1.2" xref="S3.p6.2.m2.3.3.1.1.1.2.cmml">n</mi></mrow><mo id="S3.p6.2.m2.3.3.1.1.3" xref="S3.p6.2.m2.3.3.1.2.cmml">,</mo><mn id="S3.p6.2.m2.2.2" xref="S3.p6.2.m2.2.2.cmml">1</mn><mo id="S3.p6.2.m2.3.3.1.1.4" stretchy="false" xref="S3.p6.2.m2.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.2.m2.3b"><apply id="S3.p6.2.m2.3.3.cmml" xref="S3.p6.2.m2.3.3"><in id="S3.p6.2.m2.3.3.2.cmml" xref="S3.p6.2.m2.3.3.2"></in><apply id="S3.p6.2.m2.3.3.3.cmml" xref="S3.p6.2.m2.3.3.3"><times id="S3.p6.2.m2.3.3.3.1.cmml" xref="S3.p6.2.m2.3.3.3.1"></times><apply id="S3.p6.2.m2.3.3.3.2.cmml" xref="S3.p6.2.m2.3.3.3.2"><csymbol cd="ambiguous" id="S3.p6.2.m2.3.3.3.2.1.cmml" xref="S3.p6.2.m2.3.3.3.2">subscript</csymbol><ci id="S3.p6.2.m2.3.3.3.2.2.cmml" xref="S3.p6.2.m2.3.3.3.2.2">𝑣</ci><ci id="S3.p6.2.m2.3.3.3.2.3.cmml" xref="S3.p6.2.m2.3.3.3.2.3">𝑖</ci></apply><ci id="S3.p6.2.m2.1.1.cmml" xref="S3.p6.2.m2.1.1">𝑗</ci></apply><set id="S3.p6.2.m2.3.3.1.2.cmml" xref="S3.p6.2.m2.3.3.1.1"><apply id="S3.p6.2.m2.3.3.1.1.1.cmml" xref="S3.p6.2.m2.3.3.1.1.1"><minus id="S3.p6.2.m2.3.3.1.1.1.1.cmml" xref="S3.p6.2.m2.3.3.1.1.1"></minus><ci id="S3.p6.2.m2.3.3.1.1.1.2.cmml" xref="S3.p6.2.m2.3.3.1.1.1.2">𝑛</ci></apply><cn id="S3.p6.2.m2.2.2.cmml" type="integer" xref="S3.p6.2.m2.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.2.m2.3c">v_{i}(j)\in\{-n,1\}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.2.m2.3d">italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j ) ∈ { - italic_n , 1 }</annotation></semantics></math> for all <math alttext="i,j\in N" class="ltx_Math" display="inline" id="S3.p6.3.m3.2"><semantics id="S3.p6.3.m3.2a"><mrow id="S3.p6.3.m3.2.3" xref="S3.p6.3.m3.2.3.cmml"><mrow id="S3.p6.3.m3.2.3.2.2" xref="S3.p6.3.m3.2.3.2.1.cmml"><mi id="S3.p6.3.m3.1.1" xref="S3.p6.3.m3.1.1.cmml">i</mi><mo id="S3.p6.3.m3.2.3.2.2.1" xref="S3.p6.3.m3.2.3.2.1.cmml">,</mo><mi id="S3.p6.3.m3.2.2" xref="S3.p6.3.m3.2.2.cmml">j</mi></mrow><mo id="S3.p6.3.m3.2.3.1" xref="S3.p6.3.m3.2.3.1.cmml">∈</mo><mi id="S3.p6.3.m3.2.3.3" xref="S3.p6.3.m3.2.3.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.3.m3.2b"><apply id="S3.p6.3.m3.2.3.cmml" xref="S3.p6.3.m3.2.3"><in id="S3.p6.3.m3.2.3.1.cmml" xref="S3.p6.3.m3.2.3.1"></in><list id="S3.p6.3.m3.2.3.2.1.cmml" xref="S3.p6.3.m3.2.3.2.2"><ci id="S3.p6.3.m3.1.1.cmml" xref="S3.p6.3.m3.1.1">𝑖</ci><ci id="S3.p6.3.m3.2.2.cmml" xref="S3.p6.3.m3.2.2">𝑗</ci></list><ci id="S3.p6.3.m3.2.3.3.cmml" xref="S3.p6.3.m3.2.3.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.3.m3.2c">i,j\in N</annotation><annotation encoding="application/x-llamapun" id="S3.p6.3.m3.2d">italic_i , italic_j ∈ italic_N</annotation></semantics></math>. An ASHG <math alttext="(N,v)" class="ltx_Math" display="inline" id="S3.p6.4.m4.2"><semantics id="S3.p6.4.m4.2a"><mrow id="S3.p6.4.m4.2.3.2" xref="S3.p6.4.m4.2.3.1.cmml"><mo id="S3.p6.4.m4.2.3.2.1" stretchy="false" xref="S3.p6.4.m4.2.3.1.cmml">(</mo><mi id="S3.p6.4.m4.1.1" xref="S3.p6.4.m4.1.1.cmml">N</mi><mo id="S3.p6.4.m4.2.3.2.2" xref="S3.p6.4.m4.2.3.1.cmml">,</mo><mi id="S3.p6.4.m4.2.2" xref="S3.p6.4.m4.2.2.cmml">v</mi><mo id="S3.p6.4.m4.2.3.2.3" stretchy="false" xref="S3.p6.4.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.4.m4.2b"><interval closure="open" id="S3.p6.4.m4.2.3.1.cmml" xref="S3.p6.4.m4.2.3.2"><ci id="S3.p6.4.m4.1.1.cmml" xref="S3.p6.4.m4.1.1">𝑁</ci><ci id="S3.p6.4.m4.2.2.cmml" xref="S3.p6.4.m4.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.4.m4.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S3.p6.4.m4.2d">( italic_N , italic_v )</annotation></semantics></math> is called <em class="ltx_emph ltx_font_italic" id="S3.p6.9.2">symmetric</em> if for each pair of agents <math alttext="i,j\in N" class="ltx_Math" display="inline" id="S3.p6.5.m5.2"><semantics id="S3.p6.5.m5.2a"><mrow id="S3.p6.5.m5.2.3" xref="S3.p6.5.m5.2.3.cmml"><mrow id="S3.p6.5.m5.2.3.2.2" xref="S3.p6.5.m5.2.3.2.1.cmml"><mi id="S3.p6.5.m5.1.1" xref="S3.p6.5.m5.1.1.cmml">i</mi><mo id="S3.p6.5.m5.2.3.2.2.1" xref="S3.p6.5.m5.2.3.2.1.cmml">,</mo><mi id="S3.p6.5.m5.2.2" xref="S3.p6.5.m5.2.2.cmml">j</mi></mrow><mo id="S3.p6.5.m5.2.3.1" xref="S3.p6.5.m5.2.3.1.cmml">∈</mo><mi id="S3.p6.5.m5.2.3.3" xref="S3.p6.5.m5.2.3.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.5.m5.2b"><apply id="S3.p6.5.m5.2.3.cmml" xref="S3.p6.5.m5.2.3"><in id="S3.p6.5.m5.2.3.1.cmml" xref="S3.p6.5.m5.2.3.1"></in><list id="S3.p6.5.m5.2.3.2.1.cmml" xref="S3.p6.5.m5.2.3.2.2"><ci id="S3.p6.5.m5.1.1.cmml" xref="S3.p6.5.m5.1.1">𝑖</ci><ci id="S3.p6.5.m5.2.2.cmml" xref="S3.p6.5.m5.2.2">𝑗</ci></list><ci id="S3.p6.5.m5.2.3.3.cmml" xref="S3.p6.5.m5.2.3.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.5.m5.2c">i,j\in N</annotation><annotation encoding="application/x-llamapun" id="S3.p6.5.m5.2d">italic_i , italic_j ∈ italic_N</annotation></semantics></math>, it holds that <math alttext="v_{i}(j)=v_{j}(i)" class="ltx_Math" display="inline" id="S3.p6.6.m6.2"><semantics id="S3.p6.6.m6.2a"><mrow id="S3.p6.6.m6.2.3" xref="S3.p6.6.m6.2.3.cmml"><mrow id="S3.p6.6.m6.2.3.2" xref="S3.p6.6.m6.2.3.2.cmml"><msub id="S3.p6.6.m6.2.3.2.2" xref="S3.p6.6.m6.2.3.2.2.cmml"><mi id="S3.p6.6.m6.2.3.2.2.2" xref="S3.p6.6.m6.2.3.2.2.2.cmml">v</mi><mi id="S3.p6.6.m6.2.3.2.2.3" xref="S3.p6.6.m6.2.3.2.2.3.cmml">i</mi></msub><mo id="S3.p6.6.m6.2.3.2.1" xref="S3.p6.6.m6.2.3.2.1.cmml">⁢</mo><mrow id="S3.p6.6.m6.2.3.2.3.2" xref="S3.p6.6.m6.2.3.2.cmml"><mo id="S3.p6.6.m6.2.3.2.3.2.1" stretchy="false" xref="S3.p6.6.m6.2.3.2.cmml">(</mo><mi id="S3.p6.6.m6.1.1" xref="S3.p6.6.m6.1.1.cmml">j</mi><mo id="S3.p6.6.m6.2.3.2.3.2.2" stretchy="false" xref="S3.p6.6.m6.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.p6.6.m6.2.3.1" xref="S3.p6.6.m6.2.3.1.cmml">=</mo><mrow id="S3.p6.6.m6.2.3.3" xref="S3.p6.6.m6.2.3.3.cmml"><msub id="S3.p6.6.m6.2.3.3.2" xref="S3.p6.6.m6.2.3.3.2.cmml"><mi id="S3.p6.6.m6.2.3.3.2.2" xref="S3.p6.6.m6.2.3.3.2.2.cmml">v</mi><mi id="S3.p6.6.m6.2.3.3.2.3" xref="S3.p6.6.m6.2.3.3.2.3.cmml">j</mi></msub><mo id="S3.p6.6.m6.2.3.3.1" xref="S3.p6.6.m6.2.3.3.1.cmml">⁢</mo><mrow id="S3.p6.6.m6.2.3.3.3.2" xref="S3.p6.6.m6.2.3.3.cmml"><mo id="S3.p6.6.m6.2.3.3.3.2.1" stretchy="false" xref="S3.p6.6.m6.2.3.3.cmml">(</mo><mi id="S3.p6.6.m6.2.2" xref="S3.p6.6.m6.2.2.cmml">i</mi><mo id="S3.p6.6.m6.2.3.3.3.2.2" stretchy="false" xref="S3.p6.6.m6.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.6.m6.2b"><apply id="S3.p6.6.m6.2.3.cmml" xref="S3.p6.6.m6.2.3"><eq id="S3.p6.6.m6.2.3.1.cmml" xref="S3.p6.6.m6.2.3.1"></eq><apply id="S3.p6.6.m6.2.3.2.cmml" xref="S3.p6.6.m6.2.3.2"><times id="S3.p6.6.m6.2.3.2.1.cmml" xref="S3.p6.6.m6.2.3.2.1"></times><apply id="S3.p6.6.m6.2.3.2.2.cmml" xref="S3.p6.6.m6.2.3.2.2"><csymbol cd="ambiguous" id="S3.p6.6.m6.2.3.2.2.1.cmml" xref="S3.p6.6.m6.2.3.2.2">subscript</csymbol><ci id="S3.p6.6.m6.2.3.2.2.2.cmml" xref="S3.p6.6.m6.2.3.2.2.2">𝑣</ci><ci id="S3.p6.6.m6.2.3.2.2.3.cmml" xref="S3.p6.6.m6.2.3.2.2.3">𝑖</ci></apply><ci id="S3.p6.6.m6.1.1.cmml" xref="S3.p6.6.m6.1.1">𝑗</ci></apply><apply id="S3.p6.6.m6.2.3.3.cmml" xref="S3.p6.6.m6.2.3.3"><times id="S3.p6.6.m6.2.3.3.1.cmml" xref="S3.p6.6.m6.2.3.3.1"></times><apply id="S3.p6.6.m6.2.3.3.2.cmml" xref="S3.p6.6.m6.2.3.3.2"><csymbol cd="ambiguous" id="S3.p6.6.m6.2.3.3.2.1.cmml" xref="S3.p6.6.m6.2.3.3.2">subscript</csymbol><ci id="S3.p6.6.m6.2.3.3.2.2.cmml" xref="S3.p6.6.m6.2.3.3.2.2">𝑣</ci><ci id="S3.p6.6.m6.2.3.3.2.3.cmml" xref="S3.p6.6.m6.2.3.3.2.3">𝑗</ci></apply><ci id="S3.p6.6.m6.2.2.cmml" xref="S3.p6.6.m6.2.2">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.6.m6.2c">v_{i}(j)=v_{j}(i)</annotation><annotation encoding="application/x-llamapun" id="S3.p6.6.m6.2d">italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j ) = italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_i )</annotation></semantics></math>. We write <math alttext="v(i,j)" class="ltx_Math" display="inline" id="S3.p6.7.m7.2"><semantics id="S3.p6.7.m7.2a"><mrow id="S3.p6.7.m7.2.3" xref="S3.p6.7.m7.2.3.cmml"><mi id="S3.p6.7.m7.2.3.2" xref="S3.p6.7.m7.2.3.2.cmml">v</mi><mo id="S3.p6.7.m7.2.3.1" xref="S3.p6.7.m7.2.3.1.cmml">⁢</mo><mrow id="S3.p6.7.m7.2.3.3.2" xref="S3.p6.7.m7.2.3.3.1.cmml"><mo id="S3.p6.7.m7.2.3.3.2.1" stretchy="false" xref="S3.p6.7.m7.2.3.3.1.cmml">(</mo><mi id="S3.p6.7.m7.1.1" xref="S3.p6.7.m7.1.1.cmml">i</mi><mo id="S3.p6.7.m7.2.3.3.2.2" xref="S3.p6.7.m7.2.3.3.1.cmml">,</mo><mi id="S3.p6.7.m7.2.2" xref="S3.p6.7.m7.2.2.cmml">j</mi><mo id="S3.p6.7.m7.2.3.3.2.3" stretchy="false" xref="S3.p6.7.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.7.m7.2b"><apply id="S3.p6.7.m7.2.3.cmml" xref="S3.p6.7.m7.2.3"><times id="S3.p6.7.m7.2.3.1.cmml" xref="S3.p6.7.m7.2.3.1"></times><ci id="S3.p6.7.m7.2.3.2.cmml" xref="S3.p6.7.m7.2.3.2">𝑣</ci><interval closure="open" id="S3.p6.7.m7.2.3.3.1.cmml" xref="S3.p6.7.m7.2.3.3.2"><ci id="S3.p6.7.m7.1.1.cmml" xref="S3.p6.7.m7.1.1">𝑖</ci><ci id="S3.p6.7.m7.2.2.cmml" xref="S3.p6.7.m7.2.2">𝑗</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.7.m7.2c">v(i,j)</annotation><annotation encoding="application/x-llamapun" id="S3.p6.7.m7.2d">italic_v ( italic_i , italic_j )</annotation></semantics></math> for the symmetric valuation function between <math alttext="i" class="ltx_Math" display="inline" id="S3.p6.8.m8.1"><semantics id="S3.p6.8.m8.1a"><mi id="S3.p6.8.m8.1.1" xref="S3.p6.8.m8.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p6.8.m8.1b"><ci id="S3.p6.8.m8.1.1.cmml" xref="S3.p6.8.m8.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.8.m8.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p6.8.m8.1d">italic_i</annotation></semantics></math> and <math alttext="j" class="ltx_Math" display="inline" id="S3.p6.9.m9.1"><semantics id="S3.p6.9.m9.1a"><mi id="S3.p6.9.m9.1.1" xref="S3.p6.9.m9.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.p6.9.m9.1b"><ci id="S3.p6.9.m9.1.1.cmml" xref="S3.p6.9.m9.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.9.m9.1c">j</annotation><annotation encoding="application/x-llamapun" id="S3.p6.9.m9.1d">italic_j</annotation></semantics></math>. In this paper, we restrict attention to symmetric ASHGs.<span class="ltx_note ltx_role_footnote" id="footnote2"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span> In general ASHGs, symmetry is without loss of generality when reasoning about social welfare as the welfare remains the same if we replace <math alttext="v_{i}(j)" class="ltx_Math" display="inline" id="footnote2.m1.1"><semantics id="footnote2.m1.1b"><mrow id="footnote2.m1.1.2" xref="footnote2.m1.1.2.cmml"><msub id="footnote2.m1.1.2.2" xref="footnote2.m1.1.2.2.cmml"><mi id="footnote2.m1.1.2.2.2" xref="footnote2.m1.1.2.2.2.cmml">v</mi><mi id="footnote2.m1.1.2.2.3" xref="footnote2.m1.1.2.2.3.cmml">i</mi></msub><mo id="footnote2.m1.1.2.1" xref="footnote2.m1.1.2.1.cmml">⁢</mo><mrow id="footnote2.m1.1.2.3.2" xref="footnote2.m1.1.2.cmml"><mo id="footnote2.m1.1.2.3.2.1" stretchy="false" xref="footnote2.m1.1.2.cmml">(</mo><mi id="footnote2.m1.1.1" xref="footnote2.m1.1.1.cmml">j</mi><mo id="footnote2.m1.1.2.3.2.2" stretchy="false" xref="footnote2.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m1.1c"><apply id="footnote2.m1.1.2.cmml" xref="footnote2.m1.1.2"><times id="footnote2.m1.1.2.1.cmml" xref="footnote2.m1.1.2.1"></times><apply id="footnote2.m1.1.2.2.cmml" xref="footnote2.m1.1.2.2"><csymbol cd="ambiguous" id="footnote2.m1.1.2.2.1.cmml" xref="footnote2.m1.1.2.2">subscript</csymbol><ci id="footnote2.m1.1.2.2.2.cmml" xref="footnote2.m1.1.2.2.2">𝑣</ci><ci id="footnote2.m1.1.2.2.3.cmml" xref="footnote2.m1.1.2.2.3">𝑖</ci></apply><ci id="footnote2.m1.1.1.cmml" xref="footnote2.m1.1.1">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m1.1d">v_{i}(j)</annotation><annotation encoding="application/x-llamapun" id="footnote2.m1.1e">italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math> and <math alttext="v_{j}(i)" class="ltx_Math" display="inline" id="footnote2.m2.1"><semantics id="footnote2.m2.1b"><mrow id="footnote2.m2.1.2" xref="footnote2.m2.1.2.cmml"><msub id="footnote2.m2.1.2.2" xref="footnote2.m2.1.2.2.cmml"><mi id="footnote2.m2.1.2.2.2" xref="footnote2.m2.1.2.2.2.cmml">v</mi><mi id="footnote2.m2.1.2.2.3" xref="footnote2.m2.1.2.2.3.cmml">j</mi></msub><mo id="footnote2.m2.1.2.1" xref="footnote2.m2.1.2.1.cmml">⁢</mo><mrow id="footnote2.m2.1.2.3.2" xref="footnote2.m2.1.2.cmml"><mo id="footnote2.m2.1.2.3.2.1" stretchy="false" xref="footnote2.m2.1.2.cmml">(</mo><mi id="footnote2.m2.1.1" xref="footnote2.m2.1.1.cmml">i</mi><mo id="footnote2.m2.1.2.3.2.2" stretchy="false" xref="footnote2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m2.1c"><apply id="footnote2.m2.1.2.cmml" xref="footnote2.m2.1.2"><times id="footnote2.m2.1.2.1.cmml" xref="footnote2.m2.1.2.1"></times><apply id="footnote2.m2.1.2.2.cmml" xref="footnote2.m2.1.2.2"><csymbol cd="ambiguous" id="footnote2.m2.1.2.2.1.cmml" xref="footnote2.m2.1.2.2">subscript</csymbol><ci id="footnote2.m2.1.2.2.2.cmml" xref="footnote2.m2.1.2.2.2">𝑣</ci><ci id="footnote2.m2.1.2.2.3.cmml" xref="footnote2.m2.1.2.2.3">𝑗</ci></apply><ci id="footnote2.m2.1.1.cmml" xref="footnote2.m2.1.1">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m2.1d">v_{j}(i)</annotation><annotation encoding="application/x-llamapun" id="footnote2.m2.1e">italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_i )</annotation></semantics></math> by <math alttext="\frac{v_{i}(j)+v_{j}(i)}{2}" class="ltx_Math" display="inline" id="footnote2.m3.2"><semantics id="footnote2.m3.2b"><mfrac id="footnote2.m3.2.2" xref="footnote2.m3.2.2.cmml"><mrow id="footnote2.m3.2.2.2" xref="footnote2.m3.2.2.2.cmml"><mrow id="footnote2.m3.2.2.2.4" xref="footnote2.m3.2.2.2.4.cmml"><msub id="footnote2.m3.2.2.2.4.2" xref="footnote2.m3.2.2.2.4.2.cmml"><mi id="footnote2.m3.2.2.2.4.2.2" xref="footnote2.m3.2.2.2.4.2.2.cmml">v</mi><mi id="footnote2.m3.2.2.2.4.2.3" xref="footnote2.m3.2.2.2.4.2.3.cmml">i</mi></msub><mo id="footnote2.m3.2.2.2.4.1" xref="footnote2.m3.2.2.2.4.1.cmml">⁢</mo><mrow id="footnote2.m3.2.2.2.4.3.2" xref="footnote2.m3.2.2.2.4.cmml"><mo id="footnote2.m3.2.2.2.4.3.2.1" stretchy="false" xref="footnote2.m3.2.2.2.4.cmml">(</mo><mi id="footnote2.m3.1.1.1.1" xref="footnote2.m3.1.1.1.1.cmml">j</mi><mo id="footnote2.m3.2.2.2.4.3.2.2" stretchy="false" xref="footnote2.m3.2.2.2.4.cmml">)</mo></mrow></mrow><mo id="footnote2.m3.2.2.2.3" xref="footnote2.m3.2.2.2.3.cmml">+</mo><mrow id="footnote2.m3.2.2.2.5" xref="footnote2.m3.2.2.2.5.cmml"><msub id="footnote2.m3.2.2.2.5.2" xref="footnote2.m3.2.2.2.5.2.cmml"><mi id="footnote2.m3.2.2.2.5.2.2" xref="footnote2.m3.2.2.2.5.2.2.cmml">v</mi><mi id="footnote2.m3.2.2.2.5.2.3" xref="footnote2.m3.2.2.2.5.2.3.cmml">j</mi></msub><mo id="footnote2.m3.2.2.2.5.1" xref="footnote2.m3.2.2.2.5.1.cmml">⁢</mo><mrow id="footnote2.m3.2.2.2.5.3.2" xref="footnote2.m3.2.2.2.5.cmml"><mo id="footnote2.m3.2.2.2.5.3.2.1" stretchy="false" xref="footnote2.m3.2.2.2.5.cmml">(</mo><mi id="footnote2.m3.2.2.2.2" xref="footnote2.m3.2.2.2.2.cmml">i</mi><mo id="footnote2.m3.2.2.2.5.3.2.2" stretchy="false" xref="footnote2.m3.2.2.2.5.cmml">)</mo></mrow></mrow></mrow><mn id="footnote2.m3.2.2.4" xref="footnote2.m3.2.2.4.cmml">2</mn></mfrac><annotation-xml encoding="MathML-Content" id="footnote2.m3.2c"><apply id="footnote2.m3.2.2.cmml" xref="footnote2.m3.2.2"><divide id="footnote2.m3.2.2.3.cmml" xref="footnote2.m3.2.2"></divide><apply id="footnote2.m3.2.2.2.cmml" xref="footnote2.m3.2.2.2"><plus id="footnote2.m3.2.2.2.3.cmml" xref="footnote2.m3.2.2.2.3"></plus><apply id="footnote2.m3.2.2.2.4.cmml" xref="footnote2.m3.2.2.2.4"><times id="footnote2.m3.2.2.2.4.1.cmml" xref="footnote2.m3.2.2.2.4.1"></times><apply id="footnote2.m3.2.2.2.4.2.cmml" xref="footnote2.m3.2.2.2.4.2"><csymbol cd="ambiguous" id="footnote2.m3.2.2.2.4.2.1.cmml" xref="footnote2.m3.2.2.2.4.2">subscript</csymbol><ci id="footnote2.m3.2.2.2.4.2.2.cmml" xref="footnote2.m3.2.2.2.4.2.2">𝑣</ci><ci id="footnote2.m3.2.2.2.4.2.3.cmml" xref="footnote2.m3.2.2.2.4.2.3">𝑖</ci></apply><ci id="footnote2.m3.1.1.1.1.cmml" xref="footnote2.m3.1.1.1.1">𝑗</ci></apply><apply id="footnote2.m3.2.2.2.5.cmml" xref="footnote2.m3.2.2.2.5"><times id="footnote2.m3.2.2.2.5.1.cmml" xref="footnote2.m3.2.2.2.5.1"></times><apply id="footnote2.m3.2.2.2.5.2.cmml" xref="footnote2.m3.2.2.2.5.2"><csymbol cd="ambiguous" id="footnote2.m3.2.2.2.5.2.1.cmml" xref="footnote2.m3.2.2.2.5.2">subscript</csymbol><ci id="footnote2.m3.2.2.2.5.2.2.cmml" xref="footnote2.m3.2.2.2.5.2.2">𝑣</ci><ci id="footnote2.m3.2.2.2.5.2.3.cmml" xref="footnote2.m3.2.2.2.5.2.3">𝑗</ci></apply><ci id="footnote2.m3.2.2.2.2.cmml" xref="footnote2.m3.2.2.2.2">𝑖</ci></apply></apply><cn id="footnote2.m3.2.2.4.cmml" type="integer" xref="footnote2.m3.2.2.4">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m3.2d">\frac{v_{i}(j)+v_{j}(i)}{2}</annotation><annotation encoding="application/x-llamapun" id="footnote2.m3.2e">divide start_ARG italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j ) + italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_i ) end_ARG start_ARG 2 end_ARG</annotation></semantics></math>, see, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx14" title="">Bul20</a>]</cite>. However, this is not the case for aversion-to-enemies games as the symmetrization may leave this game class.</span></span></span></p> </div> <div class="ltx_para" id="S3.p7"> <p class="ltx_p" id="S3.p7.5">Consider an ASHG <math alttext="(N,v)" class="ltx_Math" display="inline" id="S3.p7.1.m1.2"><semantics id="S3.p7.1.m1.2a"><mrow id="S3.p7.1.m1.2.3.2" xref="S3.p7.1.m1.2.3.1.cmml"><mo id="S3.p7.1.m1.2.3.2.1" stretchy="false" xref="S3.p7.1.m1.2.3.1.cmml">(</mo><mi id="S3.p7.1.m1.1.1" xref="S3.p7.1.m1.1.1.cmml">N</mi><mo id="S3.p7.1.m1.2.3.2.2" xref="S3.p7.1.m1.2.3.1.cmml">,</mo><mi id="S3.p7.1.m1.2.2" xref="S3.p7.1.m1.2.2.cmml">v</mi><mo id="S3.p7.1.m1.2.3.2.3" stretchy="false" xref="S3.p7.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p7.1.m1.2b"><interval closure="open" id="S3.p7.1.m1.2.3.1.cmml" xref="S3.p7.1.m1.2.3.2"><ci id="S3.p7.1.m1.1.1.cmml" xref="S3.p7.1.m1.1.1">𝑁</ci><ci id="S3.p7.1.m1.2.2.cmml" xref="S3.p7.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.1.m1.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S3.p7.1.m1.2d">( italic_N , italic_v )</annotation></semantics></math> and an approximation ratio <math alttext="c\geq 1" class="ltx_Math" display="inline" id="S3.p7.2.m2.1"><semantics id="S3.p7.2.m2.1a"><mrow id="S3.p7.2.m2.1.1" xref="S3.p7.2.m2.1.1.cmml"><mi id="S3.p7.2.m2.1.1.2" xref="S3.p7.2.m2.1.1.2.cmml">c</mi><mo id="S3.p7.2.m2.1.1.1" xref="S3.p7.2.m2.1.1.1.cmml">≥</mo><mn id="S3.p7.2.m2.1.1.3" xref="S3.p7.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p7.2.m2.1b"><apply id="S3.p7.2.m2.1.1.cmml" xref="S3.p7.2.m2.1.1"><geq id="S3.p7.2.m2.1.1.1.cmml" xref="S3.p7.2.m2.1.1.1"></geq><ci id="S3.p7.2.m2.1.1.2.cmml" xref="S3.p7.2.m2.1.1.2">𝑐</ci><cn id="S3.p7.2.m2.1.1.3.cmml" type="integer" xref="S3.p7.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.2.m2.1c">c\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.p7.2.m2.1d">italic_c ≥ 1</annotation></semantics></math>. A partition <math alttext="\pi" class="ltx_Math" display="inline" id="S3.p7.3.m3.1"><semantics id="S3.p7.3.m3.1a"><mi id="S3.p7.3.m3.1.1" xref="S3.p7.3.m3.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S3.p7.3.m3.1b"><ci id="S3.p7.3.m3.1.1.cmml" xref="S3.p7.3.m3.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.3.m3.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S3.p7.3.m3.1d">italic_π</annotation></semantics></math> is said to provide a <em class="ltx_emph ltx_font_italic" id="S3.p7.4.1"><math alttext="c" class="ltx_Math" display="inline" id="S3.p7.4.1.m1.1"><semantics id="S3.p7.4.1.m1.1a"><mi id="S3.p7.4.1.m1.1.1" xref="S3.p7.4.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.p7.4.1.m1.1b"><ci id="S3.p7.4.1.m1.1.1.cmml" xref="S3.p7.4.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.4.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.p7.4.1.m1.1d">italic_c</annotation></semantics></math>-approximation to maximum welfare</em> if <math alttext="c\cdot\mathcal{SW}(\pi)\geq\mathcal{SW}(\pi^{*})" class="ltx_Math" display="inline" id="S3.p7.5.m4.2"><semantics id="S3.p7.5.m4.2a"><mrow id="S3.p7.5.m4.2.2" xref="S3.p7.5.m4.2.2.cmml"><mrow id="S3.p7.5.m4.2.2.3" xref="S3.p7.5.m4.2.2.3.cmml"><mrow id="S3.p7.5.m4.2.2.3.2" xref="S3.p7.5.m4.2.2.3.2.cmml"><mi id="S3.p7.5.m4.2.2.3.2.2" xref="S3.p7.5.m4.2.2.3.2.2.cmml">c</mi><mo id="S3.p7.5.m4.2.2.3.2.1" lspace="0.222em" rspace="0.222em" xref="S3.p7.5.m4.2.2.3.2.1.cmml">⋅</mo><mi class="ltx_font_mathcaligraphic" id="S3.p7.5.m4.2.2.3.2.3" xref="S3.p7.5.m4.2.2.3.2.3.cmml">𝒮</mi></mrow><mo id="S3.p7.5.m4.2.2.3.1" xref="S3.p7.5.m4.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.p7.5.m4.2.2.3.3" xref="S3.p7.5.m4.2.2.3.3.cmml">𝒲</mi><mo id="S3.p7.5.m4.2.2.3.1a" xref="S3.p7.5.m4.2.2.3.1.cmml">⁢</mo><mrow id="S3.p7.5.m4.2.2.3.4.2" xref="S3.p7.5.m4.2.2.3.cmml"><mo id="S3.p7.5.m4.2.2.3.4.2.1" stretchy="false" xref="S3.p7.5.m4.2.2.3.cmml">(</mo><mi id="S3.p7.5.m4.1.1" xref="S3.p7.5.m4.1.1.cmml">π</mi><mo id="S3.p7.5.m4.2.2.3.4.2.2" stretchy="false" xref="S3.p7.5.m4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.p7.5.m4.2.2.2" xref="S3.p7.5.m4.2.2.2.cmml">≥</mo><mrow id="S3.p7.5.m4.2.2.1" xref="S3.p7.5.m4.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p7.5.m4.2.2.1.3" xref="S3.p7.5.m4.2.2.1.3.cmml">𝒮</mi><mo id="S3.p7.5.m4.2.2.1.2" xref="S3.p7.5.m4.2.2.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.p7.5.m4.2.2.1.4" xref="S3.p7.5.m4.2.2.1.4.cmml">𝒲</mi><mo id="S3.p7.5.m4.2.2.1.2a" xref="S3.p7.5.m4.2.2.1.2.cmml">⁢</mo><mrow id="S3.p7.5.m4.2.2.1.1.1" xref="S3.p7.5.m4.2.2.1.1.1.1.cmml"><mo id="S3.p7.5.m4.2.2.1.1.1.2" stretchy="false" xref="S3.p7.5.m4.2.2.1.1.1.1.cmml">(</mo><msup id="S3.p7.5.m4.2.2.1.1.1.1" xref="S3.p7.5.m4.2.2.1.1.1.1.cmml"><mi id="S3.p7.5.m4.2.2.1.1.1.1.2" xref="S3.p7.5.m4.2.2.1.1.1.1.2.cmml">π</mi><mo id="S3.p7.5.m4.2.2.1.1.1.1.3" xref="S3.p7.5.m4.2.2.1.1.1.1.3.cmml">∗</mo></msup><mo id="S3.p7.5.m4.2.2.1.1.1.3" stretchy="false" xref="S3.p7.5.m4.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p7.5.m4.2b"><apply id="S3.p7.5.m4.2.2.cmml" xref="S3.p7.5.m4.2.2"><geq id="S3.p7.5.m4.2.2.2.cmml" xref="S3.p7.5.m4.2.2.2"></geq><apply id="S3.p7.5.m4.2.2.3.cmml" xref="S3.p7.5.m4.2.2.3"><times id="S3.p7.5.m4.2.2.3.1.cmml" xref="S3.p7.5.m4.2.2.3.1"></times><apply id="S3.p7.5.m4.2.2.3.2.cmml" xref="S3.p7.5.m4.2.2.3.2"><ci id="S3.p7.5.m4.2.2.3.2.1.cmml" xref="S3.p7.5.m4.2.2.3.2.1">⋅</ci><ci id="S3.p7.5.m4.2.2.3.2.2.cmml" xref="S3.p7.5.m4.2.2.3.2.2">𝑐</ci><ci id="S3.p7.5.m4.2.2.3.2.3.cmml" xref="S3.p7.5.m4.2.2.3.2.3">𝒮</ci></apply><ci id="S3.p7.5.m4.2.2.3.3.cmml" xref="S3.p7.5.m4.2.2.3.3">𝒲</ci><ci id="S3.p7.5.m4.1.1.cmml" xref="S3.p7.5.m4.1.1">𝜋</ci></apply><apply id="S3.p7.5.m4.2.2.1.cmml" xref="S3.p7.5.m4.2.2.1"><times id="S3.p7.5.m4.2.2.1.2.cmml" xref="S3.p7.5.m4.2.2.1.2"></times><ci id="S3.p7.5.m4.2.2.1.3.cmml" xref="S3.p7.5.m4.2.2.1.3">𝒮</ci><ci id="S3.p7.5.m4.2.2.1.4.cmml" xref="S3.p7.5.m4.2.2.1.4">𝒲</ci><apply id="S3.p7.5.m4.2.2.1.1.1.1.cmml" xref="S3.p7.5.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.p7.5.m4.2.2.1.1.1.1.1.cmml" xref="S3.p7.5.m4.2.2.1.1.1">superscript</csymbol><ci id="S3.p7.5.m4.2.2.1.1.1.1.2.cmml" xref="S3.p7.5.m4.2.2.1.1.1.1.2">𝜋</ci><times id="S3.p7.5.m4.2.2.1.1.1.1.3.cmml" xref="S3.p7.5.m4.2.2.1.1.1.1.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.5.m4.2c">c\cdot\mathcal{SW}(\pi)\geq\mathcal{SW}(\pi^{*})</annotation><annotation encoding="application/x-llamapun" id="S3.p7.5.m4.2d">italic_c ⋅ caligraphic_S caligraphic_W ( italic_π ) ≥ caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math>. We are interested in the following computational problem.</p> </div> <figure class="ltx_table" id="S3.3"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S3.3.3"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S3.1.1.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="2" id="S3.1.1.1.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="c" class="ltx_Math" display="inline" id="S3.1.1.1.1.m1.1"><semantics id="S3.1.1.1.1.m1.1a"><mi id="S3.1.1.1.1.m1.1.1" xref="S3.1.1.1.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.1.1.1.1.m1.1b"><ci id="S3.1.1.1.1.m1.1.1.cmml" xref="S3.1.1.1.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.1.1.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.1.1.1.1.m1.1d">italic_c</annotation></semantics></math><span class="ltx_text ltx_font_bold" id="S3.1.1.1.1.1">-<span class="ltx_text ltx_font_smallcaps" id="S3.1.1.1.1.1.1">ApproxWelfare</span></span> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.2.2.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S3.2.2.2.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text ltx_font_bold" id="S3.2.2.2.2.1">Given:</span></th> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S3.2.2.2.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.2.2.2.1.1"> <span class="ltx_p" id="S3.2.2.2.1.1.1" style="width:303.5pt;">ASHG <math alttext="(N,v)" class="ltx_Math" display="inline" id="S3.2.2.2.1.1.1.m1.2"><semantics id="S3.2.2.2.1.1.1.m1.2a"><mrow id="S3.2.2.2.1.1.1.m1.2.3.2" xref="S3.2.2.2.1.1.1.m1.2.3.1.cmml"><mo id="S3.2.2.2.1.1.1.m1.2.3.2.1" stretchy="false" xref="S3.2.2.2.1.1.1.m1.2.3.1.cmml">(</mo><mi id="S3.2.2.2.1.1.1.m1.1.1" xref="S3.2.2.2.1.1.1.m1.1.1.cmml">N</mi><mo id="S3.2.2.2.1.1.1.m1.2.3.2.2" xref="S3.2.2.2.1.1.1.m1.2.3.1.cmml">,</mo><mi id="S3.2.2.2.1.1.1.m1.2.2" xref="S3.2.2.2.1.1.1.m1.2.2.cmml">v</mi><mo id="S3.2.2.2.1.1.1.m1.2.3.2.3" stretchy="false" xref="S3.2.2.2.1.1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.2.2.2.1.1.1.m1.2b"><interval closure="open" id="S3.2.2.2.1.1.1.m1.2.3.1.cmml" xref="S3.2.2.2.1.1.1.m1.2.3.2"><ci id="S3.2.2.2.1.1.1.m1.1.1.cmml" xref="S3.2.2.2.1.1.1.m1.1.1">𝑁</ci><ci id="S3.2.2.2.1.1.1.m1.2.2.cmml" xref="S3.2.2.2.1.1.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.2.2.2.1.1.1.m1.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S3.2.2.2.1.1.1.m1.2d">( italic_N , italic_v )</annotation></semantics></math>.</span> </span> </td> </tr> <tr class="ltx_tr" id="S3.3.3.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S3.3.3.3.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text ltx_font_bold" id="S3.3.3.3.2.1">Task:</span></th> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_r ltx_border_t" id="S3.3.3.3.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <span class="ltx_inline-block ltx_align_top" id="S3.3.3.3.1.1"> <span class="ltx_p" id="S3.3.3.3.1.1.1" style="width:303.5pt;">Compute a partition with a <math alttext="c" class="ltx_Math" display="inline" id="S3.3.3.3.1.1.1.m1.1"><semantics id="S3.3.3.3.1.1.1.m1.1a"><mi id="S3.3.3.3.1.1.1.m1.1.1" xref="S3.3.3.3.1.1.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.3.3.3.1.1.1.m1.1b"><ci id="S3.3.3.3.1.1.1.m1.1.1.cmml" xref="S3.3.3.3.1.1.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.3.3.3.1.1.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.3.3.3.1.1.1.m1.1d">italic_c</annotation></semantics></math>-approximation to maximum welfare.</span> </span> </td> </tr> </tbody> </table> </figure> <div class="ltx_para" id="S3.p8"> <p class="ltx_p" id="S3.p8.6">We consider both deterministic and randomized algorithms and aim at efficient algorithms. For <math alttext="c\geq 1" class="ltx_Math" display="inline" id="S3.p8.1.m1.1"><semantics id="S3.p8.1.m1.1a"><mrow id="S3.p8.1.m1.1.1" xref="S3.p8.1.m1.1.1.cmml"><mi id="S3.p8.1.m1.1.1.2" xref="S3.p8.1.m1.1.1.2.cmml">c</mi><mo id="S3.p8.1.m1.1.1.1" xref="S3.p8.1.m1.1.1.1.cmml">≥</mo><mn id="S3.p8.1.m1.1.1.3" xref="S3.p8.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p8.1.m1.1b"><apply id="S3.p8.1.m1.1.1.cmml" xref="S3.p8.1.m1.1.1"><geq id="S3.p8.1.m1.1.1.1.cmml" xref="S3.p8.1.m1.1.1.1"></geq><ci id="S3.p8.1.m1.1.1.2.cmml" xref="S3.p8.1.m1.1.1.2">𝑐</ci><cn id="S3.p8.1.m1.1.1.3.cmml" type="integer" xref="S3.p8.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.1.m1.1c">c\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.p8.1.m1.1d">italic_c ≥ 1</annotation></semantics></math>, a polynomial-time algorithm is called a <em class="ltx_emph ltx_font_italic" id="S3.p8.2.1"><math alttext="c" class="ltx_Math" display="inline" id="S3.p8.2.1.m1.1"><semantics id="S3.p8.2.1.m1.1a"><mi id="S3.p8.2.1.m1.1.1" xref="S3.p8.2.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.p8.2.1.m1.1b"><ci id="S3.p8.2.1.m1.1.1.cmml" xref="S3.p8.2.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.2.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.p8.2.1.m1.1d">italic_c</annotation></semantics></math>-approximation algorithm</em> for maximizing welfare if it solves <math alttext="c" class="ltx_Math" display="inline" id="S3.p8.3.m2.1"><semantics id="S3.p8.3.m2.1a"><mi id="S3.p8.3.m2.1.1" xref="S3.p8.3.m2.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.p8.3.m2.1b"><ci id="S3.p8.3.m2.1.1.cmml" xref="S3.p8.3.m2.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.3.m2.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.p8.3.m2.1d">italic_c</annotation></semantics></math>-<span class="ltx_text ltx_font_smallcaps" id="S3.p8.6.2">ApproxWelfare</span>. For randomized algorithms, the expected running time has to be bounded by a polynomial and the produced partition has to provide a <math alttext="c" class="ltx_Math" display="inline" id="S3.p8.4.m3.1"><semantics id="S3.p8.4.m3.1a"><mi id="S3.p8.4.m3.1.1" xref="S3.p8.4.m3.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.p8.4.m3.1b"><ci id="S3.p8.4.m3.1.1.cmml" xref="S3.p8.4.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.4.m3.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.p8.4.m3.1d">italic_c</annotation></semantics></math>-approximation to maximum welfare in expectation. Note that we allow (and frequently assume) that the factor <math alttext="c" class="ltx_Math" display="inline" id="S3.p8.5.m4.1"><semantics id="S3.p8.5.m4.1a"><mi id="S3.p8.5.m4.1.1" xref="S3.p8.5.m4.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.p8.5.m4.1b"><ci id="S3.p8.5.m4.1.1.cmml" xref="S3.p8.5.m4.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.5.m4.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.p8.5.m4.1d">italic_c</annotation></semantics></math> depends on <math alttext="n" class="ltx_Math" display="inline" id="S3.p8.6.m5.1"><semantics id="S3.p8.6.m5.1a"><mi id="S3.p8.6.m5.1.1" xref="S3.p8.6.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.p8.6.m5.1b"><ci id="S3.p8.6.m5.1.1.cmml" xref="S3.p8.6.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.6.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.p8.6.m5.1d">italic_n</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p9"> <p class="ltx_p" id="S3.p9.1">Finally, given an ASHG <math alttext="(N,v)" class="ltx_Math" display="inline" id="S3.p9.1.m1.2"><semantics id="S3.p9.1.m1.2a"><mrow id="S3.p9.1.m1.2.3.2" xref="S3.p9.1.m1.2.3.1.cmml"><mo id="S3.p9.1.m1.2.3.2.1" stretchy="false" xref="S3.p9.1.m1.2.3.1.cmml">(</mo><mi id="S3.p9.1.m1.1.1" xref="S3.p9.1.m1.1.1.cmml">N</mi><mo id="S3.p9.1.m1.2.3.2.2" xref="S3.p9.1.m1.2.3.1.cmml">,</mo><mi id="S3.p9.1.m1.2.2" xref="S3.p9.1.m1.2.2.cmml">v</mi><mo id="S3.p9.1.m1.2.3.2.3" stretchy="false" xref="S3.p9.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p9.1.m1.2b"><interval closure="open" id="S3.p9.1.m1.2.3.1.cmml" xref="S3.p9.1.m1.2.3.2"><ci id="S3.p9.1.m1.1.1.cmml" xref="S3.p9.1.m1.1.1">𝑁</ci><ci id="S3.p9.1.m1.2.2.cmml" xref="S3.p9.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.1.m1.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S3.p9.1.m1.2d">( italic_N , italic_v )</annotation></semantics></math>, we define its <em class="ltx_emph ltx_font_italic" id="S3.p9.1.1">total value</em> as</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{V}(N,v):=\sum_{i,j\in N}v_{i}(j)\text{.}" class="ltx_Math" display="block" id="S3.Ex3.m1.5"><semantics id="S3.Ex3.m1.5a"><mrow id="S3.Ex3.m1.5.6" xref="S3.Ex3.m1.5.6.cmml"><mrow id="S3.Ex3.m1.5.6.2" xref="S3.Ex3.m1.5.6.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex3.m1.5.6.2.2" xref="S3.Ex3.m1.5.6.2.2.cmml">𝒱</mi><mo id="S3.Ex3.m1.5.6.2.1" xref="S3.Ex3.m1.5.6.2.1.cmml">⁢</mo><mrow id="S3.Ex3.m1.5.6.2.3.2" xref="S3.Ex3.m1.5.6.2.3.1.cmml"><mo id="S3.Ex3.m1.5.6.2.3.2.1" stretchy="false" xref="S3.Ex3.m1.5.6.2.3.1.cmml">(</mo><mi id="S3.Ex3.m1.3.3" xref="S3.Ex3.m1.3.3.cmml">N</mi><mo id="S3.Ex3.m1.5.6.2.3.2.2" xref="S3.Ex3.m1.5.6.2.3.1.cmml">,</mo><mi id="S3.Ex3.m1.4.4" xref="S3.Ex3.m1.4.4.cmml">v</mi><mo id="S3.Ex3.m1.5.6.2.3.2.3" rspace="0.278em" stretchy="false" xref="S3.Ex3.m1.5.6.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex3.m1.5.6.1" rspace="0.111em" xref="S3.Ex3.m1.5.6.1.cmml">:=</mo><mrow id="S3.Ex3.m1.5.6.3" xref="S3.Ex3.m1.5.6.3.cmml"><munder id="S3.Ex3.m1.5.6.3.1" xref="S3.Ex3.m1.5.6.3.1.cmml"><mo id="S3.Ex3.m1.5.6.3.1.2" movablelimits="false" xref="S3.Ex3.m1.5.6.3.1.2.cmml">∑</mo><mrow id="S3.Ex3.m1.2.2.2" xref="S3.Ex3.m1.2.2.2.cmml"><mrow id="S3.Ex3.m1.2.2.2.4.2" xref="S3.Ex3.m1.2.2.2.4.1.cmml"><mi id="S3.Ex3.m1.1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.cmml">i</mi><mo id="S3.Ex3.m1.2.2.2.4.2.1" xref="S3.Ex3.m1.2.2.2.4.1.cmml">,</mo><mi id="S3.Ex3.m1.2.2.2.2" xref="S3.Ex3.m1.2.2.2.2.cmml">j</mi></mrow><mo id="S3.Ex3.m1.2.2.2.3" xref="S3.Ex3.m1.2.2.2.3.cmml">∈</mo><mi id="S3.Ex3.m1.2.2.2.5" xref="S3.Ex3.m1.2.2.2.5.cmml">N</mi></mrow></munder><mrow id="S3.Ex3.m1.5.6.3.2" xref="S3.Ex3.m1.5.6.3.2.cmml"><msub id="S3.Ex3.m1.5.6.3.2.2" xref="S3.Ex3.m1.5.6.3.2.2.cmml"><mi id="S3.Ex3.m1.5.6.3.2.2.2" xref="S3.Ex3.m1.5.6.3.2.2.2.cmml">v</mi><mi id="S3.Ex3.m1.5.6.3.2.2.3" xref="S3.Ex3.m1.5.6.3.2.2.3.cmml">i</mi></msub><mo id="S3.Ex3.m1.5.6.3.2.1" xref="S3.Ex3.m1.5.6.3.2.1.cmml">⁢</mo><mrow id="S3.Ex3.m1.5.6.3.2.3.2" xref="S3.Ex3.m1.5.6.3.2.cmml"><mo id="S3.Ex3.m1.5.6.3.2.3.2.1" stretchy="false" xref="S3.Ex3.m1.5.6.3.2.cmml">(</mo><mi id="S3.Ex3.m1.5.5" xref="S3.Ex3.m1.5.5.cmml">j</mi><mo id="S3.Ex3.m1.5.6.3.2.3.2.2" stretchy="false" xref="S3.Ex3.m1.5.6.3.2.cmml">)</mo></mrow><mo id="S3.Ex3.m1.5.6.3.2.1a" xref="S3.Ex3.m1.5.6.3.2.1.cmml">⁢</mo><mtext id="S3.Ex3.m1.5.6.3.2.4" xref="S3.Ex3.m1.5.6.3.2.4a.cmml">.</mtext></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex3.m1.5b"><apply id="S3.Ex3.m1.5.6.cmml" xref="S3.Ex3.m1.5.6"><csymbol cd="latexml" id="S3.Ex3.m1.5.6.1.cmml" xref="S3.Ex3.m1.5.6.1">assign</csymbol><apply id="S3.Ex3.m1.5.6.2.cmml" xref="S3.Ex3.m1.5.6.2"><times id="S3.Ex3.m1.5.6.2.1.cmml" xref="S3.Ex3.m1.5.6.2.1"></times><ci id="S3.Ex3.m1.5.6.2.2.cmml" xref="S3.Ex3.m1.5.6.2.2">𝒱</ci><interval closure="open" id="S3.Ex3.m1.5.6.2.3.1.cmml" xref="S3.Ex3.m1.5.6.2.3.2"><ci id="S3.Ex3.m1.3.3.cmml" xref="S3.Ex3.m1.3.3">𝑁</ci><ci id="S3.Ex3.m1.4.4.cmml" xref="S3.Ex3.m1.4.4">𝑣</ci></interval></apply><apply id="S3.Ex3.m1.5.6.3.cmml" xref="S3.Ex3.m1.5.6.3"><apply id="S3.Ex3.m1.5.6.3.1.cmml" xref="S3.Ex3.m1.5.6.3.1"><csymbol cd="ambiguous" id="S3.Ex3.m1.5.6.3.1.1.cmml" xref="S3.Ex3.m1.5.6.3.1">subscript</csymbol><sum id="S3.Ex3.m1.5.6.3.1.2.cmml" xref="S3.Ex3.m1.5.6.3.1.2"></sum><apply id="S3.Ex3.m1.2.2.2.cmml" xref="S3.Ex3.m1.2.2.2"><in id="S3.Ex3.m1.2.2.2.3.cmml" xref="S3.Ex3.m1.2.2.2.3"></in><list id="S3.Ex3.m1.2.2.2.4.1.cmml" xref="S3.Ex3.m1.2.2.2.4.2"><ci id="S3.Ex3.m1.1.1.1.1.cmml" xref="S3.Ex3.m1.1.1.1.1">𝑖</ci><ci id="S3.Ex3.m1.2.2.2.2.cmml" xref="S3.Ex3.m1.2.2.2.2">𝑗</ci></list><ci id="S3.Ex3.m1.2.2.2.5.cmml" xref="S3.Ex3.m1.2.2.2.5">𝑁</ci></apply></apply><apply id="S3.Ex3.m1.5.6.3.2.cmml" xref="S3.Ex3.m1.5.6.3.2"><times id="S3.Ex3.m1.5.6.3.2.1.cmml" xref="S3.Ex3.m1.5.6.3.2.1"></times><apply id="S3.Ex3.m1.5.6.3.2.2.cmml" xref="S3.Ex3.m1.5.6.3.2.2"><csymbol cd="ambiguous" id="S3.Ex3.m1.5.6.3.2.2.1.cmml" xref="S3.Ex3.m1.5.6.3.2.2">subscript</csymbol><ci id="S3.Ex3.m1.5.6.3.2.2.2.cmml" xref="S3.Ex3.m1.5.6.3.2.2.2">𝑣</ci><ci id="S3.Ex3.m1.5.6.3.2.2.3.cmml" xref="S3.Ex3.m1.5.6.3.2.2.3">𝑖</ci></apply><ci id="S3.Ex3.m1.5.5.cmml" xref="S3.Ex3.m1.5.5">𝑗</ci><ci id="S3.Ex3.m1.5.6.3.2.4a.cmml" xref="S3.Ex3.m1.5.6.3.2.4"><mtext id="S3.Ex3.m1.5.6.3.2.4.cmml" xref="S3.Ex3.m1.5.6.3.2.4">.</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex3.m1.5c">\mathcal{V}(N,v):=\sum_{i,j\in N}v_{i}(j)\text{.}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex3.m1.5d">caligraphic_V ( italic_N , italic_v ) := ∑ start_POSTSUBSCRIPT italic_i , italic_j ∈ italic_N end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p9.2">We will obtain good approximation guarantees by restricting attention to ASHGs with nonnegative total value.</p> </div> <div class="ltx_para" id="S3.p10"> <p class="ltx_p" id="S3.p10.3">In this paper, we use <math alttext="[k]" class="ltx_Math" display="inline" id="S3.p10.1.m1.1"><semantics id="S3.p10.1.m1.1a"><mrow id="S3.p10.1.m1.1.2.2" xref="S3.p10.1.m1.1.2.1.cmml"><mo id="S3.p10.1.m1.1.2.2.1" stretchy="false" xref="S3.p10.1.m1.1.2.1.1.cmml">[</mo><mi id="S3.p10.1.m1.1.1" xref="S3.p10.1.m1.1.1.cmml">k</mi><mo id="S3.p10.1.m1.1.2.2.2" stretchy="false" xref="S3.p10.1.m1.1.2.1.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p10.1.m1.1b"><apply id="S3.p10.1.m1.1.2.1.cmml" xref="S3.p10.1.m1.1.2.2"><csymbol cd="latexml" id="S3.p10.1.m1.1.2.1.1.cmml" xref="S3.p10.1.m1.1.2.2.1">delimited-[]</csymbol><ci id="S3.p10.1.m1.1.1.cmml" xref="S3.p10.1.m1.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p10.1.m1.1c">[k]</annotation><annotation encoding="application/x-llamapun" id="S3.p10.1.m1.1d">[ italic_k ]</annotation></semantics></math> to represent the set <math alttext="\{1,\dots,k\}" class="ltx_Math" display="inline" id="S3.p10.2.m2.3"><semantics id="S3.p10.2.m2.3a"><mrow id="S3.p10.2.m2.3.4.2" xref="S3.p10.2.m2.3.4.1.cmml"><mo id="S3.p10.2.m2.3.4.2.1" stretchy="false" xref="S3.p10.2.m2.3.4.1.cmml">{</mo><mn id="S3.p10.2.m2.1.1" xref="S3.p10.2.m2.1.1.cmml">1</mn><mo id="S3.p10.2.m2.3.4.2.2" xref="S3.p10.2.m2.3.4.1.cmml">,</mo><mi id="S3.p10.2.m2.2.2" mathvariant="normal" xref="S3.p10.2.m2.2.2.cmml">…</mi><mo id="S3.p10.2.m2.3.4.2.3" xref="S3.p10.2.m2.3.4.1.cmml">,</mo><mi id="S3.p10.2.m2.3.3" xref="S3.p10.2.m2.3.3.cmml">k</mi><mo id="S3.p10.2.m2.3.4.2.4" stretchy="false" xref="S3.p10.2.m2.3.4.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p10.2.m2.3b"><set id="S3.p10.2.m2.3.4.1.cmml" xref="S3.p10.2.m2.3.4.2"><cn id="S3.p10.2.m2.1.1.cmml" type="integer" xref="S3.p10.2.m2.1.1">1</cn><ci id="S3.p10.2.m2.2.2.cmml" xref="S3.p10.2.m2.2.2">…</ci><ci id="S3.p10.2.m2.3.3.cmml" xref="S3.p10.2.m2.3.3">𝑘</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S3.p10.2.m2.3c">\{1,\dots,k\}</annotation><annotation encoding="application/x-llamapun" id="S3.p10.2.m2.3d">{ 1 , … , italic_k }</annotation></semantics></math>. Moreover, in asymptotic statements, we state logarithms without base. They can be assumed to have base <math alttext="e" class="ltx_Math" display="inline" id="S3.p10.3.m3.1"><semantics id="S3.p10.3.m3.1a"><mi id="S3.p10.3.m3.1.1" xref="S3.p10.3.m3.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S3.p10.3.m3.1b"><ci id="S3.p10.3.m3.1.1.cmml" xref="S3.p10.3.m3.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p10.3.m3.1c">e</annotation><annotation encoding="application/x-llamapun" id="S3.p10.3.m3.1d">italic_e</annotation></semantics></math>.</p> </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Deterministic Games</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.5">Recall that Aziz et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx1" title="">ABS13</a>]</cite> show that maximizing social welfare is NP-hard. Their result even holds for symmetric valuations restricted to <math alttext="\{-1,1\}" class="ltx_Math" display="inline" id="S4.p1.1.m1.2"><semantics id="S4.p1.1.m1.2a"><mrow id="S4.p1.1.m1.2.2.1" xref="S4.p1.1.m1.2.2.2.cmml"><mo id="S4.p1.1.m1.2.2.1.2" stretchy="false" xref="S4.p1.1.m1.2.2.2.cmml">{</mo><mrow id="S4.p1.1.m1.2.2.1.1" xref="S4.p1.1.m1.2.2.1.1.cmml"><mo id="S4.p1.1.m1.2.2.1.1a" xref="S4.p1.1.m1.2.2.1.1.cmml">−</mo><mn id="S4.p1.1.m1.2.2.1.1.2" xref="S4.p1.1.m1.2.2.1.1.2.cmml">1</mn></mrow><mo id="S4.p1.1.m1.2.2.1.3" xref="S4.p1.1.m1.2.2.2.cmml">,</mo><mn id="S4.p1.1.m1.1.1" xref="S4.p1.1.m1.1.1.cmml">1</mn><mo id="S4.p1.1.m1.2.2.1.4" stretchy="false" xref="S4.p1.1.m1.2.2.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.2b"><set id="S4.p1.1.m1.2.2.2.cmml" xref="S4.p1.1.m1.2.2.1"><apply id="S4.p1.1.m1.2.2.1.1.cmml" xref="S4.p1.1.m1.2.2.1.1"><minus id="S4.p1.1.m1.2.2.1.1.1.cmml" xref="S4.p1.1.m1.2.2.1.1"></minus><cn id="S4.p1.1.m1.2.2.1.1.2.cmml" type="integer" xref="S4.p1.1.m1.2.2.1.1.2">1</cn></apply><cn id="S4.p1.1.m1.1.1.cmml" type="integer" xref="S4.p1.1.m1.1.1">1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.2c">\{-1,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.2d">{ - 1 , 1 }</annotation></semantics></math>. Moreover, Flammini et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx26" title="">FKV22</a>]</cite> prove that approximating social welfare by a factor of <math alttext="n^{1-\varepsilon}" class="ltx_Math" display="inline" id="S4.p1.2.m2.1"><semantics id="S4.p1.2.m2.1a"><msup id="S4.p1.2.m2.1.1" xref="S4.p1.2.m2.1.1.cmml"><mi id="S4.p1.2.m2.1.1.2" xref="S4.p1.2.m2.1.1.2.cmml">n</mi><mrow id="S4.p1.2.m2.1.1.3" xref="S4.p1.2.m2.1.1.3.cmml"><mn id="S4.p1.2.m2.1.1.3.2" xref="S4.p1.2.m2.1.1.3.2.cmml">1</mn><mo id="S4.p1.2.m2.1.1.3.1" xref="S4.p1.2.m2.1.1.3.1.cmml">−</mo><mi id="S4.p1.2.m2.1.1.3.3" xref="S4.p1.2.m2.1.1.3.3.cmml">ε</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.p1.2.m2.1b"><apply id="S4.p1.2.m2.1.1.cmml" xref="S4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.p1.2.m2.1.1.1.cmml" xref="S4.p1.2.m2.1.1">superscript</csymbol><ci id="S4.p1.2.m2.1.1.2.cmml" xref="S4.p1.2.m2.1.1.2">𝑛</ci><apply id="S4.p1.2.m2.1.1.3.cmml" xref="S4.p1.2.m2.1.1.3"><minus id="S4.p1.2.m2.1.1.3.1.cmml" xref="S4.p1.2.m2.1.1.3.1"></minus><cn id="S4.p1.2.m2.1.1.3.2.cmml" type="integer" xref="S4.p1.2.m2.1.1.3.2">1</cn><ci id="S4.p1.2.m2.1.1.3.3.cmml" xref="S4.p1.2.m2.1.1.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.2.m2.1c">n^{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.2.m2.1d">italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT</annotation></semantics></math> is <span class="ltx_ERROR undefined" id="S4.p1.5.1">\NP</span>-hard for aversion-to-enemies games, i.e., when valuations are in the set <math alttext="\{-n,1\}" class="ltx_Math" display="inline" id="S4.p1.3.m3.2"><semantics id="S4.p1.3.m3.2a"><mrow id="S4.p1.3.m3.2.2.1" xref="S4.p1.3.m3.2.2.2.cmml"><mo id="S4.p1.3.m3.2.2.1.2" stretchy="false" xref="S4.p1.3.m3.2.2.2.cmml">{</mo><mrow id="S4.p1.3.m3.2.2.1.1" xref="S4.p1.3.m3.2.2.1.1.cmml"><mo id="S4.p1.3.m3.2.2.1.1a" xref="S4.p1.3.m3.2.2.1.1.cmml">−</mo><mi id="S4.p1.3.m3.2.2.1.1.2" xref="S4.p1.3.m3.2.2.1.1.2.cmml">n</mi></mrow><mo id="S4.p1.3.m3.2.2.1.3" xref="S4.p1.3.m3.2.2.2.cmml">,</mo><mn id="S4.p1.3.m3.1.1" xref="S4.p1.3.m3.1.1.cmml">1</mn><mo id="S4.p1.3.m3.2.2.1.4" stretchy="false" xref="S4.p1.3.m3.2.2.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.3.m3.2b"><set id="S4.p1.3.m3.2.2.2.cmml" xref="S4.p1.3.m3.2.2.1"><apply id="S4.p1.3.m3.2.2.1.1.cmml" xref="S4.p1.3.m3.2.2.1.1"><minus id="S4.p1.3.m3.2.2.1.1.1.cmml" xref="S4.p1.3.m3.2.2.1.1"></minus><ci id="S4.p1.3.m3.2.2.1.1.2.cmml" xref="S4.p1.3.m3.2.2.1.1.2">𝑛</ci></apply><cn id="S4.p1.3.m3.1.1.cmml" type="integer" xref="S4.p1.3.m3.1.1">1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.3.m3.2c">\{-n,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.3.m3.2d">{ - italic_n , 1 }</annotation></semantics></math>. In this section, we will significantly deepen the understanding of welfare approximability around this result. First, we show that the result does not rely on unbounded negative weights by providing a reduction for valuations restricted to <math alttext="\{-v^{-},0,1\}" class="ltx_Math" display="inline" id="S4.p1.4.m4.3"><semantics id="S4.p1.4.m4.3a"><mrow id="S4.p1.4.m4.3.3.1" xref="S4.p1.4.m4.3.3.2.cmml"><mo id="S4.p1.4.m4.3.3.1.2" stretchy="false" xref="S4.p1.4.m4.3.3.2.cmml">{</mo><mrow id="S4.p1.4.m4.3.3.1.1" xref="S4.p1.4.m4.3.3.1.1.cmml"><mo id="S4.p1.4.m4.3.3.1.1a" xref="S4.p1.4.m4.3.3.1.1.cmml">−</mo><msup id="S4.p1.4.m4.3.3.1.1.2" xref="S4.p1.4.m4.3.3.1.1.2.cmml"><mi id="S4.p1.4.m4.3.3.1.1.2.2" xref="S4.p1.4.m4.3.3.1.1.2.2.cmml">v</mi><mo id="S4.p1.4.m4.3.3.1.1.2.3" xref="S4.p1.4.m4.3.3.1.1.2.3.cmml">−</mo></msup></mrow><mo id="S4.p1.4.m4.3.3.1.3" xref="S4.p1.4.m4.3.3.2.cmml">,</mo><mn id="S4.p1.4.m4.1.1" xref="S4.p1.4.m4.1.1.cmml">0</mn><mo id="S4.p1.4.m4.3.3.1.4" xref="S4.p1.4.m4.3.3.2.cmml">,</mo><mn id="S4.p1.4.m4.2.2" xref="S4.p1.4.m4.2.2.cmml">1</mn><mo id="S4.p1.4.m4.3.3.1.5" stretchy="false" xref="S4.p1.4.m4.3.3.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.4.m4.3b"><set id="S4.p1.4.m4.3.3.2.cmml" xref="S4.p1.4.m4.3.3.1"><apply id="S4.p1.4.m4.3.3.1.1.cmml" xref="S4.p1.4.m4.3.3.1.1"><minus id="S4.p1.4.m4.3.3.1.1.1.cmml" xref="S4.p1.4.m4.3.3.1.1"></minus><apply id="S4.p1.4.m4.3.3.1.1.2.cmml" xref="S4.p1.4.m4.3.3.1.1.2"><csymbol cd="ambiguous" id="S4.p1.4.m4.3.3.1.1.2.1.cmml" xref="S4.p1.4.m4.3.3.1.1.2">superscript</csymbol><ci id="S4.p1.4.m4.3.3.1.1.2.2.cmml" xref="S4.p1.4.m4.3.3.1.1.2.2">𝑣</ci><minus id="S4.p1.4.m4.3.3.1.1.2.3.cmml" xref="S4.p1.4.m4.3.3.1.1.2.3"></minus></apply></apply><cn id="S4.p1.4.m4.1.1.cmml" type="integer" xref="S4.p1.4.m4.1.1">0</cn><cn id="S4.p1.4.m4.2.2.cmml" type="integer" xref="S4.p1.4.m4.2.2">1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.4.m4.3c">\{-v^{-},0,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.4.m4.3d">{ - italic_v start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT , 0 , 1 }</annotation></semantics></math> where <math alttext="v^{-}\geq 1" class="ltx_Math" display="inline" id="S4.p1.5.m5.1"><semantics id="S4.p1.5.m5.1a"><mrow id="S4.p1.5.m5.1.1" xref="S4.p1.5.m5.1.1.cmml"><msup id="S4.p1.5.m5.1.1.2" xref="S4.p1.5.m5.1.1.2.cmml"><mi id="S4.p1.5.m5.1.1.2.2" xref="S4.p1.5.m5.1.1.2.2.cmml">v</mi><mo id="S4.p1.5.m5.1.1.2.3" xref="S4.p1.5.m5.1.1.2.3.cmml">−</mo></msup><mo id="S4.p1.5.m5.1.1.1" xref="S4.p1.5.m5.1.1.1.cmml">≥</mo><mn id="S4.p1.5.m5.1.1.3" xref="S4.p1.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.5.m5.1b"><apply id="S4.p1.5.m5.1.1.cmml" xref="S4.p1.5.m5.1.1"><geq id="S4.p1.5.m5.1.1.1.cmml" xref="S4.p1.5.m5.1.1.1"></geq><apply id="S4.p1.5.m5.1.1.2.cmml" xref="S4.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.p1.5.m5.1.1.2.1.cmml" xref="S4.p1.5.m5.1.1.2">superscript</csymbol><ci id="S4.p1.5.m5.1.1.2.2.cmml" xref="S4.p1.5.m5.1.1.2.2">𝑣</ci><minus id="S4.p1.5.m5.1.1.2.3.cmml" xref="S4.p1.5.m5.1.1.2.3"></minus></apply><cn id="S4.p1.5.m5.1.1.3.cmml" type="integer" xref="S4.p1.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.5.m5.1c">v^{-}\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.p1.5.m5.1d">italic_v start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ≥ 1</annotation></semantics></math>. In particular, this means that unbounded negative weights or negative weights of absolute value much larger than the value of positive weights are not necessary. Subsequently, we will show how to circumvent the inapproximability result for ASHG with nonnegative total value.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Welfare Inapproximability for Restricted Valuations</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">We now prove our inapproximability result.<span class="ltx_note ltx_role_footnote" id="footnote3"><sup class="ltx_note_mark">3</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">3</sup><span class="ltx_tag ltx_tag_note">3</span>We would like to thank Abheek Ghosh for the proof idea of <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1 Welfare Inapproximability for Restricted Valuations ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.1</span></a>.</span></span></span></p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.1.1.1">Theorem 4.1</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem1.p1"> <p class="ltx_p" id="S4.Thmtheorem1.p1.5"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem1.p1.5.5">Let <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.1.1.m1.1"><semantics id="S4.Thmtheorem1.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem1.p1.1.1.m1.1.1" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem1.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.2.cmml">ε</mi><mo id="S4.Thmtheorem1.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.1.cmml">></mo><mn id="S4.Thmtheorem1.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.1.1.m1.1b"><apply id="S4.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1"><gt id="S4.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.1"></gt><ci id="S4.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.2">𝜀</ci><cn id="S4.Thmtheorem1.p1.1.1.m1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.1.1.m1.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.1.1.m1.1d">italic_ε > 0</annotation></semantics></math> and <math alttext="v^{-}\geq 1" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.2.2.m2.1"><semantics id="S4.Thmtheorem1.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem1.p1.2.2.m2.1.1" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.cmml"><msup id="S4.Thmtheorem1.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.2.cmml"><mi id="S4.Thmtheorem1.p1.2.2.m2.1.1.2.2" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.2.2.cmml">v</mi><mo id="S4.Thmtheorem1.p1.2.2.m2.1.1.2.3" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.2.3.cmml">−</mo></msup><mo id="S4.Thmtheorem1.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.1.cmml">≥</mo><mn id="S4.Thmtheorem1.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.2.2.m2.1b"><apply id="S4.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1"><geq id="S4.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.1"></geq><apply id="S4.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.2.2.m2.1.1.2.1.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.2">superscript</csymbol><ci id="S4.Thmtheorem1.p1.2.2.m2.1.1.2.2.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.2.2">𝑣</ci><minus id="S4.Thmtheorem1.p1.2.2.m2.1.1.2.3.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.2.3"></minus></apply><cn id="S4.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.2.2.m2.1c">v^{-}\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.2.2.m2.1d">italic_v start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ≥ 1</annotation></semantics></math>. Then, unless ¶ <math alttext="=" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.3.3.m3.1"><semantics id="S4.Thmtheorem1.p1.3.3.m3.1a"><mo id="S4.Thmtheorem1.p1.3.3.m3.1.1" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.3.3.m3.1b"><eq id="S4.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.3.3.m3.1c">=</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.3.3.m3.1d">=</annotation></semantics></math> <span class="ltx_ERROR undefined" id="S4.Thmtheorem1.p1.5.5.1">\NP</span>, <math alttext="n^{1-\varepsilon}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.4.4.m4.1"><semantics id="S4.Thmtheorem1.p1.4.4.m4.1a"><msup id="S4.Thmtheorem1.p1.4.4.m4.1.1" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.cmml"><mi id="S4.Thmtheorem1.p1.4.4.m4.1.1.2" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.2.cmml">n</mi><mrow id="S4.Thmtheorem1.p1.4.4.m4.1.1.3" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.3.cmml"><mn id="S4.Thmtheorem1.p1.4.4.m4.1.1.3.2" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.3.2.cmml">1</mn><mo id="S4.Thmtheorem1.p1.4.4.m4.1.1.3.1" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.3.1.cmml">−</mo><mi id="S4.Thmtheorem1.p1.4.4.m4.1.1.3.3" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.3.3.cmml">ε</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.4.4.m4.1b"><apply id="S4.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.1">superscript</csymbol><ci id="S4.Thmtheorem1.p1.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.2">𝑛</ci><apply id="S4.Thmtheorem1.p1.4.4.m4.1.1.3.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.3"><minus id="S4.Thmtheorem1.p1.4.4.m4.1.1.3.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.3.1"></minus><cn id="S4.Thmtheorem1.p1.4.4.m4.1.1.3.2.cmml" type="integer" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.3.2">1</cn><ci id="S4.Thmtheorem1.p1.4.4.m4.1.1.3.3.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.4.4.m4.1c">n^{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.4.4.m4.1d">italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT</annotation></semantics></math>-<span class="ltx_text ltx_font_smallcaps" id="S4.Thmtheorem1.p1.5.5.2">ApproxWelfare</span> cannot be solved in polynomial time for symmetric ASHGs with valuations in the set <math alttext="\{-v^{-},0,1\}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.5.5.m5.3"><semantics id="S4.Thmtheorem1.p1.5.5.m5.3a"><mrow id="S4.Thmtheorem1.p1.5.5.m5.3.3.1" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.2.cmml"><mo id="S4.Thmtheorem1.p1.5.5.m5.3.3.1.2" stretchy="false" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.2.cmml">{</mo><mrow id="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.cmml"><mo id="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1a" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.cmml">−</mo><msup id="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.2" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.2.cmml"><mi id="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.2.2" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.2.2.cmml">v</mi><mo id="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.2.3" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.2.3.cmml">−</mo></msup></mrow><mo id="S4.Thmtheorem1.p1.5.5.m5.3.3.1.3" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.2.cmml">,</mo><mn id="S4.Thmtheorem1.p1.5.5.m5.1.1" xref="S4.Thmtheorem1.p1.5.5.m5.1.1.cmml">0</mn><mo id="S4.Thmtheorem1.p1.5.5.m5.3.3.1.4" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.2.cmml">,</mo><mn id="S4.Thmtheorem1.p1.5.5.m5.2.2" xref="S4.Thmtheorem1.p1.5.5.m5.2.2.cmml">1</mn><mo id="S4.Thmtheorem1.p1.5.5.m5.3.3.1.5" stretchy="false" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.5.5.m5.3b"><set id="S4.Thmtheorem1.p1.5.5.m5.3.3.2.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.1"><apply id="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1"><minus id="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1"></minus><apply id="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.2.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.2.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.2">superscript</csymbol><ci id="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.2.2.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.2.2">𝑣</ci><minus id="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.2.3.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.3.3.1.1.2.3"></minus></apply></apply><cn id="S4.Thmtheorem1.p1.5.5.m5.1.1.cmml" type="integer" xref="S4.Thmtheorem1.p1.5.5.m5.1.1">0</cn><cn id="S4.Thmtheorem1.p1.5.5.m5.2.2.cmml" type="integer" xref="S4.Thmtheorem1.p1.5.5.m5.2.2">1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.5.5.m5.3c">\{-v^{-},0,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.5.5.m5.3d">{ - italic_v start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT , 0 , 1 }</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.SS1.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS1.1.p1"> <p class="ltx_p" id="S4.SS1.1.p1.10">Let <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S4.SS1.1.p1.1.m1.1"><semantics id="S4.SS1.1.p1.1.m1.1a"><mrow id="S4.SS1.1.p1.1.m1.1.1" xref="S4.SS1.1.p1.1.m1.1.1.cmml"><mi id="S4.SS1.1.p1.1.m1.1.1.2" xref="S4.SS1.1.p1.1.m1.1.1.2.cmml">ε</mi><mo id="S4.SS1.1.p1.1.m1.1.1.1" xref="S4.SS1.1.p1.1.m1.1.1.1.cmml">></mo><mn id="S4.SS1.1.p1.1.m1.1.1.3" xref="S4.SS1.1.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.1.p1.1.m1.1b"><apply id="S4.SS1.1.p1.1.m1.1.1.cmml" xref="S4.SS1.1.p1.1.m1.1.1"><gt id="S4.SS1.1.p1.1.m1.1.1.1.cmml" xref="S4.SS1.1.p1.1.m1.1.1.1"></gt><ci id="S4.SS1.1.p1.1.m1.1.1.2.cmml" xref="S4.SS1.1.p1.1.m1.1.1.2">𝜀</ci><cn id="S4.SS1.1.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.SS1.1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.p1.1.m1.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.p1.1.m1.1d">italic_ε > 0</annotation></semantics></math> and <math alttext="v^{-}\geq 1" class="ltx_Math" display="inline" id="S4.SS1.1.p1.2.m2.1"><semantics id="S4.SS1.1.p1.2.m2.1a"><mrow id="S4.SS1.1.p1.2.m2.1.1" xref="S4.SS1.1.p1.2.m2.1.1.cmml"><msup id="S4.SS1.1.p1.2.m2.1.1.2" xref="S4.SS1.1.p1.2.m2.1.1.2.cmml"><mi id="S4.SS1.1.p1.2.m2.1.1.2.2" xref="S4.SS1.1.p1.2.m2.1.1.2.2.cmml">v</mi><mo id="S4.SS1.1.p1.2.m2.1.1.2.3" xref="S4.SS1.1.p1.2.m2.1.1.2.3.cmml">−</mo></msup><mo id="S4.SS1.1.p1.2.m2.1.1.1" xref="S4.SS1.1.p1.2.m2.1.1.1.cmml">≥</mo><mn id="S4.SS1.1.p1.2.m2.1.1.3" xref="S4.SS1.1.p1.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.1.p1.2.m2.1b"><apply id="S4.SS1.1.p1.2.m2.1.1.cmml" xref="S4.SS1.1.p1.2.m2.1.1"><geq id="S4.SS1.1.p1.2.m2.1.1.1.cmml" xref="S4.SS1.1.p1.2.m2.1.1.1"></geq><apply id="S4.SS1.1.p1.2.m2.1.1.2.cmml" xref="S4.SS1.1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.1.p1.2.m2.1.1.2.1.cmml" xref="S4.SS1.1.p1.2.m2.1.1.2">superscript</csymbol><ci id="S4.SS1.1.p1.2.m2.1.1.2.2.cmml" xref="S4.SS1.1.p1.2.m2.1.1.2.2">𝑣</ci><minus id="S4.SS1.1.p1.2.m2.1.1.2.3.cmml" xref="S4.SS1.1.p1.2.m2.1.1.2.3"></minus></apply><cn id="S4.SS1.1.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.SS1.1.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.p1.2.m2.1c">v^{-}\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.p1.2.m2.1d">italic_v start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ≥ 1</annotation></semantics></math>. We reduce from the <math alttext="n^{1-\varepsilon}" class="ltx_Math" display="inline" id="S4.SS1.1.p1.3.m3.1"><semantics id="S4.SS1.1.p1.3.m3.1a"><msup id="S4.SS1.1.p1.3.m3.1.1" xref="S4.SS1.1.p1.3.m3.1.1.cmml"><mi id="S4.SS1.1.p1.3.m3.1.1.2" xref="S4.SS1.1.p1.3.m3.1.1.2.cmml">n</mi><mrow id="S4.SS1.1.p1.3.m3.1.1.3" xref="S4.SS1.1.p1.3.m3.1.1.3.cmml"><mn id="S4.SS1.1.p1.3.m3.1.1.3.2" xref="S4.SS1.1.p1.3.m3.1.1.3.2.cmml">1</mn><mo id="S4.SS1.1.p1.3.m3.1.1.3.1" xref="S4.SS1.1.p1.3.m3.1.1.3.1.cmml">−</mo><mi id="S4.SS1.1.p1.3.m3.1.1.3.3" xref="S4.SS1.1.p1.3.m3.1.1.3.3.cmml">ε</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.1.p1.3.m3.1b"><apply id="S4.SS1.1.p1.3.m3.1.1.cmml" xref="S4.SS1.1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.1.p1.3.m3.1.1.1.cmml" xref="S4.SS1.1.p1.3.m3.1.1">superscript</csymbol><ci id="S4.SS1.1.p1.3.m3.1.1.2.cmml" xref="S4.SS1.1.p1.3.m3.1.1.2">𝑛</ci><apply id="S4.SS1.1.p1.3.m3.1.1.3.cmml" xref="S4.SS1.1.p1.3.m3.1.1.3"><minus id="S4.SS1.1.p1.3.m3.1.1.3.1.cmml" xref="S4.SS1.1.p1.3.m3.1.1.3.1"></minus><cn id="S4.SS1.1.p1.3.m3.1.1.3.2.cmml" type="integer" xref="S4.SS1.1.p1.3.m3.1.1.3.2">1</cn><ci id="S4.SS1.1.p1.3.m3.1.1.3.3.cmml" xref="S4.SS1.1.p1.3.m3.1.1.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.p1.3.m3.1c">n^{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.p1.3.m3.1d">italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT</annotation></semantics></math>-approximate <span class="ltx_text ltx_font_smallcaps" id="S4.SS1.1.p1.10.1">MaximumClique</span> problem. The input is an unweighted graph <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.1.p1.4.m4.1"><semantics id="S4.SS1.1.p1.4.m4.1a"><mi id="S4.SS1.1.p1.4.m4.1.1" xref="S4.SS1.1.p1.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.1.p1.4.m4.1b"><ci id="S4.SS1.1.p1.4.m4.1.1.cmml" xref="S4.SS1.1.p1.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.p1.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.p1.4.m4.1d">italic_G</annotation></semantics></math> and the task is to compute a clique <math alttext="C" class="ltx_Math" display="inline" id="S4.SS1.1.p1.5.m5.1"><semantics id="S4.SS1.1.p1.5.m5.1a"><mi id="S4.SS1.1.p1.5.m5.1.1" xref="S4.SS1.1.p1.5.m5.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.1.p1.5.m5.1b"><ci id="S4.SS1.1.p1.5.m5.1.1.cmml" xref="S4.SS1.1.p1.5.m5.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.p1.5.m5.1c">C</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.p1.5.m5.1d">italic_C</annotation></semantics></math> of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.1.p1.6.m6.1"><semantics id="S4.SS1.1.p1.6.m6.1a"><mi id="S4.SS1.1.p1.6.m6.1.1" xref="S4.SS1.1.p1.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.1.p1.6.m6.1b"><ci id="S4.SS1.1.p1.6.m6.1.1.cmml" xref="S4.SS1.1.p1.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.p1.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.p1.6.m6.1d">italic_G</annotation></semantics></math> with <math alttext="n^{1-\varepsilon}\cdot|C|\geq\mu^{*}" class="ltx_Math" display="inline" id="S4.SS1.1.p1.7.m7.1"><semantics id="S4.SS1.1.p1.7.m7.1a"><mrow id="S4.SS1.1.p1.7.m7.1.2" xref="S4.SS1.1.p1.7.m7.1.2.cmml"><mrow id="S4.SS1.1.p1.7.m7.1.2.2" xref="S4.SS1.1.p1.7.m7.1.2.2.cmml"><msup id="S4.SS1.1.p1.7.m7.1.2.2.2" xref="S4.SS1.1.p1.7.m7.1.2.2.2.cmml"><mi id="S4.SS1.1.p1.7.m7.1.2.2.2.2" xref="S4.SS1.1.p1.7.m7.1.2.2.2.2.cmml">n</mi><mrow id="S4.SS1.1.p1.7.m7.1.2.2.2.3" xref="S4.SS1.1.p1.7.m7.1.2.2.2.3.cmml"><mn id="S4.SS1.1.p1.7.m7.1.2.2.2.3.2" xref="S4.SS1.1.p1.7.m7.1.2.2.2.3.2.cmml">1</mn><mo id="S4.SS1.1.p1.7.m7.1.2.2.2.3.1" xref="S4.SS1.1.p1.7.m7.1.2.2.2.3.1.cmml">−</mo><mi id="S4.SS1.1.p1.7.m7.1.2.2.2.3.3" xref="S4.SS1.1.p1.7.m7.1.2.2.2.3.3.cmml">ε</mi></mrow></msup><mo id="S4.SS1.1.p1.7.m7.1.2.2.1" lspace="0.222em" rspace="0.222em" xref="S4.SS1.1.p1.7.m7.1.2.2.1.cmml">⋅</mo><mrow id="S4.SS1.1.p1.7.m7.1.2.2.3.2" xref="S4.SS1.1.p1.7.m7.1.2.2.3.1.cmml"><mo id="S4.SS1.1.p1.7.m7.1.2.2.3.2.1" stretchy="false" xref="S4.SS1.1.p1.7.m7.1.2.2.3.1.1.cmml">|</mo><mi id="S4.SS1.1.p1.7.m7.1.1" xref="S4.SS1.1.p1.7.m7.1.1.cmml">C</mi><mo id="S4.SS1.1.p1.7.m7.1.2.2.3.2.2" stretchy="false" xref="S4.SS1.1.p1.7.m7.1.2.2.3.1.1.cmml">|</mo></mrow></mrow><mo id="S4.SS1.1.p1.7.m7.1.2.1" xref="S4.SS1.1.p1.7.m7.1.2.1.cmml">≥</mo><msup id="S4.SS1.1.p1.7.m7.1.2.3" xref="S4.SS1.1.p1.7.m7.1.2.3.cmml"><mi id="S4.SS1.1.p1.7.m7.1.2.3.2" xref="S4.SS1.1.p1.7.m7.1.2.3.2.cmml">μ</mi><mo id="S4.SS1.1.p1.7.m7.1.2.3.3" xref="S4.SS1.1.p1.7.m7.1.2.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.1.p1.7.m7.1b"><apply id="S4.SS1.1.p1.7.m7.1.2.cmml" xref="S4.SS1.1.p1.7.m7.1.2"><geq id="S4.SS1.1.p1.7.m7.1.2.1.cmml" xref="S4.SS1.1.p1.7.m7.1.2.1"></geq><apply id="S4.SS1.1.p1.7.m7.1.2.2.cmml" xref="S4.SS1.1.p1.7.m7.1.2.2"><ci id="S4.SS1.1.p1.7.m7.1.2.2.1.cmml" xref="S4.SS1.1.p1.7.m7.1.2.2.1">⋅</ci><apply id="S4.SS1.1.p1.7.m7.1.2.2.2.cmml" xref="S4.SS1.1.p1.7.m7.1.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.1.p1.7.m7.1.2.2.2.1.cmml" xref="S4.SS1.1.p1.7.m7.1.2.2.2">superscript</csymbol><ci id="S4.SS1.1.p1.7.m7.1.2.2.2.2.cmml" xref="S4.SS1.1.p1.7.m7.1.2.2.2.2">𝑛</ci><apply id="S4.SS1.1.p1.7.m7.1.2.2.2.3.cmml" xref="S4.SS1.1.p1.7.m7.1.2.2.2.3"><minus id="S4.SS1.1.p1.7.m7.1.2.2.2.3.1.cmml" xref="S4.SS1.1.p1.7.m7.1.2.2.2.3.1"></minus><cn id="S4.SS1.1.p1.7.m7.1.2.2.2.3.2.cmml" type="integer" xref="S4.SS1.1.p1.7.m7.1.2.2.2.3.2">1</cn><ci id="S4.SS1.1.p1.7.m7.1.2.2.2.3.3.cmml" xref="S4.SS1.1.p1.7.m7.1.2.2.2.3.3">𝜀</ci></apply></apply><apply id="S4.SS1.1.p1.7.m7.1.2.2.3.1.cmml" xref="S4.SS1.1.p1.7.m7.1.2.2.3.2"><abs id="S4.SS1.1.p1.7.m7.1.2.2.3.1.1.cmml" xref="S4.SS1.1.p1.7.m7.1.2.2.3.2.1"></abs><ci id="S4.SS1.1.p1.7.m7.1.1.cmml" xref="S4.SS1.1.p1.7.m7.1.1">𝐶</ci></apply></apply><apply id="S4.SS1.1.p1.7.m7.1.2.3.cmml" xref="S4.SS1.1.p1.7.m7.1.2.3"><csymbol cd="ambiguous" id="S4.SS1.1.p1.7.m7.1.2.3.1.cmml" xref="S4.SS1.1.p1.7.m7.1.2.3">superscript</csymbol><ci id="S4.SS1.1.p1.7.m7.1.2.3.2.cmml" xref="S4.SS1.1.p1.7.m7.1.2.3.2">𝜇</ci><times id="S4.SS1.1.p1.7.m7.1.2.3.3.cmml" xref="S4.SS1.1.p1.7.m7.1.2.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.p1.7.m7.1c">n^{1-\varepsilon}\cdot|C|\geq\mu^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.p1.7.m7.1d">italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT ⋅ | italic_C | ≥ italic_μ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="\mu^{*}" class="ltx_Math" display="inline" id="S4.SS1.1.p1.8.m8.1"><semantics id="S4.SS1.1.p1.8.m8.1a"><msup id="S4.SS1.1.p1.8.m8.1.1" xref="S4.SS1.1.p1.8.m8.1.1.cmml"><mi id="S4.SS1.1.p1.8.m8.1.1.2" xref="S4.SS1.1.p1.8.m8.1.1.2.cmml">μ</mi><mo id="S4.SS1.1.p1.8.m8.1.1.3" xref="S4.SS1.1.p1.8.m8.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.1.p1.8.m8.1b"><apply id="S4.SS1.1.p1.8.m8.1.1.cmml" xref="S4.SS1.1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS1.1.p1.8.m8.1.1.1.cmml" xref="S4.SS1.1.p1.8.m8.1.1">superscript</csymbol><ci id="S4.SS1.1.p1.8.m8.1.1.2.cmml" xref="S4.SS1.1.p1.8.m8.1.1.2">𝜇</ci><times id="S4.SS1.1.p1.8.m8.1.1.3.cmml" xref="S4.SS1.1.p1.8.m8.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.p1.8.m8.1c">\mu^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.p1.8.m8.1d">italic_μ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is the size of a maximum clique of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.1.p1.9.m9.1"><semantics id="S4.SS1.1.p1.9.m9.1a"><mi id="S4.SS1.1.p1.9.m9.1.1" xref="S4.SS1.1.p1.9.m9.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.1.p1.9.m9.1b"><ci id="S4.SS1.1.p1.9.m9.1.1.cmml" xref="S4.SS1.1.p1.9.m9.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.p1.9.m9.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.p1.9.m9.1d">italic_G</annotation></semantics></math>. Unless ¶ <math alttext="=" class="ltx_Math" display="inline" id="S4.SS1.1.p1.10.m10.1"><semantics id="S4.SS1.1.p1.10.m10.1a"><mo id="S4.SS1.1.p1.10.m10.1.1" xref="S4.SS1.1.p1.10.m10.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.p1.10.m10.1b"><eq id="S4.SS1.1.p1.10.m10.1.1.cmml" xref="S4.SS1.1.p1.10.m10.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.p1.10.m10.1c">=</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.p1.10.m10.1d">=</annotation></semantics></math> <span class="ltx_ERROR undefined" id="S4.SS1.1.p1.10.2">\NP</span>, this problem cannot be solved in polynomial time <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx36" title="">Zuc06</a>]</cite>.</p> </div> <div class="ltx_para" id="S4.SS1.2.p2"> <p class="ltx_p" id="S4.SS1.2.p2.6">We now describe the reduction. Assume that we are given an unweighted graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S4.SS1.2.p2.1.m1.2"><semantics id="S4.SS1.2.p2.1.m1.2a"><mrow id="S4.SS1.2.p2.1.m1.2.3" xref="S4.SS1.2.p2.1.m1.2.3.cmml"><mi id="S4.SS1.2.p2.1.m1.2.3.2" xref="S4.SS1.2.p2.1.m1.2.3.2.cmml">G</mi><mo id="S4.SS1.2.p2.1.m1.2.3.1" xref="S4.SS1.2.p2.1.m1.2.3.1.cmml">=</mo><mrow id="S4.SS1.2.p2.1.m1.2.3.3.2" xref="S4.SS1.2.p2.1.m1.2.3.3.1.cmml"><mo id="S4.SS1.2.p2.1.m1.2.3.3.2.1" stretchy="false" xref="S4.SS1.2.p2.1.m1.2.3.3.1.cmml">(</mo><mi id="S4.SS1.2.p2.1.m1.1.1" xref="S4.SS1.2.p2.1.m1.1.1.cmml">V</mi><mo id="S4.SS1.2.p2.1.m1.2.3.3.2.2" xref="S4.SS1.2.p2.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.SS1.2.p2.1.m1.2.2" xref="S4.SS1.2.p2.1.m1.2.2.cmml">E</mi><mo id="S4.SS1.2.p2.1.m1.2.3.3.2.3" stretchy="false" xref="S4.SS1.2.p2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p2.1.m1.2b"><apply id="S4.SS1.2.p2.1.m1.2.3.cmml" xref="S4.SS1.2.p2.1.m1.2.3"><eq id="S4.SS1.2.p2.1.m1.2.3.1.cmml" xref="S4.SS1.2.p2.1.m1.2.3.1"></eq><ci id="S4.SS1.2.p2.1.m1.2.3.2.cmml" xref="S4.SS1.2.p2.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S4.SS1.2.p2.1.m1.2.3.3.1.cmml" xref="S4.SS1.2.p2.1.m1.2.3.3.2"><ci id="S4.SS1.2.p2.1.m1.1.1.cmml" xref="S4.SS1.2.p2.1.m1.1.1">𝑉</ci><ci id="S4.SS1.2.p2.1.m1.2.2.cmml" xref="S4.SS1.2.p2.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p2.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p2.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>. We construct an ASHG <math alttext="(N,v)" class="ltx_Math" display="inline" id="S4.SS1.2.p2.2.m2.2"><semantics id="S4.SS1.2.p2.2.m2.2a"><mrow id="S4.SS1.2.p2.2.m2.2.3.2" xref="S4.SS1.2.p2.2.m2.2.3.1.cmml"><mo id="S4.SS1.2.p2.2.m2.2.3.2.1" stretchy="false" xref="S4.SS1.2.p2.2.m2.2.3.1.cmml">(</mo><mi id="S4.SS1.2.p2.2.m2.1.1" xref="S4.SS1.2.p2.2.m2.1.1.cmml">N</mi><mo id="S4.SS1.2.p2.2.m2.2.3.2.2" xref="S4.SS1.2.p2.2.m2.2.3.1.cmml">,</mo><mi id="S4.SS1.2.p2.2.m2.2.2" xref="S4.SS1.2.p2.2.m2.2.2.cmml">v</mi><mo id="S4.SS1.2.p2.2.m2.2.3.2.3" stretchy="false" xref="S4.SS1.2.p2.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p2.2.m2.2b"><interval closure="open" id="S4.SS1.2.p2.2.m2.2.3.1.cmml" xref="S4.SS1.2.p2.2.m2.2.3.2"><ci id="S4.SS1.2.p2.2.m2.1.1.cmml" xref="S4.SS1.2.p2.2.m2.1.1">𝑁</ci><ci id="S4.SS1.2.p2.2.m2.2.2.cmml" xref="S4.SS1.2.p2.2.m2.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p2.2.m2.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p2.2.m2.2d">( italic_N , italic_v )</annotation></semantics></math> as follows. The set of players is <math alttext="N=N_{V}\cup\{z\}" class="ltx_Math" display="inline" id="S4.SS1.2.p2.3.m3.1"><semantics id="S4.SS1.2.p2.3.m3.1a"><mrow id="S4.SS1.2.p2.3.m3.1.2" xref="S4.SS1.2.p2.3.m3.1.2.cmml"><mi id="S4.SS1.2.p2.3.m3.1.2.2" xref="S4.SS1.2.p2.3.m3.1.2.2.cmml">N</mi><mo id="S4.SS1.2.p2.3.m3.1.2.1" xref="S4.SS1.2.p2.3.m3.1.2.1.cmml">=</mo><mrow id="S4.SS1.2.p2.3.m3.1.2.3" xref="S4.SS1.2.p2.3.m3.1.2.3.cmml"><msub id="S4.SS1.2.p2.3.m3.1.2.3.2" xref="S4.SS1.2.p2.3.m3.1.2.3.2.cmml"><mi id="S4.SS1.2.p2.3.m3.1.2.3.2.2" xref="S4.SS1.2.p2.3.m3.1.2.3.2.2.cmml">N</mi><mi id="S4.SS1.2.p2.3.m3.1.2.3.2.3" xref="S4.SS1.2.p2.3.m3.1.2.3.2.3.cmml">V</mi></msub><mo id="S4.SS1.2.p2.3.m3.1.2.3.1" xref="S4.SS1.2.p2.3.m3.1.2.3.1.cmml">∪</mo><mrow id="S4.SS1.2.p2.3.m3.1.2.3.3.2" xref="S4.SS1.2.p2.3.m3.1.2.3.3.1.cmml"><mo id="S4.SS1.2.p2.3.m3.1.2.3.3.2.1" stretchy="false" xref="S4.SS1.2.p2.3.m3.1.2.3.3.1.cmml">{</mo><mi id="S4.SS1.2.p2.3.m3.1.1" xref="S4.SS1.2.p2.3.m3.1.1.cmml">z</mi><mo id="S4.SS1.2.p2.3.m3.1.2.3.3.2.2" stretchy="false" xref="S4.SS1.2.p2.3.m3.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p2.3.m3.1b"><apply id="S4.SS1.2.p2.3.m3.1.2.cmml" xref="S4.SS1.2.p2.3.m3.1.2"><eq id="S4.SS1.2.p2.3.m3.1.2.1.cmml" xref="S4.SS1.2.p2.3.m3.1.2.1"></eq><ci id="S4.SS1.2.p2.3.m3.1.2.2.cmml" xref="S4.SS1.2.p2.3.m3.1.2.2">𝑁</ci><apply id="S4.SS1.2.p2.3.m3.1.2.3.cmml" xref="S4.SS1.2.p2.3.m3.1.2.3"><union id="S4.SS1.2.p2.3.m3.1.2.3.1.cmml" xref="S4.SS1.2.p2.3.m3.1.2.3.1"></union><apply id="S4.SS1.2.p2.3.m3.1.2.3.2.cmml" xref="S4.SS1.2.p2.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.2.p2.3.m3.1.2.3.2.1.cmml" xref="S4.SS1.2.p2.3.m3.1.2.3.2">subscript</csymbol><ci id="S4.SS1.2.p2.3.m3.1.2.3.2.2.cmml" xref="S4.SS1.2.p2.3.m3.1.2.3.2.2">𝑁</ci><ci id="S4.SS1.2.p2.3.m3.1.2.3.2.3.cmml" xref="S4.SS1.2.p2.3.m3.1.2.3.2.3">𝑉</ci></apply><set id="S4.SS1.2.p2.3.m3.1.2.3.3.1.cmml" xref="S4.SS1.2.p2.3.m3.1.2.3.3.2"><ci id="S4.SS1.2.p2.3.m3.1.1.cmml" xref="S4.SS1.2.p2.3.m3.1.1">𝑧</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p2.3.m3.1c">N=N_{V}\cup\{z\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p2.3.m3.1d">italic_N = italic_N start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ∪ { italic_z }</annotation></semantics></math>, where <math alttext="N_{V}=\{a_{u}\colon u\in V\}" class="ltx_Math" display="inline" id="S4.SS1.2.p2.4.m4.2"><semantics id="S4.SS1.2.p2.4.m4.2a"><mrow id="S4.SS1.2.p2.4.m4.2.2" xref="S4.SS1.2.p2.4.m4.2.2.cmml"><msub id="S4.SS1.2.p2.4.m4.2.2.4" xref="S4.SS1.2.p2.4.m4.2.2.4.cmml"><mi id="S4.SS1.2.p2.4.m4.2.2.4.2" xref="S4.SS1.2.p2.4.m4.2.2.4.2.cmml">N</mi><mi id="S4.SS1.2.p2.4.m4.2.2.4.3" xref="S4.SS1.2.p2.4.m4.2.2.4.3.cmml">V</mi></msub><mo id="S4.SS1.2.p2.4.m4.2.2.3" xref="S4.SS1.2.p2.4.m4.2.2.3.cmml">=</mo><mrow id="S4.SS1.2.p2.4.m4.2.2.2.2" xref="S4.SS1.2.p2.4.m4.2.2.2.3.cmml"><mo id="S4.SS1.2.p2.4.m4.2.2.2.2.3" stretchy="false" xref="S4.SS1.2.p2.4.m4.2.2.2.3.1.cmml">{</mo><msub id="S4.SS1.2.p2.4.m4.1.1.1.1.1" xref="S4.SS1.2.p2.4.m4.1.1.1.1.1.cmml"><mi id="S4.SS1.2.p2.4.m4.1.1.1.1.1.2" xref="S4.SS1.2.p2.4.m4.1.1.1.1.1.2.cmml">a</mi><mi id="S4.SS1.2.p2.4.m4.1.1.1.1.1.3" xref="S4.SS1.2.p2.4.m4.1.1.1.1.1.3.cmml">u</mi></msub><mo id="S4.SS1.2.p2.4.m4.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.SS1.2.p2.4.m4.2.2.2.3.1.cmml">:</mo><mrow id="S4.SS1.2.p2.4.m4.2.2.2.2.2" xref="S4.SS1.2.p2.4.m4.2.2.2.2.2.cmml"><mi id="S4.SS1.2.p2.4.m4.2.2.2.2.2.2" xref="S4.SS1.2.p2.4.m4.2.2.2.2.2.2.cmml">u</mi><mo id="S4.SS1.2.p2.4.m4.2.2.2.2.2.1" xref="S4.SS1.2.p2.4.m4.2.2.2.2.2.1.cmml">∈</mo><mi id="S4.SS1.2.p2.4.m4.2.2.2.2.2.3" xref="S4.SS1.2.p2.4.m4.2.2.2.2.2.3.cmml">V</mi></mrow><mo id="S4.SS1.2.p2.4.m4.2.2.2.2.5" stretchy="false" xref="S4.SS1.2.p2.4.m4.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p2.4.m4.2b"><apply id="S4.SS1.2.p2.4.m4.2.2.cmml" xref="S4.SS1.2.p2.4.m4.2.2"><eq id="S4.SS1.2.p2.4.m4.2.2.3.cmml" xref="S4.SS1.2.p2.4.m4.2.2.3"></eq><apply id="S4.SS1.2.p2.4.m4.2.2.4.cmml" xref="S4.SS1.2.p2.4.m4.2.2.4"><csymbol cd="ambiguous" id="S4.SS1.2.p2.4.m4.2.2.4.1.cmml" xref="S4.SS1.2.p2.4.m4.2.2.4">subscript</csymbol><ci id="S4.SS1.2.p2.4.m4.2.2.4.2.cmml" xref="S4.SS1.2.p2.4.m4.2.2.4.2">𝑁</ci><ci id="S4.SS1.2.p2.4.m4.2.2.4.3.cmml" xref="S4.SS1.2.p2.4.m4.2.2.4.3">𝑉</ci></apply><apply id="S4.SS1.2.p2.4.m4.2.2.2.3.cmml" xref="S4.SS1.2.p2.4.m4.2.2.2.2"><csymbol cd="latexml" id="S4.SS1.2.p2.4.m4.2.2.2.3.1.cmml" xref="S4.SS1.2.p2.4.m4.2.2.2.2.3">conditional-set</csymbol><apply id="S4.SS1.2.p2.4.m4.1.1.1.1.1.cmml" xref="S4.SS1.2.p2.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.2.p2.4.m4.1.1.1.1.1.1.cmml" xref="S4.SS1.2.p2.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.2.p2.4.m4.1.1.1.1.1.2.cmml" xref="S4.SS1.2.p2.4.m4.1.1.1.1.1.2">𝑎</ci><ci id="S4.SS1.2.p2.4.m4.1.1.1.1.1.3.cmml" xref="S4.SS1.2.p2.4.m4.1.1.1.1.1.3">𝑢</ci></apply><apply id="S4.SS1.2.p2.4.m4.2.2.2.2.2.cmml" xref="S4.SS1.2.p2.4.m4.2.2.2.2.2"><in id="S4.SS1.2.p2.4.m4.2.2.2.2.2.1.cmml" xref="S4.SS1.2.p2.4.m4.2.2.2.2.2.1"></in><ci id="S4.SS1.2.p2.4.m4.2.2.2.2.2.2.cmml" xref="S4.SS1.2.p2.4.m4.2.2.2.2.2.2">𝑢</ci><ci id="S4.SS1.2.p2.4.m4.2.2.2.2.2.3.cmml" xref="S4.SS1.2.p2.4.m4.2.2.2.2.2.3">𝑉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p2.4.m4.2c">N_{V}=\{a_{u}\colon u\in V\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p2.4.m4.2d">italic_N start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT = { italic_a start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT : italic_u ∈ italic_V }</annotation></semantics></math>, i.e., <math alttext="N_{V}" class="ltx_Math" display="inline" id="S4.SS1.2.p2.5.m5.1"><semantics id="S4.SS1.2.p2.5.m5.1a"><msub id="S4.SS1.2.p2.5.m5.1.1" xref="S4.SS1.2.p2.5.m5.1.1.cmml"><mi id="S4.SS1.2.p2.5.m5.1.1.2" xref="S4.SS1.2.p2.5.m5.1.1.2.cmml">N</mi><mi id="S4.SS1.2.p2.5.m5.1.1.3" xref="S4.SS1.2.p2.5.m5.1.1.3.cmml">V</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p2.5.m5.1b"><apply id="S4.SS1.2.p2.5.m5.1.1.cmml" xref="S4.SS1.2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.2.p2.5.m5.1.1.1.cmml" xref="S4.SS1.2.p2.5.m5.1.1">subscript</csymbol><ci id="S4.SS1.2.p2.5.m5.1.1.2.cmml" xref="S4.SS1.2.p2.5.m5.1.1.2">𝑁</ci><ci id="S4.SS1.2.p2.5.m5.1.1.3.cmml" xref="S4.SS1.2.p2.5.m5.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p2.5.m5.1c">N_{V}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p2.5.m5.1d">italic_N start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT</annotation></semantics></math> contains a player for each vertex of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.2.p2.6.m6.1"><semantics id="S4.SS1.2.p2.6.m6.1a"><mi id="S4.SS1.2.p2.6.m6.1.1" xref="S4.SS1.2.p2.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p2.6.m6.1b"><ci id="S4.SS1.2.p2.6.m6.1.1.cmml" xref="S4.SS1.2.p2.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p2.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p2.6.m6.1d">italic_G</annotation></semantics></math>. Symmetric valuations are given by</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="v(i,j)=\begin{cases}1&i=z,j\in N_{V}\text{,}\\ -v^{-}&i,j\in N_{V},\{i,j\}\notin E\text{, and}\\ 0&i,j\in N_{V},\{i,j\}\in E\text{.}\\ \end{cases}" class="ltx_Math" display="block" id="S4.Ex4.m1.8"><semantics id="S4.Ex4.m1.8a"><mrow id="S4.Ex4.m1.8.9" xref="S4.Ex4.m1.8.9.cmml"><mrow id="S4.Ex4.m1.8.9.2" xref="S4.Ex4.m1.8.9.2.cmml"><mi id="S4.Ex4.m1.8.9.2.2" xref="S4.Ex4.m1.8.9.2.2.cmml">v</mi><mo id="S4.Ex4.m1.8.9.2.1" xref="S4.Ex4.m1.8.9.2.1.cmml">⁢</mo><mrow id="S4.Ex4.m1.8.9.2.3.2" xref="S4.Ex4.m1.8.9.2.3.1.cmml"><mo id="S4.Ex4.m1.8.9.2.3.2.1" stretchy="false" xref="S4.Ex4.m1.8.9.2.3.1.cmml">(</mo><mi id="S4.Ex4.m1.7.7" xref="S4.Ex4.m1.7.7.cmml">i</mi><mo id="S4.Ex4.m1.8.9.2.3.2.2" xref="S4.Ex4.m1.8.9.2.3.1.cmml">,</mo><mi id="S4.Ex4.m1.8.8" xref="S4.Ex4.m1.8.8.cmml">j</mi><mo id="S4.Ex4.m1.8.9.2.3.2.3" stretchy="false" xref="S4.Ex4.m1.8.9.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex4.m1.8.9.1" xref="S4.Ex4.m1.8.9.1.cmml">=</mo><mrow id="S4.Ex4.m1.6.6" xref="S4.Ex4.m1.8.9.3.1.cmml"><mo id="S4.Ex4.m1.6.6.7" xref="S4.Ex4.m1.8.9.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" id="S4.Ex4.m1.6.6.6" rowspacing="0pt" xref="S4.Ex4.m1.8.9.3.1.cmml"><mtr id="S4.Ex4.m1.6.6.6a" xref="S4.Ex4.m1.8.9.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.Ex4.m1.6.6.6b" xref="S4.Ex4.m1.8.9.3.1.cmml"><mn id="S4.Ex4.m1.1.1.1.1.1.1" xref="S4.Ex4.m1.1.1.1.1.1.1.cmml">1</mn></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex4.m1.6.6.6c" xref="S4.Ex4.m1.8.9.3.1.cmml"><mrow id="S4.Ex4.m1.2.2.2.2.2.1.2" xref="S4.Ex4.m1.2.2.2.2.2.1.3.cmml"><mrow id="S4.Ex4.m1.2.2.2.2.2.1.1.1" xref="S4.Ex4.m1.2.2.2.2.2.1.1.1.cmml"><mi id="S4.Ex4.m1.2.2.2.2.2.1.1.1.2" xref="S4.Ex4.m1.2.2.2.2.2.1.1.1.2.cmml">i</mi><mo id="S4.Ex4.m1.2.2.2.2.2.1.1.1.1" xref="S4.Ex4.m1.2.2.2.2.2.1.1.1.1.cmml">=</mo><mi id="S4.Ex4.m1.2.2.2.2.2.1.1.1.3" xref="S4.Ex4.m1.2.2.2.2.2.1.1.1.3.cmml">z</mi></mrow><mo id="S4.Ex4.m1.2.2.2.2.2.1.2.3" xref="S4.Ex4.m1.2.2.2.2.2.1.3a.cmml">,</mo><mrow id="S4.Ex4.m1.2.2.2.2.2.1.2.2" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.cmml"><mi id="S4.Ex4.m1.2.2.2.2.2.1.2.2.2" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.2.cmml">j</mi><mo id="S4.Ex4.m1.2.2.2.2.2.1.2.2.1" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.1.cmml">∈</mo><mrow id="S4.Ex4.m1.2.2.2.2.2.1.2.2.3" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.cmml"><msub id="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.2" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.2.cmml"><mi id="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.2.2" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.2.2.cmml">N</mi><mi id="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.2.3" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.2.3.cmml">V</mi></msub><mo id="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.1" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.1.cmml">⁢</mo><mtext id="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.3" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.3a.cmml">,</mtext></mrow></mrow></mrow></mtd></mtr><mtr id="S4.Ex4.m1.6.6.6d" xref="S4.Ex4.m1.8.9.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.Ex4.m1.6.6.6e" xref="S4.Ex4.m1.8.9.3.1.cmml"><mrow id="S4.Ex4.m1.3.3.3.3.1.1" xref="S4.Ex4.m1.3.3.3.3.1.1.cmml"><mo id="S4.Ex4.m1.3.3.3.3.1.1a" xref="S4.Ex4.m1.3.3.3.3.1.1.cmml">−</mo><msup id="S4.Ex4.m1.3.3.3.3.1.1.2" xref="S4.Ex4.m1.3.3.3.3.1.1.2.cmml"><mi id="S4.Ex4.m1.3.3.3.3.1.1.2.2" xref="S4.Ex4.m1.3.3.3.3.1.1.2.2.cmml">v</mi><mo id="S4.Ex4.m1.3.3.3.3.1.1.2.3" xref="S4.Ex4.m1.3.3.3.3.1.1.2.3.cmml">−</mo></msup></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex4.m1.6.6.6f" xref="S4.Ex4.m1.8.9.3.1.cmml"><mrow id="S4.Ex4.m1.4.4.4.4.2.1.6" xref="S4.Ex4.m1.4.4.4.4.2.1.7.cmml"><mrow id="S4.Ex4.m1.4.4.4.4.2.1.5.1" xref="S4.Ex4.m1.4.4.4.4.2.1.5.1.cmml"><mrow id="S4.Ex4.m1.4.4.4.4.2.1.5.1.2.2" xref="S4.Ex4.m1.4.4.4.4.2.1.5.1.2.1.cmml"><mi id="S4.Ex4.m1.4.4.4.4.2.1.1" xref="S4.Ex4.m1.4.4.4.4.2.1.1.cmml">i</mi><mo id="S4.Ex4.m1.4.4.4.4.2.1.5.1.2.2.1" xref="S4.Ex4.m1.4.4.4.4.2.1.5.1.2.1.cmml">,</mo><mi id="S4.Ex4.m1.4.4.4.4.2.1.2" xref="S4.Ex4.m1.4.4.4.4.2.1.2.cmml">j</mi></mrow><mo id="S4.Ex4.m1.4.4.4.4.2.1.5.1.1" xref="S4.Ex4.m1.4.4.4.4.2.1.5.1.1.cmml">∈</mo><msub id="S4.Ex4.m1.4.4.4.4.2.1.5.1.3" xref="S4.Ex4.m1.4.4.4.4.2.1.5.1.3.cmml"><mi id="S4.Ex4.m1.4.4.4.4.2.1.5.1.3.2" xref="S4.Ex4.m1.4.4.4.4.2.1.5.1.3.2.cmml">N</mi><mi id="S4.Ex4.m1.4.4.4.4.2.1.5.1.3.3" xref="S4.Ex4.m1.4.4.4.4.2.1.5.1.3.3.cmml">V</mi></msub></mrow><mo id="S4.Ex4.m1.4.4.4.4.2.1.6.3" xref="S4.Ex4.m1.4.4.4.4.2.1.7a.cmml">,</mo><mrow id="S4.Ex4.m1.4.4.4.4.2.1.6.2" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.cmml"><mrow id="S4.Ex4.m1.4.4.4.4.2.1.6.2.2.2" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.2.1.cmml"><mo id="S4.Ex4.m1.4.4.4.4.2.1.6.2.2.2.1" stretchy="false" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.2.1.cmml">{</mo><mi id="S4.Ex4.m1.4.4.4.4.2.1.3" xref="S4.Ex4.m1.4.4.4.4.2.1.3.cmml">i</mi><mo id="S4.Ex4.m1.4.4.4.4.2.1.6.2.2.2.2" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.2.1.cmml">,</mo><mi id="S4.Ex4.m1.4.4.4.4.2.1.4" xref="S4.Ex4.m1.4.4.4.4.2.1.4.cmml">j</mi><mo id="S4.Ex4.m1.4.4.4.4.2.1.6.2.2.2.3" stretchy="false" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.2.1.cmml">}</mo></mrow><mo id="S4.Ex4.m1.4.4.4.4.2.1.6.2.1" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.1.cmml">∉</mo><mrow id="S4.Ex4.m1.4.4.4.4.2.1.6.2.3" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.cmml"><mi id="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.2" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.2.cmml">E</mi><mo id="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.1" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.1.cmml">⁢</mo><mtext id="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.3" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.3a.cmml">, and</mtext></mrow></mrow></mrow></mtd></mtr><mtr id="S4.Ex4.m1.6.6.6g" xref="S4.Ex4.m1.8.9.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.Ex4.m1.6.6.6h" xref="S4.Ex4.m1.8.9.3.1.cmml"><mn id="S4.Ex4.m1.5.5.5.5.1.1" xref="S4.Ex4.m1.5.5.5.5.1.1.cmml">0</mn></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex4.m1.6.6.6i" xref="S4.Ex4.m1.8.9.3.1.cmml"><mrow id="S4.Ex4.m1.6.6.6.6.2.1.6" xref="S4.Ex4.m1.6.6.6.6.2.1.7.cmml"><mrow id="S4.Ex4.m1.6.6.6.6.2.1.5.1" xref="S4.Ex4.m1.6.6.6.6.2.1.5.1.cmml"><mrow id="S4.Ex4.m1.6.6.6.6.2.1.5.1.2.2" xref="S4.Ex4.m1.6.6.6.6.2.1.5.1.2.1.cmml"><mi id="S4.Ex4.m1.6.6.6.6.2.1.1" xref="S4.Ex4.m1.6.6.6.6.2.1.1.cmml">i</mi><mo id="S4.Ex4.m1.6.6.6.6.2.1.5.1.2.2.1" xref="S4.Ex4.m1.6.6.6.6.2.1.5.1.2.1.cmml">,</mo><mi id="S4.Ex4.m1.6.6.6.6.2.1.2" xref="S4.Ex4.m1.6.6.6.6.2.1.2.cmml">j</mi></mrow><mo id="S4.Ex4.m1.6.6.6.6.2.1.5.1.1" xref="S4.Ex4.m1.6.6.6.6.2.1.5.1.1.cmml">∈</mo><msub id="S4.Ex4.m1.6.6.6.6.2.1.5.1.3" xref="S4.Ex4.m1.6.6.6.6.2.1.5.1.3.cmml"><mi id="S4.Ex4.m1.6.6.6.6.2.1.5.1.3.2" xref="S4.Ex4.m1.6.6.6.6.2.1.5.1.3.2.cmml">N</mi><mi id="S4.Ex4.m1.6.6.6.6.2.1.5.1.3.3" xref="S4.Ex4.m1.6.6.6.6.2.1.5.1.3.3.cmml">V</mi></msub></mrow><mo id="S4.Ex4.m1.6.6.6.6.2.1.6.3" xref="S4.Ex4.m1.6.6.6.6.2.1.7a.cmml">,</mo><mrow id="S4.Ex4.m1.6.6.6.6.2.1.6.2" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.cmml"><mrow id="S4.Ex4.m1.6.6.6.6.2.1.6.2.2.2" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.2.1.cmml"><mo id="S4.Ex4.m1.6.6.6.6.2.1.6.2.2.2.1" stretchy="false" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.2.1.cmml">{</mo><mi id="S4.Ex4.m1.6.6.6.6.2.1.3" xref="S4.Ex4.m1.6.6.6.6.2.1.3.cmml">i</mi><mo id="S4.Ex4.m1.6.6.6.6.2.1.6.2.2.2.2" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.2.1.cmml">,</mo><mi id="S4.Ex4.m1.6.6.6.6.2.1.4" xref="S4.Ex4.m1.6.6.6.6.2.1.4.cmml">j</mi><mo id="S4.Ex4.m1.6.6.6.6.2.1.6.2.2.2.3" stretchy="false" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.2.1.cmml">}</mo></mrow><mo id="S4.Ex4.m1.6.6.6.6.2.1.6.2.1" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.1.cmml">∈</mo><mrow id="S4.Ex4.m1.6.6.6.6.2.1.6.2.3" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.cmml"><mi id="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.2" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.2.cmml">E</mi><mo id="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.1" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.1.cmml">⁢</mo><mtext id="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.3" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.3a.cmml">.</mtext></mrow></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex4.m1.8b"><apply id="S4.Ex4.m1.8.9.cmml" xref="S4.Ex4.m1.8.9"><eq id="S4.Ex4.m1.8.9.1.cmml" xref="S4.Ex4.m1.8.9.1"></eq><apply id="S4.Ex4.m1.8.9.2.cmml" xref="S4.Ex4.m1.8.9.2"><times id="S4.Ex4.m1.8.9.2.1.cmml" xref="S4.Ex4.m1.8.9.2.1"></times><ci id="S4.Ex4.m1.8.9.2.2.cmml" xref="S4.Ex4.m1.8.9.2.2">𝑣</ci><interval closure="open" id="S4.Ex4.m1.8.9.2.3.1.cmml" xref="S4.Ex4.m1.8.9.2.3.2"><ci id="S4.Ex4.m1.7.7.cmml" xref="S4.Ex4.m1.7.7">𝑖</ci><ci id="S4.Ex4.m1.8.8.cmml" xref="S4.Ex4.m1.8.8">𝑗</ci></interval></apply><apply id="S4.Ex4.m1.8.9.3.1.cmml" xref="S4.Ex4.m1.6.6"><csymbol cd="latexml" id="S4.Ex4.m1.8.9.3.1.1.cmml" xref="S4.Ex4.m1.6.6.7">cases</csymbol><cn id="S4.Ex4.m1.1.1.1.1.1.1.cmml" type="integer" xref="S4.Ex4.m1.1.1.1.1.1.1">1</cn><apply id="S4.Ex4.m1.2.2.2.2.2.1.3.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S4.Ex4.m1.2.2.2.2.2.1.3a.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.2.3">formulae-sequence</csymbol><apply id="S4.Ex4.m1.2.2.2.2.2.1.1.1.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.1.1"><eq id="S4.Ex4.m1.2.2.2.2.2.1.1.1.1.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.1.1.1"></eq><ci id="S4.Ex4.m1.2.2.2.2.2.1.1.1.2.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.1.1.2">𝑖</ci><ci id="S4.Ex4.m1.2.2.2.2.2.1.1.1.3.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.1.1.3">𝑧</ci></apply><apply id="S4.Ex4.m1.2.2.2.2.2.1.2.2.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2"><in id="S4.Ex4.m1.2.2.2.2.2.1.2.2.1.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.1"></in><ci id="S4.Ex4.m1.2.2.2.2.2.1.2.2.2.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.2">𝑗</ci><apply id="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.3"><times id="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.1.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.1"></times><apply id="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.2.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.2"><csymbol cd="ambiguous" id="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.2.1.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.2">subscript</csymbol><ci id="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.2.2.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.2.2">𝑁</ci><ci id="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.2.3.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.2.3">𝑉</ci></apply><ci id="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.3a.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.3"><mtext id="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.3.cmml" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.3.3">,</mtext></ci></apply></apply></apply><apply id="S4.Ex4.m1.3.3.3.3.1.1.cmml" xref="S4.Ex4.m1.3.3.3.3.1.1"><minus id="S4.Ex4.m1.3.3.3.3.1.1.1.cmml" xref="S4.Ex4.m1.3.3.3.3.1.1"></minus><apply id="S4.Ex4.m1.3.3.3.3.1.1.2.cmml" xref="S4.Ex4.m1.3.3.3.3.1.1.2"><csymbol cd="ambiguous" id="S4.Ex4.m1.3.3.3.3.1.1.2.1.cmml" xref="S4.Ex4.m1.3.3.3.3.1.1.2">superscript</csymbol><ci id="S4.Ex4.m1.3.3.3.3.1.1.2.2.cmml" xref="S4.Ex4.m1.3.3.3.3.1.1.2.2">𝑣</ci><minus id="S4.Ex4.m1.3.3.3.3.1.1.2.3.cmml" xref="S4.Ex4.m1.3.3.3.3.1.1.2.3"></minus></apply></apply><apply id="S4.Ex4.m1.4.4.4.4.2.1.7.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.6"><csymbol cd="ambiguous" id="S4.Ex4.m1.4.4.4.4.2.1.7a.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.6.3">formulae-sequence</csymbol><apply id="S4.Ex4.m1.4.4.4.4.2.1.5.1.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.5.1"><in id="S4.Ex4.m1.4.4.4.4.2.1.5.1.1.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.5.1.1"></in><list id="S4.Ex4.m1.4.4.4.4.2.1.5.1.2.1.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.5.1.2.2"><ci id="S4.Ex4.m1.4.4.4.4.2.1.1.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.1">𝑖</ci><ci id="S4.Ex4.m1.4.4.4.4.2.1.2.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.2">𝑗</ci></list><apply id="S4.Ex4.m1.4.4.4.4.2.1.5.1.3.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.5.1.3"><csymbol cd="ambiguous" id="S4.Ex4.m1.4.4.4.4.2.1.5.1.3.1.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.5.1.3">subscript</csymbol><ci id="S4.Ex4.m1.4.4.4.4.2.1.5.1.3.2.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.5.1.3.2">𝑁</ci><ci id="S4.Ex4.m1.4.4.4.4.2.1.5.1.3.3.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.5.1.3.3">𝑉</ci></apply></apply><apply id="S4.Ex4.m1.4.4.4.4.2.1.6.2.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2"><notin id="S4.Ex4.m1.4.4.4.4.2.1.6.2.1.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.1"></notin><set id="S4.Ex4.m1.4.4.4.4.2.1.6.2.2.1.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.2.2"><ci id="S4.Ex4.m1.4.4.4.4.2.1.3.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.3">𝑖</ci><ci id="S4.Ex4.m1.4.4.4.4.2.1.4.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.4">𝑗</ci></set><apply id="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.3"><times id="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.1.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.1"></times><ci id="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.2.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.2">𝐸</ci><ci id="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.3a.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.3"><mtext id="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.3.cmml" xref="S4.Ex4.m1.4.4.4.4.2.1.6.2.3.3">, and</mtext></ci></apply></apply></apply><cn id="S4.Ex4.m1.5.5.5.5.1.1.cmml" type="integer" xref="S4.Ex4.m1.5.5.5.5.1.1">0</cn><apply id="S4.Ex4.m1.6.6.6.6.2.1.7.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.6"><csymbol cd="ambiguous" id="S4.Ex4.m1.6.6.6.6.2.1.7a.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.6.3">formulae-sequence</csymbol><apply id="S4.Ex4.m1.6.6.6.6.2.1.5.1.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.5.1"><in id="S4.Ex4.m1.6.6.6.6.2.1.5.1.1.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.5.1.1"></in><list id="S4.Ex4.m1.6.6.6.6.2.1.5.1.2.1.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.5.1.2.2"><ci id="S4.Ex4.m1.6.6.6.6.2.1.1.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.1">𝑖</ci><ci id="S4.Ex4.m1.6.6.6.6.2.1.2.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.2">𝑗</ci></list><apply id="S4.Ex4.m1.6.6.6.6.2.1.5.1.3.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.5.1.3"><csymbol cd="ambiguous" id="S4.Ex4.m1.6.6.6.6.2.1.5.1.3.1.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.5.1.3">subscript</csymbol><ci id="S4.Ex4.m1.6.6.6.6.2.1.5.1.3.2.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.5.1.3.2">𝑁</ci><ci id="S4.Ex4.m1.6.6.6.6.2.1.5.1.3.3.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.5.1.3.3">𝑉</ci></apply></apply><apply id="S4.Ex4.m1.6.6.6.6.2.1.6.2.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2"><in id="S4.Ex4.m1.6.6.6.6.2.1.6.2.1.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.1"></in><set id="S4.Ex4.m1.6.6.6.6.2.1.6.2.2.1.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.2.2"><ci id="S4.Ex4.m1.6.6.6.6.2.1.3.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.3">𝑖</ci><ci id="S4.Ex4.m1.6.6.6.6.2.1.4.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.4">𝑗</ci></set><apply id="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.3"><times id="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.1.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.1"></times><ci id="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.2.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.2">𝐸</ci><ci id="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.3a.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.3"><mtext id="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.3.cmml" xref="S4.Ex4.m1.6.6.6.6.2.1.6.2.3.3">.</mtext></ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex4.m1.8c">v(i,j)=\begin{cases}1&i=z,j\in N_{V}\text{,}\\ -v^{-}&i,j\in N_{V},\{i,j\}\notin E\text{, and}\\ 0&i,j\in N_{V},\{i,j\}\in E\text{.}\\ \end{cases}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex4.m1.8d">italic_v ( italic_i , italic_j ) = { start_ROW start_CELL 1 end_CELL start_CELL italic_i = italic_z , italic_j ∈ italic_N start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL - italic_v start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_CELL start_CELL italic_i , italic_j ∈ italic_N start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT , { italic_i , italic_j } ∉ italic_E , and end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_i , italic_j ∈ italic_N start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT , { italic_i , italic_j } ∈ italic_E . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS1.3.p3"> <p class="ltx_p" id="S4.SS1.3.p3.37">Let <math alttext="\pi" class="ltx_Math" display="inline" id="S4.SS1.3.p3.1.m1.1"><semantics id="S4.SS1.3.p3.1.m1.1a"><mi id="S4.SS1.3.p3.1.m1.1.1" xref="S4.SS1.3.p3.1.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.1.m1.1b"><ci id="S4.SS1.3.p3.1.m1.1.1.cmml" xref="S4.SS1.3.p3.1.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.1.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.1.m1.1d">italic_π</annotation></semantics></math> be a partition of <math alttext="N" class="ltx_Math" display="inline" id="S4.SS1.3.p3.2.m2.1"><semantics id="S4.SS1.3.p3.2.m2.1a"><mi id="S4.SS1.3.p3.2.m2.1.1" xref="S4.SS1.3.p3.2.m2.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.2.m2.1b"><ci id="S4.SS1.3.p3.2.m2.1.1.cmml" xref="S4.SS1.3.p3.2.m2.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.2.m2.1c">N</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.2.m2.1d">italic_N</annotation></semantics></math> and let <math alttext="C_{1}=\{u\in V\colon a_{u}\in\pi(z)\}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.3.m3.3"><semantics id="S4.SS1.3.p3.3.m3.3a"><mrow id="S4.SS1.3.p3.3.m3.3.3" xref="S4.SS1.3.p3.3.m3.3.3.cmml"><msub id="S4.SS1.3.p3.3.m3.3.3.4" xref="S4.SS1.3.p3.3.m3.3.3.4.cmml"><mi id="S4.SS1.3.p3.3.m3.3.3.4.2" xref="S4.SS1.3.p3.3.m3.3.3.4.2.cmml">C</mi><mn id="S4.SS1.3.p3.3.m3.3.3.4.3" xref="S4.SS1.3.p3.3.m3.3.3.4.3.cmml">1</mn></msub><mo id="S4.SS1.3.p3.3.m3.3.3.3" xref="S4.SS1.3.p3.3.m3.3.3.3.cmml">=</mo><mrow id="S4.SS1.3.p3.3.m3.3.3.2.2" xref="S4.SS1.3.p3.3.m3.3.3.2.3.cmml"><mo id="S4.SS1.3.p3.3.m3.3.3.2.2.3" stretchy="false" xref="S4.SS1.3.p3.3.m3.3.3.2.3.1.cmml">{</mo><mrow id="S4.SS1.3.p3.3.m3.2.2.1.1.1" xref="S4.SS1.3.p3.3.m3.2.2.1.1.1.cmml"><mi id="S4.SS1.3.p3.3.m3.2.2.1.1.1.2" xref="S4.SS1.3.p3.3.m3.2.2.1.1.1.2.cmml">u</mi><mo id="S4.SS1.3.p3.3.m3.2.2.1.1.1.1" xref="S4.SS1.3.p3.3.m3.2.2.1.1.1.1.cmml">∈</mo><mi id="S4.SS1.3.p3.3.m3.2.2.1.1.1.3" xref="S4.SS1.3.p3.3.m3.2.2.1.1.1.3.cmml">V</mi></mrow><mo id="S4.SS1.3.p3.3.m3.3.3.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.SS1.3.p3.3.m3.3.3.2.3.1.cmml">:</mo><mrow id="S4.SS1.3.p3.3.m3.3.3.2.2.2" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.cmml"><msub id="S4.SS1.3.p3.3.m3.3.3.2.2.2.2" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.2.cmml"><mi id="S4.SS1.3.p3.3.m3.3.3.2.2.2.2.2" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.2.2.cmml">a</mi><mi id="S4.SS1.3.p3.3.m3.3.3.2.2.2.2.3" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.2.3.cmml">u</mi></msub><mo id="S4.SS1.3.p3.3.m3.3.3.2.2.2.1" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.1.cmml">∈</mo><mrow id="S4.SS1.3.p3.3.m3.3.3.2.2.2.3" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.cmml"><mi id="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.2" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.2.cmml">π</mi><mo id="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.1" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.3.2" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.cmml"><mo id="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.3.2.1" stretchy="false" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.cmml">(</mo><mi id="S4.SS1.3.p3.3.m3.1.1" xref="S4.SS1.3.p3.3.m3.1.1.cmml">z</mi><mo id="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.3.2.2" stretchy="false" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS1.3.p3.3.m3.3.3.2.2.5" stretchy="false" xref="S4.SS1.3.p3.3.m3.3.3.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.3.m3.3b"><apply id="S4.SS1.3.p3.3.m3.3.3.cmml" xref="S4.SS1.3.p3.3.m3.3.3"><eq id="S4.SS1.3.p3.3.m3.3.3.3.cmml" xref="S4.SS1.3.p3.3.m3.3.3.3"></eq><apply id="S4.SS1.3.p3.3.m3.3.3.4.cmml" xref="S4.SS1.3.p3.3.m3.3.3.4"><csymbol cd="ambiguous" id="S4.SS1.3.p3.3.m3.3.3.4.1.cmml" xref="S4.SS1.3.p3.3.m3.3.3.4">subscript</csymbol><ci id="S4.SS1.3.p3.3.m3.3.3.4.2.cmml" xref="S4.SS1.3.p3.3.m3.3.3.4.2">𝐶</ci><cn id="S4.SS1.3.p3.3.m3.3.3.4.3.cmml" type="integer" xref="S4.SS1.3.p3.3.m3.3.3.4.3">1</cn></apply><apply id="S4.SS1.3.p3.3.m3.3.3.2.3.cmml" xref="S4.SS1.3.p3.3.m3.3.3.2.2"><csymbol cd="latexml" id="S4.SS1.3.p3.3.m3.3.3.2.3.1.cmml" xref="S4.SS1.3.p3.3.m3.3.3.2.2.3">conditional-set</csymbol><apply id="S4.SS1.3.p3.3.m3.2.2.1.1.1.cmml" xref="S4.SS1.3.p3.3.m3.2.2.1.1.1"><in id="S4.SS1.3.p3.3.m3.2.2.1.1.1.1.cmml" xref="S4.SS1.3.p3.3.m3.2.2.1.1.1.1"></in><ci id="S4.SS1.3.p3.3.m3.2.2.1.1.1.2.cmml" xref="S4.SS1.3.p3.3.m3.2.2.1.1.1.2">𝑢</ci><ci id="S4.SS1.3.p3.3.m3.2.2.1.1.1.3.cmml" xref="S4.SS1.3.p3.3.m3.2.2.1.1.1.3">𝑉</ci></apply><apply id="S4.SS1.3.p3.3.m3.3.3.2.2.2.cmml" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2"><in id="S4.SS1.3.p3.3.m3.3.3.2.2.2.1.cmml" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.1"></in><apply id="S4.SS1.3.p3.3.m3.3.3.2.2.2.2.cmml" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.3.p3.3.m3.3.3.2.2.2.2.1.cmml" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.2">subscript</csymbol><ci id="S4.SS1.3.p3.3.m3.3.3.2.2.2.2.2.cmml" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.2.2">𝑎</ci><ci id="S4.SS1.3.p3.3.m3.3.3.2.2.2.2.3.cmml" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.2.3">𝑢</ci></apply><apply id="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.cmml" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.3"><times id="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.1.cmml" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.1"></times><ci id="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.2.cmml" xref="S4.SS1.3.p3.3.m3.3.3.2.2.2.3.2">𝜋</ci><ci id="S4.SS1.3.p3.3.m3.1.1.cmml" xref="S4.SS1.3.p3.3.m3.1.1">𝑧</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.3.m3.3c">C_{1}=\{u\in V\colon a_{u}\in\pi(z)\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.3.m3.3d">italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = { italic_u ∈ italic_V : italic_a start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ∈ italic_π ( italic_z ) }</annotation></semantics></math>. Hence, <math alttext="C_{1}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.4.m4.1"><semantics id="S4.SS1.3.p3.4.m4.1a"><msub id="S4.SS1.3.p3.4.m4.1.1" xref="S4.SS1.3.p3.4.m4.1.1.cmml"><mi id="S4.SS1.3.p3.4.m4.1.1.2" xref="S4.SS1.3.p3.4.m4.1.1.2.cmml">C</mi><mn id="S4.SS1.3.p3.4.m4.1.1.3" xref="S4.SS1.3.p3.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.4.m4.1b"><apply id="S4.SS1.3.p3.4.m4.1.1.cmml" xref="S4.SS1.3.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.4.m4.1.1.1.cmml" xref="S4.SS1.3.p3.4.m4.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.4.m4.1.1.2.cmml" xref="S4.SS1.3.p3.4.m4.1.1.2">𝐶</ci><cn id="S4.SS1.3.p3.4.m4.1.1.3.cmml" type="integer" xref="S4.SS1.3.p3.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.4.m4.1c">C_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.4.m4.1d">italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is a vertex set in <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.3.p3.5.m5.1"><semantics id="S4.SS1.3.p3.5.m5.1a"><mi id="S4.SS1.3.p3.5.m5.1.1" xref="S4.SS1.3.p3.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.5.m5.1b"><ci id="S4.SS1.3.p3.5.m5.1.1.cmml" xref="S4.SS1.3.p3.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.5.m5.1d">italic_G</annotation></semantics></math>. We create a sequence of vertex sets until we end with a clique of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.3.p3.6.m6.1"><semantics id="S4.SS1.3.p3.6.m6.1a"><mi id="S4.SS1.3.p3.6.m6.1.1" xref="S4.SS1.3.p3.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.6.m6.1b"><ci id="S4.SS1.3.p3.6.m6.1.1.cmml" xref="S4.SS1.3.p3.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.6.m6.1d">italic_G</annotation></semantics></math>. For <math alttext="i\geq 1" class="ltx_Math" display="inline" id="S4.SS1.3.p3.7.m7.1"><semantics id="S4.SS1.3.p3.7.m7.1a"><mrow id="S4.SS1.3.p3.7.m7.1.1" xref="S4.SS1.3.p3.7.m7.1.1.cmml"><mi id="S4.SS1.3.p3.7.m7.1.1.2" xref="S4.SS1.3.p3.7.m7.1.1.2.cmml">i</mi><mo id="S4.SS1.3.p3.7.m7.1.1.1" xref="S4.SS1.3.p3.7.m7.1.1.1.cmml">≥</mo><mn id="S4.SS1.3.p3.7.m7.1.1.3" xref="S4.SS1.3.p3.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.7.m7.1b"><apply id="S4.SS1.3.p3.7.m7.1.1.cmml" xref="S4.SS1.3.p3.7.m7.1.1"><geq id="S4.SS1.3.p3.7.m7.1.1.1.cmml" xref="S4.SS1.3.p3.7.m7.1.1.1"></geq><ci id="S4.SS1.3.p3.7.m7.1.1.2.cmml" xref="S4.SS1.3.p3.7.m7.1.1.2">𝑖</ci><cn id="S4.SS1.3.p3.7.m7.1.1.3.cmml" type="integer" xref="S4.SS1.3.p3.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.7.m7.1c">i\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.7.m7.1d">italic_i ≥ 1</annotation></semantics></math>, assume that we have constructed a set <math alttext="C_{i}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.8.m8.1"><semantics id="S4.SS1.3.p3.8.m8.1a"><msub id="S4.SS1.3.p3.8.m8.1.1" xref="S4.SS1.3.p3.8.m8.1.1.cmml"><mi id="S4.SS1.3.p3.8.m8.1.1.2" xref="S4.SS1.3.p3.8.m8.1.1.2.cmml">C</mi><mi id="S4.SS1.3.p3.8.m8.1.1.3" xref="S4.SS1.3.p3.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.8.m8.1b"><apply id="S4.SS1.3.p3.8.m8.1.1.cmml" xref="S4.SS1.3.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.8.m8.1.1.1.cmml" xref="S4.SS1.3.p3.8.m8.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.8.m8.1.1.2.cmml" xref="S4.SS1.3.p3.8.m8.1.1.2">𝐶</ci><ci id="S4.SS1.3.p3.8.m8.1.1.3.cmml" xref="S4.SS1.3.p3.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.8.m8.1c">C_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.8.m8.1d">italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. We stop if <math alttext="C_{i}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.9.m9.1"><semantics id="S4.SS1.3.p3.9.m9.1a"><msub id="S4.SS1.3.p3.9.m9.1.1" xref="S4.SS1.3.p3.9.m9.1.1.cmml"><mi id="S4.SS1.3.p3.9.m9.1.1.2" xref="S4.SS1.3.p3.9.m9.1.1.2.cmml">C</mi><mi id="S4.SS1.3.p3.9.m9.1.1.3" xref="S4.SS1.3.p3.9.m9.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.9.m9.1b"><apply id="S4.SS1.3.p3.9.m9.1.1.cmml" xref="S4.SS1.3.p3.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.9.m9.1.1.1.cmml" xref="S4.SS1.3.p3.9.m9.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.9.m9.1.1.2.cmml" xref="S4.SS1.3.p3.9.m9.1.1.2">𝐶</ci><ci id="S4.SS1.3.p3.9.m9.1.1.3.cmml" xref="S4.SS1.3.p3.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.9.m9.1c">C_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.9.m9.1d">italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is a clique. Otherwise, we find a vertex <math alttext="u_{i}\in C_{i}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.10.m10.1"><semantics id="S4.SS1.3.p3.10.m10.1a"><mrow id="S4.SS1.3.p3.10.m10.1.1" xref="S4.SS1.3.p3.10.m10.1.1.cmml"><msub id="S4.SS1.3.p3.10.m10.1.1.2" xref="S4.SS1.3.p3.10.m10.1.1.2.cmml"><mi id="S4.SS1.3.p3.10.m10.1.1.2.2" xref="S4.SS1.3.p3.10.m10.1.1.2.2.cmml">u</mi><mi id="S4.SS1.3.p3.10.m10.1.1.2.3" xref="S4.SS1.3.p3.10.m10.1.1.2.3.cmml">i</mi></msub><mo id="S4.SS1.3.p3.10.m10.1.1.1" xref="S4.SS1.3.p3.10.m10.1.1.1.cmml">∈</mo><msub id="S4.SS1.3.p3.10.m10.1.1.3" xref="S4.SS1.3.p3.10.m10.1.1.3.cmml"><mi id="S4.SS1.3.p3.10.m10.1.1.3.2" xref="S4.SS1.3.p3.10.m10.1.1.3.2.cmml">C</mi><mi id="S4.SS1.3.p3.10.m10.1.1.3.3" xref="S4.SS1.3.p3.10.m10.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.10.m10.1b"><apply id="S4.SS1.3.p3.10.m10.1.1.cmml" xref="S4.SS1.3.p3.10.m10.1.1"><in id="S4.SS1.3.p3.10.m10.1.1.1.cmml" xref="S4.SS1.3.p3.10.m10.1.1.1"></in><apply id="S4.SS1.3.p3.10.m10.1.1.2.cmml" xref="S4.SS1.3.p3.10.m10.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.3.p3.10.m10.1.1.2.1.cmml" xref="S4.SS1.3.p3.10.m10.1.1.2">subscript</csymbol><ci id="S4.SS1.3.p3.10.m10.1.1.2.2.cmml" xref="S4.SS1.3.p3.10.m10.1.1.2.2">𝑢</ci><ci id="S4.SS1.3.p3.10.m10.1.1.2.3.cmml" xref="S4.SS1.3.p3.10.m10.1.1.2.3">𝑖</ci></apply><apply id="S4.SS1.3.p3.10.m10.1.1.3.cmml" xref="S4.SS1.3.p3.10.m10.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.3.p3.10.m10.1.1.3.1.cmml" xref="S4.SS1.3.p3.10.m10.1.1.3">subscript</csymbol><ci id="S4.SS1.3.p3.10.m10.1.1.3.2.cmml" xref="S4.SS1.3.p3.10.m10.1.1.3.2">𝐶</ci><ci id="S4.SS1.3.p3.10.m10.1.1.3.3.cmml" xref="S4.SS1.3.p3.10.m10.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.10.m10.1c">u_{i}\in C_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.10.m10.1d">italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="u_{i}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.11.m11.1"><semantics id="S4.SS1.3.p3.11.m11.1a"><msub id="S4.SS1.3.p3.11.m11.1.1" xref="S4.SS1.3.p3.11.m11.1.1.cmml"><mi id="S4.SS1.3.p3.11.m11.1.1.2" xref="S4.SS1.3.p3.11.m11.1.1.2.cmml">u</mi><mi id="S4.SS1.3.p3.11.m11.1.1.3" xref="S4.SS1.3.p3.11.m11.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.11.m11.1b"><apply id="S4.SS1.3.p3.11.m11.1.1.cmml" xref="S4.SS1.3.p3.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.11.m11.1.1.1.cmml" xref="S4.SS1.3.p3.11.m11.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.11.m11.1.1.2.cmml" xref="S4.SS1.3.p3.11.m11.1.1.2">𝑢</ci><ci id="S4.SS1.3.p3.11.m11.1.1.3.cmml" xref="S4.SS1.3.p3.11.m11.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.11.m11.1c">u_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.11.m11.1d">italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is not adjacent to all other vertices in <math alttext="C_{i}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.12.m12.1"><semantics id="S4.SS1.3.p3.12.m12.1a"><msub id="S4.SS1.3.p3.12.m12.1.1" xref="S4.SS1.3.p3.12.m12.1.1.cmml"><mi id="S4.SS1.3.p3.12.m12.1.1.2" xref="S4.SS1.3.p3.12.m12.1.1.2.cmml">C</mi><mi id="S4.SS1.3.p3.12.m12.1.1.3" xref="S4.SS1.3.p3.12.m12.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.12.m12.1b"><apply id="S4.SS1.3.p3.12.m12.1.1.cmml" xref="S4.SS1.3.p3.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.12.m12.1.1.1.cmml" xref="S4.SS1.3.p3.12.m12.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.12.m12.1.1.2.cmml" xref="S4.SS1.3.p3.12.m12.1.1.2">𝐶</ci><ci id="S4.SS1.3.p3.12.m12.1.1.3.cmml" xref="S4.SS1.3.p3.12.m12.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.12.m12.1c">C_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.12.m12.1d">italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and set <math alttext="C_{i+1}=C_{i}\setminus\{u_{i}\}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.13.m13.1"><semantics id="S4.SS1.3.p3.13.m13.1a"><mrow id="S4.SS1.3.p3.13.m13.1.1" xref="S4.SS1.3.p3.13.m13.1.1.cmml"><msub id="S4.SS1.3.p3.13.m13.1.1.3" xref="S4.SS1.3.p3.13.m13.1.1.3.cmml"><mi id="S4.SS1.3.p3.13.m13.1.1.3.2" xref="S4.SS1.3.p3.13.m13.1.1.3.2.cmml">C</mi><mrow id="S4.SS1.3.p3.13.m13.1.1.3.3" xref="S4.SS1.3.p3.13.m13.1.1.3.3.cmml"><mi id="S4.SS1.3.p3.13.m13.1.1.3.3.2" xref="S4.SS1.3.p3.13.m13.1.1.3.3.2.cmml">i</mi><mo id="S4.SS1.3.p3.13.m13.1.1.3.3.1" xref="S4.SS1.3.p3.13.m13.1.1.3.3.1.cmml">+</mo><mn id="S4.SS1.3.p3.13.m13.1.1.3.3.3" xref="S4.SS1.3.p3.13.m13.1.1.3.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS1.3.p3.13.m13.1.1.2" xref="S4.SS1.3.p3.13.m13.1.1.2.cmml">=</mo><mrow id="S4.SS1.3.p3.13.m13.1.1.1" xref="S4.SS1.3.p3.13.m13.1.1.1.cmml"><msub id="S4.SS1.3.p3.13.m13.1.1.1.3" xref="S4.SS1.3.p3.13.m13.1.1.1.3.cmml"><mi id="S4.SS1.3.p3.13.m13.1.1.1.3.2" xref="S4.SS1.3.p3.13.m13.1.1.1.3.2.cmml">C</mi><mi id="S4.SS1.3.p3.13.m13.1.1.1.3.3" xref="S4.SS1.3.p3.13.m13.1.1.1.3.3.cmml">i</mi></msub><mo id="S4.SS1.3.p3.13.m13.1.1.1.2" xref="S4.SS1.3.p3.13.m13.1.1.1.2.cmml">∖</mo><mrow id="S4.SS1.3.p3.13.m13.1.1.1.1.1" xref="S4.SS1.3.p3.13.m13.1.1.1.1.2.cmml"><mo id="S4.SS1.3.p3.13.m13.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.3.p3.13.m13.1.1.1.1.2.cmml">{</mo><msub id="S4.SS1.3.p3.13.m13.1.1.1.1.1.1" xref="S4.SS1.3.p3.13.m13.1.1.1.1.1.1.cmml"><mi id="S4.SS1.3.p3.13.m13.1.1.1.1.1.1.2" xref="S4.SS1.3.p3.13.m13.1.1.1.1.1.1.2.cmml">u</mi><mi id="S4.SS1.3.p3.13.m13.1.1.1.1.1.1.3" xref="S4.SS1.3.p3.13.m13.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.SS1.3.p3.13.m13.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.3.p3.13.m13.1.1.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.13.m13.1b"><apply id="S4.SS1.3.p3.13.m13.1.1.cmml" xref="S4.SS1.3.p3.13.m13.1.1"><eq id="S4.SS1.3.p3.13.m13.1.1.2.cmml" xref="S4.SS1.3.p3.13.m13.1.1.2"></eq><apply id="S4.SS1.3.p3.13.m13.1.1.3.cmml" xref="S4.SS1.3.p3.13.m13.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.3.p3.13.m13.1.1.3.1.cmml" xref="S4.SS1.3.p3.13.m13.1.1.3">subscript</csymbol><ci id="S4.SS1.3.p3.13.m13.1.1.3.2.cmml" xref="S4.SS1.3.p3.13.m13.1.1.3.2">𝐶</ci><apply id="S4.SS1.3.p3.13.m13.1.1.3.3.cmml" xref="S4.SS1.3.p3.13.m13.1.1.3.3"><plus id="S4.SS1.3.p3.13.m13.1.1.3.3.1.cmml" xref="S4.SS1.3.p3.13.m13.1.1.3.3.1"></plus><ci id="S4.SS1.3.p3.13.m13.1.1.3.3.2.cmml" xref="S4.SS1.3.p3.13.m13.1.1.3.3.2">𝑖</ci><cn id="S4.SS1.3.p3.13.m13.1.1.3.3.3.cmml" type="integer" xref="S4.SS1.3.p3.13.m13.1.1.3.3.3">1</cn></apply></apply><apply id="S4.SS1.3.p3.13.m13.1.1.1.cmml" xref="S4.SS1.3.p3.13.m13.1.1.1"><setdiff id="S4.SS1.3.p3.13.m13.1.1.1.2.cmml" xref="S4.SS1.3.p3.13.m13.1.1.1.2"></setdiff><apply id="S4.SS1.3.p3.13.m13.1.1.1.3.cmml" xref="S4.SS1.3.p3.13.m13.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.3.p3.13.m13.1.1.1.3.1.cmml" xref="S4.SS1.3.p3.13.m13.1.1.1.3">subscript</csymbol><ci id="S4.SS1.3.p3.13.m13.1.1.1.3.2.cmml" xref="S4.SS1.3.p3.13.m13.1.1.1.3.2">𝐶</ci><ci id="S4.SS1.3.p3.13.m13.1.1.1.3.3.cmml" xref="S4.SS1.3.p3.13.m13.1.1.1.3.3">𝑖</ci></apply><set id="S4.SS1.3.p3.13.m13.1.1.1.1.2.cmml" xref="S4.SS1.3.p3.13.m13.1.1.1.1.1"><apply id="S4.SS1.3.p3.13.m13.1.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.13.m13.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.13.m13.1.1.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.13.m13.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.13.m13.1.1.1.1.1.1.2.cmml" xref="S4.SS1.3.p3.13.m13.1.1.1.1.1.1.2">𝑢</ci><ci id="S4.SS1.3.p3.13.m13.1.1.1.1.1.1.3.cmml" xref="S4.SS1.3.p3.13.m13.1.1.1.1.1.1.3">𝑖</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.13.m13.1c">C_{i+1}=C_{i}\setminus\{u_{i}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.13.m13.1d">italic_C start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∖ { italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT }</annotation></semantics></math>. Since the number of vertices of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.3.p3.14.m14.1"><semantics id="S4.SS1.3.p3.14.m14.1a"><mi id="S4.SS1.3.p3.14.m14.1.1" xref="S4.SS1.3.p3.14.m14.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.14.m14.1b"><ci id="S4.SS1.3.p3.14.m14.1.1.cmml" xref="S4.SS1.3.p3.14.m14.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.14.m14.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.14.m14.1d">italic_G</annotation></semantics></math> is finite, this stops after <math alttext="k\leq|V|" class="ltx_Math" display="inline" id="S4.SS1.3.p3.15.m15.1"><semantics id="S4.SS1.3.p3.15.m15.1a"><mrow id="S4.SS1.3.p3.15.m15.1.2" xref="S4.SS1.3.p3.15.m15.1.2.cmml"><mi id="S4.SS1.3.p3.15.m15.1.2.2" xref="S4.SS1.3.p3.15.m15.1.2.2.cmml">k</mi><mo id="S4.SS1.3.p3.15.m15.1.2.1" xref="S4.SS1.3.p3.15.m15.1.2.1.cmml">≤</mo><mrow id="S4.SS1.3.p3.15.m15.1.2.3.2" xref="S4.SS1.3.p3.15.m15.1.2.3.1.cmml"><mo id="S4.SS1.3.p3.15.m15.1.2.3.2.1" stretchy="false" xref="S4.SS1.3.p3.15.m15.1.2.3.1.1.cmml">|</mo><mi id="S4.SS1.3.p3.15.m15.1.1" xref="S4.SS1.3.p3.15.m15.1.1.cmml">V</mi><mo id="S4.SS1.3.p3.15.m15.1.2.3.2.2" stretchy="false" xref="S4.SS1.3.p3.15.m15.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.15.m15.1b"><apply id="S4.SS1.3.p3.15.m15.1.2.cmml" xref="S4.SS1.3.p3.15.m15.1.2"><leq id="S4.SS1.3.p3.15.m15.1.2.1.cmml" xref="S4.SS1.3.p3.15.m15.1.2.1"></leq><ci id="S4.SS1.3.p3.15.m15.1.2.2.cmml" xref="S4.SS1.3.p3.15.m15.1.2.2">𝑘</ci><apply id="S4.SS1.3.p3.15.m15.1.2.3.1.cmml" xref="S4.SS1.3.p3.15.m15.1.2.3.2"><abs id="S4.SS1.3.p3.15.m15.1.2.3.1.1.cmml" xref="S4.SS1.3.p3.15.m15.1.2.3.2.1"></abs><ci id="S4.SS1.3.p3.15.m15.1.1.cmml" xref="S4.SS1.3.p3.15.m15.1.1">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.15.m15.1c">k\leq|V|</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.15.m15.1d">italic_k ≤ | italic_V |</annotation></semantics></math> steps with a clique <math alttext="C_{k}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.16.m16.1"><semantics id="S4.SS1.3.p3.16.m16.1a"><msub id="S4.SS1.3.p3.16.m16.1.1" xref="S4.SS1.3.p3.16.m16.1.1.cmml"><mi id="S4.SS1.3.p3.16.m16.1.1.2" xref="S4.SS1.3.p3.16.m16.1.1.2.cmml">C</mi><mi id="S4.SS1.3.p3.16.m16.1.1.3" xref="S4.SS1.3.p3.16.m16.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.16.m16.1b"><apply id="S4.SS1.3.p3.16.m16.1.1.cmml" xref="S4.SS1.3.p3.16.m16.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.16.m16.1.1.1.cmml" xref="S4.SS1.3.p3.16.m16.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.16.m16.1.1.2.cmml" xref="S4.SS1.3.p3.16.m16.1.1.2">𝐶</ci><ci id="S4.SS1.3.p3.16.m16.1.1.3.cmml" xref="S4.SS1.3.p3.16.m16.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.16.m16.1c">C_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.16.m16.1d">italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. For <math alttext="1\leq\ell\leq k" class="ltx_Math" display="inline" id="S4.SS1.3.p3.17.m17.1"><semantics id="S4.SS1.3.p3.17.m17.1a"><mrow id="S4.SS1.3.p3.17.m17.1.1" xref="S4.SS1.3.p3.17.m17.1.1.cmml"><mn id="S4.SS1.3.p3.17.m17.1.1.2" xref="S4.SS1.3.p3.17.m17.1.1.2.cmml">1</mn><mo id="S4.SS1.3.p3.17.m17.1.1.3" xref="S4.SS1.3.p3.17.m17.1.1.3.cmml">≤</mo><mi id="S4.SS1.3.p3.17.m17.1.1.4" mathvariant="normal" xref="S4.SS1.3.p3.17.m17.1.1.4.cmml">ℓ</mi><mo id="S4.SS1.3.p3.17.m17.1.1.5" xref="S4.SS1.3.p3.17.m17.1.1.5.cmml">≤</mo><mi id="S4.SS1.3.p3.17.m17.1.1.6" xref="S4.SS1.3.p3.17.m17.1.1.6.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.17.m17.1b"><apply id="S4.SS1.3.p3.17.m17.1.1.cmml" xref="S4.SS1.3.p3.17.m17.1.1"><and id="S4.SS1.3.p3.17.m17.1.1a.cmml" xref="S4.SS1.3.p3.17.m17.1.1"></and><apply id="S4.SS1.3.p3.17.m17.1.1b.cmml" xref="S4.SS1.3.p3.17.m17.1.1"><leq id="S4.SS1.3.p3.17.m17.1.1.3.cmml" xref="S4.SS1.3.p3.17.m17.1.1.3"></leq><cn id="S4.SS1.3.p3.17.m17.1.1.2.cmml" type="integer" xref="S4.SS1.3.p3.17.m17.1.1.2">1</cn><ci id="S4.SS1.3.p3.17.m17.1.1.4.cmml" xref="S4.SS1.3.p3.17.m17.1.1.4">ℓ</ci></apply><apply id="S4.SS1.3.p3.17.m17.1.1c.cmml" xref="S4.SS1.3.p3.17.m17.1.1"><leq id="S4.SS1.3.p3.17.m17.1.1.5.cmml" xref="S4.SS1.3.p3.17.m17.1.1.5"></leq><share href="https://arxiv.org/html/2503.06017v1#S4.SS1.3.p3.17.m17.1.1.4.cmml" id="S4.SS1.3.p3.17.m17.1.1d.cmml" xref="S4.SS1.3.p3.17.m17.1.1"></share><ci id="S4.SS1.3.p3.17.m17.1.1.6.cmml" xref="S4.SS1.3.p3.17.m17.1.1.6">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.17.m17.1c">1\leq\ell\leq k</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.17.m17.1d">1 ≤ roman_ℓ ≤ italic_k</annotation></semantics></math>, we define the partition <math alttext="\pi^{\ell}=\{\{a_{u}\colon u\in C_{\ell}\}\cup\{z\}\}\cup\{\{a_{u}\}\colon u% \in V\setminus C_{k}\}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.18.m18.4"><semantics id="S4.SS1.3.p3.18.m18.4a"><mrow id="S4.SS1.3.p3.18.m18.4.4" xref="S4.SS1.3.p3.18.m18.4.4.cmml"><msup id="S4.SS1.3.p3.18.m18.4.4.5" xref="S4.SS1.3.p3.18.m18.4.4.5.cmml"><mi id="S4.SS1.3.p3.18.m18.4.4.5.2" xref="S4.SS1.3.p3.18.m18.4.4.5.2.cmml">π</mi><mi id="S4.SS1.3.p3.18.m18.4.4.5.3" mathvariant="normal" xref="S4.SS1.3.p3.18.m18.4.4.5.3.cmml">ℓ</mi></msup><mo id="S4.SS1.3.p3.18.m18.4.4.4" xref="S4.SS1.3.p3.18.m18.4.4.4.cmml">=</mo><mrow id="S4.SS1.3.p3.18.m18.4.4.3" xref="S4.SS1.3.p3.18.m18.4.4.3.cmml"><mrow id="S4.SS1.3.p3.18.m18.2.2.1.1.1" xref="S4.SS1.3.p3.18.m18.2.2.1.1.2.cmml"><mo id="S4.SS1.3.p3.18.m18.2.2.1.1.1.2" stretchy="false" xref="S4.SS1.3.p3.18.m18.2.2.1.1.2.cmml">{</mo><mrow id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.cmml"><mrow id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.3.cmml"><mo id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.3" stretchy="false" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.3.1.cmml">{</mo><msub id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.1.1.1" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.1.1.1.cmml"><mi id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.1.1.1.2" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.1.1.1.3" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.1.1.1.3.cmml">u</mi></msub><mo id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.3.1.cmml">:</mo><mrow id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.cmml"><mi id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.2" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.2.cmml">u</mi><mo id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.1" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.1.cmml">∈</mo><msub id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.3" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.3.cmml"><mi id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.3.2" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.3.2.cmml">C</mi><mi id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.3.3" mathvariant="normal" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.3.3.cmml">ℓ</mi></msub></mrow><mo id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.5" stretchy="false" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.3.1.cmml">}</mo></mrow><mo id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.3" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.3.cmml">∪</mo><mrow id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.4.2" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.4.1.cmml"><mo id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.4.2.1" stretchy="false" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.4.1.cmml">{</mo><mi id="S4.SS1.3.p3.18.m18.1.1" xref="S4.SS1.3.p3.18.m18.1.1.cmml">z</mi><mo id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.4.2.2" stretchy="false" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.4.1.cmml">}</mo></mrow></mrow><mo id="S4.SS1.3.p3.18.m18.2.2.1.1.1.3" stretchy="false" xref="S4.SS1.3.p3.18.m18.2.2.1.1.2.cmml">}</mo></mrow><mo id="S4.SS1.3.p3.18.m18.4.4.3.4" xref="S4.SS1.3.p3.18.m18.4.4.3.4.cmml">∪</mo><mrow id="S4.SS1.3.p3.18.m18.4.4.3.3.2" xref="S4.SS1.3.p3.18.m18.4.4.3.3.3.cmml"><mo id="S4.SS1.3.p3.18.m18.4.4.3.3.2.3" stretchy="false" xref="S4.SS1.3.p3.18.m18.4.4.3.3.3.1.cmml">{</mo><mrow id="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1" xref="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.2.cmml"><mo id="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.2" stretchy="false" xref="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.2.cmml">{</mo><msub id="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.1" xref="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.1.cmml"><mi id="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.1.2" xref="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.1.2.cmml">a</mi><mi id="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.1.3" xref="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.1.3.cmml">u</mi></msub><mo id="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.3" rspace="0.278em" stretchy="false" xref="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.2.cmml">}</mo></mrow><mo id="S4.SS1.3.p3.18.m18.4.4.3.3.2.4" rspace="0.278em" xref="S4.SS1.3.p3.18.m18.4.4.3.3.3.1.cmml">:</mo><mrow id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.cmml"><mi id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.2" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.2.cmml">u</mi><mo id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.1" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.1.cmml">∈</mo><mrow id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.cmml"><mi id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.2" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.2.cmml">V</mi><mo id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.1" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.1.cmml">∖</mo><msub id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.3" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.3.cmml"><mi id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.3.2" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.3.2.cmml">C</mi><mi id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.3.3" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.3.3.cmml">k</mi></msub></mrow></mrow><mo id="S4.SS1.3.p3.18.m18.4.4.3.3.2.5" stretchy="false" xref="S4.SS1.3.p3.18.m18.4.4.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.18.m18.4b"><apply id="S4.SS1.3.p3.18.m18.4.4.cmml" xref="S4.SS1.3.p3.18.m18.4.4"><eq id="S4.SS1.3.p3.18.m18.4.4.4.cmml" xref="S4.SS1.3.p3.18.m18.4.4.4"></eq><apply id="S4.SS1.3.p3.18.m18.4.4.5.cmml" xref="S4.SS1.3.p3.18.m18.4.4.5"><csymbol cd="ambiguous" id="S4.SS1.3.p3.18.m18.4.4.5.1.cmml" xref="S4.SS1.3.p3.18.m18.4.4.5">superscript</csymbol><ci id="S4.SS1.3.p3.18.m18.4.4.5.2.cmml" xref="S4.SS1.3.p3.18.m18.4.4.5.2">𝜋</ci><ci id="S4.SS1.3.p3.18.m18.4.4.5.3.cmml" xref="S4.SS1.3.p3.18.m18.4.4.5.3">ℓ</ci></apply><apply id="S4.SS1.3.p3.18.m18.4.4.3.cmml" xref="S4.SS1.3.p3.18.m18.4.4.3"><union id="S4.SS1.3.p3.18.m18.4.4.3.4.cmml" xref="S4.SS1.3.p3.18.m18.4.4.3.4"></union><set id="S4.SS1.3.p3.18.m18.2.2.1.1.2.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1"><apply id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1"><union id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.3.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.3"></union><apply id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.3.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2"><csymbol cd="latexml" id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.3.1.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.3">conditional-set</csymbol><apply id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.1.1.1.2.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.1.1.1.2">𝑎</ci><ci id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.1.1.1.3.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.1.1.1.3">𝑢</ci></apply><apply id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2"><in id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.1.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.1"></in><ci id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.2.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.2">𝑢</ci><apply id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.3.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.3"><csymbol cd="ambiguous" id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.3.1.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.3">subscript</csymbol><ci id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.3.2.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.3.2">𝐶</ci><ci id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.3.3.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.2.2.2.3.3">ℓ</ci></apply></apply></apply><set id="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.4.1.cmml" xref="S4.SS1.3.p3.18.m18.2.2.1.1.1.1.4.2"><ci id="S4.SS1.3.p3.18.m18.1.1.cmml" xref="S4.SS1.3.p3.18.m18.1.1">𝑧</ci></set></apply></set><apply id="S4.SS1.3.p3.18.m18.4.4.3.3.3.cmml" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2"><csymbol cd="latexml" id="S4.SS1.3.p3.18.m18.4.4.3.3.3.1.cmml" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.3">conditional-set</csymbol><set id="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.2.cmml" xref="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1"><apply id="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.1.cmml" xref="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.1.2.cmml" xref="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.1.2">𝑎</ci><ci id="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.1.3.cmml" xref="S4.SS1.3.p3.18.m18.3.3.2.2.1.1.1.1.3">𝑢</ci></apply></set><apply id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.cmml" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2"><in id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.1.cmml" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.1"></in><ci id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.2.cmml" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.2">𝑢</ci><apply id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.cmml" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3"><setdiff id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.1.cmml" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.1"></setdiff><ci id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.2.cmml" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.2">𝑉</ci><apply id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.3.cmml" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.3"><csymbol cd="ambiguous" id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.3.1.cmml" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.3">subscript</csymbol><ci id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.3.2.cmml" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.3.2">𝐶</ci><ci id="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.3.3.cmml" xref="S4.SS1.3.p3.18.m18.4.4.3.3.2.2.3.3.3">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.18.m18.4c">\pi^{\ell}=\{\{a_{u}\colon u\in C_{\ell}\}\cup\{z\}\}\cup\{\{a_{u}\}\colon u% \in V\setminus C_{k}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.18.m18.4d">italic_π start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT = { { italic_a start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT : italic_u ∈ italic_C start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT } ∪ { italic_z } } ∪ { { italic_a start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT } : italic_u ∈ italic_V ∖ italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }</annotation></semantics></math>. We now show that, for <math alttext="1\leq\ell\leq k-1" class="ltx_Math" display="inline" id="S4.SS1.3.p3.19.m19.1"><semantics id="S4.SS1.3.p3.19.m19.1a"><mrow id="S4.SS1.3.p3.19.m19.1.1" xref="S4.SS1.3.p3.19.m19.1.1.cmml"><mn id="S4.SS1.3.p3.19.m19.1.1.2" xref="S4.SS1.3.p3.19.m19.1.1.2.cmml">1</mn><mo id="S4.SS1.3.p3.19.m19.1.1.3" xref="S4.SS1.3.p3.19.m19.1.1.3.cmml">≤</mo><mi id="S4.SS1.3.p3.19.m19.1.1.4" mathvariant="normal" xref="S4.SS1.3.p3.19.m19.1.1.4.cmml">ℓ</mi><mo id="S4.SS1.3.p3.19.m19.1.1.5" xref="S4.SS1.3.p3.19.m19.1.1.5.cmml">≤</mo><mrow id="S4.SS1.3.p3.19.m19.1.1.6" xref="S4.SS1.3.p3.19.m19.1.1.6.cmml"><mi id="S4.SS1.3.p3.19.m19.1.1.6.2" xref="S4.SS1.3.p3.19.m19.1.1.6.2.cmml">k</mi><mo id="S4.SS1.3.p3.19.m19.1.1.6.1" xref="S4.SS1.3.p3.19.m19.1.1.6.1.cmml">−</mo><mn id="S4.SS1.3.p3.19.m19.1.1.6.3" xref="S4.SS1.3.p3.19.m19.1.1.6.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.19.m19.1b"><apply id="S4.SS1.3.p3.19.m19.1.1.cmml" xref="S4.SS1.3.p3.19.m19.1.1"><and id="S4.SS1.3.p3.19.m19.1.1a.cmml" xref="S4.SS1.3.p3.19.m19.1.1"></and><apply id="S4.SS1.3.p3.19.m19.1.1b.cmml" xref="S4.SS1.3.p3.19.m19.1.1"><leq id="S4.SS1.3.p3.19.m19.1.1.3.cmml" xref="S4.SS1.3.p3.19.m19.1.1.3"></leq><cn id="S4.SS1.3.p3.19.m19.1.1.2.cmml" type="integer" xref="S4.SS1.3.p3.19.m19.1.1.2">1</cn><ci id="S4.SS1.3.p3.19.m19.1.1.4.cmml" xref="S4.SS1.3.p3.19.m19.1.1.4">ℓ</ci></apply><apply id="S4.SS1.3.p3.19.m19.1.1c.cmml" xref="S4.SS1.3.p3.19.m19.1.1"><leq id="S4.SS1.3.p3.19.m19.1.1.5.cmml" xref="S4.SS1.3.p3.19.m19.1.1.5"></leq><share href="https://arxiv.org/html/2503.06017v1#S4.SS1.3.p3.19.m19.1.1.4.cmml" id="S4.SS1.3.p3.19.m19.1.1d.cmml" xref="S4.SS1.3.p3.19.m19.1.1"></share><apply id="S4.SS1.3.p3.19.m19.1.1.6.cmml" xref="S4.SS1.3.p3.19.m19.1.1.6"><minus id="S4.SS1.3.p3.19.m19.1.1.6.1.cmml" xref="S4.SS1.3.p3.19.m19.1.1.6.1"></minus><ci id="S4.SS1.3.p3.19.m19.1.1.6.2.cmml" xref="S4.SS1.3.p3.19.m19.1.1.6.2">𝑘</ci><cn id="S4.SS1.3.p3.19.m19.1.1.6.3.cmml" type="integer" xref="S4.SS1.3.p3.19.m19.1.1.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.19.m19.1c">1\leq\ell\leq k-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.19.m19.1d">1 ≤ roman_ℓ ≤ italic_k - 1</annotation></semantics></math>, it holds that <math alttext="\mathcal{SW}(\pi^{\ell})\leq\mathcal{SW}(\pi^{\ell+1})" class="ltx_Math" display="inline" id="S4.SS1.3.p3.20.m20.2"><semantics id="S4.SS1.3.p3.20.m20.2a"><mrow id="S4.SS1.3.p3.20.m20.2.2" xref="S4.SS1.3.p3.20.m20.2.2.cmml"><mrow id="S4.SS1.3.p3.20.m20.1.1.1" xref="S4.SS1.3.p3.20.m20.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.3.p3.20.m20.1.1.1.3" xref="S4.SS1.3.p3.20.m20.1.1.1.3.cmml">𝒮</mi><mo id="S4.SS1.3.p3.20.m20.1.1.1.2" xref="S4.SS1.3.p3.20.m20.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.3.p3.20.m20.1.1.1.4" xref="S4.SS1.3.p3.20.m20.1.1.1.4.cmml">𝒲</mi><mo id="S4.SS1.3.p3.20.m20.1.1.1.2a" xref="S4.SS1.3.p3.20.m20.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.3.p3.20.m20.1.1.1.1.1" xref="S4.SS1.3.p3.20.m20.1.1.1.1.1.1.cmml"><mo id="S4.SS1.3.p3.20.m20.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.3.p3.20.m20.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS1.3.p3.20.m20.1.1.1.1.1.1" xref="S4.SS1.3.p3.20.m20.1.1.1.1.1.1.cmml"><mi id="S4.SS1.3.p3.20.m20.1.1.1.1.1.1.2" xref="S4.SS1.3.p3.20.m20.1.1.1.1.1.1.2.cmml">π</mi><mi id="S4.SS1.3.p3.20.m20.1.1.1.1.1.1.3" mathvariant="normal" xref="S4.SS1.3.p3.20.m20.1.1.1.1.1.1.3.cmml">ℓ</mi></msup><mo id="S4.SS1.3.p3.20.m20.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.3.p3.20.m20.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.3.p3.20.m20.2.2.3" xref="S4.SS1.3.p3.20.m20.2.2.3.cmml">≤</mo><mrow id="S4.SS1.3.p3.20.m20.2.2.2" xref="S4.SS1.3.p3.20.m20.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.3.p3.20.m20.2.2.2.3" xref="S4.SS1.3.p3.20.m20.2.2.2.3.cmml">𝒮</mi><mo id="S4.SS1.3.p3.20.m20.2.2.2.2" xref="S4.SS1.3.p3.20.m20.2.2.2.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.3.p3.20.m20.2.2.2.4" xref="S4.SS1.3.p3.20.m20.2.2.2.4.cmml">𝒲</mi><mo id="S4.SS1.3.p3.20.m20.2.2.2.2a" xref="S4.SS1.3.p3.20.m20.2.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.3.p3.20.m20.2.2.2.1.1" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.cmml"><mo id="S4.SS1.3.p3.20.m20.2.2.2.1.1.2" stretchy="false" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.cmml">(</mo><msup id="S4.SS1.3.p3.20.m20.2.2.2.1.1.1" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.cmml"><mi id="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.2" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.2.cmml">π</mi><mrow id="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3.cmml"><mi id="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3.2" mathvariant="normal" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3.2.cmml">ℓ</mi><mo id="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3.1" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3.1.cmml">+</mo><mn id="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3.3" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S4.SS1.3.p3.20.m20.2.2.2.1.1.3" stretchy="false" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.20.m20.2b"><apply id="S4.SS1.3.p3.20.m20.2.2.cmml" xref="S4.SS1.3.p3.20.m20.2.2"><leq id="S4.SS1.3.p3.20.m20.2.2.3.cmml" xref="S4.SS1.3.p3.20.m20.2.2.3"></leq><apply id="S4.SS1.3.p3.20.m20.1.1.1.cmml" xref="S4.SS1.3.p3.20.m20.1.1.1"><times id="S4.SS1.3.p3.20.m20.1.1.1.2.cmml" xref="S4.SS1.3.p3.20.m20.1.1.1.2"></times><ci id="S4.SS1.3.p3.20.m20.1.1.1.3.cmml" xref="S4.SS1.3.p3.20.m20.1.1.1.3">𝒮</ci><ci id="S4.SS1.3.p3.20.m20.1.1.1.4.cmml" xref="S4.SS1.3.p3.20.m20.1.1.1.4">𝒲</ci><apply id="S4.SS1.3.p3.20.m20.1.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.20.m20.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.20.m20.1.1.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.20.m20.1.1.1.1.1">superscript</csymbol><ci id="S4.SS1.3.p3.20.m20.1.1.1.1.1.1.2.cmml" xref="S4.SS1.3.p3.20.m20.1.1.1.1.1.1.2">𝜋</ci><ci id="S4.SS1.3.p3.20.m20.1.1.1.1.1.1.3.cmml" xref="S4.SS1.3.p3.20.m20.1.1.1.1.1.1.3">ℓ</ci></apply></apply><apply id="S4.SS1.3.p3.20.m20.2.2.2.cmml" xref="S4.SS1.3.p3.20.m20.2.2.2"><times id="S4.SS1.3.p3.20.m20.2.2.2.2.cmml" xref="S4.SS1.3.p3.20.m20.2.2.2.2"></times><ci id="S4.SS1.3.p3.20.m20.2.2.2.3.cmml" xref="S4.SS1.3.p3.20.m20.2.2.2.3">𝒮</ci><ci id="S4.SS1.3.p3.20.m20.2.2.2.4.cmml" xref="S4.SS1.3.p3.20.m20.2.2.2.4">𝒲</ci><apply id="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.cmml" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.1.cmml" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1">superscript</csymbol><ci id="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.2.cmml" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.2">𝜋</ci><apply id="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3.cmml" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3"><plus id="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3.1.cmml" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3.1"></plus><ci id="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3.2.cmml" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3.2">ℓ</ci><cn id="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S4.SS1.3.p3.20.m20.2.2.2.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.20.m20.2c">\mathcal{SW}(\pi^{\ell})\leq\mathcal{SW}(\pi^{\ell+1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.20.m20.2d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ) ≤ caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT roman_ℓ + 1 end_POSTSUPERSCRIPT )</annotation></semantics></math>. Indeed, <math alttext="v(u_{\ell},u)=1" class="ltx_Math" display="inline" id="S4.SS1.3.p3.21.m21.2"><semantics id="S4.SS1.3.p3.21.m21.2a"><mrow id="S4.SS1.3.p3.21.m21.2.2" xref="S4.SS1.3.p3.21.m21.2.2.cmml"><mrow id="S4.SS1.3.p3.21.m21.2.2.1" xref="S4.SS1.3.p3.21.m21.2.2.1.cmml"><mi id="S4.SS1.3.p3.21.m21.2.2.1.3" xref="S4.SS1.3.p3.21.m21.2.2.1.3.cmml">v</mi><mo id="S4.SS1.3.p3.21.m21.2.2.1.2" xref="S4.SS1.3.p3.21.m21.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS1.3.p3.21.m21.2.2.1.1.1" xref="S4.SS1.3.p3.21.m21.2.2.1.1.2.cmml"><mo id="S4.SS1.3.p3.21.m21.2.2.1.1.1.2" stretchy="false" xref="S4.SS1.3.p3.21.m21.2.2.1.1.2.cmml">(</mo><msub id="S4.SS1.3.p3.21.m21.2.2.1.1.1.1" xref="S4.SS1.3.p3.21.m21.2.2.1.1.1.1.cmml"><mi id="S4.SS1.3.p3.21.m21.2.2.1.1.1.1.2" xref="S4.SS1.3.p3.21.m21.2.2.1.1.1.1.2.cmml">u</mi><mi id="S4.SS1.3.p3.21.m21.2.2.1.1.1.1.3" mathvariant="normal" xref="S4.SS1.3.p3.21.m21.2.2.1.1.1.1.3.cmml">ℓ</mi></msub><mo id="S4.SS1.3.p3.21.m21.2.2.1.1.1.3" xref="S4.SS1.3.p3.21.m21.2.2.1.1.2.cmml">,</mo><mi id="S4.SS1.3.p3.21.m21.1.1" xref="S4.SS1.3.p3.21.m21.1.1.cmml">u</mi><mo id="S4.SS1.3.p3.21.m21.2.2.1.1.1.4" stretchy="false" xref="S4.SS1.3.p3.21.m21.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.3.p3.21.m21.2.2.2" xref="S4.SS1.3.p3.21.m21.2.2.2.cmml">=</mo><mn id="S4.SS1.3.p3.21.m21.2.2.3" xref="S4.SS1.3.p3.21.m21.2.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.21.m21.2b"><apply id="S4.SS1.3.p3.21.m21.2.2.cmml" xref="S4.SS1.3.p3.21.m21.2.2"><eq id="S4.SS1.3.p3.21.m21.2.2.2.cmml" xref="S4.SS1.3.p3.21.m21.2.2.2"></eq><apply id="S4.SS1.3.p3.21.m21.2.2.1.cmml" xref="S4.SS1.3.p3.21.m21.2.2.1"><times id="S4.SS1.3.p3.21.m21.2.2.1.2.cmml" xref="S4.SS1.3.p3.21.m21.2.2.1.2"></times><ci id="S4.SS1.3.p3.21.m21.2.2.1.3.cmml" xref="S4.SS1.3.p3.21.m21.2.2.1.3">𝑣</ci><interval closure="open" id="S4.SS1.3.p3.21.m21.2.2.1.1.2.cmml" xref="S4.SS1.3.p3.21.m21.2.2.1.1.1"><apply id="S4.SS1.3.p3.21.m21.2.2.1.1.1.1.cmml" xref="S4.SS1.3.p3.21.m21.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.21.m21.2.2.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.21.m21.2.2.1.1.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.21.m21.2.2.1.1.1.1.2.cmml" xref="S4.SS1.3.p3.21.m21.2.2.1.1.1.1.2">𝑢</ci><ci id="S4.SS1.3.p3.21.m21.2.2.1.1.1.1.3.cmml" xref="S4.SS1.3.p3.21.m21.2.2.1.1.1.1.3">ℓ</ci></apply><ci id="S4.SS1.3.p3.21.m21.1.1.cmml" xref="S4.SS1.3.p3.21.m21.1.1">𝑢</ci></interval></apply><cn id="S4.SS1.3.p3.21.m21.2.2.3.cmml" type="integer" xref="S4.SS1.3.p3.21.m21.2.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.21.m21.2c">v(u_{\ell},u)=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.21.m21.2d">italic_v ( italic_u start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT , italic_u ) = 1</annotation></semantics></math> if <math alttext="u=z" class="ltx_Math" display="inline" id="S4.SS1.3.p3.22.m22.1"><semantics id="S4.SS1.3.p3.22.m22.1a"><mrow id="S4.SS1.3.p3.22.m22.1.1" xref="S4.SS1.3.p3.22.m22.1.1.cmml"><mi id="S4.SS1.3.p3.22.m22.1.1.2" xref="S4.SS1.3.p3.22.m22.1.1.2.cmml">u</mi><mo id="S4.SS1.3.p3.22.m22.1.1.1" xref="S4.SS1.3.p3.22.m22.1.1.1.cmml">=</mo><mi id="S4.SS1.3.p3.22.m22.1.1.3" xref="S4.SS1.3.p3.22.m22.1.1.3.cmml">z</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.22.m22.1b"><apply id="S4.SS1.3.p3.22.m22.1.1.cmml" xref="S4.SS1.3.p3.22.m22.1.1"><eq id="S4.SS1.3.p3.22.m22.1.1.1.cmml" xref="S4.SS1.3.p3.22.m22.1.1.1"></eq><ci id="S4.SS1.3.p3.22.m22.1.1.2.cmml" xref="S4.SS1.3.p3.22.m22.1.1.2">𝑢</ci><ci id="S4.SS1.3.p3.22.m22.1.1.3.cmml" xref="S4.SS1.3.p3.22.m22.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.22.m22.1c">u=z</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.22.m22.1d">italic_u = italic_z</annotation></semantics></math> and <math alttext="v(u_{\ell},u)\leq 0" class="ltx_Math" display="inline" id="S4.SS1.3.p3.23.m23.2"><semantics id="S4.SS1.3.p3.23.m23.2a"><mrow id="S4.SS1.3.p3.23.m23.2.2" xref="S4.SS1.3.p3.23.m23.2.2.cmml"><mrow id="S4.SS1.3.p3.23.m23.2.2.1" xref="S4.SS1.3.p3.23.m23.2.2.1.cmml"><mi id="S4.SS1.3.p3.23.m23.2.2.1.3" xref="S4.SS1.3.p3.23.m23.2.2.1.3.cmml">v</mi><mo id="S4.SS1.3.p3.23.m23.2.2.1.2" xref="S4.SS1.3.p3.23.m23.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS1.3.p3.23.m23.2.2.1.1.1" xref="S4.SS1.3.p3.23.m23.2.2.1.1.2.cmml"><mo id="S4.SS1.3.p3.23.m23.2.2.1.1.1.2" stretchy="false" xref="S4.SS1.3.p3.23.m23.2.2.1.1.2.cmml">(</mo><msub id="S4.SS1.3.p3.23.m23.2.2.1.1.1.1" xref="S4.SS1.3.p3.23.m23.2.2.1.1.1.1.cmml"><mi id="S4.SS1.3.p3.23.m23.2.2.1.1.1.1.2" xref="S4.SS1.3.p3.23.m23.2.2.1.1.1.1.2.cmml">u</mi><mi id="S4.SS1.3.p3.23.m23.2.2.1.1.1.1.3" mathvariant="normal" xref="S4.SS1.3.p3.23.m23.2.2.1.1.1.1.3.cmml">ℓ</mi></msub><mo id="S4.SS1.3.p3.23.m23.2.2.1.1.1.3" xref="S4.SS1.3.p3.23.m23.2.2.1.1.2.cmml">,</mo><mi id="S4.SS1.3.p3.23.m23.1.1" xref="S4.SS1.3.p3.23.m23.1.1.cmml">u</mi><mo id="S4.SS1.3.p3.23.m23.2.2.1.1.1.4" stretchy="false" xref="S4.SS1.3.p3.23.m23.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.3.p3.23.m23.2.2.2" xref="S4.SS1.3.p3.23.m23.2.2.2.cmml">≤</mo><mn id="S4.SS1.3.p3.23.m23.2.2.3" xref="S4.SS1.3.p3.23.m23.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.23.m23.2b"><apply id="S4.SS1.3.p3.23.m23.2.2.cmml" xref="S4.SS1.3.p3.23.m23.2.2"><leq id="S4.SS1.3.p3.23.m23.2.2.2.cmml" xref="S4.SS1.3.p3.23.m23.2.2.2"></leq><apply id="S4.SS1.3.p3.23.m23.2.2.1.cmml" xref="S4.SS1.3.p3.23.m23.2.2.1"><times id="S4.SS1.3.p3.23.m23.2.2.1.2.cmml" xref="S4.SS1.3.p3.23.m23.2.2.1.2"></times><ci id="S4.SS1.3.p3.23.m23.2.2.1.3.cmml" xref="S4.SS1.3.p3.23.m23.2.2.1.3">𝑣</ci><interval closure="open" id="S4.SS1.3.p3.23.m23.2.2.1.1.2.cmml" xref="S4.SS1.3.p3.23.m23.2.2.1.1.1"><apply id="S4.SS1.3.p3.23.m23.2.2.1.1.1.1.cmml" xref="S4.SS1.3.p3.23.m23.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.23.m23.2.2.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.23.m23.2.2.1.1.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.23.m23.2.2.1.1.1.1.2.cmml" xref="S4.SS1.3.p3.23.m23.2.2.1.1.1.1.2">𝑢</ci><ci id="S4.SS1.3.p3.23.m23.2.2.1.1.1.1.3.cmml" xref="S4.SS1.3.p3.23.m23.2.2.1.1.1.1.3">ℓ</ci></apply><ci id="S4.SS1.3.p3.23.m23.1.1.cmml" xref="S4.SS1.3.p3.23.m23.1.1">𝑢</ci></interval></apply><cn id="S4.SS1.3.p3.23.m23.2.2.3.cmml" type="integer" xref="S4.SS1.3.p3.23.m23.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.23.m23.2c">v(u_{\ell},u)\leq 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.23.m23.2d">italic_v ( italic_u start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT , italic_u ) ≤ 0</annotation></semantics></math> for all <math alttext="u\in\pi^{\ell}(u_{\ell})\setminus\{u_{\ell},z\}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.24.m24.3"><semantics id="S4.SS1.3.p3.24.m24.3a"><mrow id="S4.SS1.3.p3.24.m24.3.3" xref="S4.SS1.3.p3.24.m24.3.3.cmml"><mi id="S4.SS1.3.p3.24.m24.3.3.4" xref="S4.SS1.3.p3.24.m24.3.3.4.cmml">u</mi><mo id="S4.SS1.3.p3.24.m24.3.3.3" xref="S4.SS1.3.p3.24.m24.3.3.3.cmml">∈</mo><mrow id="S4.SS1.3.p3.24.m24.3.3.2" xref="S4.SS1.3.p3.24.m24.3.3.2.cmml"><mrow id="S4.SS1.3.p3.24.m24.2.2.1.1" xref="S4.SS1.3.p3.24.m24.2.2.1.1.cmml"><msup id="S4.SS1.3.p3.24.m24.2.2.1.1.3" xref="S4.SS1.3.p3.24.m24.2.2.1.1.3.cmml"><mi id="S4.SS1.3.p3.24.m24.2.2.1.1.3.2" xref="S4.SS1.3.p3.24.m24.2.2.1.1.3.2.cmml">π</mi><mi id="S4.SS1.3.p3.24.m24.2.2.1.1.3.3" mathvariant="normal" xref="S4.SS1.3.p3.24.m24.2.2.1.1.3.3.cmml">ℓ</mi></msup><mo id="S4.SS1.3.p3.24.m24.2.2.1.1.2" xref="S4.SS1.3.p3.24.m24.2.2.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.3.p3.24.m24.2.2.1.1.1.1" xref="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.1.cmml"><mo id="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.2" stretchy="false" xref="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.1.cmml">(</mo><msub id="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.1" xref="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.1.cmml"><mi id="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.1.2" xref="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.1.2.cmml">u</mi><mi id="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.1.3" mathvariant="normal" xref="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.1.3.cmml">ℓ</mi></msub><mo id="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.3" stretchy="false" xref="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.3.p3.24.m24.3.3.2.3" xref="S4.SS1.3.p3.24.m24.3.3.2.3.cmml">∖</mo><mrow id="S4.SS1.3.p3.24.m24.3.3.2.2.1" xref="S4.SS1.3.p3.24.m24.3.3.2.2.2.cmml"><mo id="S4.SS1.3.p3.24.m24.3.3.2.2.1.2" stretchy="false" xref="S4.SS1.3.p3.24.m24.3.3.2.2.2.cmml">{</mo><msub id="S4.SS1.3.p3.24.m24.3.3.2.2.1.1" xref="S4.SS1.3.p3.24.m24.3.3.2.2.1.1.cmml"><mi id="S4.SS1.3.p3.24.m24.3.3.2.2.1.1.2" xref="S4.SS1.3.p3.24.m24.3.3.2.2.1.1.2.cmml">u</mi><mi id="S4.SS1.3.p3.24.m24.3.3.2.2.1.1.3" mathvariant="normal" xref="S4.SS1.3.p3.24.m24.3.3.2.2.1.1.3.cmml">ℓ</mi></msub><mo id="S4.SS1.3.p3.24.m24.3.3.2.2.1.3" xref="S4.SS1.3.p3.24.m24.3.3.2.2.2.cmml">,</mo><mi id="S4.SS1.3.p3.24.m24.1.1" xref="S4.SS1.3.p3.24.m24.1.1.cmml">z</mi><mo id="S4.SS1.3.p3.24.m24.3.3.2.2.1.4" stretchy="false" xref="S4.SS1.3.p3.24.m24.3.3.2.2.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.24.m24.3b"><apply id="S4.SS1.3.p3.24.m24.3.3.cmml" xref="S4.SS1.3.p3.24.m24.3.3"><in id="S4.SS1.3.p3.24.m24.3.3.3.cmml" xref="S4.SS1.3.p3.24.m24.3.3.3"></in><ci id="S4.SS1.3.p3.24.m24.3.3.4.cmml" xref="S4.SS1.3.p3.24.m24.3.3.4">𝑢</ci><apply id="S4.SS1.3.p3.24.m24.3.3.2.cmml" xref="S4.SS1.3.p3.24.m24.3.3.2"><setdiff id="S4.SS1.3.p3.24.m24.3.3.2.3.cmml" xref="S4.SS1.3.p3.24.m24.3.3.2.3"></setdiff><apply id="S4.SS1.3.p3.24.m24.2.2.1.1.cmml" xref="S4.SS1.3.p3.24.m24.2.2.1.1"><times id="S4.SS1.3.p3.24.m24.2.2.1.1.2.cmml" xref="S4.SS1.3.p3.24.m24.2.2.1.1.2"></times><apply id="S4.SS1.3.p3.24.m24.2.2.1.1.3.cmml" xref="S4.SS1.3.p3.24.m24.2.2.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.3.p3.24.m24.2.2.1.1.3.1.cmml" xref="S4.SS1.3.p3.24.m24.2.2.1.1.3">superscript</csymbol><ci id="S4.SS1.3.p3.24.m24.2.2.1.1.3.2.cmml" xref="S4.SS1.3.p3.24.m24.2.2.1.1.3.2">𝜋</ci><ci id="S4.SS1.3.p3.24.m24.2.2.1.1.3.3.cmml" xref="S4.SS1.3.p3.24.m24.2.2.1.1.3.3">ℓ</ci></apply><apply id="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.24.m24.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.24.m24.2.2.1.1.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.1.2.cmml" xref="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.1.2">𝑢</ci><ci id="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.1.3.cmml" xref="S4.SS1.3.p3.24.m24.2.2.1.1.1.1.1.3">ℓ</ci></apply></apply><set id="S4.SS1.3.p3.24.m24.3.3.2.2.2.cmml" xref="S4.SS1.3.p3.24.m24.3.3.2.2.1"><apply id="S4.SS1.3.p3.24.m24.3.3.2.2.1.1.cmml" xref="S4.SS1.3.p3.24.m24.3.3.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.24.m24.3.3.2.2.1.1.1.cmml" xref="S4.SS1.3.p3.24.m24.3.3.2.2.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.24.m24.3.3.2.2.1.1.2.cmml" xref="S4.SS1.3.p3.24.m24.3.3.2.2.1.1.2">𝑢</ci><ci id="S4.SS1.3.p3.24.m24.3.3.2.2.1.1.3.cmml" xref="S4.SS1.3.p3.24.m24.3.3.2.2.1.1.3">ℓ</ci></apply><ci id="S4.SS1.3.p3.24.m24.1.1.cmml" xref="S4.SS1.3.p3.24.m24.1.1">𝑧</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.24.m24.3c">u\in\pi^{\ell}(u_{\ell})\setminus\{u_{\ell},z\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.24.m24.3d">italic_u ∈ italic_π start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ( italic_u start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ) ∖ { italic_u start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT , italic_z }</annotation></semantics></math>. Moreover, since <math alttext="C_{\ell}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.25.m25.1"><semantics id="S4.SS1.3.p3.25.m25.1a"><msub id="S4.SS1.3.p3.25.m25.1.1" xref="S4.SS1.3.p3.25.m25.1.1.cmml"><mi id="S4.SS1.3.p3.25.m25.1.1.2" xref="S4.SS1.3.p3.25.m25.1.1.2.cmml">C</mi><mi id="S4.SS1.3.p3.25.m25.1.1.3" mathvariant="normal" xref="S4.SS1.3.p3.25.m25.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.25.m25.1b"><apply id="S4.SS1.3.p3.25.m25.1.1.cmml" xref="S4.SS1.3.p3.25.m25.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.25.m25.1.1.1.cmml" xref="S4.SS1.3.p3.25.m25.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.25.m25.1.1.2.cmml" xref="S4.SS1.3.p3.25.m25.1.1.2">𝐶</ci><ci id="S4.SS1.3.p3.25.m25.1.1.3.cmml" xref="S4.SS1.3.p3.25.m25.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.25.m25.1c">C_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.25.m25.1d">italic_C start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> is not a clique, there exists an agent <math alttext="u\in\pi^{\ell}(u_{\ell})\setminus\{u_{\ell},z\}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.26.m26.3"><semantics id="S4.SS1.3.p3.26.m26.3a"><mrow id="S4.SS1.3.p3.26.m26.3.3" xref="S4.SS1.3.p3.26.m26.3.3.cmml"><mi id="S4.SS1.3.p3.26.m26.3.3.4" xref="S4.SS1.3.p3.26.m26.3.3.4.cmml">u</mi><mo id="S4.SS1.3.p3.26.m26.3.3.3" xref="S4.SS1.3.p3.26.m26.3.3.3.cmml">∈</mo><mrow id="S4.SS1.3.p3.26.m26.3.3.2" xref="S4.SS1.3.p3.26.m26.3.3.2.cmml"><mrow id="S4.SS1.3.p3.26.m26.2.2.1.1" xref="S4.SS1.3.p3.26.m26.2.2.1.1.cmml"><msup id="S4.SS1.3.p3.26.m26.2.2.1.1.3" xref="S4.SS1.3.p3.26.m26.2.2.1.1.3.cmml"><mi id="S4.SS1.3.p3.26.m26.2.2.1.1.3.2" xref="S4.SS1.3.p3.26.m26.2.2.1.1.3.2.cmml">π</mi><mi id="S4.SS1.3.p3.26.m26.2.2.1.1.3.3" mathvariant="normal" xref="S4.SS1.3.p3.26.m26.2.2.1.1.3.3.cmml">ℓ</mi></msup><mo id="S4.SS1.3.p3.26.m26.2.2.1.1.2" xref="S4.SS1.3.p3.26.m26.2.2.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.3.p3.26.m26.2.2.1.1.1.1" xref="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.1.cmml"><mo id="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.2" stretchy="false" xref="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.1.cmml">(</mo><msub id="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.1" xref="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.1.cmml"><mi id="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.1.2" xref="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.1.2.cmml">u</mi><mi id="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.1.3" mathvariant="normal" xref="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.1.3.cmml">ℓ</mi></msub><mo id="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.3" stretchy="false" xref="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.3.p3.26.m26.3.3.2.3" xref="S4.SS1.3.p3.26.m26.3.3.2.3.cmml">∖</mo><mrow id="S4.SS1.3.p3.26.m26.3.3.2.2.1" xref="S4.SS1.3.p3.26.m26.3.3.2.2.2.cmml"><mo id="S4.SS1.3.p3.26.m26.3.3.2.2.1.2" stretchy="false" xref="S4.SS1.3.p3.26.m26.3.3.2.2.2.cmml">{</mo><msub id="S4.SS1.3.p3.26.m26.3.3.2.2.1.1" xref="S4.SS1.3.p3.26.m26.3.3.2.2.1.1.cmml"><mi id="S4.SS1.3.p3.26.m26.3.3.2.2.1.1.2" xref="S4.SS1.3.p3.26.m26.3.3.2.2.1.1.2.cmml">u</mi><mi id="S4.SS1.3.p3.26.m26.3.3.2.2.1.1.3" mathvariant="normal" xref="S4.SS1.3.p3.26.m26.3.3.2.2.1.1.3.cmml">ℓ</mi></msub><mo id="S4.SS1.3.p3.26.m26.3.3.2.2.1.3" xref="S4.SS1.3.p3.26.m26.3.3.2.2.2.cmml">,</mo><mi id="S4.SS1.3.p3.26.m26.1.1" xref="S4.SS1.3.p3.26.m26.1.1.cmml">z</mi><mo id="S4.SS1.3.p3.26.m26.3.3.2.2.1.4" stretchy="false" xref="S4.SS1.3.p3.26.m26.3.3.2.2.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.26.m26.3b"><apply id="S4.SS1.3.p3.26.m26.3.3.cmml" xref="S4.SS1.3.p3.26.m26.3.3"><in id="S4.SS1.3.p3.26.m26.3.3.3.cmml" xref="S4.SS1.3.p3.26.m26.3.3.3"></in><ci id="S4.SS1.3.p3.26.m26.3.3.4.cmml" xref="S4.SS1.3.p3.26.m26.3.3.4">𝑢</ci><apply id="S4.SS1.3.p3.26.m26.3.3.2.cmml" xref="S4.SS1.3.p3.26.m26.3.3.2"><setdiff id="S4.SS1.3.p3.26.m26.3.3.2.3.cmml" xref="S4.SS1.3.p3.26.m26.3.3.2.3"></setdiff><apply id="S4.SS1.3.p3.26.m26.2.2.1.1.cmml" xref="S4.SS1.3.p3.26.m26.2.2.1.1"><times id="S4.SS1.3.p3.26.m26.2.2.1.1.2.cmml" xref="S4.SS1.3.p3.26.m26.2.2.1.1.2"></times><apply id="S4.SS1.3.p3.26.m26.2.2.1.1.3.cmml" xref="S4.SS1.3.p3.26.m26.2.2.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.3.p3.26.m26.2.2.1.1.3.1.cmml" xref="S4.SS1.3.p3.26.m26.2.2.1.1.3">superscript</csymbol><ci id="S4.SS1.3.p3.26.m26.2.2.1.1.3.2.cmml" xref="S4.SS1.3.p3.26.m26.2.2.1.1.3.2">𝜋</ci><ci id="S4.SS1.3.p3.26.m26.2.2.1.1.3.3.cmml" xref="S4.SS1.3.p3.26.m26.2.2.1.1.3.3">ℓ</ci></apply><apply id="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.26.m26.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.26.m26.2.2.1.1.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.1.2.cmml" xref="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.1.2">𝑢</ci><ci id="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.1.3.cmml" xref="S4.SS1.3.p3.26.m26.2.2.1.1.1.1.1.3">ℓ</ci></apply></apply><set id="S4.SS1.3.p3.26.m26.3.3.2.2.2.cmml" xref="S4.SS1.3.p3.26.m26.3.3.2.2.1"><apply id="S4.SS1.3.p3.26.m26.3.3.2.2.1.1.cmml" xref="S4.SS1.3.p3.26.m26.3.3.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.26.m26.3.3.2.2.1.1.1.cmml" xref="S4.SS1.3.p3.26.m26.3.3.2.2.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.26.m26.3.3.2.2.1.1.2.cmml" xref="S4.SS1.3.p3.26.m26.3.3.2.2.1.1.2">𝑢</ci><ci id="S4.SS1.3.p3.26.m26.3.3.2.2.1.1.3.cmml" xref="S4.SS1.3.p3.26.m26.3.3.2.2.1.1.3">ℓ</ci></apply><ci id="S4.SS1.3.p3.26.m26.1.1.cmml" xref="S4.SS1.3.p3.26.m26.1.1">𝑧</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.26.m26.3c">u\in\pi^{\ell}(u_{\ell})\setminus\{u_{\ell},z\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.26.m26.3d">italic_u ∈ italic_π start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ( italic_u start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ) ∖ { italic_u start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT , italic_z }</annotation></semantics></math> with <math alttext="v(u_{\ell},u)=-v^{-}\leq-1" class="ltx_Math" display="inline" id="S4.SS1.3.p3.27.m27.2"><semantics id="S4.SS1.3.p3.27.m27.2a"><mrow id="S4.SS1.3.p3.27.m27.2.2" xref="S4.SS1.3.p3.27.m27.2.2.cmml"><mrow id="S4.SS1.3.p3.27.m27.2.2.1" xref="S4.SS1.3.p3.27.m27.2.2.1.cmml"><mi id="S4.SS1.3.p3.27.m27.2.2.1.3" xref="S4.SS1.3.p3.27.m27.2.2.1.3.cmml">v</mi><mo id="S4.SS1.3.p3.27.m27.2.2.1.2" xref="S4.SS1.3.p3.27.m27.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS1.3.p3.27.m27.2.2.1.1.1" xref="S4.SS1.3.p3.27.m27.2.2.1.1.2.cmml"><mo id="S4.SS1.3.p3.27.m27.2.2.1.1.1.2" stretchy="false" xref="S4.SS1.3.p3.27.m27.2.2.1.1.2.cmml">(</mo><msub id="S4.SS1.3.p3.27.m27.2.2.1.1.1.1" xref="S4.SS1.3.p3.27.m27.2.2.1.1.1.1.cmml"><mi id="S4.SS1.3.p3.27.m27.2.2.1.1.1.1.2" xref="S4.SS1.3.p3.27.m27.2.2.1.1.1.1.2.cmml">u</mi><mi id="S4.SS1.3.p3.27.m27.2.2.1.1.1.1.3" mathvariant="normal" xref="S4.SS1.3.p3.27.m27.2.2.1.1.1.1.3.cmml">ℓ</mi></msub><mo id="S4.SS1.3.p3.27.m27.2.2.1.1.1.3" xref="S4.SS1.3.p3.27.m27.2.2.1.1.2.cmml">,</mo><mi id="S4.SS1.3.p3.27.m27.1.1" xref="S4.SS1.3.p3.27.m27.1.1.cmml">u</mi><mo id="S4.SS1.3.p3.27.m27.2.2.1.1.1.4" stretchy="false" xref="S4.SS1.3.p3.27.m27.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.3.p3.27.m27.2.2.3" xref="S4.SS1.3.p3.27.m27.2.2.3.cmml">=</mo><mrow id="S4.SS1.3.p3.27.m27.2.2.4" xref="S4.SS1.3.p3.27.m27.2.2.4.cmml"><mo id="S4.SS1.3.p3.27.m27.2.2.4a" xref="S4.SS1.3.p3.27.m27.2.2.4.cmml">−</mo><msup id="S4.SS1.3.p3.27.m27.2.2.4.2" xref="S4.SS1.3.p3.27.m27.2.2.4.2.cmml"><mi id="S4.SS1.3.p3.27.m27.2.2.4.2.2" xref="S4.SS1.3.p3.27.m27.2.2.4.2.2.cmml">v</mi><mo id="S4.SS1.3.p3.27.m27.2.2.4.2.3" xref="S4.SS1.3.p3.27.m27.2.2.4.2.3.cmml">−</mo></msup></mrow><mo id="S4.SS1.3.p3.27.m27.2.2.5" xref="S4.SS1.3.p3.27.m27.2.2.5.cmml">≤</mo><mrow id="S4.SS1.3.p3.27.m27.2.2.6" xref="S4.SS1.3.p3.27.m27.2.2.6.cmml"><mo id="S4.SS1.3.p3.27.m27.2.2.6a" xref="S4.SS1.3.p3.27.m27.2.2.6.cmml">−</mo><mn id="S4.SS1.3.p3.27.m27.2.2.6.2" xref="S4.SS1.3.p3.27.m27.2.2.6.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.27.m27.2b"><apply id="S4.SS1.3.p3.27.m27.2.2.cmml" xref="S4.SS1.3.p3.27.m27.2.2"><and id="S4.SS1.3.p3.27.m27.2.2a.cmml" xref="S4.SS1.3.p3.27.m27.2.2"></and><apply id="S4.SS1.3.p3.27.m27.2.2b.cmml" xref="S4.SS1.3.p3.27.m27.2.2"><eq id="S4.SS1.3.p3.27.m27.2.2.3.cmml" xref="S4.SS1.3.p3.27.m27.2.2.3"></eq><apply id="S4.SS1.3.p3.27.m27.2.2.1.cmml" xref="S4.SS1.3.p3.27.m27.2.2.1"><times id="S4.SS1.3.p3.27.m27.2.2.1.2.cmml" xref="S4.SS1.3.p3.27.m27.2.2.1.2"></times><ci id="S4.SS1.3.p3.27.m27.2.2.1.3.cmml" xref="S4.SS1.3.p3.27.m27.2.2.1.3">𝑣</ci><interval closure="open" id="S4.SS1.3.p3.27.m27.2.2.1.1.2.cmml" xref="S4.SS1.3.p3.27.m27.2.2.1.1.1"><apply id="S4.SS1.3.p3.27.m27.2.2.1.1.1.1.cmml" xref="S4.SS1.3.p3.27.m27.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.27.m27.2.2.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.27.m27.2.2.1.1.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.27.m27.2.2.1.1.1.1.2.cmml" xref="S4.SS1.3.p3.27.m27.2.2.1.1.1.1.2">𝑢</ci><ci id="S4.SS1.3.p3.27.m27.2.2.1.1.1.1.3.cmml" xref="S4.SS1.3.p3.27.m27.2.2.1.1.1.1.3">ℓ</ci></apply><ci id="S4.SS1.3.p3.27.m27.1.1.cmml" xref="S4.SS1.3.p3.27.m27.1.1">𝑢</ci></interval></apply><apply id="S4.SS1.3.p3.27.m27.2.2.4.cmml" xref="S4.SS1.3.p3.27.m27.2.2.4"><minus id="S4.SS1.3.p3.27.m27.2.2.4.1.cmml" xref="S4.SS1.3.p3.27.m27.2.2.4"></minus><apply id="S4.SS1.3.p3.27.m27.2.2.4.2.cmml" xref="S4.SS1.3.p3.27.m27.2.2.4.2"><csymbol cd="ambiguous" id="S4.SS1.3.p3.27.m27.2.2.4.2.1.cmml" xref="S4.SS1.3.p3.27.m27.2.2.4.2">superscript</csymbol><ci id="S4.SS1.3.p3.27.m27.2.2.4.2.2.cmml" xref="S4.SS1.3.p3.27.m27.2.2.4.2.2">𝑣</ci><minus id="S4.SS1.3.p3.27.m27.2.2.4.2.3.cmml" xref="S4.SS1.3.p3.27.m27.2.2.4.2.3"></minus></apply></apply></apply><apply id="S4.SS1.3.p3.27.m27.2.2c.cmml" xref="S4.SS1.3.p3.27.m27.2.2"><leq id="S4.SS1.3.p3.27.m27.2.2.5.cmml" xref="S4.SS1.3.p3.27.m27.2.2.5"></leq><share href="https://arxiv.org/html/2503.06017v1#S4.SS1.3.p3.27.m27.2.2.4.cmml" id="S4.SS1.3.p3.27.m27.2.2d.cmml" xref="S4.SS1.3.p3.27.m27.2.2"></share><apply id="S4.SS1.3.p3.27.m27.2.2.6.cmml" xref="S4.SS1.3.p3.27.m27.2.2.6"><minus id="S4.SS1.3.p3.27.m27.2.2.6.1.cmml" xref="S4.SS1.3.p3.27.m27.2.2.6"></minus><cn id="S4.SS1.3.p3.27.m27.2.2.6.2.cmml" type="integer" xref="S4.SS1.3.p3.27.m27.2.2.6.2">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.27.m27.2c">v(u_{\ell},u)=-v^{-}\leq-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.27.m27.2d">italic_v ( italic_u start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT , italic_u ) = - italic_v start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ≤ - 1</annotation></semantics></math>. Hence, removing <math alttext="u_{\ell}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.28.m28.1"><semantics id="S4.SS1.3.p3.28.m28.1a"><msub id="S4.SS1.3.p3.28.m28.1.1" xref="S4.SS1.3.p3.28.m28.1.1.cmml"><mi id="S4.SS1.3.p3.28.m28.1.1.2" xref="S4.SS1.3.p3.28.m28.1.1.2.cmml">u</mi><mi id="S4.SS1.3.p3.28.m28.1.1.3" mathvariant="normal" xref="S4.SS1.3.p3.28.m28.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.28.m28.1b"><apply id="S4.SS1.3.p3.28.m28.1.1.cmml" xref="S4.SS1.3.p3.28.m28.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.28.m28.1.1.1.cmml" xref="S4.SS1.3.p3.28.m28.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.28.m28.1.1.2.cmml" xref="S4.SS1.3.p3.28.m28.1.1.2">𝑢</ci><ci id="S4.SS1.3.p3.28.m28.1.1.3.cmml" xref="S4.SS1.3.p3.28.m28.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.28.m28.1c">u_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.28.m28.1d">italic_u start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> from their coalition and forming a singleton coalition can only increase the social welfare. In addition, it holds that <math alttext="\mathcal{SW}(\pi)\leq\mathcal{SW}(\pi^{1})" class="ltx_Math" display="inline" id="S4.SS1.3.p3.29.m29.2"><semantics id="S4.SS1.3.p3.29.m29.2a"><mrow id="S4.SS1.3.p3.29.m29.2.2" xref="S4.SS1.3.p3.29.m29.2.2.cmml"><mrow id="S4.SS1.3.p3.29.m29.2.2.3" xref="S4.SS1.3.p3.29.m29.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.3.p3.29.m29.2.2.3.2" xref="S4.SS1.3.p3.29.m29.2.2.3.2.cmml">𝒮</mi><mo id="S4.SS1.3.p3.29.m29.2.2.3.1" xref="S4.SS1.3.p3.29.m29.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.3.p3.29.m29.2.2.3.3" xref="S4.SS1.3.p3.29.m29.2.2.3.3.cmml">𝒲</mi><mo id="S4.SS1.3.p3.29.m29.2.2.3.1a" xref="S4.SS1.3.p3.29.m29.2.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.3.p3.29.m29.2.2.3.4.2" xref="S4.SS1.3.p3.29.m29.2.2.3.cmml"><mo id="S4.SS1.3.p3.29.m29.2.2.3.4.2.1" stretchy="false" xref="S4.SS1.3.p3.29.m29.2.2.3.cmml">(</mo><mi id="S4.SS1.3.p3.29.m29.1.1" xref="S4.SS1.3.p3.29.m29.1.1.cmml">π</mi><mo id="S4.SS1.3.p3.29.m29.2.2.3.4.2.2" stretchy="false" xref="S4.SS1.3.p3.29.m29.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.SS1.3.p3.29.m29.2.2.2" xref="S4.SS1.3.p3.29.m29.2.2.2.cmml">≤</mo><mrow id="S4.SS1.3.p3.29.m29.2.2.1" xref="S4.SS1.3.p3.29.m29.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.3.p3.29.m29.2.2.1.3" xref="S4.SS1.3.p3.29.m29.2.2.1.3.cmml">𝒮</mi><mo id="S4.SS1.3.p3.29.m29.2.2.1.2" xref="S4.SS1.3.p3.29.m29.2.2.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.3.p3.29.m29.2.2.1.4" xref="S4.SS1.3.p3.29.m29.2.2.1.4.cmml">𝒲</mi><mo id="S4.SS1.3.p3.29.m29.2.2.1.2a" xref="S4.SS1.3.p3.29.m29.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS1.3.p3.29.m29.2.2.1.1.1" xref="S4.SS1.3.p3.29.m29.2.2.1.1.1.1.cmml"><mo id="S4.SS1.3.p3.29.m29.2.2.1.1.1.2" stretchy="false" xref="S4.SS1.3.p3.29.m29.2.2.1.1.1.1.cmml">(</mo><msup id="S4.SS1.3.p3.29.m29.2.2.1.1.1.1" xref="S4.SS1.3.p3.29.m29.2.2.1.1.1.1.cmml"><mi id="S4.SS1.3.p3.29.m29.2.2.1.1.1.1.2" xref="S4.SS1.3.p3.29.m29.2.2.1.1.1.1.2.cmml">π</mi><mn id="S4.SS1.3.p3.29.m29.2.2.1.1.1.1.3" xref="S4.SS1.3.p3.29.m29.2.2.1.1.1.1.3.cmml">1</mn></msup><mo id="S4.SS1.3.p3.29.m29.2.2.1.1.1.3" stretchy="false" xref="S4.SS1.3.p3.29.m29.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.29.m29.2b"><apply id="S4.SS1.3.p3.29.m29.2.2.cmml" xref="S4.SS1.3.p3.29.m29.2.2"><leq id="S4.SS1.3.p3.29.m29.2.2.2.cmml" xref="S4.SS1.3.p3.29.m29.2.2.2"></leq><apply id="S4.SS1.3.p3.29.m29.2.2.3.cmml" xref="S4.SS1.3.p3.29.m29.2.2.3"><times id="S4.SS1.3.p3.29.m29.2.2.3.1.cmml" xref="S4.SS1.3.p3.29.m29.2.2.3.1"></times><ci id="S4.SS1.3.p3.29.m29.2.2.3.2.cmml" xref="S4.SS1.3.p3.29.m29.2.2.3.2">𝒮</ci><ci id="S4.SS1.3.p3.29.m29.2.2.3.3.cmml" xref="S4.SS1.3.p3.29.m29.2.2.3.3">𝒲</ci><ci id="S4.SS1.3.p3.29.m29.1.1.cmml" xref="S4.SS1.3.p3.29.m29.1.1">𝜋</ci></apply><apply id="S4.SS1.3.p3.29.m29.2.2.1.cmml" xref="S4.SS1.3.p3.29.m29.2.2.1"><times id="S4.SS1.3.p3.29.m29.2.2.1.2.cmml" xref="S4.SS1.3.p3.29.m29.2.2.1.2"></times><ci id="S4.SS1.3.p3.29.m29.2.2.1.3.cmml" xref="S4.SS1.3.p3.29.m29.2.2.1.3">𝒮</ci><ci id="S4.SS1.3.p3.29.m29.2.2.1.4.cmml" xref="S4.SS1.3.p3.29.m29.2.2.1.4">𝒲</ci><apply id="S4.SS1.3.p3.29.m29.2.2.1.1.1.1.cmml" xref="S4.SS1.3.p3.29.m29.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.29.m29.2.2.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.29.m29.2.2.1.1.1">superscript</csymbol><ci id="S4.SS1.3.p3.29.m29.2.2.1.1.1.1.2.cmml" xref="S4.SS1.3.p3.29.m29.2.2.1.1.1.1.2">𝜋</ci><cn id="S4.SS1.3.p3.29.m29.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.SS1.3.p3.29.m29.2.2.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.29.m29.2c">\mathcal{SW}(\pi)\leq\mathcal{SW}(\pi^{1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.29.m29.2d">caligraphic_S caligraphic_W ( italic_π ) ≤ caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT )</annotation></semantics></math> since <math alttext="\pi^{1}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.30.m30.1"><semantics id="S4.SS1.3.p3.30.m30.1a"><msup id="S4.SS1.3.p3.30.m30.1.1" xref="S4.SS1.3.p3.30.m30.1.1.cmml"><mi id="S4.SS1.3.p3.30.m30.1.1.2" xref="S4.SS1.3.p3.30.m30.1.1.2.cmml">π</mi><mn id="S4.SS1.3.p3.30.m30.1.1.3" xref="S4.SS1.3.p3.30.m30.1.1.3.cmml">1</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.30.m30.1b"><apply id="S4.SS1.3.p3.30.m30.1.1.cmml" xref="S4.SS1.3.p3.30.m30.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.30.m30.1.1.1.cmml" xref="S4.SS1.3.p3.30.m30.1.1">superscript</csymbol><ci id="S4.SS1.3.p3.30.m30.1.1.2.cmml" xref="S4.SS1.3.p3.30.m30.1.1.2">𝜋</ci><cn id="S4.SS1.3.p3.30.m30.1.1.3.cmml" type="integer" xref="S4.SS1.3.p3.30.m30.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.30.m30.1c">\pi^{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.30.m30.1d">italic_π start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT</annotation></semantics></math> can only differ from <math alttext="\pi" class="ltx_Math" display="inline" id="S4.SS1.3.p3.31.m31.1"><semantics id="S4.SS1.3.p3.31.m31.1a"><mi id="S4.SS1.3.p3.31.m31.1.1" xref="S4.SS1.3.p3.31.m31.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.31.m31.1b"><ci id="S4.SS1.3.p3.31.m31.1.1.cmml" xref="S4.SS1.3.p3.31.m31.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.31.m31.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.31.m31.1d">italic_π</annotation></semantics></math> by dissolving nonsingleton coalitions not containing <math alttext="z" class="ltx_Math" display="inline" id="S4.SS1.3.p3.32.m32.1"><semantics id="S4.SS1.3.p3.32.m32.1a"><mi id="S4.SS1.3.p3.32.m32.1.1" xref="S4.SS1.3.p3.32.m32.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.32.m32.1b"><ci id="S4.SS1.3.p3.32.m32.1.1.cmml" xref="S4.SS1.3.p3.32.m32.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.32.m32.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.32.m32.1d">italic_z</annotation></semantics></math>, which can only increase the welfare. Finally, <math alttext="\mathcal{SW}(\pi^{k})=|C_{k}|" class="ltx_Math" display="inline" id="S4.SS1.3.p3.33.m33.2"><semantics id="S4.SS1.3.p3.33.m33.2a"><mrow id="S4.SS1.3.p3.33.m33.2.2" xref="S4.SS1.3.p3.33.m33.2.2.cmml"><mrow id="S4.SS1.3.p3.33.m33.1.1.1" xref="S4.SS1.3.p3.33.m33.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.3.p3.33.m33.1.1.1.3" xref="S4.SS1.3.p3.33.m33.1.1.1.3.cmml">𝒮</mi><mo id="S4.SS1.3.p3.33.m33.1.1.1.2" xref="S4.SS1.3.p3.33.m33.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.3.p3.33.m33.1.1.1.4" xref="S4.SS1.3.p3.33.m33.1.1.1.4.cmml">𝒲</mi><mo id="S4.SS1.3.p3.33.m33.1.1.1.2a" xref="S4.SS1.3.p3.33.m33.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.3.p3.33.m33.1.1.1.1.1" xref="S4.SS1.3.p3.33.m33.1.1.1.1.1.1.cmml"><mo id="S4.SS1.3.p3.33.m33.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.3.p3.33.m33.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS1.3.p3.33.m33.1.1.1.1.1.1" xref="S4.SS1.3.p3.33.m33.1.1.1.1.1.1.cmml"><mi id="S4.SS1.3.p3.33.m33.1.1.1.1.1.1.2" xref="S4.SS1.3.p3.33.m33.1.1.1.1.1.1.2.cmml">π</mi><mi id="S4.SS1.3.p3.33.m33.1.1.1.1.1.1.3" xref="S4.SS1.3.p3.33.m33.1.1.1.1.1.1.3.cmml">k</mi></msup><mo id="S4.SS1.3.p3.33.m33.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.3.p3.33.m33.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.3.p3.33.m33.2.2.3" xref="S4.SS1.3.p3.33.m33.2.2.3.cmml">=</mo><mrow id="S4.SS1.3.p3.33.m33.2.2.2.1" xref="S4.SS1.3.p3.33.m33.2.2.2.2.cmml"><mo id="S4.SS1.3.p3.33.m33.2.2.2.1.2" stretchy="false" xref="S4.SS1.3.p3.33.m33.2.2.2.2.1.cmml">|</mo><msub id="S4.SS1.3.p3.33.m33.2.2.2.1.1" xref="S4.SS1.3.p3.33.m33.2.2.2.1.1.cmml"><mi id="S4.SS1.3.p3.33.m33.2.2.2.1.1.2" xref="S4.SS1.3.p3.33.m33.2.2.2.1.1.2.cmml">C</mi><mi id="S4.SS1.3.p3.33.m33.2.2.2.1.1.3" xref="S4.SS1.3.p3.33.m33.2.2.2.1.1.3.cmml">k</mi></msub><mo id="S4.SS1.3.p3.33.m33.2.2.2.1.3" stretchy="false" xref="S4.SS1.3.p3.33.m33.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.33.m33.2b"><apply id="S4.SS1.3.p3.33.m33.2.2.cmml" xref="S4.SS1.3.p3.33.m33.2.2"><eq id="S4.SS1.3.p3.33.m33.2.2.3.cmml" xref="S4.SS1.3.p3.33.m33.2.2.3"></eq><apply id="S4.SS1.3.p3.33.m33.1.1.1.cmml" xref="S4.SS1.3.p3.33.m33.1.1.1"><times id="S4.SS1.3.p3.33.m33.1.1.1.2.cmml" xref="S4.SS1.3.p3.33.m33.1.1.1.2"></times><ci id="S4.SS1.3.p3.33.m33.1.1.1.3.cmml" xref="S4.SS1.3.p3.33.m33.1.1.1.3">𝒮</ci><ci id="S4.SS1.3.p3.33.m33.1.1.1.4.cmml" xref="S4.SS1.3.p3.33.m33.1.1.1.4">𝒲</ci><apply id="S4.SS1.3.p3.33.m33.1.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.33.m33.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.33.m33.1.1.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.33.m33.1.1.1.1.1">superscript</csymbol><ci id="S4.SS1.3.p3.33.m33.1.1.1.1.1.1.2.cmml" xref="S4.SS1.3.p3.33.m33.1.1.1.1.1.1.2">𝜋</ci><ci id="S4.SS1.3.p3.33.m33.1.1.1.1.1.1.3.cmml" xref="S4.SS1.3.p3.33.m33.1.1.1.1.1.1.3">𝑘</ci></apply></apply><apply id="S4.SS1.3.p3.33.m33.2.2.2.2.cmml" xref="S4.SS1.3.p3.33.m33.2.2.2.1"><abs id="S4.SS1.3.p3.33.m33.2.2.2.2.1.cmml" xref="S4.SS1.3.p3.33.m33.2.2.2.1.2"></abs><apply id="S4.SS1.3.p3.33.m33.2.2.2.1.1.cmml" xref="S4.SS1.3.p3.33.m33.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.33.m33.2.2.2.1.1.1.cmml" xref="S4.SS1.3.p3.33.m33.2.2.2.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.33.m33.2.2.2.1.1.2.cmml" xref="S4.SS1.3.p3.33.m33.2.2.2.1.1.2">𝐶</ci><ci id="S4.SS1.3.p3.33.m33.2.2.2.1.1.3.cmml" xref="S4.SS1.3.p3.33.m33.2.2.2.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.33.m33.2c">\mathcal{SW}(\pi^{k})=|C_{k}|</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.33.m33.2d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) = | italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |</annotation></semantics></math> as the only nonsingleton coalition in <math alttext="\pi^{k}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.34.m34.1"><semantics id="S4.SS1.3.p3.34.m34.1a"><msup id="S4.SS1.3.p3.34.m34.1.1" xref="S4.SS1.3.p3.34.m34.1.1.cmml"><mi id="S4.SS1.3.p3.34.m34.1.1.2" xref="S4.SS1.3.p3.34.m34.1.1.2.cmml">π</mi><mi id="S4.SS1.3.p3.34.m34.1.1.3" xref="S4.SS1.3.p3.34.m34.1.1.3.cmml">k</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.34.m34.1b"><apply id="S4.SS1.3.p3.34.m34.1.1.cmml" xref="S4.SS1.3.p3.34.m34.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.34.m34.1.1.1.cmml" xref="S4.SS1.3.p3.34.m34.1.1">superscript</csymbol><ci id="S4.SS1.3.p3.34.m34.1.1.2.cmml" xref="S4.SS1.3.p3.34.m34.1.1.2">𝜋</ci><ci id="S4.SS1.3.p3.34.m34.1.1.3.cmml" xref="S4.SS1.3.p3.34.m34.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.34.m34.1c">\pi^{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.34.m34.1d">italic_π start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT</annotation></semantics></math> is <math alttext="\pi^{k}(z)" class="ltx_Math" display="inline" id="S4.SS1.3.p3.35.m35.1"><semantics id="S4.SS1.3.p3.35.m35.1a"><mrow id="S4.SS1.3.p3.35.m35.1.2" xref="S4.SS1.3.p3.35.m35.1.2.cmml"><msup id="S4.SS1.3.p3.35.m35.1.2.2" xref="S4.SS1.3.p3.35.m35.1.2.2.cmml"><mi id="S4.SS1.3.p3.35.m35.1.2.2.2" xref="S4.SS1.3.p3.35.m35.1.2.2.2.cmml">π</mi><mi id="S4.SS1.3.p3.35.m35.1.2.2.3" xref="S4.SS1.3.p3.35.m35.1.2.2.3.cmml">k</mi></msup><mo id="S4.SS1.3.p3.35.m35.1.2.1" xref="S4.SS1.3.p3.35.m35.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.3.p3.35.m35.1.2.3.2" xref="S4.SS1.3.p3.35.m35.1.2.cmml"><mo id="S4.SS1.3.p3.35.m35.1.2.3.2.1" stretchy="false" xref="S4.SS1.3.p3.35.m35.1.2.cmml">(</mo><mi id="S4.SS1.3.p3.35.m35.1.1" xref="S4.SS1.3.p3.35.m35.1.1.cmml">z</mi><mo id="S4.SS1.3.p3.35.m35.1.2.3.2.2" stretchy="false" xref="S4.SS1.3.p3.35.m35.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.35.m35.1b"><apply id="S4.SS1.3.p3.35.m35.1.2.cmml" xref="S4.SS1.3.p3.35.m35.1.2"><times id="S4.SS1.3.p3.35.m35.1.2.1.cmml" xref="S4.SS1.3.p3.35.m35.1.2.1"></times><apply id="S4.SS1.3.p3.35.m35.1.2.2.cmml" xref="S4.SS1.3.p3.35.m35.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.3.p3.35.m35.1.2.2.1.cmml" xref="S4.SS1.3.p3.35.m35.1.2.2">superscript</csymbol><ci id="S4.SS1.3.p3.35.m35.1.2.2.2.cmml" xref="S4.SS1.3.p3.35.m35.1.2.2.2">𝜋</ci><ci id="S4.SS1.3.p3.35.m35.1.2.2.3.cmml" xref="S4.SS1.3.p3.35.m35.1.2.2.3">𝑘</ci></apply><ci id="S4.SS1.3.p3.35.m35.1.1.cmml" xref="S4.SS1.3.p3.35.m35.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.35.m35.1c">\pi^{k}(z)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.35.m35.1d">italic_π start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_z )</annotation></semantics></math> which contains exactly <math alttext="|C_{k}|" class="ltx_Math" display="inline" id="S4.SS1.3.p3.36.m36.1"><semantics id="S4.SS1.3.p3.36.m36.1a"><mrow id="S4.SS1.3.p3.36.m36.1.1.1" xref="S4.SS1.3.p3.36.m36.1.1.2.cmml"><mo id="S4.SS1.3.p3.36.m36.1.1.1.2" stretchy="false" xref="S4.SS1.3.p3.36.m36.1.1.2.1.cmml">|</mo><msub id="S4.SS1.3.p3.36.m36.1.1.1.1" xref="S4.SS1.3.p3.36.m36.1.1.1.1.cmml"><mi id="S4.SS1.3.p3.36.m36.1.1.1.1.2" xref="S4.SS1.3.p3.36.m36.1.1.1.1.2.cmml">C</mi><mi id="S4.SS1.3.p3.36.m36.1.1.1.1.3" xref="S4.SS1.3.p3.36.m36.1.1.1.1.3.cmml">k</mi></msub><mo id="S4.SS1.3.p3.36.m36.1.1.1.3" stretchy="false" xref="S4.SS1.3.p3.36.m36.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.36.m36.1b"><apply id="S4.SS1.3.p3.36.m36.1.1.2.cmml" xref="S4.SS1.3.p3.36.m36.1.1.1"><abs id="S4.SS1.3.p3.36.m36.1.1.2.1.cmml" xref="S4.SS1.3.p3.36.m36.1.1.1.2"></abs><apply id="S4.SS1.3.p3.36.m36.1.1.1.1.cmml" xref="S4.SS1.3.p3.36.m36.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.3.p3.36.m36.1.1.1.1.1.cmml" xref="S4.SS1.3.p3.36.m36.1.1.1.1">subscript</csymbol><ci id="S4.SS1.3.p3.36.m36.1.1.1.1.2.cmml" xref="S4.SS1.3.p3.36.m36.1.1.1.1.2">𝐶</ci><ci id="S4.SS1.3.p3.36.m36.1.1.1.1.3.cmml" xref="S4.SS1.3.p3.36.m36.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.36.m36.1c">|C_{k}|</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.36.m36.1d">| italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |</annotation></semantics></math> other agents forming a clique in <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.3.p3.37.m37.1"><semantics id="S4.SS1.3.p3.37.m37.1a"><mi id="S4.SS1.3.p3.37.m37.1.1" xref="S4.SS1.3.p3.37.m37.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.37.m37.1b"><ci id="S4.SS1.3.p3.37.m37.1.1.cmml" xref="S4.SS1.3.p3.37.m37.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.37.m37.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.37.m37.1d">italic_G</annotation></semantics></math>. Hence,</p> <table class="ltx_equation ltx_eqn_table" id="S4.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{SW}(\pi)\leq\mathcal{SW}(\pi^{k})=|C_{k}|\text{.}" class="ltx_Math" display="block" id="S4.E1.m1.3"><semantics id="S4.E1.m1.3a"><mrow id="S4.E1.m1.3.3" xref="S4.E1.m1.3.3.cmml"><mrow id="S4.E1.m1.3.3.4" xref="S4.E1.m1.3.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E1.m1.3.3.4.2" xref="S4.E1.m1.3.3.4.2.cmml">𝒮</mi><mo id="S4.E1.m1.3.3.4.1" xref="S4.E1.m1.3.3.4.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.E1.m1.3.3.4.3" xref="S4.E1.m1.3.3.4.3.cmml">𝒲</mi><mo id="S4.E1.m1.3.3.4.1a" xref="S4.E1.m1.3.3.4.1.cmml">⁢</mo><mrow id="S4.E1.m1.3.3.4.4.2" xref="S4.E1.m1.3.3.4.cmml"><mo id="S4.E1.m1.3.3.4.4.2.1" stretchy="false" xref="S4.E1.m1.3.3.4.cmml">(</mo><mi id="S4.E1.m1.1.1" xref="S4.E1.m1.1.1.cmml">π</mi><mo id="S4.E1.m1.3.3.4.4.2.2" stretchy="false" xref="S4.E1.m1.3.3.4.cmml">)</mo></mrow></mrow><mo id="S4.E1.m1.3.3.5" xref="S4.E1.m1.3.3.5.cmml">≤</mo><mrow id="S4.E1.m1.2.2.1" xref="S4.E1.m1.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E1.m1.2.2.1.3" xref="S4.E1.m1.2.2.1.3.cmml">𝒮</mi><mo id="S4.E1.m1.2.2.1.2" xref="S4.E1.m1.2.2.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.E1.m1.2.2.1.4" xref="S4.E1.m1.2.2.1.4.cmml">𝒲</mi><mo id="S4.E1.m1.2.2.1.2a" xref="S4.E1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S4.E1.m1.2.2.1.1.1" xref="S4.E1.m1.2.2.1.1.1.1.cmml"><mo id="S4.E1.m1.2.2.1.1.1.2" stretchy="false" xref="S4.E1.m1.2.2.1.1.1.1.cmml">(</mo><msup id="S4.E1.m1.2.2.1.1.1.1" xref="S4.E1.m1.2.2.1.1.1.1.cmml"><mi id="S4.E1.m1.2.2.1.1.1.1.2" xref="S4.E1.m1.2.2.1.1.1.1.2.cmml">π</mi><mi id="S4.E1.m1.2.2.1.1.1.1.3" xref="S4.E1.m1.2.2.1.1.1.1.3.cmml">k</mi></msup><mo id="S4.E1.m1.2.2.1.1.1.3" stretchy="false" xref="S4.E1.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E1.m1.3.3.6" xref="S4.E1.m1.3.3.6.cmml">=</mo><mrow id="S4.E1.m1.3.3.2" xref="S4.E1.m1.3.3.2.cmml"><mrow id="S4.E1.m1.3.3.2.1.1" xref="S4.E1.m1.3.3.2.1.2.cmml"><mo id="S4.E1.m1.3.3.2.1.1.2" stretchy="false" xref="S4.E1.m1.3.3.2.1.2.1.cmml">|</mo><msub id="S4.E1.m1.3.3.2.1.1.1" xref="S4.E1.m1.3.3.2.1.1.1.cmml"><mi id="S4.E1.m1.3.3.2.1.1.1.2" xref="S4.E1.m1.3.3.2.1.1.1.2.cmml">C</mi><mi id="S4.E1.m1.3.3.2.1.1.1.3" xref="S4.E1.m1.3.3.2.1.1.1.3.cmml">k</mi></msub><mo id="S4.E1.m1.3.3.2.1.1.3" stretchy="false" xref="S4.E1.m1.3.3.2.1.2.1.cmml">|</mo></mrow><mo id="S4.E1.m1.3.3.2.2" xref="S4.E1.m1.3.3.2.2.cmml">⁢</mo><mtext id="S4.E1.m1.3.3.2.3" xref="S4.E1.m1.3.3.2.3a.cmml">.</mtext></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E1.m1.3b"><apply id="S4.E1.m1.3.3.cmml" xref="S4.E1.m1.3.3"><and id="S4.E1.m1.3.3a.cmml" xref="S4.E1.m1.3.3"></and><apply id="S4.E1.m1.3.3b.cmml" xref="S4.E1.m1.3.3"><leq id="S4.E1.m1.3.3.5.cmml" xref="S4.E1.m1.3.3.5"></leq><apply id="S4.E1.m1.3.3.4.cmml" xref="S4.E1.m1.3.3.4"><times id="S4.E1.m1.3.3.4.1.cmml" xref="S4.E1.m1.3.3.4.1"></times><ci id="S4.E1.m1.3.3.4.2.cmml" xref="S4.E1.m1.3.3.4.2">𝒮</ci><ci id="S4.E1.m1.3.3.4.3.cmml" xref="S4.E1.m1.3.3.4.3">𝒲</ci><ci id="S4.E1.m1.1.1.cmml" xref="S4.E1.m1.1.1">𝜋</ci></apply><apply id="S4.E1.m1.2.2.1.cmml" xref="S4.E1.m1.2.2.1"><times id="S4.E1.m1.2.2.1.2.cmml" xref="S4.E1.m1.2.2.1.2"></times><ci id="S4.E1.m1.2.2.1.3.cmml" xref="S4.E1.m1.2.2.1.3">𝒮</ci><ci id="S4.E1.m1.2.2.1.4.cmml" xref="S4.E1.m1.2.2.1.4">𝒲</ci><apply id="S4.E1.m1.2.2.1.1.1.1.cmml" xref="S4.E1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.E1.m1.2.2.1.1.1.1.1.cmml" xref="S4.E1.m1.2.2.1.1.1">superscript</csymbol><ci id="S4.E1.m1.2.2.1.1.1.1.2.cmml" xref="S4.E1.m1.2.2.1.1.1.1.2">𝜋</ci><ci id="S4.E1.m1.2.2.1.1.1.1.3.cmml" xref="S4.E1.m1.2.2.1.1.1.1.3">𝑘</ci></apply></apply></apply><apply id="S4.E1.m1.3.3c.cmml" xref="S4.E1.m1.3.3"><eq id="S4.E1.m1.3.3.6.cmml" xref="S4.E1.m1.3.3.6"></eq><share href="https://arxiv.org/html/2503.06017v1#S4.E1.m1.2.2.1.cmml" id="S4.E1.m1.3.3d.cmml" xref="S4.E1.m1.3.3"></share><apply id="S4.E1.m1.3.3.2.cmml" xref="S4.E1.m1.3.3.2"><times id="S4.E1.m1.3.3.2.2.cmml" xref="S4.E1.m1.3.3.2.2"></times><apply id="S4.E1.m1.3.3.2.1.2.cmml" xref="S4.E1.m1.3.3.2.1.1"><abs id="S4.E1.m1.3.3.2.1.2.1.cmml" xref="S4.E1.m1.3.3.2.1.1.2"></abs><apply id="S4.E1.m1.3.3.2.1.1.1.cmml" xref="S4.E1.m1.3.3.2.1.1.1"><csymbol cd="ambiguous" id="S4.E1.m1.3.3.2.1.1.1.1.cmml" xref="S4.E1.m1.3.3.2.1.1.1">subscript</csymbol><ci id="S4.E1.m1.3.3.2.1.1.1.2.cmml" xref="S4.E1.m1.3.3.2.1.1.1.2">𝐶</ci><ci id="S4.E1.m1.3.3.2.1.1.1.3.cmml" xref="S4.E1.m1.3.3.2.1.1.1.3">𝑘</ci></apply></apply><ci id="S4.E1.m1.3.3.2.3a.cmml" xref="S4.E1.m1.3.3.2.3"><mtext id="S4.E1.m1.3.3.2.3.cmml" xref="S4.E1.m1.3.3.2.3">.</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E1.m1.3c">\mathcal{SW}(\pi)\leq\mathcal{SW}(\pi^{k})=|C_{k}|\text{.}</annotation><annotation encoding="application/x-llamapun" id="S4.E1.m1.3d">caligraphic_S caligraphic_W ( italic_π ) ≤ caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) = | italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS1.4.p4"> <p class="ltx_p" id="S4.SS1.4.p4.6">Next, let <math alttext="C^{*}" class="ltx_Math" display="inline" id="S4.SS1.4.p4.1.m1.1"><semantics id="S4.SS1.4.p4.1.m1.1a"><msup id="S4.SS1.4.p4.1.m1.1.1" xref="S4.SS1.4.p4.1.m1.1.1.cmml"><mi id="S4.SS1.4.p4.1.m1.1.1.2" xref="S4.SS1.4.p4.1.m1.1.1.2.cmml">C</mi><mo id="S4.SS1.4.p4.1.m1.1.1.3" xref="S4.SS1.4.p4.1.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p4.1.m1.1b"><apply id="S4.SS1.4.p4.1.m1.1.1.cmml" xref="S4.SS1.4.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p4.1.m1.1.1.1.cmml" xref="S4.SS1.4.p4.1.m1.1.1">superscript</csymbol><ci id="S4.SS1.4.p4.1.m1.1.1.2.cmml" xref="S4.SS1.4.p4.1.m1.1.1.2">𝐶</ci><times id="S4.SS1.4.p4.1.m1.1.1.3.cmml" xref="S4.SS1.4.p4.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p4.1.m1.1c">C^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p4.1.m1.1d">italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be a maximum clique in <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.4.p4.2.m2.1"><semantics id="S4.SS1.4.p4.2.m2.1a"><mi id="S4.SS1.4.p4.2.m2.1.1" xref="S4.SS1.4.p4.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p4.2.m2.1b"><ci id="S4.SS1.4.p4.2.m2.1.1.cmml" xref="S4.SS1.4.p4.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p4.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p4.2.m2.1d">italic_G</annotation></semantics></math>. Consider <math alttext="\pi^{\prime}=\{\{z\}\cup\{a_{u}\colon u\in C^{*}\}\}\cup\{\{a_{u}\}\colon u\in V% \setminus C^{*}\}" class="ltx_Math" display="inline" id="S4.SS1.4.p4.3.m3.4"><semantics id="S4.SS1.4.p4.3.m3.4a"><mrow id="S4.SS1.4.p4.3.m3.4.4" xref="S4.SS1.4.p4.3.m3.4.4.cmml"><msup id="S4.SS1.4.p4.3.m3.4.4.5" xref="S4.SS1.4.p4.3.m3.4.4.5.cmml"><mi id="S4.SS1.4.p4.3.m3.4.4.5.2" xref="S4.SS1.4.p4.3.m3.4.4.5.2.cmml">π</mi><mo id="S4.SS1.4.p4.3.m3.4.4.5.3" xref="S4.SS1.4.p4.3.m3.4.4.5.3.cmml">′</mo></msup><mo id="S4.SS1.4.p4.3.m3.4.4.4" xref="S4.SS1.4.p4.3.m3.4.4.4.cmml">=</mo><mrow id="S4.SS1.4.p4.3.m3.4.4.3" xref="S4.SS1.4.p4.3.m3.4.4.3.cmml"><mrow id="S4.SS1.4.p4.3.m3.2.2.1.1.1" xref="S4.SS1.4.p4.3.m3.2.2.1.1.2.cmml"><mo id="S4.SS1.4.p4.3.m3.2.2.1.1.1.2" stretchy="false" xref="S4.SS1.4.p4.3.m3.2.2.1.1.2.cmml">{</mo><mrow id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.cmml"><mrow id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.4.2" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.4.1.cmml"><mo id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.4.2.1" stretchy="false" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.4.1.cmml">{</mo><mi id="S4.SS1.4.p4.3.m3.1.1" xref="S4.SS1.4.p4.3.m3.1.1.cmml">z</mi><mo id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.4.2.2" stretchy="false" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.4.1.cmml">}</mo></mrow><mo id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.3" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.3.cmml">∪</mo><mrow id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.3.cmml"><mo id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.3" stretchy="false" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.3.1.cmml">{</mo><msub id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.1.1.1" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.1.1.1.cmml"><mi id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.1.1.1.2" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.1.1.1.3" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.1.1.1.3.cmml">u</mi></msub><mo id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.3.1.cmml">:</mo><mrow id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.cmml"><mi id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.2" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.2.cmml">u</mi><mo id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.1" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.1.cmml">∈</mo><msup id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.3" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.3.cmml"><mi id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.3.2" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.3.2.cmml">C</mi><mo id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.3.3" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.3.3.cmml">∗</mo></msup></mrow><mo id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.5" stretchy="false" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.3.1.cmml">}</mo></mrow></mrow><mo id="S4.SS1.4.p4.3.m3.2.2.1.1.1.3" stretchy="false" xref="S4.SS1.4.p4.3.m3.2.2.1.1.2.cmml">}</mo></mrow><mo id="S4.SS1.4.p4.3.m3.4.4.3.4" xref="S4.SS1.4.p4.3.m3.4.4.3.4.cmml">∪</mo><mrow id="S4.SS1.4.p4.3.m3.4.4.3.3.2" xref="S4.SS1.4.p4.3.m3.4.4.3.3.3.cmml"><mo id="S4.SS1.4.p4.3.m3.4.4.3.3.2.3" stretchy="false" xref="S4.SS1.4.p4.3.m3.4.4.3.3.3.1.cmml">{</mo><mrow id="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1" xref="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.2.cmml"><mo id="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.2" stretchy="false" xref="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.2.cmml">{</mo><msub id="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.1" xref="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.1.cmml"><mi id="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.1.2" xref="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.1.2.cmml">a</mi><mi id="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.1.3" xref="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.1.3.cmml">u</mi></msub><mo id="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.3" rspace="0.278em" stretchy="false" xref="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.2.cmml">}</mo></mrow><mo id="S4.SS1.4.p4.3.m3.4.4.3.3.2.4" rspace="0.278em" xref="S4.SS1.4.p4.3.m3.4.4.3.3.3.1.cmml">:</mo><mrow id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.cmml"><mi id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.2" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.2.cmml">u</mi><mo id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.1" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.1.cmml">∈</mo><mrow id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.cmml"><mi id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.2" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.2.cmml">V</mi><mo id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.1" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.1.cmml">∖</mo><msup id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.3" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.3.cmml"><mi id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.3.2" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.3.2.cmml">C</mi><mo id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.3.3" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.3.3.cmml">∗</mo></msup></mrow></mrow><mo id="S4.SS1.4.p4.3.m3.4.4.3.3.2.5" stretchy="false" xref="S4.SS1.4.p4.3.m3.4.4.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p4.3.m3.4b"><apply id="S4.SS1.4.p4.3.m3.4.4.cmml" xref="S4.SS1.4.p4.3.m3.4.4"><eq id="S4.SS1.4.p4.3.m3.4.4.4.cmml" xref="S4.SS1.4.p4.3.m3.4.4.4"></eq><apply id="S4.SS1.4.p4.3.m3.4.4.5.cmml" xref="S4.SS1.4.p4.3.m3.4.4.5"><csymbol cd="ambiguous" id="S4.SS1.4.p4.3.m3.4.4.5.1.cmml" xref="S4.SS1.4.p4.3.m3.4.4.5">superscript</csymbol><ci id="S4.SS1.4.p4.3.m3.4.4.5.2.cmml" xref="S4.SS1.4.p4.3.m3.4.4.5.2">𝜋</ci><ci id="S4.SS1.4.p4.3.m3.4.4.5.3.cmml" xref="S4.SS1.4.p4.3.m3.4.4.5.3">′</ci></apply><apply id="S4.SS1.4.p4.3.m3.4.4.3.cmml" xref="S4.SS1.4.p4.3.m3.4.4.3"><union id="S4.SS1.4.p4.3.m3.4.4.3.4.cmml" xref="S4.SS1.4.p4.3.m3.4.4.3.4"></union><set id="S4.SS1.4.p4.3.m3.2.2.1.1.2.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1"><apply id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1"><union id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.3.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.3"></union><set id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.4.1.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.4.2"><ci id="S4.SS1.4.p4.3.m3.1.1.cmml" xref="S4.SS1.4.p4.3.m3.1.1">𝑧</ci></set><apply id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.3.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2"><csymbol cd="latexml" id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.3.1.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.3">conditional-set</csymbol><apply id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.1.1.1.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.1.1.1.2.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.1.1.1.2">𝑎</ci><ci id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.1.1.1.3.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.1.1.1.3">𝑢</ci></apply><apply id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2"><in id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.1.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.1"></in><ci id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.2.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.2">𝑢</ci><apply id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.3.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.3"><csymbol cd="ambiguous" id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.3.1.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.3">superscript</csymbol><ci id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.3.2.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.3.2">𝐶</ci><times id="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.3.3.cmml" xref="S4.SS1.4.p4.3.m3.2.2.1.1.1.1.2.2.2.3.3"></times></apply></apply></apply></apply></set><apply id="S4.SS1.4.p4.3.m3.4.4.3.3.3.cmml" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2"><csymbol cd="latexml" id="S4.SS1.4.p4.3.m3.4.4.3.3.3.1.cmml" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.3">conditional-set</csymbol><set id="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.2.cmml" xref="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1"><apply id="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.1.cmml" xref="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.1.1.cmml" xref="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.1">subscript</csymbol><ci id="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.1.2.cmml" xref="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.1.2">𝑎</ci><ci id="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.1.3.cmml" xref="S4.SS1.4.p4.3.m3.3.3.2.2.1.1.1.1.3">𝑢</ci></apply></set><apply id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.cmml" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2"><in id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.1.cmml" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.1"></in><ci id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.2.cmml" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.2">𝑢</ci><apply id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.cmml" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3"><setdiff id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.1.cmml" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.1"></setdiff><ci id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.2.cmml" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.2">𝑉</ci><apply id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.3.cmml" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.3"><csymbol cd="ambiguous" id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.3.1.cmml" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.3">superscript</csymbol><ci id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.3.2.cmml" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.3.2">𝐶</ci><times id="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.3.3.cmml" xref="S4.SS1.4.p4.3.m3.4.4.3.3.2.2.3.3.3"></times></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p4.3.m3.4c">\pi^{\prime}=\{\{z\}\cup\{a_{u}\colon u\in C^{*}\}\}\cup\{\{a_{u}\}\colon u\in V% \setminus C^{*}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p4.3.m3.4d">italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = { { italic_z } ∪ { italic_a start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT : italic_u ∈ italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT } } ∪ { { italic_a start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT } : italic_u ∈ italic_V ∖ italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT }</annotation></semantics></math> is a partition in <math alttext="(N,v)" class="ltx_Math" display="inline" id="S4.SS1.4.p4.4.m4.2"><semantics id="S4.SS1.4.p4.4.m4.2a"><mrow id="S4.SS1.4.p4.4.m4.2.3.2" xref="S4.SS1.4.p4.4.m4.2.3.1.cmml"><mo id="S4.SS1.4.p4.4.m4.2.3.2.1" stretchy="false" xref="S4.SS1.4.p4.4.m4.2.3.1.cmml">(</mo><mi id="S4.SS1.4.p4.4.m4.1.1" xref="S4.SS1.4.p4.4.m4.1.1.cmml">N</mi><mo id="S4.SS1.4.p4.4.m4.2.3.2.2" xref="S4.SS1.4.p4.4.m4.2.3.1.cmml">,</mo><mi id="S4.SS1.4.p4.4.m4.2.2" xref="S4.SS1.4.p4.4.m4.2.2.cmml">v</mi><mo id="S4.SS1.4.p4.4.m4.2.3.2.3" stretchy="false" xref="S4.SS1.4.p4.4.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p4.4.m4.2b"><interval closure="open" id="S4.SS1.4.p4.4.m4.2.3.1.cmml" xref="S4.SS1.4.p4.4.m4.2.3.2"><ci id="S4.SS1.4.p4.4.m4.1.1.cmml" xref="S4.SS1.4.p4.4.m4.1.1">𝑁</ci><ci id="S4.SS1.4.p4.4.m4.2.2.cmml" xref="S4.SS1.4.p4.4.m4.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p4.4.m4.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p4.4.m4.2d">( italic_N , italic_v )</annotation></semantics></math> with <math alttext="\mathcal{SW}(\pi^{\prime})=|C^{*}|" class="ltx_Math" display="inline" id="S4.SS1.4.p4.5.m5.2"><semantics id="S4.SS1.4.p4.5.m5.2a"><mrow id="S4.SS1.4.p4.5.m5.2.2" xref="S4.SS1.4.p4.5.m5.2.2.cmml"><mrow id="S4.SS1.4.p4.5.m5.1.1.1" xref="S4.SS1.4.p4.5.m5.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.4.p4.5.m5.1.1.1.3" xref="S4.SS1.4.p4.5.m5.1.1.1.3.cmml">𝒮</mi><mo id="S4.SS1.4.p4.5.m5.1.1.1.2" xref="S4.SS1.4.p4.5.m5.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.4.p4.5.m5.1.1.1.4" xref="S4.SS1.4.p4.5.m5.1.1.1.4.cmml">𝒲</mi><mo id="S4.SS1.4.p4.5.m5.1.1.1.2a" xref="S4.SS1.4.p4.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS1.4.p4.5.m5.1.1.1.1.1" xref="S4.SS1.4.p4.5.m5.1.1.1.1.1.1.cmml"><mo id="S4.SS1.4.p4.5.m5.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.4.p4.5.m5.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS1.4.p4.5.m5.1.1.1.1.1.1" xref="S4.SS1.4.p4.5.m5.1.1.1.1.1.1.cmml"><mi id="S4.SS1.4.p4.5.m5.1.1.1.1.1.1.2" xref="S4.SS1.4.p4.5.m5.1.1.1.1.1.1.2.cmml">π</mi><mo id="S4.SS1.4.p4.5.m5.1.1.1.1.1.1.3" xref="S4.SS1.4.p4.5.m5.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.SS1.4.p4.5.m5.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.4.p4.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.4.p4.5.m5.2.2.3" xref="S4.SS1.4.p4.5.m5.2.2.3.cmml">=</mo><mrow id="S4.SS1.4.p4.5.m5.2.2.2.1" xref="S4.SS1.4.p4.5.m5.2.2.2.2.cmml"><mo id="S4.SS1.4.p4.5.m5.2.2.2.1.2" stretchy="false" xref="S4.SS1.4.p4.5.m5.2.2.2.2.1.cmml">|</mo><msup id="S4.SS1.4.p4.5.m5.2.2.2.1.1" xref="S4.SS1.4.p4.5.m5.2.2.2.1.1.cmml"><mi id="S4.SS1.4.p4.5.m5.2.2.2.1.1.2" xref="S4.SS1.4.p4.5.m5.2.2.2.1.1.2.cmml">C</mi><mo id="S4.SS1.4.p4.5.m5.2.2.2.1.1.3" xref="S4.SS1.4.p4.5.m5.2.2.2.1.1.3.cmml">∗</mo></msup><mo id="S4.SS1.4.p4.5.m5.2.2.2.1.3" stretchy="false" xref="S4.SS1.4.p4.5.m5.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p4.5.m5.2b"><apply id="S4.SS1.4.p4.5.m5.2.2.cmml" xref="S4.SS1.4.p4.5.m5.2.2"><eq id="S4.SS1.4.p4.5.m5.2.2.3.cmml" xref="S4.SS1.4.p4.5.m5.2.2.3"></eq><apply id="S4.SS1.4.p4.5.m5.1.1.1.cmml" xref="S4.SS1.4.p4.5.m5.1.1.1"><times id="S4.SS1.4.p4.5.m5.1.1.1.2.cmml" xref="S4.SS1.4.p4.5.m5.1.1.1.2"></times><ci id="S4.SS1.4.p4.5.m5.1.1.1.3.cmml" xref="S4.SS1.4.p4.5.m5.1.1.1.3">𝒮</ci><ci id="S4.SS1.4.p4.5.m5.1.1.1.4.cmml" xref="S4.SS1.4.p4.5.m5.1.1.1.4">𝒲</ci><apply id="S4.SS1.4.p4.5.m5.1.1.1.1.1.1.cmml" xref="S4.SS1.4.p4.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p4.5.m5.1.1.1.1.1.1.1.cmml" xref="S4.SS1.4.p4.5.m5.1.1.1.1.1">superscript</csymbol><ci id="S4.SS1.4.p4.5.m5.1.1.1.1.1.1.2.cmml" xref="S4.SS1.4.p4.5.m5.1.1.1.1.1.1.2">𝜋</ci><ci id="S4.SS1.4.p4.5.m5.1.1.1.1.1.1.3.cmml" xref="S4.SS1.4.p4.5.m5.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="S4.SS1.4.p4.5.m5.2.2.2.2.cmml" xref="S4.SS1.4.p4.5.m5.2.2.2.1"><abs id="S4.SS1.4.p4.5.m5.2.2.2.2.1.cmml" xref="S4.SS1.4.p4.5.m5.2.2.2.1.2"></abs><apply id="S4.SS1.4.p4.5.m5.2.2.2.1.1.cmml" xref="S4.SS1.4.p4.5.m5.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p4.5.m5.2.2.2.1.1.1.cmml" xref="S4.SS1.4.p4.5.m5.2.2.2.1.1">superscript</csymbol><ci id="S4.SS1.4.p4.5.m5.2.2.2.1.1.2.cmml" xref="S4.SS1.4.p4.5.m5.2.2.2.1.1.2">𝐶</ci><times id="S4.SS1.4.p4.5.m5.2.2.2.1.1.3.cmml" xref="S4.SS1.4.p4.5.m5.2.2.2.1.1.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p4.5.m5.2c">\mathcal{SW}(\pi^{\prime})=|C^{*}|</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p4.5.m5.2d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = | italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT |</annotation></semantics></math>. Hence, for the partition <math alttext="\pi^{*}" class="ltx_Math" display="inline" id="S4.SS1.4.p4.6.m6.1"><semantics id="S4.SS1.4.p4.6.m6.1a"><msup id="S4.SS1.4.p4.6.m6.1.1" xref="S4.SS1.4.p4.6.m6.1.1.cmml"><mi id="S4.SS1.4.p4.6.m6.1.1.2" xref="S4.SS1.4.p4.6.m6.1.1.2.cmml">π</mi><mo id="S4.SS1.4.p4.6.m6.1.1.3" xref="S4.SS1.4.p4.6.m6.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p4.6.m6.1b"><apply id="S4.SS1.4.p4.6.m6.1.1.cmml" xref="S4.SS1.4.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p4.6.m6.1.1.1.cmml" xref="S4.SS1.4.p4.6.m6.1.1">superscript</csymbol><ci id="S4.SS1.4.p4.6.m6.1.1.2.cmml" xref="S4.SS1.4.p4.6.m6.1.1.2">𝜋</ci><times id="S4.SS1.4.p4.6.m6.1.1.3.cmml" xref="S4.SS1.4.p4.6.m6.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p4.6.m6.1c">\pi^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p4.6.m6.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> maximizing welfare, it holds that</p> <table class="ltx_equation ltx_eqn_table" id="S4.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{SW}(\pi^{*})\geq\mathcal{SW}(\pi^{\prime})=|C^{*}|\text{.}" class="ltx_Math" display="block" id="S4.E2.m1.3"><semantics id="S4.E2.m1.3a"><mrow id="S4.E2.m1.3.3" xref="S4.E2.m1.3.3.cmml"><mrow id="S4.E2.m1.1.1.1" xref="S4.E2.m1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E2.m1.1.1.1.3" xref="S4.E2.m1.1.1.1.3.cmml">𝒮</mi><mo id="S4.E2.m1.1.1.1.2" xref="S4.E2.m1.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.E2.m1.1.1.1.4" xref="S4.E2.m1.1.1.1.4.cmml">𝒲</mi><mo id="S4.E2.m1.1.1.1.2a" xref="S4.E2.m1.1.1.1.2.cmml">⁢</mo><mrow id="S4.E2.m1.1.1.1.1.1" xref="S4.E2.m1.1.1.1.1.1.1.cmml"><mo id="S4.E2.m1.1.1.1.1.1.2" stretchy="false" xref="S4.E2.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.E2.m1.1.1.1.1.1.1" xref="S4.E2.m1.1.1.1.1.1.1.cmml"><mi id="S4.E2.m1.1.1.1.1.1.1.2" xref="S4.E2.m1.1.1.1.1.1.1.2.cmml">π</mi><mo id="S4.E2.m1.1.1.1.1.1.1.3" xref="S4.E2.m1.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.E2.m1.1.1.1.1.1.3" stretchy="false" xref="S4.E2.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E2.m1.3.3.5" xref="S4.E2.m1.3.3.5.cmml">≥</mo><mrow id="S4.E2.m1.2.2.2" xref="S4.E2.m1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E2.m1.2.2.2.3" xref="S4.E2.m1.2.2.2.3.cmml">𝒮</mi><mo id="S4.E2.m1.2.2.2.2" xref="S4.E2.m1.2.2.2.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.E2.m1.2.2.2.4" xref="S4.E2.m1.2.2.2.4.cmml">𝒲</mi><mo id="S4.E2.m1.2.2.2.2a" xref="S4.E2.m1.2.2.2.2.cmml">⁢</mo><mrow id="S4.E2.m1.2.2.2.1.1" xref="S4.E2.m1.2.2.2.1.1.1.cmml"><mo id="S4.E2.m1.2.2.2.1.1.2" stretchy="false" xref="S4.E2.m1.2.2.2.1.1.1.cmml">(</mo><msup id="S4.E2.m1.2.2.2.1.1.1" xref="S4.E2.m1.2.2.2.1.1.1.cmml"><mi id="S4.E2.m1.2.2.2.1.1.1.2" xref="S4.E2.m1.2.2.2.1.1.1.2.cmml">π</mi><mo id="S4.E2.m1.2.2.2.1.1.1.3" xref="S4.E2.m1.2.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S4.E2.m1.2.2.2.1.1.3" stretchy="false" xref="S4.E2.m1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E2.m1.3.3.6" xref="S4.E2.m1.3.3.6.cmml">=</mo><mrow id="S4.E2.m1.3.3.3" xref="S4.E2.m1.3.3.3.cmml"><mrow id="S4.E2.m1.3.3.3.1.1" xref="S4.E2.m1.3.3.3.1.2.cmml"><mo id="S4.E2.m1.3.3.3.1.1.2" stretchy="false" xref="S4.E2.m1.3.3.3.1.2.1.cmml">|</mo><msup id="S4.E2.m1.3.3.3.1.1.1" xref="S4.E2.m1.3.3.3.1.1.1.cmml"><mi id="S4.E2.m1.3.3.3.1.1.1.2" xref="S4.E2.m1.3.3.3.1.1.1.2.cmml">C</mi><mo id="S4.E2.m1.3.3.3.1.1.1.3" xref="S4.E2.m1.3.3.3.1.1.1.3.cmml">∗</mo></msup><mo id="S4.E2.m1.3.3.3.1.1.3" stretchy="false" xref="S4.E2.m1.3.3.3.1.2.1.cmml">|</mo></mrow><mo id="S4.E2.m1.3.3.3.2" xref="S4.E2.m1.3.3.3.2.cmml">⁢</mo><mtext id="S4.E2.m1.3.3.3.3" xref="S4.E2.m1.3.3.3.3a.cmml">.</mtext></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E2.m1.3b"><apply id="S4.E2.m1.3.3.cmml" xref="S4.E2.m1.3.3"><and id="S4.E2.m1.3.3a.cmml" xref="S4.E2.m1.3.3"></and><apply id="S4.E2.m1.3.3b.cmml" xref="S4.E2.m1.3.3"><geq id="S4.E2.m1.3.3.5.cmml" xref="S4.E2.m1.3.3.5"></geq><apply id="S4.E2.m1.1.1.1.cmml" xref="S4.E2.m1.1.1.1"><times id="S4.E2.m1.1.1.1.2.cmml" xref="S4.E2.m1.1.1.1.2"></times><ci id="S4.E2.m1.1.1.1.3.cmml" xref="S4.E2.m1.1.1.1.3">𝒮</ci><ci id="S4.E2.m1.1.1.1.4.cmml" xref="S4.E2.m1.1.1.1.4">𝒲</ci><apply id="S4.E2.m1.1.1.1.1.1.1.cmml" xref="S4.E2.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E2.m1.1.1.1.1.1.1.1.cmml" xref="S4.E2.m1.1.1.1.1.1">superscript</csymbol><ci id="S4.E2.m1.1.1.1.1.1.1.2.cmml" xref="S4.E2.m1.1.1.1.1.1.1.2">𝜋</ci><times id="S4.E2.m1.1.1.1.1.1.1.3.cmml" xref="S4.E2.m1.1.1.1.1.1.1.3"></times></apply></apply><apply id="S4.E2.m1.2.2.2.cmml" xref="S4.E2.m1.2.2.2"><times id="S4.E2.m1.2.2.2.2.cmml" xref="S4.E2.m1.2.2.2.2"></times><ci id="S4.E2.m1.2.2.2.3.cmml" xref="S4.E2.m1.2.2.2.3">𝒮</ci><ci id="S4.E2.m1.2.2.2.4.cmml" xref="S4.E2.m1.2.2.2.4">𝒲</ci><apply id="S4.E2.m1.2.2.2.1.1.1.cmml" xref="S4.E2.m1.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.E2.m1.2.2.2.1.1.1.1.cmml" xref="S4.E2.m1.2.2.2.1.1">superscript</csymbol><ci id="S4.E2.m1.2.2.2.1.1.1.2.cmml" xref="S4.E2.m1.2.2.2.1.1.1.2">𝜋</ci><ci id="S4.E2.m1.2.2.2.1.1.1.3.cmml" xref="S4.E2.m1.2.2.2.1.1.1.3">′</ci></apply></apply></apply><apply id="S4.E2.m1.3.3c.cmml" xref="S4.E2.m1.3.3"><eq id="S4.E2.m1.3.3.6.cmml" xref="S4.E2.m1.3.3.6"></eq><share href="https://arxiv.org/html/2503.06017v1#S4.E2.m1.2.2.2.cmml" id="S4.E2.m1.3.3d.cmml" xref="S4.E2.m1.3.3"></share><apply id="S4.E2.m1.3.3.3.cmml" xref="S4.E2.m1.3.3.3"><times id="S4.E2.m1.3.3.3.2.cmml" xref="S4.E2.m1.3.3.3.2"></times><apply id="S4.E2.m1.3.3.3.1.2.cmml" xref="S4.E2.m1.3.3.3.1.1"><abs id="S4.E2.m1.3.3.3.1.2.1.cmml" xref="S4.E2.m1.3.3.3.1.1.2"></abs><apply id="S4.E2.m1.3.3.3.1.1.1.cmml" xref="S4.E2.m1.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.E2.m1.3.3.3.1.1.1.1.cmml" xref="S4.E2.m1.3.3.3.1.1.1">superscript</csymbol><ci id="S4.E2.m1.3.3.3.1.1.1.2.cmml" xref="S4.E2.m1.3.3.3.1.1.1.2">𝐶</ci><times id="S4.E2.m1.3.3.3.1.1.1.3.cmml" xref="S4.E2.m1.3.3.3.1.1.1.3"></times></apply></apply><ci id="S4.E2.m1.3.3.3.3a.cmml" xref="S4.E2.m1.3.3.3.3"><mtext id="S4.E2.m1.3.3.3.3.cmml" xref="S4.E2.m1.3.3.3.3">.</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E2.m1.3c">\mathcal{SW}(\pi^{*})\geq\mathcal{SW}(\pi^{\prime})=|C^{*}|\text{.}</annotation><annotation encoding="application/x-llamapun" id="S4.E2.m1.3d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ≥ caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = | italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS1.5.p5"> <p class="ltx_p" id="S4.SS1.5.p5.3">Now assume that we have a polynomial-time algorithm computing a partition <math alttext="\pi" class="ltx_Math" display="inline" id="S4.SS1.5.p5.1.m1.1"><semantics id="S4.SS1.5.p5.1.m1.1a"><mi id="S4.SS1.5.p5.1.m1.1.1" xref="S4.SS1.5.p5.1.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p5.1.m1.1b"><ci id="S4.SS1.5.p5.1.m1.1.1.cmml" xref="S4.SS1.5.p5.1.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p5.1.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p5.1.m1.1d">italic_π</annotation></semantics></math> with <math alttext="n^{1-\varepsilon}\cdot\mathcal{SW}(\pi)\geq\mathcal{SW}(\pi^{*})" class="ltx_Math" display="inline" id="S4.SS1.5.p5.2.m2.2"><semantics id="S4.SS1.5.p5.2.m2.2a"><mrow id="S4.SS1.5.p5.2.m2.2.2" xref="S4.SS1.5.p5.2.m2.2.2.cmml"><mrow id="S4.SS1.5.p5.2.m2.2.2.3" xref="S4.SS1.5.p5.2.m2.2.2.3.cmml"><mrow id="S4.SS1.5.p5.2.m2.2.2.3.2" xref="S4.SS1.5.p5.2.m2.2.2.3.2.cmml"><msup id="S4.SS1.5.p5.2.m2.2.2.3.2.2" xref="S4.SS1.5.p5.2.m2.2.2.3.2.2.cmml"><mi id="S4.SS1.5.p5.2.m2.2.2.3.2.2.2" xref="S4.SS1.5.p5.2.m2.2.2.3.2.2.2.cmml">n</mi><mrow id="S4.SS1.5.p5.2.m2.2.2.3.2.2.3" xref="S4.SS1.5.p5.2.m2.2.2.3.2.2.3.cmml"><mn id="S4.SS1.5.p5.2.m2.2.2.3.2.2.3.2" xref="S4.SS1.5.p5.2.m2.2.2.3.2.2.3.2.cmml">1</mn><mo id="S4.SS1.5.p5.2.m2.2.2.3.2.2.3.1" xref="S4.SS1.5.p5.2.m2.2.2.3.2.2.3.1.cmml">−</mo><mi id="S4.SS1.5.p5.2.m2.2.2.3.2.2.3.3" xref="S4.SS1.5.p5.2.m2.2.2.3.2.2.3.3.cmml">ε</mi></mrow></msup><mo id="S4.SS1.5.p5.2.m2.2.2.3.2.1" lspace="0.222em" rspace="0.222em" xref="S4.SS1.5.p5.2.m2.2.2.3.2.1.cmml">⋅</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.5.p5.2.m2.2.2.3.2.3" xref="S4.SS1.5.p5.2.m2.2.2.3.2.3.cmml">𝒮</mi></mrow><mo id="S4.SS1.5.p5.2.m2.2.2.3.1" xref="S4.SS1.5.p5.2.m2.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.5.p5.2.m2.2.2.3.3" xref="S4.SS1.5.p5.2.m2.2.2.3.3.cmml">𝒲</mi><mo id="S4.SS1.5.p5.2.m2.2.2.3.1a" xref="S4.SS1.5.p5.2.m2.2.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.5.p5.2.m2.2.2.3.4.2" xref="S4.SS1.5.p5.2.m2.2.2.3.cmml"><mo id="S4.SS1.5.p5.2.m2.2.2.3.4.2.1" stretchy="false" xref="S4.SS1.5.p5.2.m2.2.2.3.cmml">(</mo><mi id="S4.SS1.5.p5.2.m2.1.1" xref="S4.SS1.5.p5.2.m2.1.1.cmml">π</mi><mo id="S4.SS1.5.p5.2.m2.2.2.3.4.2.2" stretchy="false" xref="S4.SS1.5.p5.2.m2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.SS1.5.p5.2.m2.2.2.2" xref="S4.SS1.5.p5.2.m2.2.2.2.cmml">≥</mo><mrow id="S4.SS1.5.p5.2.m2.2.2.1" xref="S4.SS1.5.p5.2.m2.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.5.p5.2.m2.2.2.1.3" xref="S4.SS1.5.p5.2.m2.2.2.1.3.cmml">𝒮</mi><mo id="S4.SS1.5.p5.2.m2.2.2.1.2" xref="S4.SS1.5.p5.2.m2.2.2.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.5.p5.2.m2.2.2.1.4" xref="S4.SS1.5.p5.2.m2.2.2.1.4.cmml">𝒲</mi><mo id="S4.SS1.5.p5.2.m2.2.2.1.2a" xref="S4.SS1.5.p5.2.m2.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS1.5.p5.2.m2.2.2.1.1.1" xref="S4.SS1.5.p5.2.m2.2.2.1.1.1.1.cmml"><mo id="S4.SS1.5.p5.2.m2.2.2.1.1.1.2" stretchy="false" xref="S4.SS1.5.p5.2.m2.2.2.1.1.1.1.cmml">(</mo><msup id="S4.SS1.5.p5.2.m2.2.2.1.1.1.1" xref="S4.SS1.5.p5.2.m2.2.2.1.1.1.1.cmml"><mi id="S4.SS1.5.p5.2.m2.2.2.1.1.1.1.2" xref="S4.SS1.5.p5.2.m2.2.2.1.1.1.1.2.cmml">π</mi><mo id="S4.SS1.5.p5.2.m2.2.2.1.1.1.1.3" xref="S4.SS1.5.p5.2.m2.2.2.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.SS1.5.p5.2.m2.2.2.1.1.1.3" stretchy="false" xref="S4.SS1.5.p5.2.m2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p5.2.m2.2b"><apply id="S4.SS1.5.p5.2.m2.2.2.cmml" xref="S4.SS1.5.p5.2.m2.2.2"><geq id="S4.SS1.5.p5.2.m2.2.2.2.cmml" xref="S4.SS1.5.p5.2.m2.2.2.2"></geq><apply id="S4.SS1.5.p5.2.m2.2.2.3.cmml" xref="S4.SS1.5.p5.2.m2.2.2.3"><times id="S4.SS1.5.p5.2.m2.2.2.3.1.cmml" xref="S4.SS1.5.p5.2.m2.2.2.3.1"></times><apply id="S4.SS1.5.p5.2.m2.2.2.3.2.cmml" xref="S4.SS1.5.p5.2.m2.2.2.3.2"><ci id="S4.SS1.5.p5.2.m2.2.2.3.2.1.cmml" xref="S4.SS1.5.p5.2.m2.2.2.3.2.1">⋅</ci><apply id="S4.SS1.5.p5.2.m2.2.2.3.2.2.cmml" xref="S4.SS1.5.p5.2.m2.2.2.3.2.2"><csymbol cd="ambiguous" id="S4.SS1.5.p5.2.m2.2.2.3.2.2.1.cmml" xref="S4.SS1.5.p5.2.m2.2.2.3.2.2">superscript</csymbol><ci id="S4.SS1.5.p5.2.m2.2.2.3.2.2.2.cmml" xref="S4.SS1.5.p5.2.m2.2.2.3.2.2.2">𝑛</ci><apply id="S4.SS1.5.p5.2.m2.2.2.3.2.2.3.cmml" xref="S4.SS1.5.p5.2.m2.2.2.3.2.2.3"><minus id="S4.SS1.5.p5.2.m2.2.2.3.2.2.3.1.cmml" xref="S4.SS1.5.p5.2.m2.2.2.3.2.2.3.1"></minus><cn id="S4.SS1.5.p5.2.m2.2.2.3.2.2.3.2.cmml" type="integer" xref="S4.SS1.5.p5.2.m2.2.2.3.2.2.3.2">1</cn><ci id="S4.SS1.5.p5.2.m2.2.2.3.2.2.3.3.cmml" xref="S4.SS1.5.p5.2.m2.2.2.3.2.2.3.3">𝜀</ci></apply></apply><ci id="S4.SS1.5.p5.2.m2.2.2.3.2.3.cmml" xref="S4.SS1.5.p5.2.m2.2.2.3.2.3">𝒮</ci></apply><ci id="S4.SS1.5.p5.2.m2.2.2.3.3.cmml" xref="S4.SS1.5.p5.2.m2.2.2.3.3">𝒲</ci><ci id="S4.SS1.5.p5.2.m2.1.1.cmml" xref="S4.SS1.5.p5.2.m2.1.1">𝜋</ci></apply><apply id="S4.SS1.5.p5.2.m2.2.2.1.cmml" xref="S4.SS1.5.p5.2.m2.2.2.1"><times id="S4.SS1.5.p5.2.m2.2.2.1.2.cmml" xref="S4.SS1.5.p5.2.m2.2.2.1.2"></times><ci id="S4.SS1.5.p5.2.m2.2.2.1.3.cmml" xref="S4.SS1.5.p5.2.m2.2.2.1.3">𝒮</ci><ci id="S4.SS1.5.p5.2.m2.2.2.1.4.cmml" xref="S4.SS1.5.p5.2.m2.2.2.1.4">𝒲</ci><apply id="S4.SS1.5.p5.2.m2.2.2.1.1.1.1.cmml" xref="S4.SS1.5.p5.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.5.p5.2.m2.2.2.1.1.1.1.1.cmml" xref="S4.SS1.5.p5.2.m2.2.2.1.1.1">superscript</csymbol><ci id="S4.SS1.5.p5.2.m2.2.2.1.1.1.1.2.cmml" xref="S4.SS1.5.p5.2.m2.2.2.1.1.1.1.2">𝜋</ci><times id="S4.SS1.5.p5.2.m2.2.2.1.1.1.1.3.cmml" xref="S4.SS1.5.p5.2.m2.2.2.1.1.1.1.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p5.2.m2.2c">n^{1-\varepsilon}\cdot\mathcal{SW}(\pi)\geq\mathcal{SW}(\pi^{*})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p5.2.m2.2d">italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT ⋅ caligraphic_S caligraphic_W ( italic_π ) ≥ caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math>. Clearly, the procedure described above to construct <math alttext="C_{k}" class="ltx_Math" display="inline" id="S4.SS1.5.p5.3.m3.1"><semantics id="S4.SS1.5.p5.3.m3.1a"><msub id="S4.SS1.5.p5.3.m3.1.1" xref="S4.SS1.5.p5.3.m3.1.1.cmml"><mi id="S4.SS1.5.p5.3.m3.1.1.2" xref="S4.SS1.5.p5.3.m3.1.1.2.cmml">C</mi><mi id="S4.SS1.5.p5.3.m3.1.1.3" xref="S4.SS1.5.p5.3.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p5.3.m3.1b"><apply id="S4.SS1.5.p5.3.m3.1.1.cmml" xref="S4.SS1.5.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.5.p5.3.m3.1.1.1.cmml" xref="S4.SS1.5.p5.3.m3.1.1">subscript</csymbol><ci id="S4.SS1.5.p5.3.m3.1.1.2.cmml" xref="S4.SS1.5.p5.3.m3.1.1.2">𝐶</ci><ci id="S4.SS1.5.p5.3.m3.1.1.3.cmml" xref="S4.SS1.5.p5.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p5.3.m3.1c">C_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p5.3.m3.1d">italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> runs in polynomial time as well. It holds that</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="n^{1-\varepsilon}\cdot|C_{k}|\overset{\textnormal{Eq.~{}(\ref{eq:ReducedClique% })}}{\geq}n^{1-\varepsilon}\cdot\mathcal{SW}(\pi)\geq\mathcal{SW}(\pi^{*})% \overset{\textnormal{Eq.~{}(\ref{eq:MaxClique})}}{\geq}|C^{*}|\text{.}" class="ltx_Math" display="block" id="S4.Ex5.m1.4"><semantics id="S4.Ex5.m1.4a"><mrow id="S4.Ex5.m1.4.4" xref="S4.Ex5.m1.4.4.cmml"><mrow id="S4.Ex5.m1.2.2.1" xref="S4.Ex5.m1.2.2.1.cmml"><mrow id="S4.Ex5.m1.2.2.1.1" xref="S4.Ex5.m1.2.2.1.1.cmml"><mrow id="S4.Ex5.m1.2.2.1.1.1" xref="S4.Ex5.m1.2.2.1.1.1.cmml"><mrow id="S4.Ex5.m1.2.2.1.1.1.1" xref="S4.Ex5.m1.2.2.1.1.1.1.cmml"><msup id="S4.Ex5.m1.2.2.1.1.1.1.3" xref="S4.Ex5.m1.2.2.1.1.1.1.3.cmml"><mi id="S4.Ex5.m1.2.2.1.1.1.1.3.2" xref="S4.Ex5.m1.2.2.1.1.1.1.3.2.cmml">n</mi><mrow id="S4.Ex5.m1.2.2.1.1.1.1.3.3" xref="S4.Ex5.m1.2.2.1.1.1.1.3.3.cmml"><mn id="S4.Ex5.m1.2.2.1.1.1.1.3.3.2" xref="S4.Ex5.m1.2.2.1.1.1.1.3.3.2.cmml">1</mn><mo id="S4.Ex5.m1.2.2.1.1.1.1.3.3.1" xref="S4.Ex5.m1.2.2.1.1.1.1.3.3.1.cmml">−</mo><mi id="S4.Ex5.m1.2.2.1.1.1.1.3.3.3" xref="S4.Ex5.m1.2.2.1.1.1.1.3.3.3.cmml">ε</mi></mrow></msup><mo id="S4.Ex5.m1.2.2.1.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S4.Ex5.m1.2.2.1.1.1.1.2.cmml">⋅</mo><mrow id="S4.Ex5.m1.2.2.1.1.1.1.1.1" xref="S4.Ex5.m1.2.2.1.1.1.1.1.2.cmml"><mo id="S4.Ex5.m1.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex5.m1.2.2.1.1.1.1.1.2.1.cmml">|</mo><msub id="S4.Ex5.m1.2.2.1.1.1.1.1.1.1" xref="S4.Ex5.m1.2.2.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex5.m1.2.2.1.1.1.1.1.1.1.2" xref="S4.Ex5.m1.2.2.1.1.1.1.1.1.1.2.cmml">C</mi><mi id="S4.Ex5.m1.2.2.1.1.1.1.1.1.1.3" xref="S4.Ex5.m1.2.2.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S4.Ex5.m1.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex5.m1.2.2.1.1.1.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="S4.Ex5.m1.2.2.1.1.1.2" xref="S4.Ex5.m1.2.2.1.1.1.2.cmml">⁢</mo><mover accent="true" id="S4.Ex5.m1.2.2.1.1.1.3" xref="S4.Ex5.m1.2.2.1.1.1.3.cmml"><mo id="S4.Ex5.m1.2.2.1.1.1.3.2" xref="S4.Ex5.m1.2.2.1.1.1.3.2.cmml">≥</mo><mrow id="S4.Ex5.m1.2.2.1.1.1.3.1" xref="S4.Ex5.m1.2.2.1.1.1.3.1f.cmml"><mtext id="S4.Ex5.m1.2.2.1.1.1.3.1a" xref="S4.Ex5.m1.2.2.1.1.1.3.1f.cmml">Eq. (</mtext><mtext id="S4.Ex5.m1.2.2.1.1.1.3.1b" xref="S4.Ex5.m1.2.2.1.1.1.3.1f.cmml"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.E1" title="Equation 1 ‣ Proof. ‣ 4.1 Welfare Inapproximability for Restricted Valuations ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">1</span></a></mtext><mtext id="S4.Ex5.m1.2.2.1.1.1.3.1e" xref="S4.Ex5.m1.2.2.1.1.1.3.1f.cmml">)</mtext></mrow></mover><mo id="S4.Ex5.m1.2.2.1.1.1.2a" xref="S4.Ex5.m1.2.2.1.1.1.2.cmml">⁢</mo><msup id="S4.Ex5.m1.2.2.1.1.1.4" xref="S4.Ex5.m1.2.2.1.1.1.4.cmml"><mi id="S4.Ex5.m1.2.2.1.1.1.4.2" xref="S4.Ex5.m1.2.2.1.1.1.4.2.cmml">n</mi><mrow id="S4.Ex5.m1.2.2.1.1.1.4.3" xref="S4.Ex5.m1.2.2.1.1.1.4.3.cmml"><mn id="S4.Ex5.m1.2.2.1.1.1.4.3.2" xref="S4.Ex5.m1.2.2.1.1.1.4.3.2.cmml">1</mn><mo id="S4.Ex5.m1.2.2.1.1.1.4.3.1" xref="S4.Ex5.m1.2.2.1.1.1.4.3.1.cmml">−</mo><mi id="S4.Ex5.m1.2.2.1.1.1.4.3.3" xref="S4.Ex5.m1.2.2.1.1.1.4.3.3.cmml">ε</mi></mrow></msup></mrow><mo id="S4.Ex5.m1.2.2.1.1.2" lspace="0.222em" rspace="0.222em" xref="S4.Ex5.m1.2.2.1.1.2.cmml">⋅</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex5.m1.2.2.1.1.3" xref="S4.Ex5.m1.2.2.1.1.3.cmml">𝒮</mi></mrow><mo id="S4.Ex5.m1.2.2.1.2" xref="S4.Ex5.m1.2.2.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex5.m1.2.2.1.3" xref="S4.Ex5.m1.2.2.1.3.cmml">𝒲</mi><mo id="S4.Ex5.m1.2.2.1.2a" xref="S4.Ex5.m1.2.2.1.2.cmml">⁢</mo><mrow id="S4.Ex5.m1.2.2.1.4.2" xref="S4.Ex5.m1.2.2.1.cmml"><mo id="S4.Ex5.m1.2.2.1.4.2.1" stretchy="false" xref="S4.Ex5.m1.2.2.1.cmml">(</mo><mi id="S4.Ex5.m1.1.1" xref="S4.Ex5.m1.1.1.cmml">π</mi><mo id="S4.Ex5.m1.2.2.1.4.2.2" stretchy="false" xref="S4.Ex5.m1.2.2.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex5.m1.4.4.4" xref="S4.Ex5.m1.4.4.4.cmml">≥</mo><mrow id="S4.Ex5.m1.4.4.3" xref="S4.Ex5.m1.4.4.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex5.m1.4.4.3.4" xref="S4.Ex5.m1.4.4.3.4.cmml">𝒮</mi><mo id="S4.Ex5.m1.4.4.3.3" xref="S4.Ex5.m1.4.4.3.3.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex5.m1.4.4.3.5" xref="S4.Ex5.m1.4.4.3.5.cmml">𝒲</mi><mo id="S4.Ex5.m1.4.4.3.3a" xref="S4.Ex5.m1.4.4.3.3.cmml">⁢</mo><mrow id="S4.Ex5.m1.3.3.2.1.1" xref="S4.Ex5.m1.3.3.2.1.1.1.cmml"><mo id="S4.Ex5.m1.3.3.2.1.1.2" stretchy="false" xref="S4.Ex5.m1.3.3.2.1.1.1.cmml">(</mo><msup id="S4.Ex5.m1.3.3.2.1.1.1" xref="S4.Ex5.m1.3.3.2.1.1.1.cmml"><mi id="S4.Ex5.m1.3.3.2.1.1.1.2" xref="S4.Ex5.m1.3.3.2.1.1.1.2.cmml">π</mi><mo id="S4.Ex5.m1.3.3.2.1.1.1.3" xref="S4.Ex5.m1.3.3.2.1.1.1.3.cmml">∗</mo></msup><mo id="S4.Ex5.m1.3.3.2.1.1.3" stretchy="false" xref="S4.Ex5.m1.3.3.2.1.1.1.cmml">)</mo></mrow><mo id="S4.Ex5.m1.4.4.3.3b" xref="S4.Ex5.m1.4.4.3.3.cmml">⁢</mo><mover accent="true" id="S4.Ex5.m1.4.4.3.6" xref="S4.Ex5.m1.4.4.3.6.cmml"><mo id="S4.Ex5.m1.4.4.3.6.2" xref="S4.Ex5.m1.4.4.3.6.2.cmml">≥</mo><mrow id="S4.Ex5.m1.4.4.3.6.1" xref="S4.Ex5.m1.4.4.3.6.1f.cmml"><mtext id="S4.Ex5.m1.4.4.3.6.1a" xref="S4.Ex5.m1.4.4.3.6.1f.cmml">Eq. (</mtext><mtext id="S4.Ex5.m1.4.4.3.6.1b" xref="S4.Ex5.m1.4.4.3.6.1f.cmml"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.E2" title="Equation 2 ‣ Proof. ‣ 4.1 Welfare Inapproximability for Restricted Valuations ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">2</span></a></mtext><mtext id="S4.Ex5.m1.4.4.3.6.1e" xref="S4.Ex5.m1.4.4.3.6.1f.cmml">)</mtext></mrow></mover><mo id="S4.Ex5.m1.4.4.3.3c" xref="S4.Ex5.m1.4.4.3.3.cmml">⁢</mo><mrow id="S4.Ex5.m1.4.4.3.2.1" xref="S4.Ex5.m1.4.4.3.2.2.cmml"><mo id="S4.Ex5.m1.4.4.3.2.1.2" stretchy="false" xref="S4.Ex5.m1.4.4.3.2.2.1.cmml">|</mo><msup id="S4.Ex5.m1.4.4.3.2.1.1" xref="S4.Ex5.m1.4.4.3.2.1.1.cmml"><mi id="S4.Ex5.m1.4.4.3.2.1.1.2" xref="S4.Ex5.m1.4.4.3.2.1.1.2.cmml">C</mi><mo id="S4.Ex5.m1.4.4.3.2.1.1.3" xref="S4.Ex5.m1.4.4.3.2.1.1.3.cmml">∗</mo></msup><mo id="S4.Ex5.m1.4.4.3.2.1.3" stretchy="false" xref="S4.Ex5.m1.4.4.3.2.2.1.cmml">|</mo></mrow><mo id="S4.Ex5.m1.4.4.3.3d" xref="S4.Ex5.m1.4.4.3.3.cmml">⁢</mo><mtext id="S4.Ex5.m1.4.4.3.7" xref="S4.Ex5.m1.4.4.3.7a.cmml">.</mtext></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex5.m1.4b"><apply id="S4.Ex5.m1.4.4.cmml" xref="S4.Ex5.m1.4.4"><geq id="S4.Ex5.m1.4.4.4.cmml" xref="S4.Ex5.m1.4.4.4"></geq><apply id="S4.Ex5.m1.2.2.1.cmml" xref="S4.Ex5.m1.2.2.1"><times id="S4.Ex5.m1.2.2.1.2.cmml" xref="S4.Ex5.m1.2.2.1.2"></times><apply id="S4.Ex5.m1.2.2.1.1.cmml" xref="S4.Ex5.m1.2.2.1.1"><ci id="S4.Ex5.m1.2.2.1.1.2.cmml" xref="S4.Ex5.m1.2.2.1.1.2">⋅</ci><apply id="S4.Ex5.m1.2.2.1.1.1.cmml" xref="S4.Ex5.m1.2.2.1.1.1"><times id="S4.Ex5.m1.2.2.1.1.1.2.cmml" xref="S4.Ex5.m1.2.2.1.1.1.2"></times><apply id="S4.Ex5.m1.2.2.1.1.1.1.cmml" xref="S4.Ex5.m1.2.2.1.1.1.1"><ci id="S4.Ex5.m1.2.2.1.1.1.1.2.cmml" xref="S4.Ex5.m1.2.2.1.1.1.1.2">⋅</ci><apply id="S4.Ex5.m1.2.2.1.1.1.1.3.cmml" xref="S4.Ex5.m1.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Ex5.m1.2.2.1.1.1.1.3.1.cmml" xref="S4.Ex5.m1.2.2.1.1.1.1.3">superscript</csymbol><ci id="S4.Ex5.m1.2.2.1.1.1.1.3.2.cmml" xref="S4.Ex5.m1.2.2.1.1.1.1.3.2">𝑛</ci><apply id="S4.Ex5.m1.2.2.1.1.1.1.3.3.cmml" xref="S4.Ex5.m1.2.2.1.1.1.1.3.3"><minus id="S4.Ex5.m1.2.2.1.1.1.1.3.3.1.cmml" xref="S4.Ex5.m1.2.2.1.1.1.1.3.3.1"></minus><cn id="S4.Ex5.m1.2.2.1.1.1.1.3.3.2.cmml" type="integer" xref="S4.Ex5.m1.2.2.1.1.1.1.3.3.2">1</cn><ci id="S4.Ex5.m1.2.2.1.1.1.1.3.3.3.cmml" xref="S4.Ex5.m1.2.2.1.1.1.1.3.3.3">𝜀</ci></apply></apply><apply id="S4.Ex5.m1.2.2.1.1.1.1.1.2.cmml" xref="S4.Ex5.m1.2.2.1.1.1.1.1.1"><abs id="S4.Ex5.m1.2.2.1.1.1.1.1.2.1.cmml" xref="S4.Ex5.m1.2.2.1.1.1.1.1.1.2"></abs><apply id="S4.Ex5.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S4.Ex5.m1.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex5.m1.2.2.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex5.m1.2.2.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex5.m1.2.2.1.1.1.1.1.1.1.2.cmml" xref="S4.Ex5.m1.2.2.1.1.1.1.1.1.1.2">𝐶</ci><ci id="S4.Ex5.m1.2.2.1.1.1.1.1.1.1.3.cmml" xref="S4.Ex5.m1.2.2.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply><apply id="S4.Ex5.m1.2.2.1.1.1.3.cmml" xref="S4.Ex5.m1.2.2.1.1.1.3"><ci id="S4.Ex5.m1.2.2.1.1.1.3.1f.cmml" xref="S4.Ex5.m1.2.2.1.1.1.3.1"><mrow id="S4.Ex5.m1.2.2.1.1.1.3.1.cmml" xref="S4.Ex5.m1.2.2.1.1.1.3.1"><mtext id="S4.Ex5.m1.2.2.1.1.1.3.1a.cmml" xref="S4.Ex5.m1.2.2.1.1.1.3.1">Eq. (</mtext><mtext id="S4.Ex5.m1.2.2.1.1.1.3.1b.cmml" xref="S4.Ex5.m1.2.2.1.1.1.3.1"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.E1" title="Equation 1 ‣ Proof. ‣ 4.1 Welfare Inapproximability for Restricted Valuations ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">1</span></a></mtext><mtext id="S4.Ex5.m1.2.2.1.1.1.3.1e.cmml" xref="S4.Ex5.m1.2.2.1.1.1.3.1">)</mtext></mrow></ci><geq id="S4.Ex5.m1.2.2.1.1.1.3.2.cmml" xref="S4.Ex5.m1.2.2.1.1.1.3.2"></geq></apply><apply id="S4.Ex5.m1.2.2.1.1.1.4.cmml" xref="S4.Ex5.m1.2.2.1.1.1.4"><csymbol cd="ambiguous" id="S4.Ex5.m1.2.2.1.1.1.4.1.cmml" xref="S4.Ex5.m1.2.2.1.1.1.4">superscript</csymbol><ci id="S4.Ex5.m1.2.2.1.1.1.4.2.cmml" xref="S4.Ex5.m1.2.2.1.1.1.4.2">𝑛</ci><apply id="S4.Ex5.m1.2.2.1.1.1.4.3.cmml" xref="S4.Ex5.m1.2.2.1.1.1.4.3"><minus id="S4.Ex5.m1.2.2.1.1.1.4.3.1.cmml" xref="S4.Ex5.m1.2.2.1.1.1.4.3.1"></minus><cn id="S4.Ex5.m1.2.2.1.1.1.4.3.2.cmml" type="integer" xref="S4.Ex5.m1.2.2.1.1.1.4.3.2">1</cn><ci id="S4.Ex5.m1.2.2.1.1.1.4.3.3.cmml" xref="S4.Ex5.m1.2.2.1.1.1.4.3.3">𝜀</ci></apply></apply></apply><ci id="S4.Ex5.m1.2.2.1.1.3.cmml" xref="S4.Ex5.m1.2.2.1.1.3">𝒮</ci></apply><ci id="S4.Ex5.m1.2.2.1.3.cmml" xref="S4.Ex5.m1.2.2.1.3">𝒲</ci><ci id="S4.Ex5.m1.1.1.cmml" xref="S4.Ex5.m1.1.1">𝜋</ci></apply><apply id="S4.Ex5.m1.4.4.3.cmml" xref="S4.Ex5.m1.4.4.3"><times id="S4.Ex5.m1.4.4.3.3.cmml" xref="S4.Ex5.m1.4.4.3.3"></times><ci id="S4.Ex5.m1.4.4.3.4.cmml" xref="S4.Ex5.m1.4.4.3.4">𝒮</ci><ci id="S4.Ex5.m1.4.4.3.5.cmml" xref="S4.Ex5.m1.4.4.3.5">𝒲</ci><apply id="S4.Ex5.m1.3.3.2.1.1.1.cmml" xref="S4.Ex5.m1.3.3.2.1.1"><csymbol cd="ambiguous" id="S4.Ex5.m1.3.3.2.1.1.1.1.cmml" xref="S4.Ex5.m1.3.3.2.1.1">superscript</csymbol><ci id="S4.Ex5.m1.3.3.2.1.1.1.2.cmml" xref="S4.Ex5.m1.3.3.2.1.1.1.2">𝜋</ci><times id="S4.Ex5.m1.3.3.2.1.1.1.3.cmml" xref="S4.Ex5.m1.3.3.2.1.1.1.3"></times></apply><apply id="S4.Ex5.m1.4.4.3.6.cmml" xref="S4.Ex5.m1.4.4.3.6"><ci id="S4.Ex5.m1.4.4.3.6.1f.cmml" xref="S4.Ex5.m1.4.4.3.6.1"><mrow id="S4.Ex5.m1.4.4.3.6.1.cmml" xref="S4.Ex5.m1.4.4.3.6.1"><mtext id="S4.Ex5.m1.4.4.3.6.1a.cmml" xref="S4.Ex5.m1.4.4.3.6.1">Eq. (</mtext><mtext id="S4.Ex5.m1.4.4.3.6.1b.cmml" xref="S4.Ex5.m1.4.4.3.6.1"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.E2" title="Equation 2 ‣ Proof. ‣ 4.1 Welfare Inapproximability for Restricted Valuations ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">2</span></a></mtext><mtext id="S4.Ex5.m1.4.4.3.6.1e.cmml" xref="S4.Ex5.m1.4.4.3.6.1">)</mtext></mrow></ci><geq id="S4.Ex5.m1.4.4.3.6.2.cmml" xref="S4.Ex5.m1.4.4.3.6.2"></geq></apply><apply id="S4.Ex5.m1.4.4.3.2.2.cmml" xref="S4.Ex5.m1.4.4.3.2.1"><abs id="S4.Ex5.m1.4.4.3.2.2.1.cmml" xref="S4.Ex5.m1.4.4.3.2.1.2"></abs><apply id="S4.Ex5.m1.4.4.3.2.1.1.cmml" xref="S4.Ex5.m1.4.4.3.2.1.1"><csymbol cd="ambiguous" id="S4.Ex5.m1.4.4.3.2.1.1.1.cmml" xref="S4.Ex5.m1.4.4.3.2.1.1">superscript</csymbol><ci id="S4.Ex5.m1.4.4.3.2.1.1.2.cmml" xref="S4.Ex5.m1.4.4.3.2.1.1.2">𝐶</ci><times id="S4.Ex5.m1.4.4.3.2.1.1.3.cmml" xref="S4.Ex5.m1.4.4.3.2.1.1.3"></times></apply></apply><ci id="S4.Ex5.m1.4.4.3.7a.cmml" xref="S4.Ex5.m1.4.4.3.7"><mtext id="S4.Ex5.m1.4.4.3.7.cmml" xref="S4.Ex5.m1.4.4.3.7">.</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex5.m1.4c">n^{1-\varepsilon}\cdot|C_{k}|\overset{\textnormal{Eq.~{}(\ref{eq:ReducedClique% })}}{\geq}n^{1-\varepsilon}\cdot\mathcal{SW}(\pi)\geq\mathcal{SW}(\pi^{*})% \overset{\textnormal{Eq.~{}(\ref{eq:MaxClique})}}{\geq}|C^{*}|\text{.}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex5.m1.4d">italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT ⋅ | italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | overEq. () start_ARG ≥ end_ARG italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT ⋅ caligraphic_S caligraphic_W ( italic_π ) ≥ caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) overEq. () start_ARG ≥ end_ARG | italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS1.6.p6"> <p class="ltx_p" id="S4.SS1.6.p6.2">Hence, we have found a polynomial-time algorithm to approximate the maximum clique within a factor of <math alttext="n^{1-\varepsilon}" class="ltx_Math" display="inline" id="S4.SS1.6.p6.1.m1.1"><semantics id="S4.SS1.6.p6.1.m1.1a"><msup id="S4.SS1.6.p6.1.m1.1.1" xref="S4.SS1.6.p6.1.m1.1.1.cmml"><mi id="S4.SS1.6.p6.1.m1.1.1.2" xref="S4.SS1.6.p6.1.m1.1.1.2.cmml">n</mi><mrow id="S4.SS1.6.p6.1.m1.1.1.3" xref="S4.SS1.6.p6.1.m1.1.1.3.cmml"><mn id="S4.SS1.6.p6.1.m1.1.1.3.2" xref="S4.SS1.6.p6.1.m1.1.1.3.2.cmml">1</mn><mo id="S4.SS1.6.p6.1.m1.1.1.3.1" xref="S4.SS1.6.p6.1.m1.1.1.3.1.cmml">−</mo><mi id="S4.SS1.6.p6.1.m1.1.1.3.3" xref="S4.SS1.6.p6.1.m1.1.1.3.3.cmml">ε</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p6.1.m1.1b"><apply id="S4.SS1.6.p6.1.m1.1.1.cmml" xref="S4.SS1.6.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.6.p6.1.m1.1.1.1.cmml" xref="S4.SS1.6.p6.1.m1.1.1">superscript</csymbol><ci id="S4.SS1.6.p6.1.m1.1.1.2.cmml" xref="S4.SS1.6.p6.1.m1.1.1.2">𝑛</ci><apply id="S4.SS1.6.p6.1.m1.1.1.3.cmml" xref="S4.SS1.6.p6.1.m1.1.1.3"><minus id="S4.SS1.6.p6.1.m1.1.1.3.1.cmml" xref="S4.SS1.6.p6.1.m1.1.1.3.1"></minus><cn id="S4.SS1.6.p6.1.m1.1.1.3.2.cmml" type="integer" xref="S4.SS1.6.p6.1.m1.1.1.3.2">1</cn><ci id="S4.SS1.6.p6.1.m1.1.1.3.3.cmml" xref="S4.SS1.6.p6.1.m1.1.1.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p6.1.m1.1c">n^{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p6.1.m1.1d">italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT</annotation></semantics></math>. As argued in the beginning, this can only happen if ¶ <math alttext="=" class="ltx_Math" display="inline" id="S4.SS1.6.p6.2.m2.1"><semantics id="S4.SS1.6.p6.2.m2.1a"><mo id="S4.SS1.6.p6.2.m2.1.1" xref="S4.SS1.6.p6.2.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p6.2.m2.1b"><eq id="S4.SS1.6.p6.2.m2.1.1.cmml" xref="S4.SS1.6.p6.2.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p6.2.m2.1c">=</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p6.2.m2.1d">=</annotation></semantics></math> <span class="ltx_ERROR undefined" id="S4.SS1.6.p6.2.1">\NP</span>, completing our proof. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.9">Notably, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1 Welfare Inapproximability for Restricted Valuations ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.1</span></a> immediately implies hardness of <math alttext="n^{1-\varepsilon}" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><msup id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml"><mi id="S4.SS1.p2.1.m1.1.1.2" xref="S4.SS1.p2.1.m1.1.1.2.cmml">n</mi><mrow id="S4.SS1.p2.1.m1.1.1.3" xref="S4.SS1.p2.1.m1.1.1.3.cmml"><mn id="S4.SS1.p2.1.m1.1.1.3.2" xref="S4.SS1.p2.1.m1.1.1.3.2.cmml">1</mn><mo id="S4.SS1.p2.1.m1.1.1.3.1" xref="S4.SS1.p2.1.m1.1.1.3.1.cmml">−</mo><mi id="S4.SS1.p2.1.m1.1.1.3.3" xref="S4.SS1.p2.1.m1.1.1.3.3.cmml">ε</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><apply id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.1.m1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">superscript</csymbol><ci id="S4.SS1.p2.1.m1.1.1.2.cmml" xref="S4.SS1.p2.1.m1.1.1.2">𝑛</ci><apply id="S4.SS1.p2.1.m1.1.1.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3"><minus id="S4.SS1.p2.1.m1.1.1.3.1.cmml" xref="S4.SS1.p2.1.m1.1.1.3.1"></minus><cn id="S4.SS1.p2.1.m1.1.1.3.2.cmml" type="integer" xref="S4.SS1.p2.1.m1.1.1.3.2">1</cn><ci id="S4.SS1.p2.1.m1.1.1.3.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">n^{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">italic_n start_POSTSUPERSCRIPT 1 - italic_ε end_POSTSUPERSCRIPT</annotation></semantics></math>-<span class="ltx_text ltx_font_smallcaps" id="S4.SS1.p2.9.1">ApproxWelfare</span> for ASHGs with nonsymmetric valuations restricted to <math alttext="\{-1,1\}" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.2"><semantics id="S4.SS1.p2.2.m2.2a"><mrow id="S4.SS1.p2.2.m2.2.2.1" xref="S4.SS1.p2.2.m2.2.2.2.cmml"><mo id="S4.SS1.p2.2.m2.2.2.1.2" stretchy="false" xref="S4.SS1.p2.2.m2.2.2.2.cmml">{</mo><mrow id="S4.SS1.p2.2.m2.2.2.1.1" xref="S4.SS1.p2.2.m2.2.2.1.1.cmml"><mo id="S4.SS1.p2.2.m2.2.2.1.1a" xref="S4.SS1.p2.2.m2.2.2.1.1.cmml">−</mo><mn id="S4.SS1.p2.2.m2.2.2.1.1.2" xref="S4.SS1.p2.2.m2.2.2.1.1.2.cmml">1</mn></mrow><mo id="S4.SS1.p2.2.m2.2.2.1.3" xref="S4.SS1.p2.2.m2.2.2.2.cmml">,</mo><mn id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml">1</mn><mo id="S4.SS1.p2.2.m2.2.2.1.4" stretchy="false" xref="S4.SS1.p2.2.m2.2.2.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.2b"><set id="S4.SS1.p2.2.m2.2.2.2.cmml" xref="S4.SS1.p2.2.m2.2.2.1"><apply id="S4.SS1.p2.2.m2.2.2.1.1.cmml" xref="S4.SS1.p2.2.m2.2.2.1.1"><minus id="S4.SS1.p2.2.m2.2.2.1.1.1.cmml" xref="S4.SS1.p2.2.m2.2.2.1.1"></minus><cn id="S4.SS1.p2.2.m2.2.2.1.1.2.cmml" type="integer" xref="S4.SS1.p2.2.m2.2.2.1.1.2">1</cn></apply><cn id="S4.SS1.p2.2.m2.1.1.cmml" type="integer" xref="S4.SS1.p2.2.m2.1.1">1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.2c">\{-1,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.2d">{ - 1 , 1 }</annotation></semantics></math>. We can simply replace valuations <math alttext="v(i,j)=0" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.2"><semantics id="S4.SS1.p2.3.m3.2a"><mrow id="S4.SS1.p2.3.m3.2.3" xref="S4.SS1.p2.3.m3.2.3.cmml"><mrow id="S4.SS1.p2.3.m3.2.3.2" xref="S4.SS1.p2.3.m3.2.3.2.cmml"><mi id="S4.SS1.p2.3.m3.2.3.2.2" xref="S4.SS1.p2.3.m3.2.3.2.2.cmml">v</mi><mo id="S4.SS1.p2.3.m3.2.3.2.1" xref="S4.SS1.p2.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS1.p2.3.m3.2.3.2.3.2" xref="S4.SS1.p2.3.m3.2.3.2.3.1.cmml"><mo id="S4.SS1.p2.3.m3.2.3.2.3.2.1" stretchy="false" xref="S4.SS1.p2.3.m3.2.3.2.3.1.cmml">(</mo><mi id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml">i</mi><mo id="S4.SS1.p2.3.m3.2.3.2.3.2.2" xref="S4.SS1.p2.3.m3.2.3.2.3.1.cmml">,</mo><mi id="S4.SS1.p2.3.m3.2.2" xref="S4.SS1.p2.3.m3.2.2.cmml">j</mi><mo id="S4.SS1.p2.3.m3.2.3.2.3.2.3" stretchy="false" xref="S4.SS1.p2.3.m3.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p2.3.m3.2.3.1" xref="S4.SS1.p2.3.m3.2.3.1.cmml">=</mo><mn id="S4.SS1.p2.3.m3.2.3.3" xref="S4.SS1.p2.3.m3.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.2b"><apply id="S4.SS1.p2.3.m3.2.3.cmml" xref="S4.SS1.p2.3.m3.2.3"><eq id="S4.SS1.p2.3.m3.2.3.1.cmml" xref="S4.SS1.p2.3.m3.2.3.1"></eq><apply id="S4.SS1.p2.3.m3.2.3.2.cmml" xref="S4.SS1.p2.3.m3.2.3.2"><times id="S4.SS1.p2.3.m3.2.3.2.1.cmml" xref="S4.SS1.p2.3.m3.2.3.2.1"></times><ci id="S4.SS1.p2.3.m3.2.3.2.2.cmml" xref="S4.SS1.p2.3.m3.2.3.2.2">𝑣</ci><interval closure="open" id="S4.SS1.p2.3.m3.2.3.2.3.1.cmml" xref="S4.SS1.p2.3.m3.2.3.2.3.2"><ci id="S4.SS1.p2.3.m3.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1">𝑖</ci><ci id="S4.SS1.p2.3.m3.2.2.cmml" xref="S4.SS1.p2.3.m3.2.2">𝑗</ci></interval></apply><cn id="S4.SS1.p2.3.m3.2.3.3.cmml" type="integer" xref="S4.SS1.p2.3.m3.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.2c">v(i,j)=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.2d">italic_v ( italic_i , italic_j ) = 0</annotation></semantics></math> by <math alttext="v_{i}(j)=1" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.1"><semantics id="S4.SS1.p2.4.m4.1a"><mrow id="S4.SS1.p2.4.m4.1.2" xref="S4.SS1.p2.4.m4.1.2.cmml"><mrow id="S4.SS1.p2.4.m4.1.2.2" xref="S4.SS1.p2.4.m4.1.2.2.cmml"><msub id="S4.SS1.p2.4.m4.1.2.2.2" xref="S4.SS1.p2.4.m4.1.2.2.2.cmml"><mi id="S4.SS1.p2.4.m4.1.2.2.2.2" xref="S4.SS1.p2.4.m4.1.2.2.2.2.cmml">v</mi><mi id="S4.SS1.p2.4.m4.1.2.2.2.3" xref="S4.SS1.p2.4.m4.1.2.2.2.3.cmml">i</mi></msub><mo id="S4.SS1.p2.4.m4.1.2.2.1" xref="S4.SS1.p2.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS1.p2.4.m4.1.2.2.3.2" xref="S4.SS1.p2.4.m4.1.2.2.cmml"><mo id="S4.SS1.p2.4.m4.1.2.2.3.2.1" stretchy="false" xref="S4.SS1.p2.4.m4.1.2.2.cmml">(</mo><mi id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml">j</mi><mo id="S4.SS1.p2.4.m4.1.2.2.3.2.2" stretchy="false" xref="S4.SS1.p2.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p2.4.m4.1.2.1" xref="S4.SS1.p2.4.m4.1.2.1.cmml">=</mo><mn id="S4.SS1.p2.4.m4.1.2.3" xref="S4.SS1.p2.4.m4.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.1b"><apply id="S4.SS1.p2.4.m4.1.2.cmml" xref="S4.SS1.p2.4.m4.1.2"><eq id="S4.SS1.p2.4.m4.1.2.1.cmml" xref="S4.SS1.p2.4.m4.1.2.1"></eq><apply id="S4.SS1.p2.4.m4.1.2.2.cmml" xref="S4.SS1.p2.4.m4.1.2.2"><times id="S4.SS1.p2.4.m4.1.2.2.1.cmml" xref="S4.SS1.p2.4.m4.1.2.2.1"></times><apply id="S4.SS1.p2.4.m4.1.2.2.2.cmml" xref="S4.SS1.p2.4.m4.1.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.1.2.2.2.1.cmml" xref="S4.SS1.p2.4.m4.1.2.2.2">subscript</csymbol><ci id="S4.SS1.p2.4.m4.1.2.2.2.2.cmml" xref="S4.SS1.p2.4.m4.1.2.2.2.2">𝑣</ci><ci id="S4.SS1.p2.4.m4.1.2.2.2.3.cmml" xref="S4.SS1.p2.4.m4.1.2.2.2.3">𝑖</ci></apply><ci id="S4.SS1.p2.4.m4.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1">𝑗</ci></apply><cn id="S4.SS1.p2.4.m4.1.2.3.cmml" type="integer" xref="S4.SS1.p2.4.m4.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.1c">v_{i}(j)=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.1d">italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j ) = 1</annotation></semantics></math> and <math alttext="v_{j}(i)=-1" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m5.1"><semantics id="S4.SS1.p2.5.m5.1a"><mrow id="S4.SS1.p2.5.m5.1.2" xref="S4.SS1.p2.5.m5.1.2.cmml"><mrow id="S4.SS1.p2.5.m5.1.2.2" xref="S4.SS1.p2.5.m5.1.2.2.cmml"><msub id="S4.SS1.p2.5.m5.1.2.2.2" xref="S4.SS1.p2.5.m5.1.2.2.2.cmml"><mi id="S4.SS1.p2.5.m5.1.2.2.2.2" xref="S4.SS1.p2.5.m5.1.2.2.2.2.cmml">v</mi><mi id="S4.SS1.p2.5.m5.1.2.2.2.3" xref="S4.SS1.p2.5.m5.1.2.2.2.3.cmml">j</mi></msub><mo id="S4.SS1.p2.5.m5.1.2.2.1" xref="S4.SS1.p2.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS1.p2.5.m5.1.2.2.3.2" xref="S4.SS1.p2.5.m5.1.2.2.cmml"><mo id="S4.SS1.p2.5.m5.1.2.2.3.2.1" stretchy="false" xref="S4.SS1.p2.5.m5.1.2.2.cmml">(</mo><mi id="S4.SS1.p2.5.m5.1.1" xref="S4.SS1.p2.5.m5.1.1.cmml">i</mi><mo id="S4.SS1.p2.5.m5.1.2.2.3.2.2" stretchy="false" xref="S4.SS1.p2.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p2.5.m5.1.2.1" xref="S4.SS1.p2.5.m5.1.2.1.cmml">=</mo><mrow id="S4.SS1.p2.5.m5.1.2.3" xref="S4.SS1.p2.5.m5.1.2.3.cmml"><mo id="S4.SS1.p2.5.m5.1.2.3a" xref="S4.SS1.p2.5.m5.1.2.3.cmml">−</mo><mn id="S4.SS1.p2.5.m5.1.2.3.2" xref="S4.SS1.p2.5.m5.1.2.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m5.1b"><apply id="S4.SS1.p2.5.m5.1.2.cmml" xref="S4.SS1.p2.5.m5.1.2"><eq id="S4.SS1.p2.5.m5.1.2.1.cmml" xref="S4.SS1.p2.5.m5.1.2.1"></eq><apply id="S4.SS1.p2.5.m5.1.2.2.cmml" xref="S4.SS1.p2.5.m5.1.2.2"><times id="S4.SS1.p2.5.m5.1.2.2.1.cmml" xref="S4.SS1.p2.5.m5.1.2.2.1"></times><apply id="S4.SS1.p2.5.m5.1.2.2.2.cmml" xref="S4.SS1.p2.5.m5.1.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p2.5.m5.1.2.2.2.1.cmml" xref="S4.SS1.p2.5.m5.1.2.2.2">subscript</csymbol><ci id="S4.SS1.p2.5.m5.1.2.2.2.2.cmml" xref="S4.SS1.p2.5.m5.1.2.2.2.2">𝑣</ci><ci id="S4.SS1.p2.5.m5.1.2.2.2.3.cmml" xref="S4.SS1.p2.5.m5.1.2.2.2.3">𝑗</ci></apply><ci id="S4.SS1.p2.5.m5.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1">𝑖</ci></apply><apply id="S4.SS1.p2.5.m5.1.2.3.cmml" xref="S4.SS1.p2.5.m5.1.2.3"><minus id="S4.SS1.p2.5.m5.1.2.3.1.cmml" xref="S4.SS1.p2.5.m5.1.2.3"></minus><cn id="S4.SS1.p2.5.m5.1.2.3.2.cmml" type="integer" xref="S4.SS1.p2.5.m5.1.2.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m5.1c">v_{j}(i)=-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m5.1d">italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_i ) = - 1</annotation></semantics></math> and obtain a reduced instance in which all partitions have the identical welfare. However, it remains an open problem to resolve the complexity of <math alttext="c" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m6.1"><semantics id="S4.SS1.p2.6.m6.1a"><mi id="S4.SS1.p2.6.m6.1.1" xref="S4.SS1.p2.6.m6.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m6.1b"><ci id="S4.SS1.p2.6.m6.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.6.m6.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m6.1d">italic_c</annotation></semantics></math>-<span class="ltx_text ltx_font_smallcaps" id="S4.SS1.p2.9.2">ApproxWelfare</span> for symmetric ASHGs with valuations restricted to <math alttext="\{-1,1\}" class="ltx_Math" display="inline" id="S4.SS1.p2.7.m7.2"><semantics id="S4.SS1.p2.7.m7.2a"><mrow id="S4.SS1.p2.7.m7.2.2.1" xref="S4.SS1.p2.7.m7.2.2.2.cmml"><mo id="S4.SS1.p2.7.m7.2.2.1.2" stretchy="false" xref="S4.SS1.p2.7.m7.2.2.2.cmml">{</mo><mrow id="S4.SS1.p2.7.m7.2.2.1.1" xref="S4.SS1.p2.7.m7.2.2.1.1.cmml"><mo id="S4.SS1.p2.7.m7.2.2.1.1a" xref="S4.SS1.p2.7.m7.2.2.1.1.cmml">−</mo><mn id="S4.SS1.p2.7.m7.2.2.1.1.2" xref="S4.SS1.p2.7.m7.2.2.1.1.2.cmml">1</mn></mrow><mo id="S4.SS1.p2.7.m7.2.2.1.3" xref="S4.SS1.p2.7.m7.2.2.2.cmml">,</mo><mn id="S4.SS1.p2.7.m7.1.1" xref="S4.SS1.p2.7.m7.1.1.cmml">1</mn><mo id="S4.SS1.p2.7.m7.2.2.1.4" stretchy="false" xref="S4.SS1.p2.7.m7.2.2.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.7.m7.2b"><set id="S4.SS1.p2.7.m7.2.2.2.cmml" xref="S4.SS1.p2.7.m7.2.2.1"><apply id="S4.SS1.p2.7.m7.2.2.1.1.cmml" xref="S4.SS1.p2.7.m7.2.2.1.1"><minus id="S4.SS1.p2.7.m7.2.2.1.1.1.cmml" xref="S4.SS1.p2.7.m7.2.2.1.1"></minus><cn id="S4.SS1.p2.7.m7.2.2.1.1.2.cmml" type="integer" xref="S4.SS1.p2.7.m7.2.2.1.1.2">1</cn></apply><cn id="S4.SS1.p2.7.m7.1.1.cmml" type="integer" xref="S4.SS1.p2.7.m7.1.1">1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.7.m7.2c">\{-1,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.7.m7.2d">{ - 1 , 1 }</annotation></semantics></math>, even if <math alttext="c>1" class="ltx_Math" display="inline" id="S4.SS1.p2.8.m8.1"><semantics id="S4.SS1.p2.8.m8.1a"><mrow id="S4.SS1.p2.8.m8.1.1" xref="S4.SS1.p2.8.m8.1.1.cmml"><mi id="S4.SS1.p2.8.m8.1.1.2" xref="S4.SS1.p2.8.m8.1.1.2.cmml">c</mi><mo id="S4.SS1.p2.8.m8.1.1.1" xref="S4.SS1.p2.8.m8.1.1.1.cmml">></mo><mn id="S4.SS1.p2.8.m8.1.1.3" xref="S4.SS1.p2.8.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.8.m8.1b"><apply id="S4.SS1.p2.8.m8.1.1.cmml" xref="S4.SS1.p2.8.m8.1.1"><gt id="S4.SS1.p2.8.m8.1.1.1.cmml" xref="S4.SS1.p2.8.m8.1.1.1"></gt><ci id="S4.SS1.p2.8.m8.1.1.2.cmml" xref="S4.SS1.p2.8.m8.1.1.2">𝑐</ci><cn id="S4.SS1.p2.8.m8.1.1.3.cmml" type="integer" xref="S4.SS1.p2.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.8.m8.1c">c>1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.8.m8.1d">italic_c > 1</annotation></semantics></math> is assumed to be a constant not dependent on <math alttext="n" class="ltx_Math" display="inline" id="S4.SS1.p2.9.m9.1"><semantics id="S4.SS1.p2.9.m9.1a"><mi id="S4.SS1.p2.9.m9.1.1" xref="S4.SS1.p2.9.m9.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.9.m9.1b"><ci id="S4.SS1.p2.9.m9.1.1.cmml" xref="S4.SS1.p2.9.m9.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.9.m9.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.9.m9.1d">italic_n</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>Logarithmic Approximation for Nonnegative Total Value</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.2">We will now show that we can get beyond the inapproximability result of <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1 Welfare Inapproximability for Restricted Valuations ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.1</span></a> if we restrict attention to ASHGs with a nonnegative total value. For this, we will draw a connection to a related problem from the literature on correlation clustering. Instances of correlation clustering usually only use binary information, i.e., whether two objects should belong to the same or different clusters. The goal is then to optimize one of two objectives: maximizing agreements, i.e., the number of pairs whose pairwise relationship is classified correctly, and minimizing disagreements, i.e., the number of pairs classified incorrectly. In addition, one can consider the combination of these two objectives, where agreements should be maximized while disagreements should simultaneously be minimized. In the spirit of hedonic games, we capture a weighted version of this objective as a notion of welfare. Given an ASHG <math alttext="(N,v)" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.2"><semantics id="S4.SS2.p1.1.m1.2a"><mrow id="S4.SS2.p1.1.m1.2.3.2" xref="S4.SS2.p1.1.m1.2.3.1.cmml"><mo id="S4.SS2.p1.1.m1.2.3.2.1" stretchy="false" xref="S4.SS2.p1.1.m1.2.3.1.cmml">(</mo><mi id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml">N</mi><mo id="S4.SS2.p1.1.m1.2.3.2.2" xref="S4.SS2.p1.1.m1.2.3.1.cmml">,</mo><mi id="S4.SS2.p1.1.m1.2.2" xref="S4.SS2.p1.1.m1.2.2.cmml">v</mi><mo id="S4.SS2.p1.1.m1.2.3.2.3" stretchy="false" xref="S4.SS2.p1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.2b"><interval closure="open" id="S4.SS2.p1.1.m1.2.3.1.cmml" xref="S4.SS2.p1.1.m1.2.3.2"><ci id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">𝑁</ci><ci id="S4.SS2.p1.1.m1.2.2.cmml" xref="S4.SS2.p1.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.2d">( italic_N , italic_v )</annotation></semantics></math> and a partition <math alttext="\pi" class="ltx_Math" display="inline" id="S4.SS2.p1.2.m2.1"><semantics id="S4.SS2.p1.2.m2.1a"><mi id="S4.SS2.p1.2.m2.1.1" xref="S4.SS2.p1.2.m2.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.1b"><ci id="S4.SS2.p1.2.m2.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.1d">italic_π</annotation></semantics></math>, we define its <em class="ltx_emph ltx_font_italic" id="S4.SS2.p1.2.1">correlation welfare</em> as</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{CW}(\pi):=\frac{1}{2}\left[\sum_{i\in N}\left(\sum_{j\in\pi(i)}v_{i}(% j)-\sum_{j\in N\setminus\pi(i)}v_{i}(j)\right)\right]\text{.}" class="ltx_Math" display="block" id="S4.Ex6.m1.6"><semantics id="S4.Ex6.m1.6a"><mrow id="S4.Ex6.m1.6.6" xref="S4.Ex6.m1.6.6.cmml"><mrow id="S4.Ex6.m1.6.6.3" xref="S4.Ex6.m1.6.6.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex6.m1.6.6.3.2" xref="S4.Ex6.m1.6.6.3.2.cmml">𝒞</mi><mo id="S4.Ex6.m1.6.6.3.1" xref="S4.Ex6.m1.6.6.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex6.m1.6.6.3.3" xref="S4.Ex6.m1.6.6.3.3.cmml">𝒲</mi><mo id="S4.Ex6.m1.6.6.3.1a" xref="S4.Ex6.m1.6.6.3.1.cmml">⁢</mo><mrow id="S4.Ex6.m1.6.6.3.4.2" xref="S4.Ex6.m1.6.6.3.cmml"><mo id="S4.Ex6.m1.6.6.3.4.2.1" stretchy="false" xref="S4.Ex6.m1.6.6.3.cmml">(</mo><mi id="S4.Ex6.m1.3.3" xref="S4.Ex6.m1.3.3.cmml">π</mi><mo id="S4.Ex6.m1.6.6.3.4.2.2" rspace="0.278em" stretchy="false" xref="S4.Ex6.m1.6.6.3.cmml">)</mo></mrow></mrow><mo id="S4.Ex6.m1.6.6.2" rspace="0.278em" xref="S4.Ex6.m1.6.6.2.cmml">:=</mo><mrow id="S4.Ex6.m1.6.6.1" xref="S4.Ex6.m1.6.6.1.cmml"><mfrac id="S4.Ex6.m1.6.6.1.3" xref="S4.Ex6.m1.6.6.1.3.cmml"><mn id="S4.Ex6.m1.6.6.1.3.2" xref="S4.Ex6.m1.6.6.1.3.2.cmml">1</mn><mn id="S4.Ex6.m1.6.6.1.3.3" xref="S4.Ex6.m1.6.6.1.3.3.cmml">2</mn></mfrac><mo id="S4.Ex6.m1.6.6.1.2" xref="S4.Ex6.m1.6.6.1.2.cmml">⁢</mo><mrow id="S4.Ex6.m1.6.6.1.1.1" xref="S4.Ex6.m1.6.6.1.1.2.cmml"><mo id="S4.Ex6.m1.6.6.1.1.1.2" xref="S4.Ex6.m1.6.6.1.1.2.1.cmml">[</mo><mrow id="S4.Ex6.m1.6.6.1.1.1.1" xref="S4.Ex6.m1.6.6.1.1.1.1.cmml"><munder id="S4.Ex6.m1.6.6.1.1.1.1.2" xref="S4.Ex6.m1.6.6.1.1.1.1.2.cmml"><mo id="S4.Ex6.m1.6.6.1.1.1.1.2.2" lspace="0em" movablelimits="false" rspace="0em" xref="S4.Ex6.m1.6.6.1.1.1.1.2.2.cmml">∑</mo><mrow id="S4.Ex6.m1.6.6.1.1.1.1.2.3" xref="S4.Ex6.m1.6.6.1.1.1.1.2.3.cmml"><mi id="S4.Ex6.m1.6.6.1.1.1.1.2.3.2" xref="S4.Ex6.m1.6.6.1.1.1.1.2.3.2.cmml">i</mi><mo id="S4.Ex6.m1.6.6.1.1.1.1.2.3.1" xref="S4.Ex6.m1.6.6.1.1.1.1.2.3.1.cmml">∈</mo><mi id="S4.Ex6.m1.6.6.1.1.1.1.2.3.3" xref="S4.Ex6.m1.6.6.1.1.1.1.2.3.3.cmml">N</mi></mrow></munder><mrow id="S4.Ex6.m1.6.6.1.1.1.1.1.1" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex6.m1.6.6.1.1.1.1.1.1.2" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.cmml"><mrow id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.cmml"><munder id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.1" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.1.cmml"><mo id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.1.2" lspace="0em" movablelimits="false" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.1.2.cmml">∑</mo><mrow id="S4.Ex6.m1.1.1.1" xref="S4.Ex6.m1.1.1.1.cmml"><mi id="S4.Ex6.m1.1.1.1.3" xref="S4.Ex6.m1.1.1.1.3.cmml">j</mi><mo id="S4.Ex6.m1.1.1.1.2" xref="S4.Ex6.m1.1.1.1.2.cmml">∈</mo><mrow id="S4.Ex6.m1.1.1.1.4" xref="S4.Ex6.m1.1.1.1.4.cmml"><mi id="S4.Ex6.m1.1.1.1.4.2" xref="S4.Ex6.m1.1.1.1.4.2.cmml">π</mi><mo id="S4.Ex6.m1.1.1.1.4.1" xref="S4.Ex6.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S4.Ex6.m1.1.1.1.4.3.2" xref="S4.Ex6.m1.1.1.1.4.cmml"><mo id="S4.Ex6.m1.1.1.1.4.3.2.1" stretchy="false" xref="S4.Ex6.m1.1.1.1.4.cmml">(</mo><mi id="S4.Ex6.m1.1.1.1.1" xref="S4.Ex6.m1.1.1.1.1.cmml">i</mi><mo id="S4.Ex6.m1.1.1.1.4.3.2.2" stretchy="false" xref="S4.Ex6.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.cmml"><msub id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.2" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.2.cmml"><mi id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.2.2" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.2.2.cmml">v</mi><mi id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.2.3" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.2.3.cmml">i</mi></msub><mo id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.1" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.1.cmml">⁢</mo><mrow id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.3.2" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.cmml"><mo id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.3.2.1" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.cmml">(</mo><mi id="S4.Ex6.m1.4.4" xref="S4.Ex6.m1.4.4.cmml">j</mi><mo id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.3.2.2" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.1" rspace="0.055em" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.cmml"><munder id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.1" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.1.cmml"><mo id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.1.2" movablelimits="false" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.1.2.cmml">∑</mo><mrow id="S4.Ex6.m1.2.2.1" xref="S4.Ex6.m1.2.2.1.cmml"><mi id="S4.Ex6.m1.2.2.1.3" xref="S4.Ex6.m1.2.2.1.3.cmml">j</mi><mo id="S4.Ex6.m1.2.2.1.2" xref="S4.Ex6.m1.2.2.1.2.cmml">∈</mo><mrow id="S4.Ex6.m1.2.2.1.4" xref="S4.Ex6.m1.2.2.1.4.cmml"><mi id="S4.Ex6.m1.2.2.1.4.2" xref="S4.Ex6.m1.2.2.1.4.2.cmml">N</mi><mo id="S4.Ex6.m1.2.2.1.4.1" xref="S4.Ex6.m1.2.2.1.4.1.cmml">∖</mo><mrow id="S4.Ex6.m1.2.2.1.4.3" xref="S4.Ex6.m1.2.2.1.4.3.cmml"><mi id="S4.Ex6.m1.2.2.1.4.3.2" xref="S4.Ex6.m1.2.2.1.4.3.2.cmml">π</mi><mo id="S4.Ex6.m1.2.2.1.4.3.1" xref="S4.Ex6.m1.2.2.1.4.3.1.cmml">⁢</mo><mrow id="S4.Ex6.m1.2.2.1.4.3.3.2" xref="S4.Ex6.m1.2.2.1.4.3.cmml"><mo id="S4.Ex6.m1.2.2.1.4.3.3.2.1" stretchy="false" xref="S4.Ex6.m1.2.2.1.4.3.cmml">(</mo><mi id="S4.Ex6.m1.2.2.1.1" xref="S4.Ex6.m1.2.2.1.1.cmml">i</mi><mo id="S4.Ex6.m1.2.2.1.4.3.3.2.2" stretchy="false" xref="S4.Ex6.m1.2.2.1.4.3.cmml">)</mo></mrow></mrow></mrow></mrow></munder><mrow id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.cmml"><msub id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.2" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.2.cmml"><mi id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.2.2" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.2.2.cmml">v</mi><mi id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.2.3" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.2.3.cmml">i</mi></msub><mo id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.1" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.3.2" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.cmml"><mo id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.3.2.1" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.cmml">(</mo><mi id="S4.Ex6.m1.5.5" xref="S4.Ex6.m1.5.5.cmml">j</mi><mo id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.3.2.2" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S4.Ex6.m1.6.6.1.1.1.1.1.1.3" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex6.m1.6.6.1.1.1.3" xref="S4.Ex6.m1.6.6.1.1.2.1.cmml">]</mo></mrow><mo id="S4.Ex6.m1.6.6.1.2a" xref="S4.Ex6.m1.6.6.1.2.cmml">⁢</mo><mtext id="S4.Ex6.m1.6.6.1.4" xref="S4.Ex6.m1.6.6.1.4a.cmml">.</mtext></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex6.m1.6b"><apply id="S4.Ex6.m1.6.6.cmml" xref="S4.Ex6.m1.6.6"><csymbol cd="latexml" id="S4.Ex6.m1.6.6.2.cmml" xref="S4.Ex6.m1.6.6.2">assign</csymbol><apply id="S4.Ex6.m1.6.6.3.cmml" xref="S4.Ex6.m1.6.6.3"><times id="S4.Ex6.m1.6.6.3.1.cmml" xref="S4.Ex6.m1.6.6.3.1"></times><ci id="S4.Ex6.m1.6.6.3.2.cmml" xref="S4.Ex6.m1.6.6.3.2">𝒞</ci><ci id="S4.Ex6.m1.6.6.3.3.cmml" xref="S4.Ex6.m1.6.6.3.3">𝒲</ci><ci id="S4.Ex6.m1.3.3.cmml" xref="S4.Ex6.m1.3.3">𝜋</ci></apply><apply id="S4.Ex6.m1.6.6.1.cmml" xref="S4.Ex6.m1.6.6.1"><times id="S4.Ex6.m1.6.6.1.2.cmml" xref="S4.Ex6.m1.6.6.1.2"></times><apply id="S4.Ex6.m1.6.6.1.3.cmml" xref="S4.Ex6.m1.6.6.1.3"><divide id="S4.Ex6.m1.6.6.1.3.1.cmml" xref="S4.Ex6.m1.6.6.1.3"></divide><cn id="S4.Ex6.m1.6.6.1.3.2.cmml" type="integer" xref="S4.Ex6.m1.6.6.1.3.2">1</cn><cn id="S4.Ex6.m1.6.6.1.3.3.cmml" type="integer" xref="S4.Ex6.m1.6.6.1.3.3">2</cn></apply><apply id="S4.Ex6.m1.6.6.1.1.2.cmml" xref="S4.Ex6.m1.6.6.1.1.1"><csymbol cd="latexml" id="S4.Ex6.m1.6.6.1.1.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.2">delimited-[]</csymbol><apply id="S4.Ex6.m1.6.6.1.1.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1"><apply id="S4.Ex6.m1.6.6.1.1.1.1.2.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.1.1.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.2">subscript</csymbol><sum id="S4.Ex6.m1.6.6.1.1.1.1.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.2.2"></sum><apply id="S4.Ex6.m1.6.6.1.1.1.1.2.3.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.2.3"><in id="S4.Ex6.m1.6.6.1.1.1.1.2.3.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.2.3.1"></in><ci id="S4.Ex6.m1.6.6.1.1.1.1.2.3.2.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.2.3.2">𝑖</ci><ci id="S4.Ex6.m1.6.6.1.1.1.1.2.3.3.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.2.3.3">𝑁</ci></apply></apply><apply id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1"><minus id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.1"></minus><apply id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2"><apply id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.1"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.1">subscript</csymbol><sum id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.1.2.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.1.2"></sum><apply id="S4.Ex6.m1.1.1.1.cmml" xref="S4.Ex6.m1.1.1.1"><in id="S4.Ex6.m1.1.1.1.2.cmml" xref="S4.Ex6.m1.1.1.1.2"></in><ci id="S4.Ex6.m1.1.1.1.3.cmml" xref="S4.Ex6.m1.1.1.1.3">𝑗</ci><apply id="S4.Ex6.m1.1.1.1.4.cmml" xref="S4.Ex6.m1.1.1.1.4"><times id="S4.Ex6.m1.1.1.1.4.1.cmml" xref="S4.Ex6.m1.1.1.1.4.1"></times><ci id="S4.Ex6.m1.1.1.1.4.2.cmml" xref="S4.Ex6.m1.1.1.1.4.2">𝜋</ci><ci id="S4.Ex6.m1.1.1.1.1.cmml" xref="S4.Ex6.m1.1.1.1.1">𝑖</ci></apply></apply></apply><apply id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2"><times id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.1"></times><apply id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.2.2">𝑣</ci><ci id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.2.3.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.2.2.2.3">𝑖</ci></apply><ci id="S4.Ex6.m1.4.4.cmml" xref="S4.Ex6.m1.4.4">𝑗</ci></apply></apply><apply id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3"><apply id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.1">subscript</csymbol><sum id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.1.2.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.1.2"></sum><apply id="S4.Ex6.m1.2.2.1.cmml" xref="S4.Ex6.m1.2.2.1"><in id="S4.Ex6.m1.2.2.1.2.cmml" xref="S4.Ex6.m1.2.2.1.2"></in><ci id="S4.Ex6.m1.2.2.1.3.cmml" xref="S4.Ex6.m1.2.2.1.3">𝑗</ci><apply id="S4.Ex6.m1.2.2.1.4.cmml" xref="S4.Ex6.m1.2.2.1.4"><setdiff id="S4.Ex6.m1.2.2.1.4.1.cmml" xref="S4.Ex6.m1.2.2.1.4.1"></setdiff><ci id="S4.Ex6.m1.2.2.1.4.2.cmml" xref="S4.Ex6.m1.2.2.1.4.2">𝑁</ci><apply id="S4.Ex6.m1.2.2.1.4.3.cmml" xref="S4.Ex6.m1.2.2.1.4.3"><times id="S4.Ex6.m1.2.2.1.4.3.1.cmml" xref="S4.Ex6.m1.2.2.1.4.3.1"></times><ci id="S4.Ex6.m1.2.2.1.4.3.2.cmml" xref="S4.Ex6.m1.2.2.1.4.3.2">𝜋</ci><ci id="S4.Ex6.m1.2.2.1.1.cmml" xref="S4.Ex6.m1.2.2.1.1">𝑖</ci></apply></apply></apply></apply><apply id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2"><times id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.1"></times><apply id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.2">subscript</csymbol><ci id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.2.2">𝑣</ci><ci id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.2.3.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.3.2.2.3">𝑖</ci></apply><ci id="S4.Ex6.m1.5.5.cmml" xref="S4.Ex6.m1.5.5">𝑗</ci></apply></apply></apply></apply></apply><ci id="S4.Ex6.m1.6.6.1.4a.cmml" xref="S4.Ex6.m1.6.6.1.4"><mtext id="S4.Ex6.m1.6.6.1.4.cmml" xref="S4.Ex6.m1.6.6.1.4">.</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex6.m1.6c">\mathcal{CW}(\pi):=\frac{1}{2}\left[\sum_{i\in N}\left(\sum_{j\in\pi(i)}v_{i}(% j)-\sum_{j\in N\setminus\pi(i)}v_{i}(j)\right)\right]\text{.}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex6.m1.6d">caligraphic_C caligraphic_W ( italic_π ) := divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ ∑ start_POSTSUBSCRIPT italic_i ∈ italic_N end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_j ∈ italic_π ( italic_i ) end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j ) - ∑ start_POSTSUBSCRIPT italic_j ∈ italic_N ∖ italic_π ( italic_i ) end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j ) ) ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.7">Charikar and Wirth <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx18" title="">CW04</a>]</cite> present a randomized <math alttext="\mathcal{O}(\log n)" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mrow id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p2.1.m1.1.1.3" xref="S4.SS2.p2.1.m1.1.1.3.cmml">𝒪</mi><mo id="S4.SS2.p2.1.m1.1.1.2" xref="S4.SS2.p2.1.m1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p2.1.m1.1.1.1.1" xref="S4.SS2.p2.1.m1.1.1.1.1.1.cmml"><mo id="S4.SS2.p2.1.m1.1.1.1.1.2" stretchy="false" xref="S4.SS2.p2.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p2.1.m1.1.1.1.1.1" xref="S4.SS2.p2.1.m1.1.1.1.1.1.cmml"><mi id="S4.SS2.p2.1.m1.1.1.1.1.1.1" xref="S4.SS2.p2.1.m1.1.1.1.1.1.1.cmml">log</mi><mo id="S4.SS2.p2.1.m1.1.1.1.1.1a" lspace="0.167em" xref="S4.SS2.p2.1.m1.1.1.1.1.1.cmml">⁡</mo><mi id="S4.SS2.p2.1.m1.1.1.1.1.1.2" xref="S4.SS2.p2.1.m1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S4.SS2.p2.1.m1.1.1.1.1.3" stretchy="false" xref="S4.SS2.p2.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><apply id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1"><times id="S4.SS2.p2.1.m1.1.1.2.cmml" xref="S4.SS2.p2.1.m1.1.1.2"></times><ci id="S4.SS2.p2.1.m1.1.1.3.cmml" xref="S4.SS2.p2.1.m1.1.1.3">𝒪</ci><apply id="S4.SS2.p2.1.m1.1.1.1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1"><log id="S4.SS2.p2.1.m1.1.1.1.1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.1.1"></log><ci id="S4.SS2.p2.1.m1.1.1.1.1.1.2.cmml" xref="S4.SS2.p2.1.m1.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">\mathcal{O}(\log n)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">caligraphic_O ( roman_log italic_n )</annotation></semantics></math>-approximation algorithm for maximizing<span class="ltx_note ltx_role_footnote" id="footnote4"><sup class="ltx_note_mark">4</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">4</sup><span class="ltx_tag ltx_tag_note">4</span>In their terminology, this is the problem of maximizing <span class="ltx_text ltx_font_smallcaps" id="footnote4.1">MaxQP</span>.</span></span></span> <math alttext="\mathcal{CW}" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.1"><semantics id="S4.SS2.p2.2.m2.1a"><mrow id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p2.2.m2.1.1.2" xref="S4.SS2.p2.2.m2.1.1.2.cmml">𝒞</mi><mo id="S4.SS2.p2.2.m2.1.1.1" xref="S4.SS2.p2.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p2.2.m2.1.1.3" xref="S4.SS2.p2.2.m2.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.1b"><apply id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1"><times id="S4.SS2.p2.2.m2.1.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1.1"></times><ci id="S4.SS2.p2.2.m2.1.1.2.cmml" xref="S4.SS2.p2.2.m2.1.1.2">𝒞</ci><ci id="S4.SS2.p2.2.m2.1.1.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.1c">\mathcal{CW}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.1d">caligraphic_C caligraphic_W</annotation></semantics></math> subject to <math alttext="\pi\in\Pi_{N}^{(2)}" class="ltx_Math" display="inline" id="S4.SS2.p2.3.m3.1"><semantics id="S4.SS2.p2.3.m3.1a"><mrow id="S4.SS2.p2.3.m3.1.2" xref="S4.SS2.p2.3.m3.1.2.cmml"><mi id="S4.SS2.p2.3.m3.1.2.2" xref="S4.SS2.p2.3.m3.1.2.2.cmml">π</mi><mo id="S4.SS2.p2.3.m3.1.2.1" xref="S4.SS2.p2.3.m3.1.2.1.cmml">∈</mo><msubsup id="S4.SS2.p2.3.m3.1.2.3" xref="S4.SS2.p2.3.m3.1.2.3.cmml"><mi id="S4.SS2.p2.3.m3.1.2.3.2.2" mathvariant="normal" xref="S4.SS2.p2.3.m3.1.2.3.2.2.cmml">Π</mi><mi id="S4.SS2.p2.3.m3.1.2.3.2.3" xref="S4.SS2.p2.3.m3.1.2.3.2.3.cmml">N</mi><mrow id="S4.SS2.p2.3.m3.1.1.1.3" xref="S4.SS2.p2.3.m3.1.2.3.cmml"><mo id="S4.SS2.p2.3.m3.1.1.1.3.1" stretchy="false" xref="S4.SS2.p2.3.m3.1.2.3.cmml">(</mo><mn id="S4.SS2.p2.3.m3.1.1.1.1" xref="S4.SS2.p2.3.m3.1.1.1.1.cmml">2</mn><mo id="S4.SS2.p2.3.m3.1.1.1.3.2" stretchy="false" xref="S4.SS2.p2.3.m3.1.2.3.cmml">)</mo></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m3.1b"><apply id="S4.SS2.p2.3.m3.1.2.cmml" xref="S4.SS2.p2.3.m3.1.2"><in id="S4.SS2.p2.3.m3.1.2.1.cmml" xref="S4.SS2.p2.3.m3.1.2.1"></in><ci id="S4.SS2.p2.3.m3.1.2.2.cmml" xref="S4.SS2.p2.3.m3.1.2.2">𝜋</ci><apply id="S4.SS2.p2.3.m3.1.2.3.cmml" xref="S4.SS2.p2.3.m3.1.2.3"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m3.1.2.3.1.cmml" xref="S4.SS2.p2.3.m3.1.2.3">superscript</csymbol><apply id="S4.SS2.p2.3.m3.1.2.3.2.cmml" xref="S4.SS2.p2.3.m3.1.2.3"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m3.1.2.3.2.1.cmml" xref="S4.SS2.p2.3.m3.1.2.3">subscript</csymbol><ci id="S4.SS2.p2.3.m3.1.2.3.2.2.cmml" xref="S4.SS2.p2.3.m3.1.2.3.2.2">Π</ci><ci id="S4.SS2.p2.3.m3.1.2.3.2.3.cmml" xref="S4.SS2.p2.3.m3.1.2.3.2.3">𝑁</ci></apply><cn id="S4.SS2.p2.3.m3.1.1.1.1.cmml" type="integer" xref="S4.SS2.p2.3.m3.1.1.1.1">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m3.1c">\pi\in\Pi_{N}^{(2)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.3.m3.1d">italic_π ∈ roman_Π start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT</annotation></semantics></math>. They then show that this extends to maximizing <math alttext="\mathcal{CW}" class="ltx_Math" display="inline" id="S4.SS2.p2.4.m4.1"><semantics id="S4.SS2.p2.4.m4.1a"><mrow id="S4.SS2.p2.4.m4.1.1" xref="S4.SS2.p2.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p2.4.m4.1.1.2" xref="S4.SS2.p2.4.m4.1.1.2.cmml">𝒞</mi><mo id="S4.SS2.p2.4.m4.1.1.1" xref="S4.SS2.p2.4.m4.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p2.4.m4.1.1.3" xref="S4.SS2.p2.4.m4.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.4.m4.1b"><apply id="S4.SS2.p2.4.m4.1.1.cmml" xref="S4.SS2.p2.4.m4.1.1"><times id="S4.SS2.p2.4.m4.1.1.1.cmml" xref="S4.SS2.p2.4.m4.1.1.1"></times><ci id="S4.SS2.p2.4.m4.1.1.2.cmml" xref="S4.SS2.p2.4.m4.1.1.2">𝒞</ci><ci id="S4.SS2.p2.4.m4.1.1.3.cmml" xref="S4.SS2.p2.4.m4.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.4.m4.1c">\mathcal{CW}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.4.m4.1d">caligraphic_C caligraphic_W</annotation></semantics></math> within <math alttext="\Pi_{N}" class="ltx_Math" display="inline" id="S4.SS2.p2.5.m5.1"><semantics id="S4.SS2.p2.5.m5.1a"><msub id="S4.SS2.p2.5.m5.1.1" xref="S4.SS2.p2.5.m5.1.1.cmml"><mi id="S4.SS2.p2.5.m5.1.1.2" mathvariant="normal" xref="S4.SS2.p2.5.m5.1.1.2.cmml">Π</mi><mi id="S4.SS2.p2.5.m5.1.1.3" xref="S4.SS2.p2.5.m5.1.1.3.cmml">N</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.5.m5.1b"><apply id="S4.SS2.p2.5.m5.1.1.cmml" xref="S4.SS2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.5.m5.1.1.1.cmml" xref="S4.SS2.p2.5.m5.1.1">subscript</csymbol><ci id="S4.SS2.p2.5.m5.1.1.2.cmml" xref="S4.SS2.p2.5.m5.1.1.2">Π</ci><ci id="S4.SS2.p2.5.m5.1.1.3.cmml" xref="S4.SS2.p2.5.m5.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.5.m5.1c">\Pi_{N}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.5.m5.1d">roman_Π start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT</annotation></semantics></math> in the case of valuation functions in the range <math alttext="\{-1,1\}" class="ltx_Math" display="inline" id="S4.SS2.p2.6.m6.2"><semantics id="S4.SS2.p2.6.m6.2a"><mrow id="S4.SS2.p2.6.m6.2.2.1" xref="S4.SS2.p2.6.m6.2.2.2.cmml"><mo id="S4.SS2.p2.6.m6.2.2.1.2" stretchy="false" xref="S4.SS2.p2.6.m6.2.2.2.cmml">{</mo><mrow id="S4.SS2.p2.6.m6.2.2.1.1" xref="S4.SS2.p2.6.m6.2.2.1.1.cmml"><mo id="S4.SS2.p2.6.m6.2.2.1.1a" xref="S4.SS2.p2.6.m6.2.2.1.1.cmml">−</mo><mn id="S4.SS2.p2.6.m6.2.2.1.1.2" xref="S4.SS2.p2.6.m6.2.2.1.1.2.cmml">1</mn></mrow><mo id="S4.SS2.p2.6.m6.2.2.1.3" xref="S4.SS2.p2.6.m6.2.2.2.cmml">,</mo><mn id="S4.SS2.p2.6.m6.1.1" xref="S4.SS2.p2.6.m6.1.1.cmml">1</mn><mo id="S4.SS2.p2.6.m6.2.2.1.4" stretchy="false" xref="S4.SS2.p2.6.m6.2.2.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.6.m6.2b"><set id="S4.SS2.p2.6.m6.2.2.2.cmml" xref="S4.SS2.p2.6.m6.2.2.1"><apply id="S4.SS2.p2.6.m6.2.2.1.1.cmml" xref="S4.SS2.p2.6.m6.2.2.1.1"><minus id="S4.SS2.p2.6.m6.2.2.1.1.1.cmml" xref="S4.SS2.p2.6.m6.2.2.1.1"></minus><cn id="S4.SS2.p2.6.m6.2.2.1.1.2.cmml" type="integer" xref="S4.SS2.p2.6.m6.2.2.1.1.2">1</cn></apply><cn id="S4.SS2.p2.6.m6.1.1.cmml" type="integer" xref="S4.SS2.p2.6.m6.1.1">1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.6.m6.2c">\{-1,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.6.m6.2d">{ - 1 , 1 }</annotation></semantics></math>. It is easy to see that the same technique applies for the general range of valuation functions (cf. <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.Thmtheorem6" title="Lemma 4.6. ‣ 4.2 Logarithmic Approximation for Nonnegative Total Value ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.6</span></a>). The goal of this section is to extend the approximation guarantee to maximizing <math alttext="\mathcal{SW}" class="ltx_Math" display="inline" id="S4.SS2.p2.7.m7.1"><semantics id="S4.SS2.p2.7.m7.1a"><mrow id="S4.SS2.p2.7.m7.1.1" xref="S4.SS2.p2.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p2.7.m7.1.1.2" xref="S4.SS2.p2.7.m7.1.1.2.cmml">𝒮</mi><mo id="S4.SS2.p2.7.m7.1.1.1" xref="S4.SS2.p2.7.m7.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p2.7.m7.1.1.3" xref="S4.SS2.p2.7.m7.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.7.m7.1b"><apply id="S4.SS2.p2.7.m7.1.1.cmml" xref="S4.SS2.p2.7.m7.1.1"><times id="S4.SS2.p2.7.m7.1.1.1.cmml" xref="S4.SS2.p2.7.m7.1.1.1"></times><ci id="S4.SS2.p2.7.m7.1.1.2.cmml" xref="S4.SS2.p2.7.m7.1.1.2">𝒮</ci><ci id="S4.SS2.p2.7.m7.1.1.3.cmml" xref="S4.SS2.p2.7.m7.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.7.m7.1c">\mathcal{SW}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.7.m7.1d">caligraphic_S caligraphic_W</annotation></semantics></math> for ASHGs with nonnegative total value.</p> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.2">First, by plugging in definitions, we immediately obtain the following relationship between <math alttext="\mathcal{SW}" class="ltx_Math" display="inline" id="S4.SS2.p3.1.m1.1"><semantics id="S4.SS2.p3.1.m1.1a"><mrow id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p3.1.m1.1.1.2" xref="S4.SS2.p3.1.m1.1.1.2.cmml">𝒮</mi><mo id="S4.SS2.p3.1.m1.1.1.1" xref="S4.SS2.p3.1.m1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p3.1.m1.1.1.3" xref="S4.SS2.p3.1.m1.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><apply id="S4.SS2.p3.1.m1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1"><times id="S4.SS2.p3.1.m1.1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1.1"></times><ci id="S4.SS2.p3.1.m1.1.1.2.cmml" xref="S4.SS2.p3.1.m1.1.1.2">𝒮</ci><ci id="S4.SS2.p3.1.m1.1.1.3.cmml" xref="S4.SS2.p3.1.m1.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.m1.1c">\mathcal{SW}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.1.m1.1d">caligraphic_S caligraphic_W</annotation></semantics></math> and <math alttext="\mathcal{CW}" class="ltx_Math" display="inline" id="S4.SS2.p3.2.m2.1"><semantics id="S4.SS2.p3.2.m2.1a"><mrow id="S4.SS2.p3.2.m2.1.1" xref="S4.SS2.p3.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p3.2.m2.1.1.2" xref="S4.SS2.p3.2.m2.1.1.2.cmml">𝒞</mi><mo id="S4.SS2.p3.2.m2.1.1.1" xref="S4.SS2.p3.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p3.2.m2.1.1.3" xref="S4.SS2.p3.2.m2.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.2.m2.1b"><apply id="S4.SS2.p3.2.m2.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1"><times id="S4.SS2.p3.2.m2.1.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1.1"></times><ci id="S4.SS2.p3.2.m2.1.1.2.cmml" xref="S4.SS2.p3.2.m2.1.1.2">𝒞</ci><ci id="S4.SS2.p3.2.m2.1.1.3.cmml" xref="S4.SS2.p3.2.m2.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.2.m2.1c">\mathcal{CW}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.2.m2.1d">caligraphic_C caligraphic_W</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S4.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.1.1.1">Proposition 4.2</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem2.p1"> <p class="ltx_p" id="S4.Thmtheorem2.p1.3"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.p1.3.3">Consider an ASHG <math alttext="(N,v)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.1.1.m1.2"><semantics id="S4.Thmtheorem2.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem2.p1.1.1.m1.2.3.2" xref="S4.Thmtheorem2.p1.1.1.m1.2.3.1.cmml"><mo id="S4.Thmtheorem2.p1.1.1.m1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.1.1.m1.2.3.1.cmml">(</mo><mi id="S4.Thmtheorem2.p1.1.1.m1.1.1" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.cmml">N</mi><mo id="S4.Thmtheorem2.p1.1.1.m1.2.3.2.2" xref="S4.Thmtheorem2.p1.1.1.m1.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem2.p1.1.1.m1.2.2" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.cmml">v</mi><mo id="S4.Thmtheorem2.p1.1.1.m1.2.3.2.3" stretchy="false" xref="S4.Thmtheorem2.p1.1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.1.1.m1.2b"><interval closure="open" id="S4.Thmtheorem2.p1.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.2.3.2"><ci id="S4.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1">𝑁</ci><ci id="S4.Thmtheorem2.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.1.1.m1.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.1.1.m1.2d">( italic_N , italic_v )</annotation></semantics></math> and a partition <math alttext="\pi" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.2.2.m2.1"><semantics id="S4.Thmtheorem2.p1.2.2.m2.1a"><mi id="S4.Thmtheorem2.p1.2.2.m2.1.1" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.2.2.m2.1b"><ci id="S4.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.2.2.m2.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.2.2.m2.1d">italic_π</annotation></semantics></math>. Then it holds that <math alttext="\mathcal{CW}(\pi)+\frac{1}{2}\mathcal{V}(N,v)=\mathcal{SW}(\pi)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.3.3.m3.4"><semantics id="S4.Thmtheorem2.p1.3.3.m3.4a"><mrow id="S4.Thmtheorem2.p1.3.3.m3.4.5" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.cmml"><mrow id="S4.Thmtheorem2.p1.3.3.m3.4.5.2" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.cmml"><mrow id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.2" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.2.cmml">𝒞</mi><mo id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.1" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.3" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.3.cmml">𝒲</mi><mo id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.1a" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.4.2" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.cmml"><mo id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.4.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.cmml">(</mo><mi id="S4.Thmtheorem2.p1.3.3.m3.1.1" xref="S4.Thmtheorem2.p1.3.3.m3.1.1.cmml">π</mi><mo id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.4.2.2" stretchy="false" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.1" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.1.cmml">+</mo><mrow id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.cmml"><mfrac id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.2" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.2.cmml"><mn id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.2.2" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.2.2.cmml">1</mn><mn id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.2.3" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.2.3.cmml">2</mn></mfrac><mo id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.1" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.3" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.3.cmml">𝒱</mi><mo id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.1a" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.4.2" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.4.1.cmml"><mo id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.4.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.4.1.cmml">(</mo><mi id="S4.Thmtheorem2.p1.3.3.m3.2.2" xref="S4.Thmtheorem2.p1.3.3.m3.2.2.cmml">N</mi><mo id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.4.2.2" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.4.1.cmml">,</mo><mi id="S4.Thmtheorem2.p1.3.3.m3.3.3" xref="S4.Thmtheorem2.p1.3.3.m3.3.3.cmml">v</mi><mo id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.4.2.3" stretchy="false" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.4.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Thmtheorem2.p1.3.3.m3.4.5.1" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.1.cmml">=</mo><mrow id="S4.Thmtheorem2.p1.3.3.m3.4.5.3" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem2.p1.3.3.m3.4.5.3.2" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.3.2.cmml">𝒮</mi><mo id="S4.Thmtheorem2.p1.3.3.m3.4.5.3.1" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem2.p1.3.3.m3.4.5.3.3" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.3.3.cmml">𝒲</mi><mo id="S4.Thmtheorem2.p1.3.3.m3.4.5.3.1a" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.3.3.m3.4.5.3.4.2" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.3.cmml"><mo id="S4.Thmtheorem2.p1.3.3.m3.4.5.3.4.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.3.cmml">(</mo><mi id="S4.Thmtheorem2.p1.3.3.m3.4.4" xref="S4.Thmtheorem2.p1.3.3.m3.4.4.cmml">π</mi><mo id="S4.Thmtheorem2.p1.3.3.m3.4.5.3.4.2.2" stretchy="false" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.3.3.m3.4b"><apply id="S4.Thmtheorem2.p1.3.3.m3.4.5.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5"><eq id="S4.Thmtheorem2.p1.3.3.m3.4.5.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.1"></eq><apply id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2"><plus id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.1"></plus><apply id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2"><times id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.1"></times><ci id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.2.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.2">𝒞</ci><ci id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.3.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.2.3">𝒲</ci><ci id="S4.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.1.1">𝜋</ci></apply><apply id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3"><times id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.1"></times><apply id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.2.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.2"><divide id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.2.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.2"></divide><cn id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.2.2.cmml" type="integer" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.2.2">1</cn><cn id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.2.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.2.3">2</cn></apply><ci id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.3.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.3">𝒱</ci><interval closure="open" id="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.4.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.2.3.4.2"><ci id="S4.Thmtheorem2.p1.3.3.m3.2.2.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.2.2">𝑁</ci><ci id="S4.Thmtheorem2.p1.3.3.m3.3.3.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.3.3">𝑣</ci></interval></apply></apply><apply id="S4.Thmtheorem2.p1.3.3.m3.4.5.3.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.3"><times id="S4.Thmtheorem2.p1.3.3.m3.4.5.3.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.3.1"></times><ci id="S4.Thmtheorem2.p1.3.3.m3.4.5.3.2.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.3.2">𝒮</ci><ci id="S4.Thmtheorem2.p1.3.3.m3.4.5.3.3.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.5.3.3">𝒲</ci><ci id="S4.Thmtheorem2.p1.3.3.m3.4.4.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.4.4">𝜋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.3.3.m3.4c">\mathcal{CW}(\pi)+\frac{1}{2}\mathcal{V}(N,v)=\mathcal{SW}(\pi)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.3.3.m3.4d">caligraphic_C caligraphic_W ( italic_π ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG caligraphic_V ( italic_N , italic_v ) = caligraphic_S caligraphic_W ( italic_π )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.SS2.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.1.p1"> <p class="ltx_p" id="S4.SS2.1.p1.2">Let <math alttext="(N,v)" class="ltx_Math" display="inline" id="S4.SS2.1.p1.1.m1.2"><semantics id="S4.SS2.1.p1.1.m1.2a"><mrow id="S4.SS2.1.p1.1.m1.2.3.2" xref="S4.SS2.1.p1.1.m1.2.3.1.cmml"><mo id="S4.SS2.1.p1.1.m1.2.3.2.1" stretchy="false" xref="S4.SS2.1.p1.1.m1.2.3.1.cmml">(</mo><mi id="S4.SS2.1.p1.1.m1.1.1" xref="S4.SS2.1.p1.1.m1.1.1.cmml">N</mi><mo id="S4.SS2.1.p1.1.m1.2.3.2.2" xref="S4.SS2.1.p1.1.m1.2.3.1.cmml">,</mo><mi id="S4.SS2.1.p1.1.m1.2.2" xref="S4.SS2.1.p1.1.m1.2.2.cmml">v</mi><mo id="S4.SS2.1.p1.1.m1.2.3.2.3" stretchy="false" xref="S4.SS2.1.p1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.1.m1.2b"><interval closure="open" id="S4.SS2.1.p1.1.m1.2.3.1.cmml" xref="S4.SS2.1.p1.1.m1.2.3.2"><ci id="S4.SS2.1.p1.1.m1.1.1.cmml" xref="S4.SS2.1.p1.1.m1.1.1">𝑁</ci><ci id="S4.SS2.1.p1.1.m1.2.2.cmml" xref="S4.SS2.1.p1.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.1.m1.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.1.m1.2d">( italic_N , italic_v )</annotation></semantics></math> be an ASHG together with a partition <math alttext="\pi\in\Pi_{N}" class="ltx_Math" display="inline" id="S4.SS2.1.p1.2.m2.1"><semantics id="S4.SS2.1.p1.2.m2.1a"><mrow id="S4.SS2.1.p1.2.m2.1.1" xref="S4.SS2.1.p1.2.m2.1.1.cmml"><mi id="S4.SS2.1.p1.2.m2.1.1.2" xref="S4.SS2.1.p1.2.m2.1.1.2.cmml">π</mi><mo id="S4.SS2.1.p1.2.m2.1.1.1" xref="S4.SS2.1.p1.2.m2.1.1.1.cmml">∈</mo><msub id="S4.SS2.1.p1.2.m2.1.1.3" xref="S4.SS2.1.p1.2.m2.1.1.3.cmml"><mi id="S4.SS2.1.p1.2.m2.1.1.3.2" mathvariant="normal" xref="S4.SS2.1.p1.2.m2.1.1.3.2.cmml">Π</mi><mi id="S4.SS2.1.p1.2.m2.1.1.3.3" xref="S4.SS2.1.p1.2.m2.1.1.3.3.cmml">N</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.2.m2.1b"><apply id="S4.SS2.1.p1.2.m2.1.1.cmml" xref="S4.SS2.1.p1.2.m2.1.1"><in id="S4.SS2.1.p1.2.m2.1.1.1.cmml" xref="S4.SS2.1.p1.2.m2.1.1.1"></in><ci id="S4.SS2.1.p1.2.m2.1.1.2.cmml" xref="S4.SS2.1.p1.2.m2.1.1.2">𝜋</ci><apply id="S4.SS2.1.p1.2.m2.1.1.3.cmml" xref="S4.SS2.1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.1.p1.2.m2.1.1.3.1.cmml" xref="S4.SS2.1.p1.2.m2.1.1.3">subscript</csymbol><ci id="S4.SS2.1.p1.2.m2.1.1.3.2.cmml" xref="S4.SS2.1.p1.2.m2.1.1.3.2">Π</ci><ci id="S4.SS2.1.p1.2.m2.1.1.3.3.cmml" xref="S4.SS2.1.p1.2.m2.1.1.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.2.m2.1c">\pi\in\Pi_{N}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.2.m2.1d">italic_π ∈ roman_Π start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT</annotation></semantics></math>. Then,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx1"> <tbody id="S4.Ex7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mathcal{CW}(\pi)+\frac{1}{2}\mathcal{V}(N,v)" class="ltx_Math" display="inline" id="S4.Ex7.m1.3"><semantics id="S4.Ex7.m1.3a"><mrow id="S4.Ex7.m1.3.4" xref="S4.Ex7.m1.3.4.cmml"><mrow id="S4.Ex7.m1.3.4.2" xref="S4.Ex7.m1.3.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex7.m1.3.4.2.2" xref="S4.Ex7.m1.3.4.2.2.cmml">𝒞</mi><mo id="S4.Ex7.m1.3.4.2.1" xref="S4.Ex7.m1.3.4.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex7.m1.3.4.2.3" xref="S4.Ex7.m1.3.4.2.3.cmml">𝒲</mi><mo id="S4.Ex7.m1.3.4.2.1a" xref="S4.Ex7.m1.3.4.2.1.cmml">⁢</mo><mrow id="S4.Ex7.m1.3.4.2.4.2" xref="S4.Ex7.m1.3.4.2.cmml"><mo id="S4.Ex7.m1.3.4.2.4.2.1" stretchy="false" xref="S4.Ex7.m1.3.4.2.cmml">(</mo><mi id="S4.Ex7.m1.1.1" xref="S4.Ex7.m1.1.1.cmml">π</mi><mo id="S4.Ex7.m1.3.4.2.4.2.2" stretchy="false" xref="S4.Ex7.m1.3.4.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex7.m1.3.4.1" xref="S4.Ex7.m1.3.4.1.cmml">+</mo><mrow id="S4.Ex7.m1.3.4.3" xref="S4.Ex7.m1.3.4.3.cmml"><mstyle displaystyle="true" id="S4.Ex7.m1.3.4.3.2" xref="S4.Ex7.m1.3.4.3.2.cmml"><mfrac id="S4.Ex7.m1.3.4.3.2a" xref="S4.Ex7.m1.3.4.3.2.cmml"><mn id="S4.Ex7.m1.3.4.3.2.2" xref="S4.Ex7.m1.3.4.3.2.2.cmml">1</mn><mn id="S4.Ex7.m1.3.4.3.2.3" xref="S4.Ex7.m1.3.4.3.2.3.cmml">2</mn></mfrac></mstyle><mo id="S4.Ex7.m1.3.4.3.1" xref="S4.Ex7.m1.3.4.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex7.m1.3.4.3.3" xref="S4.Ex7.m1.3.4.3.3.cmml">𝒱</mi><mo id="S4.Ex7.m1.3.4.3.1a" xref="S4.Ex7.m1.3.4.3.1.cmml">⁢</mo><mrow id="S4.Ex7.m1.3.4.3.4.2" xref="S4.Ex7.m1.3.4.3.4.1.cmml"><mo id="S4.Ex7.m1.3.4.3.4.2.1" stretchy="false" xref="S4.Ex7.m1.3.4.3.4.1.cmml">(</mo><mi id="S4.Ex7.m1.2.2" xref="S4.Ex7.m1.2.2.cmml">N</mi><mo id="S4.Ex7.m1.3.4.3.4.2.2" xref="S4.Ex7.m1.3.4.3.4.1.cmml">,</mo><mi id="S4.Ex7.m1.3.3" xref="S4.Ex7.m1.3.3.cmml">v</mi><mo id="S4.Ex7.m1.3.4.3.4.2.3" stretchy="false" xref="S4.Ex7.m1.3.4.3.4.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex7.m1.3b"><apply id="S4.Ex7.m1.3.4.cmml" xref="S4.Ex7.m1.3.4"><plus id="S4.Ex7.m1.3.4.1.cmml" xref="S4.Ex7.m1.3.4.1"></plus><apply id="S4.Ex7.m1.3.4.2.cmml" xref="S4.Ex7.m1.3.4.2"><times id="S4.Ex7.m1.3.4.2.1.cmml" xref="S4.Ex7.m1.3.4.2.1"></times><ci id="S4.Ex7.m1.3.4.2.2.cmml" xref="S4.Ex7.m1.3.4.2.2">𝒞</ci><ci id="S4.Ex7.m1.3.4.2.3.cmml" xref="S4.Ex7.m1.3.4.2.3">𝒲</ci><ci id="S4.Ex7.m1.1.1.cmml" xref="S4.Ex7.m1.1.1">𝜋</ci></apply><apply id="S4.Ex7.m1.3.4.3.cmml" xref="S4.Ex7.m1.3.4.3"><times id="S4.Ex7.m1.3.4.3.1.cmml" xref="S4.Ex7.m1.3.4.3.1"></times><apply id="S4.Ex7.m1.3.4.3.2.cmml" xref="S4.Ex7.m1.3.4.3.2"><divide id="S4.Ex7.m1.3.4.3.2.1.cmml" xref="S4.Ex7.m1.3.4.3.2"></divide><cn id="S4.Ex7.m1.3.4.3.2.2.cmml" type="integer" xref="S4.Ex7.m1.3.4.3.2.2">1</cn><cn id="S4.Ex7.m1.3.4.3.2.3.cmml" type="integer" xref="S4.Ex7.m1.3.4.3.2.3">2</cn></apply><ci id="S4.Ex7.m1.3.4.3.3.cmml" xref="S4.Ex7.m1.3.4.3.3">𝒱</ci><interval closure="open" id="S4.Ex7.m1.3.4.3.4.1.cmml" xref="S4.Ex7.m1.3.4.3.4.2"><ci id="S4.Ex7.m1.2.2.cmml" xref="S4.Ex7.m1.2.2">𝑁</ci><ci id="S4.Ex7.m1.3.3.cmml" xref="S4.Ex7.m1.3.3">𝑣</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex7.m1.3c">\displaystyle\mathcal{CW}(\pi)+\frac{1}{2}\mathcal{V}(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex7.m1.3d">caligraphic_C caligraphic_W ( italic_π ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG caligraphic_V ( italic_N , italic_v )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{1}{2}\left[\sum_{i\in N}\left(\sum_{j\in\pi(i)}v_{i}(j)-% \sum_{j\in N\setminus\pi(i)}v_{i}(j)\right)\right]+\frac{1}{2}\sum_{i\in N}% \sum_{j\in N}v_{i}(j)" class="ltx_Math" display="inline" id="S4.Ex8.m1.6"><semantics id="S4.Ex8.m1.6a"><mrow id="S4.Ex8.m1.6.6" xref="S4.Ex8.m1.6.6.cmml"><mi id="S4.Ex8.m1.6.6.3" xref="S4.Ex8.m1.6.6.3.cmml"></mi><mo id="S4.Ex8.m1.6.6.2" xref="S4.Ex8.m1.6.6.2.cmml">=</mo><mrow id="S4.Ex8.m1.6.6.1" xref="S4.Ex8.m1.6.6.1.cmml"><mrow id="S4.Ex8.m1.6.6.1.1" xref="S4.Ex8.m1.6.6.1.1.cmml"><mstyle displaystyle="true" id="S4.Ex8.m1.6.6.1.1.3" xref="S4.Ex8.m1.6.6.1.1.3.cmml"><mfrac id="S4.Ex8.m1.6.6.1.1.3a" xref="S4.Ex8.m1.6.6.1.1.3.cmml"><mn id="S4.Ex8.m1.6.6.1.1.3.2" xref="S4.Ex8.m1.6.6.1.1.3.2.cmml">1</mn><mn id="S4.Ex8.m1.6.6.1.1.3.3" xref="S4.Ex8.m1.6.6.1.1.3.3.cmml">2</mn></mfrac></mstyle><mo id="S4.Ex8.m1.6.6.1.1.2" xref="S4.Ex8.m1.6.6.1.1.2.cmml">⁢</mo><mrow id="S4.Ex8.m1.6.6.1.1.1.1" xref="S4.Ex8.m1.6.6.1.1.1.2.cmml"><mo id="S4.Ex8.m1.6.6.1.1.1.1.2" xref="S4.Ex8.m1.6.6.1.1.1.2.1.cmml">[</mo><mrow id="S4.Ex8.m1.6.6.1.1.1.1.1" xref="S4.Ex8.m1.6.6.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="S4.Ex8.m1.6.6.1.1.1.1.1.2" xref="S4.Ex8.m1.6.6.1.1.1.1.1.2.cmml"><munder id="S4.Ex8.m1.6.6.1.1.1.1.1.2a" xref="S4.Ex8.m1.6.6.1.1.1.1.1.2.cmml"><mo id="S4.Ex8.m1.6.6.1.1.1.1.1.2.2" movablelimits="false" xref="S4.Ex8.m1.6.6.1.1.1.1.1.2.2.cmml">∑</mo><mrow id="S4.Ex8.m1.6.6.1.1.1.1.1.2.3" xref="S4.Ex8.m1.6.6.1.1.1.1.1.2.3.cmml"><mi id="S4.Ex8.m1.6.6.1.1.1.1.1.2.3.2" xref="S4.Ex8.m1.6.6.1.1.1.1.1.2.3.2.cmml">i</mi><mo id="S4.Ex8.m1.6.6.1.1.1.1.1.2.3.1" xref="S4.Ex8.m1.6.6.1.1.1.1.1.2.3.1.cmml">∈</mo><mi id="S4.Ex8.m1.6.6.1.1.1.1.1.2.3.3" xref="S4.Ex8.m1.6.6.1.1.1.1.1.2.3.3.cmml">N</mi></mrow></munder></mstyle><mrow id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.2" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.cmml"><mrow id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.cmml"><mstyle displaystyle="true" id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.1" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.1.cmml"><munder id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.1a" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.1.cmml"><mo id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.1.2" movablelimits="false" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.1.2.cmml">∑</mo><mrow id="S4.Ex8.m1.1.1.1" xref="S4.Ex8.m1.1.1.1.cmml"><mi id="S4.Ex8.m1.1.1.1.3" xref="S4.Ex8.m1.1.1.1.3.cmml">j</mi><mo id="S4.Ex8.m1.1.1.1.2" xref="S4.Ex8.m1.1.1.1.2.cmml">∈</mo><mrow id="S4.Ex8.m1.1.1.1.4" xref="S4.Ex8.m1.1.1.1.4.cmml"><mi id="S4.Ex8.m1.1.1.1.4.2" xref="S4.Ex8.m1.1.1.1.4.2.cmml">π</mi><mo id="S4.Ex8.m1.1.1.1.4.1" xref="S4.Ex8.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S4.Ex8.m1.1.1.1.4.3.2" xref="S4.Ex8.m1.1.1.1.4.cmml"><mo id="S4.Ex8.m1.1.1.1.4.3.2.1" stretchy="false" xref="S4.Ex8.m1.1.1.1.4.cmml">(</mo><mi id="S4.Ex8.m1.1.1.1.1" xref="S4.Ex8.m1.1.1.1.1.cmml">i</mi><mo id="S4.Ex8.m1.1.1.1.4.3.2.2" stretchy="false" xref="S4.Ex8.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder></mstyle><mrow id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.cmml"><msub id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.2" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.2.cmml"><mi id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.2.2" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.2.2.cmml">v</mi><mi id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.2.3" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.2.3.cmml">i</mi></msub><mo id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.1" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.1.cmml">⁢</mo><mrow id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.3.2" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.cmml"><mo id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.3.2.1" stretchy="false" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.cmml">(</mo><mi id="S4.Ex8.m1.3.3" xref="S4.Ex8.m1.3.3.cmml">j</mi><mo id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.3.2.2" stretchy="false" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.1" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.cmml"><mstyle displaystyle="true" id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.1" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.1.cmml"><munder id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.1a" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.1.cmml"><mo id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.1.2" movablelimits="false" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.1.2.cmml">∑</mo><mrow id="S4.Ex8.m1.2.2.1" xref="S4.Ex8.m1.2.2.1.cmml"><mi id="S4.Ex8.m1.2.2.1.3" xref="S4.Ex8.m1.2.2.1.3.cmml">j</mi><mo id="S4.Ex8.m1.2.2.1.2" xref="S4.Ex8.m1.2.2.1.2.cmml">∈</mo><mrow id="S4.Ex8.m1.2.2.1.4" xref="S4.Ex8.m1.2.2.1.4.cmml"><mi id="S4.Ex8.m1.2.2.1.4.2" xref="S4.Ex8.m1.2.2.1.4.2.cmml">N</mi><mo id="S4.Ex8.m1.2.2.1.4.1" xref="S4.Ex8.m1.2.2.1.4.1.cmml">∖</mo><mrow id="S4.Ex8.m1.2.2.1.4.3" xref="S4.Ex8.m1.2.2.1.4.3.cmml"><mi id="S4.Ex8.m1.2.2.1.4.3.2" xref="S4.Ex8.m1.2.2.1.4.3.2.cmml">π</mi><mo id="S4.Ex8.m1.2.2.1.4.3.1" xref="S4.Ex8.m1.2.2.1.4.3.1.cmml">⁢</mo><mrow id="S4.Ex8.m1.2.2.1.4.3.3.2" xref="S4.Ex8.m1.2.2.1.4.3.cmml"><mo id="S4.Ex8.m1.2.2.1.4.3.3.2.1" stretchy="false" xref="S4.Ex8.m1.2.2.1.4.3.cmml">(</mo><mi id="S4.Ex8.m1.2.2.1.1" xref="S4.Ex8.m1.2.2.1.1.cmml">i</mi><mo id="S4.Ex8.m1.2.2.1.4.3.3.2.2" stretchy="false" xref="S4.Ex8.m1.2.2.1.4.3.cmml">)</mo></mrow></mrow></mrow></mrow></munder></mstyle><mrow id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.cmml"><msub id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.2" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.2.cmml"><mi id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.2.2" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.2.2.cmml">v</mi><mi id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.2.3" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.2.3.cmml">i</mi></msub><mo id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.1" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.3.2" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.cmml"><mo id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.3.2.1" stretchy="false" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.cmml">(</mo><mi id="S4.Ex8.m1.4.4" xref="S4.Ex8.m1.4.4.cmml">j</mi><mo id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.3.2.2" stretchy="false" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.3" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex8.m1.6.6.1.1.1.1.3" xref="S4.Ex8.m1.6.6.1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S4.Ex8.m1.6.6.1.2" xref="S4.Ex8.m1.6.6.1.2.cmml">+</mo><mrow id="S4.Ex8.m1.6.6.1.3" xref="S4.Ex8.m1.6.6.1.3.cmml"><mstyle displaystyle="true" id="S4.Ex8.m1.6.6.1.3.2" xref="S4.Ex8.m1.6.6.1.3.2.cmml"><mfrac id="S4.Ex8.m1.6.6.1.3.2a" xref="S4.Ex8.m1.6.6.1.3.2.cmml"><mn id="S4.Ex8.m1.6.6.1.3.2.2" xref="S4.Ex8.m1.6.6.1.3.2.2.cmml">1</mn><mn id="S4.Ex8.m1.6.6.1.3.2.3" xref="S4.Ex8.m1.6.6.1.3.2.3.cmml">2</mn></mfrac></mstyle><mo id="S4.Ex8.m1.6.6.1.3.1" xref="S4.Ex8.m1.6.6.1.3.1.cmml">⁢</mo><mrow id="S4.Ex8.m1.6.6.1.3.3" xref="S4.Ex8.m1.6.6.1.3.3.cmml"><mstyle displaystyle="true" id="S4.Ex8.m1.6.6.1.3.3.1" xref="S4.Ex8.m1.6.6.1.3.3.1.cmml"><munder id="S4.Ex8.m1.6.6.1.3.3.1a" xref="S4.Ex8.m1.6.6.1.3.3.1.cmml"><mo id="S4.Ex8.m1.6.6.1.3.3.1.2" movablelimits="false" xref="S4.Ex8.m1.6.6.1.3.3.1.2.cmml">∑</mo><mrow id="S4.Ex8.m1.6.6.1.3.3.1.3" xref="S4.Ex8.m1.6.6.1.3.3.1.3.cmml"><mi id="S4.Ex8.m1.6.6.1.3.3.1.3.2" xref="S4.Ex8.m1.6.6.1.3.3.1.3.2.cmml">i</mi><mo id="S4.Ex8.m1.6.6.1.3.3.1.3.1" xref="S4.Ex8.m1.6.6.1.3.3.1.3.1.cmml">∈</mo><mi id="S4.Ex8.m1.6.6.1.3.3.1.3.3" xref="S4.Ex8.m1.6.6.1.3.3.1.3.3.cmml">N</mi></mrow></munder></mstyle><mrow id="S4.Ex8.m1.6.6.1.3.3.2" xref="S4.Ex8.m1.6.6.1.3.3.2.cmml"><mstyle displaystyle="true" id="S4.Ex8.m1.6.6.1.3.3.2.1" xref="S4.Ex8.m1.6.6.1.3.3.2.1.cmml"><munder id="S4.Ex8.m1.6.6.1.3.3.2.1a" xref="S4.Ex8.m1.6.6.1.3.3.2.1.cmml"><mo id="S4.Ex8.m1.6.6.1.3.3.2.1.2" movablelimits="false" xref="S4.Ex8.m1.6.6.1.3.3.2.1.2.cmml">∑</mo><mrow id="S4.Ex8.m1.6.6.1.3.3.2.1.3" xref="S4.Ex8.m1.6.6.1.3.3.2.1.3.cmml"><mi id="S4.Ex8.m1.6.6.1.3.3.2.1.3.2" xref="S4.Ex8.m1.6.6.1.3.3.2.1.3.2.cmml">j</mi><mo id="S4.Ex8.m1.6.6.1.3.3.2.1.3.1" xref="S4.Ex8.m1.6.6.1.3.3.2.1.3.1.cmml">∈</mo><mi id="S4.Ex8.m1.6.6.1.3.3.2.1.3.3" xref="S4.Ex8.m1.6.6.1.3.3.2.1.3.3.cmml">N</mi></mrow></munder></mstyle><mrow id="S4.Ex8.m1.6.6.1.3.3.2.2" xref="S4.Ex8.m1.6.6.1.3.3.2.2.cmml"><msub id="S4.Ex8.m1.6.6.1.3.3.2.2.2" xref="S4.Ex8.m1.6.6.1.3.3.2.2.2.cmml"><mi id="S4.Ex8.m1.6.6.1.3.3.2.2.2.2" xref="S4.Ex8.m1.6.6.1.3.3.2.2.2.2.cmml">v</mi><mi id="S4.Ex8.m1.6.6.1.3.3.2.2.2.3" xref="S4.Ex8.m1.6.6.1.3.3.2.2.2.3.cmml">i</mi></msub><mo id="S4.Ex8.m1.6.6.1.3.3.2.2.1" xref="S4.Ex8.m1.6.6.1.3.3.2.2.1.cmml">⁢</mo><mrow id="S4.Ex8.m1.6.6.1.3.3.2.2.3.2" xref="S4.Ex8.m1.6.6.1.3.3.2.2.cmml"><mo id="S4.Ex8.m1.6.6.1.3.3.2.2.3.2.1" stretchy="false" xref="S4.Ex8.m1.6.6.1.3.3.2.2.cmml">(</mo><mi id="S4.Ex8.m1.5.5" xref="S4.Ex8.m1.5.5.cmml">j</mi><mo id="S4.Ex8.m1.6.6.1.3.3.2.2.3.2.2" stretchy="false" xref="S4.Ex8.m1.6.6.1.3.3.2.2.cmml">)</mo></mrow></mrow></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex8.m1.6b"><apply id="S4.Ex8.m1.6.6.cmml" xref="S4.Ex8.m1.6.6"><eq id="S4.Ex8.m1.6.6.2.cmml" xref="S4.Ex8.m1.6.6.2"></eq><csymbol cd="latexml" id="S4.Ex8.m1.6.6.3.cmml" xref="S4.Ex8.m1.6.6.3">absent</csymbol><apply id="S4.Ex8.m1.6.6.1.cmml" xref="S4.Ex8.m1.6.6.1"><plus id="S4.Ex8.m1.6.6.1.2.cmml" xref="S4.Ex8.m1.6.6.1.2"></plus><apply id="S4.Ex8.m1.6.6.1.1.cmml" xref="S4.Ex8.m1.6.6.1.1"><times id="S4.Ex8.m1.6.6.1.1.2.cmml" xref="S4.Ex8.m1.6.6.1.1.2"></times><apply id="S4.Ex8.m1.6.6.1.1.3.cmml" xref="S4.Ex8.m1.6.6.1.1.3"><divide id="S4.Ex8.m1.6.6.1.1.3.1.cmml" xref="S4.Ex8.m1.6.6.1.1.3"></divide><cn id="S4.Ex8.m1.6.6.1.1.3.2.cmml" type="integer" xref="S4.Ex8.m1.6.6.1.1.3.2">1</cn><cn id="S4.Ex8.m1.6.6.1.1.3.3.cmml" type="integer" xref="S4.Ex8.m1.6.6.1.1.3.3">2</cn></apply><apply id="S4.Ex8.m1.6.6.1.1.1.2.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1"><csymbol cd="latexml" id="S4.Ex8.m1.6.6.1.1.1.2.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.2">delimited-[]</csymbol><apply id="S4.Ex8.m1.6.6.1.1.1.1.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1"><apply id="S4.Ex8.m1.6.6.1.1.1.1.1.2.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.Ex8.m1.6.6.1.1.1.1.1.2.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.2">subscript</csymbol><sum id="S4.Ex8.m1.6.6.1.1.1.1.1.2.2.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.2.2"></sum><apply id="S4.Ex8.m1.6.6.1.1.1.1.1.2.3.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.2.3"><in id="S4.Ex8.m1.6.6.1.1.1.1.1.2.3.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.2.3.1"></in><ci id="S4.Ex8.m1.6.6.1.1.1.1.1.2.3.2.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.2.3.2">𝑖</ci><ci id="S4.Ex8.m1.6.6.1.1.1.1.1.2.3.3.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.2.3.3">𝑁</ci></apply></apply><apply id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1"><minus id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.1"></minus><apply id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2"><apply id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.1"><csymbol cd="ambiguous" id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.1.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.1">subscript</csymbol><sum id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.1.2.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.1.2"></sum><apply id="S4.Ex8.m1.1.1.1.cmml" xref="S4.Ex8.m1.1.1.1"><in id="S4.Ex8.m1.1.1.1.2.cmml" xref="S4.Ex8.m1.1.1.1.2"></in><ci id="S4.Ex8.m1.1.1.1.3.cmml" xref="S4.Ex8.m1.1.1.1.3">𝑗</ci><apply id="S4.Ex8.m1.1.1.1.4.cmml" xref="S4.Ex8.m1.1.1.1.4"><times id="S4.Ex8.m1.1.1.1.4.1.cmml" xref="S4.Ex8.m1.1.1.1.4.1"></times><ci id="S4.Ex8.m1.1.1.1.4.2.cmml" xref="S4.Ex8.m1.1.1.1.4.2">𝜋</ci><ci id="S4.Ex8.m1.1.1.1.1.cmml" xref="S4.Ex8.m1.1.1.1.1">𝑖</ci></apply></apply></apply><apply id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2"><times id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.1"></times><apply id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.2.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.2.2.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.2.2">𝑣</ci><ci id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.2.3.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.2.2.2.3">𝑖</ci></apply><ci id="S4.Ex8.m1.3.3.cmml" xref="S4.Ex8.m1.3.3">𝑗</ci></apply></apply><apply id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3"><apply id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.1.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.1">subscript</csymbol><sum id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.1.2.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.1.2"></sum><apply id="S4.Ex8.m1.2.2.1.cmml" xref="S4.Ex8.m1.2.2.1"><in id="S4.Ex8.m1.2.2.1.2.cmml" xref="S4.Ex8.m1.2.2.1.2"></in><ci id="S4.Ex8.m1.2.2.1.3.cmml" xref="S4.Ex8.m1.2.2.1.3">𝑗</ci><apply id="S4.Ex8.m1.2.2.1.4.cmml" xref="S4.Ex8.m1.2.2.1.4"><setdiff id="S4.Ex8.m1.2.2.1.4.1.cmml" xref="S4.Ex8.m1.2.2.1.4.1"></setdiff><ci id="S4.Ex8.m1.2.2.1.4.2.cmml" xref="S4.Ex8.m1.2.2.1.4.2">𝑁</ci><apply id="S4.Ex8.m1.2.2.1.4.3.cmml" xref="S4.Ex8.m1.2.2.1.4.3"><times id="S4.Ex8.m1.2.2.1.4.3.1.cmml" xref="S4.Ex8.m1.2.2.1.4.3.1"></times><ci id="S4.Ex8.m1.2.2.1.4.3.2.cmml" xref="S4.Ex8.m1.2.2.1.4.3.2">𝜋</ci><ci id="S4.Ex8.m1.2.2.1.1.cmml" xref="S4.Ex8.m1.2.2.1.1">𝑖</ci></apply></apply></apply></apply><apply id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2"><times id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.1"></times><apply id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.2.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.2.1.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.2">subscript</csymbol><ci id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.2.2.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.2.2">𝑣</ci><ci id="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.2.3.cmml" xref="S4.Ex8.m1.6.6.1.1.1.1.1.1.1.1.3.2.2.3">𝑖</ci></apply><ci id="S4.Ex8.m1.4.4.cmml" xref="S4.Ex8.m1.4.4">𝑗</ci></apply></apply></apply></apply></apply></apply><apply id="S4.Ex8.m1.6.6.1.3.cmml" xref="S4.Ex8.m1.6.6.1.3"><times id="S4.Ex8.m1.6.6.1.3.1.cmml" xref="S4.Ex8.m1.6.6.1.3.1"></times><apply id="S4.Ex8.m1.6.6.1.3.2.cmml" xref="S4.Ex8.m1.6.6.1.3.2"><divide id="S4.Ex8.m1.6.6.1.3.2.1.cmml" xref="S4.Ex8.m1.6.6.1.3.2"></divide><cn id="S4.Ex8.m1.6.6.1.3.2.2.cmml" type="integer" xref="S4.Ex8.m1.6.6.1.3.2.2">1</cn><cn id="S4.Ex8.m1.6.6.1.3.2.3.cmml" type="integer" xref="S4.Ex8.m1.6.6.1.3.2.3">2</cn></apply><apply id="S4.Ex8.m1.6.6.1.3.3.cmml" xref="S4.Ex8.m1.6.6.1.3.3"><apply id="S4.Ex8.m1.6.6.1.3.3.1.cmml" xref="S4.Ex8.m1.6.6.1.3.3.1"><csymbol cd="ambiguous" id="S4.Ex8.m1.6.6.1.3.3.1.1.cmml" xref="S4.Ex8.m1.6.6.1.3.3.1">subscript</csymbol><sum id="S4.Ex8.m1.6.6.1.3.3.1.2.cmml" xref="S4.Ex8.m1.6.6.1.3.3.1.2"></sum><apply id="S4.Ex8.m1.6.6.1.3.3.1.3.cmml" xref="S4.Ex8.m1.6.6.1.3.3.1.3"><in id="S4.Ex8.m1.6.6.1.3.3.1.3.1.cmml" xref="S4.Ex8.m1.6.6.1.3.3.1.3.1"></in><ci id="S4.Ex8.m1.6.6.1.3.3.1.3.2.cmml" xref="S4.Ex8.m1.6.6.1.3.3.1.3.2">𝑖</ci><ci id="S4.Ex8.m1.6.6.1.3.3.1.3.3.cmml" xref="S4.Ex8.m1.6.6.1.3.3.1.3.3">𝑁</ci></apply></apply><apply id="S4.Ex8.m1.6.6.1.3.3.2.cmml" xref="S4.Ex8.m1.6.6.1.3.3.2"><apply id="S4.Ex8.m1.6.6.1.3.3.2.1.cmml" xref="S4.Ex8.m1.6.6.1.3.3.2.1"><csymbol cd="ambiguous" id="S4.Ex8.m1.6.6.1.3.3.2.1.1.cmml" xref="S4.Ex8.m1.6.6.1.3.3.2.1">subscript</csymbol><sum id="S4.Ex8.m1.6.6.1.3.3.2.1.2.cmml" xref="S4.Ex8.m1.6.6.1.3.3.2.1.2"></sum><apply id="S4.Ex8.m1.6.6.1.3.3.2.1.3.cmml" xref="S4.Ex8.m1.6.6.1.3.3.2.1.3"><in id="S4.Ex8.m1.6.6.1.3.3.2.1.3.1.cmml" xref="S4.Ex8.m1.6.6.1.3.3.2.1.3.1"></in><ci id="S4.Ex8.m1.6.6.1.3.3.2.1.3.2.cmml" xref="S4.Ex8.m1.6.6.1.3.3.2.1.3.2">𝑗</ci><ci id="S4.Ex8.m1.6.6.1.3.3.2.1.3.3.cmml" xref="S4.Ex8.m1.6.6.1.3.3.2.1.3.3">𝑁</ci></apply></apply><apply id="S4.Ex8.m1.6.6.1.3.3.2.2.cmml" xref="S4.Ex8.m1.6.6.1.3.3.2.2"><times id="S4.Ex8.m1.6.6.1.3.3.2.2.1.cmml" xref="S4.Ex8.m1.6.6.1.3.3.2.2.1"></times><apply id="S4.Ex8.m1.6.6.1.3.3.2.2.2.cmml" xref="S4.Ex8.m1.6.6.1.3.3.2.2.2"><csymbol cd="ambiguous" id="S4.Ex8.m1.6.6.1.3.3.2.2.2.1.cmml" xref="S4.Ex8.m1.6.6.1.3.3.2.2.2">subscript</csymbol><ci id="S4.Ex8.m1.6.6.1.3.3.2.2.2.2.cmml" xref="S4.Ex8.m1.6.6.1.3.3.2.2.2.2">𝑣</ci><ci id="S4.Ex8.m1.6.6.1.3.3.2.2.2.3.cmml" xref="S4.Ex8.m1.6.6.1.3.3.2.2.2.3">𝑖</ci></apply><ci id="S4.Ex8.m1.5.5.cmml" xref="S4.Ex8.m1.5.5">𝑗</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex8.m1.6c">\displaystyle=\frac{1}{2}\left[\sum_{i\in N}\left(\sum_{j\in\pi(i)}v_{i}(j)-% \sum_{j\in N\setminus\pi(i)}v_{i}(j)\right)\right]+\frac{1}{2}\sum_{i\in N}% \sum_{j\in N}v_{i}(j)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex8.m1.6d">= divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ ∑ start_POSTSUBSCRIPT italic_i ∈ italic_N end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_j ∈ italic_π ( italic_i ) end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j ) - ∑ start_POSTSUBSCRIPT italic_j ∈ italic_N ∖ italic_π ( italic_i ) end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j ) ) ] + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_i ∈ italic_N end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_j ∈ italic_N end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{i\in N}\sum_{j\in\pi(i)}v_{i}(j)=\mathcal{SW}(\pi)\text{.}" class="ltx_Math" display="inline" id="S4.Ex9.m1.3"><semantics id="S4.Ex9.m1.3a"><mrow id="S4.Ex9.m1.3.4" xref="S4.Ex9.m1.3.4.cmml"><mi id="S4.Ex9.m1.3.4.2" xref="S4.Ex9.m1.3.4.2.cmml"></mi><mo id="S4.Ex9.m1.3.4.3" xref="S4.Ex9.m1.3.4.3.cmml">=</mo><mrow id="S4.Ex9.m1.3.4.4" xref="S4.Ex9.m1.3.4.4.cmml"><mstyle displaystyle="true" id="S4.Ex9.m1.3.4.4.1" xref="S4.Ex9.m1.3.4.4.1.cmml"><munder id="S4.Ex9.m1.3.4.4.1a" xref="S4.Ex9.m1.3.4.4.1.cmml"><mo id="S4.Ex9.m1.3.4.4.1.2" movablelimits="false" xref="S4.Ex9.m1.3.4.4.1.2.cmml">∑</mo><mrow id="S4.Ex9.m1.3.4.4.1.3" xref="S4.Ex9.m1.3.4.4.1.3.cmml"><mi id="S4.Ex9.m1.3.4.4.1.3.2" xref="S4.Ex9.m1.3.4.4.1.3.2.cmml">i</mi><mo id="S4.Ex9.m1.3.4.4.1.3.1" xref="S4.Ex9.m1.3.4.4.1.3.1.cmml">∈</mo><mi id="S4.Ex9.m1.3.4.4.1.3.3" xref="S4.Ex9.m1.3.4.4.1.3.3.cmml">N</mi></mrow></munder></mstyle><mrow id="S4.Ex9.m1.3.4.4.2" xref="S4.Ex9.m1.3.4.4.2.cmml"><mstyle displaystyle="true" id="S4.Ex9.m1.3.4.4.2.1" xref="S4.Ex9.m1.3.4.4.2.1.cmml"><munder id="S4.Ex9.m1.3.4.4.2.1a" xref="S4.Ex9.m1.3.4.4.2.1.cmml"><mo id="S4.Ex9.m1.3.4.4.2.1.2" movablelimits="false" xref="S4.Ex9.m1.3.4.4.2.1.2.cmml">∑</mo><mrow id="S4.Ex9.m1.1.1.1" xref="S4.Ex9.m1.1.1.1.cmml"><mi id="S4.Ex9.m1.1.1.1.3" xref="S4.Ex9.m1.1.1.1.3.cmml">j</mi><mo id="S4.Ex9.m1.1.1.1.2" xref="S4.Ex9.m1.1.1.1.2.cmml">∈</mo><mrow id="S4.Ex9.m1.1.1.1.4" xref="S4.Ex9.m1.1.1.1.4.cmml"><mi id="S4.Ex9.m1.1.1.1.4.2" xref="S4.Ex9.m1.1.1.1.4.2.cmml">π</mi><mo id="S4.Ex9.m1.1.1.1.4.1" xref="S4.Ex9.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S4.Ex9.m1.1.1.1.4.3.2" xref="S4.Ex9.m1.1.1.1.4.cmml"><mo id="S4.Ex9.m1.1.1.1.4.3.2.1" stretchy="false" xref="S4.Ex9.m1.1.1.1.4.cmml">(</mo><mi id="S4.Ex9.m1.1.1.1.1" xref="S4.Ex9.m1.1.1.1.1.cmml">i</mi><mo id="S4.Ex9.m1.1.1.1.4.3.2.2" stretchy="false" xref="S4.Ex9.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder></mstyle><mrow id="S4.Ex9.m1.3.4.4.2.2" xref="S4.Ex9.m1.3.4.4.2.2.cmml"><msub id="S4.Ex9.m1.3.4.4.2.2.2" xref="S4.Ex9.m1.3.4.4.2.2.2.cmml"><mi id="S4.Ex9.m1.3.4.4.2.2.2.2" xref="S4.Ex9.m1.3.4.4.2.2.2.2.cmml">v</mi><mi id="S4.Ex9.m1.3.4.4.2.2.2.3" xref="S4.Ex9.m1.3.4.4.2.2.2.3.cmml">i</mi></msub><mo id="S4.Ex9.m1.3.4.4.2.2.1" xref="S4.Ex9.m1.3.4.4.2.2.1.cmml">⁢</mo><mrow id="S4.Ex9.m1.3.4.4.2.2.3.2" xref="S4.Ex9.m1.3.4.4.2.2.cmml"><mo id="S4.Ex9.m1.3.4.4.2.2.3.2.1" stretchy="false" xref="S4.Ex9.m1.3.4.4.2.2.cmml">(</mo><mi id="S4.Ex9.m1.2.2" xref="S4.Ex9.m1.2.2.cmml">j</mi><mo id="S4.Ex9.m1.3.4.4.2.2.3.2.2" stretchy="false" xref="S4.Ex9.m1.3.4.4.2.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S4.Ex9.m1.3.4.5" xref="S4.Ex9.m1.3.4.5.cmml">=</mo><mrow id="S4.Ex9.m1.3.4.6" xref="S4.Ex9.m1.3.4.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex9.m1.3.4.6.2" xref="S4.Ex9.m1.3.4.6.2.cmml">𝒮</mi><mo id="S4.Ex9.m1.3.4.6.1" xref="S4.Ex9.m1.3.4.6.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex9.m1.3.4.6.3" xref="S4.Ex9.m1.3.4.6.3.cmml">𝒲</mi><mo id="S4.Ex9.m1.3.4.6.1a" xref="S4.Ex9.m1.3.4.6.1.cmml">⁢</mo><mrow id="S4.Ex9.m1.3.4.6.4.2" xref="S4.Ex9.m1.3.4.6.cmml"><mo id="S4.Ex9.m1.3.4.6.4.2.1" stretchy="false" xref="S4.Ex9.m1.3.4.6.cmml">(</mo><mi id="S4.Ex9.m1.3.3" xref="S4.Ex9.m1.3.3.cmml">π</mi><mo id="S4.Ex9.m1.3.4.6.4.2.2" stretchy="false" xref="S4.Ex9.m1.3.4.6.cmml">)</mo></mrow><mo id="S4.Ex9.m1.3.4.6.1b" xref="S4.Ex9.m1.3.4.6.1.cmml">⁢</mo><mtext id="S4.Ex9.m1.3.4.6.5" xref="S4.Ex9.m1.3.4.6.5a.cmml">.</mtext></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex9.m1.3b"><apply id="S4.Ex9.m1.3.4.cmml" xref="S4.Ex9.m1.3.4"><and id="S4.Ex9.m1.3.4a.cmml" xref="S4.Ex9.m1.3.4"></and><apply id="S4.Ex9.m1.3.4b.cmml" xref="S4.Ex9.m1.3.4"><eq id="S4.Ex9.m1.3.4.3.cmml" xref="S4.Ex9.m1.3.4.3"></eq><csymbol cd="latexml" id="S4.Ex9.m1.3.4.2.cmml" xref="S4.Ex9.m1.3.4.2">absent</csymbol><apply id="S4.Ex9.m1.3.4.4.cmml" xref="S4.Ex9.m1.3.4.4"><apply id="S4.Ex9.m1.3.4.4.1.cmml" xref="S4.Ex9.m1.3.4.4.1"><csymbol cd="ambiguous" id="S4.Ex9.m1.3.4.4.1.1.cmml" xref="S4.Ex9.m1.3.4.4.1">subscript</csymbol><sum id="S4.Ex9.m1.3.4.4.1.2.cmml" xref="S4.Ex9.m1.3.4.4.1.2"></sum><apply id="S4.Ex9.m1.3.4.4.1.3.cmml" xref="S4.Ex9.m1.3.4.4.1.3"><in id="S4.Ex9.m1.3.4.4.1.3.1.cmml" xref="S4.Ex9.m1.3.4.4.1.3.1"></in><ci id="S4.Ex9.m1.3.4.4.1.3.2.cmml" xref="S4.Ex9.m1.3.4.4.1.3.2">𝑖</ci><ci id="S4.Ex9.m1.3.4.4.1.3.3.cmml" xref="S4.Ex9.m1.3.4.4.1.3.3">𝑁</ci></apply></apply><apply id="S4.Ex9.m1.3.4.4.2.cmml" xref="S4.Ex9.m1.3.4.4.2"><apply id="S4.Ex9.m1.3.4.4.2.1.cmml" xref="S4.Ex9.m1.3.4.4.2.1"><csymbol cd="ambiguous" id="S4.Ex9.m1.3.4.4.2.1.1.cmml" xref="S4.Ex9.m1.3.4.4.2.1">subscript</csymbol><sum id="S4.Ex9.m1.3.4.4.2.1.2.cmml" xref="S4.Ex9.m1.3.4.4.2.1.2"></sum><apply id="S4.Ex9.m1.1.1.1.cmml" xref="S4.Ex9.m1.1.1.1"><in id="S4.Ex9.m1.1.1.1.2.cmml" xref="S4.Ex9.m1.1.1.1.2"></in><ci id="S4.Ex9.m1.1.1.1.3.cmml" xref="S4.Ex9.m1.1.1.1.3">𝑗</ci><apply id="S4.Ex9.m1.1.1.1.4.cmml" xref="S4.Ex9.m1.1.1.1.4"><times id="S4.Ex9.m1.1.1.1.4.1.cmml" xref="S4.Ex9.m1.1.1.1.4.1"></times><ci id="S4.Ex9.m1.1.1.1.4.2.cmml" xref="S4.Ex9.m1.1.1.1.4.2">𝜋</ci><ci id="S4.Ex9.m1.1.1.1.1.cmml" xref="S4.Ex9.m1.1.1.1.1">𝑖</ci></apply></apply></apply><apply id="S4.Ex9.m1.3.4.4.2.2.cmml" xref="S4.Ex9.m1.3.4.4.2.2"><times id="S4.Ex9.m1.3.4.4.2.2.1.cmml" xref="S4.Ex9.m1.3.4.4.2.2.1"></times><apply id="S4.Ex9.m1.3.4.4.2.2.2.cmml" xref="S4.Ex9.m1.3.4.4.2.2.2"><csymbol cd="ambiguous" id="S4.Ex9.m1.3.4.4.2.2.2.1.cmml" xref="S4.Ex9.m1.3.4.4.2.2.2">subscript</csymbol><ci id="S4.Ex9.m1.3.4.4.2.2.2.2.cmml" xref="S4.Ex9.m1.3.4.4.2.2.2.2">𝑣</ci><ci id="S4.Ex9.m1.3.4.4.2.2.2.3.cmml" xref="S4.Ex9.m1.3.4.4.2.2.2.3">𝑖</ci></apply><ci id="S4.Ex9.m1.2.2.cmml" xref="S4.Ex9.m1.2.2">𝑗</ci></apply></apply></apply></apply><apply id="S4.Ex9.m1.3.4c.cmml" xref="S4.Ex9.m1.3.4"><eq id="S4.Ex9.m1.3.4.5.cmml" xref="S4.Ex9.m1.3.4.5"></eq><share href="https://arxiv.org/html/2503.06017v1#S4.Ex9.m1.3.4.4.cmml" id="S4.Ex9.m1.3.4d.cmml" xref="S4.Ex9.m1.3.4"></share><apply id="S4.Ex9.m1.3.4.6.cmml" xref="S4.Ex9.m1.3.4.6"><times id="S4.Ex9.m1.3.4.6.1.cmml" xref="S4.Ex9.m1.3.4.6.1"></times><ci id="S4.Ex9.m1.3.4.6.2.cmml" xref="S4.Ex9.m1.3.4.6.2">𝒮</ci><ci id="S4.Ex9.m1.3.4.6.3.cmml" xref="S4.Ex9.m1.3.4.6.3">𝒲</ci><ci id="S4.Ex9.m1.3.3.cmml" xref="S4.Ex9.m1.3.3">𝜋</ci><ci id="S4.Ex9.m1.3.4.6.5a.cmml" xref="S4.Ex9.m1.3.4.6.5"><mtext id="S4.Ex9.m1.3.4.6.5.cmml" xref="S4.Ex9.m1.3.4.6.5">.</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex9.m1.3c">\displaystyle=\sum_{i\in N}\sum_{j\in\pi(i)}v_{i}(j)=\mathcal{SW}(\pi)\text{.}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex9.m1.3d">= ∑ start_POSTSUBSCRIPT italic_i ∈ italic_N end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_j ∈ italic_π ( italic_i ) end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j ) = caligraphic_S caligraphic_W ( italic_π ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.1.p1.3">This proves the desired equation. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS2.p4"> <p class="ltx_p" id="S4.SS2.p4.2">As a consequence, we obtain that the same partitions maximize <math alttext="\mathcal{SW}" class="ltx_Math" display="inline" id="S4.SS2.p4.1.m1.1"><semantics id="S4.SS2.p4.1.m1.1a"><mrow id="S4.SS2.p4.1.m1.1.1" xref="S4.SS2.p4.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p4.1.m1.1.1.2" xref="S4.SS2.p4.1.m1.1.1.2.cmml">𝒮</mi><mo id="S4.SS2.p4.1.m1.1.1.1" xref="S4.SS2.p4.1.m1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p4.1.m1.1.1.3" xref="S4.SS2.p4.1.m1.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.1.m1.1b"><apply id="S4.SS2.p4.1.m1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1"><times id="S4.SS2.p4.1.m1.1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1.1"></times><ci id="S4.SS2.p4.1.m1.1.1.2.cmml" xref="S4.SS2.p4.1.m1.1.1.2">𝒮</ci><ci id="S4.SS2.p4.1.m1.1.1.3.cmml" xref="S4.SS2.p4.1.m1.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.1.m1.1c">\mathcal{SW}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.1.m1.1d">caligraphic_S caligraphic_W</annotation></semantics></math> and <math alttext="\mathcal{CW}" class="ltx_Math" display="inline" id="S4.SS2.p4.2.m2.1"><semantics id="S4.SS2.p4.2.m2.1a"><mrow id="S4.SS2.p4.2.m2.1.1" xref="S4.SS2.p4.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p4.2.m2.1.1.2" xref="S4.SS2.p4.2.m2.1.1.2.cmml">𝒞</mi><mo id="S4.SS2.p4.2.m2.1.1.1" xref="S4.SS2.p4.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p4.2.m2.1.1.3" xref="S4.SS2.p4.2.m2.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.2.m2.1b"><apply id="S4.SS2.p4.2.m2.1.1.cmml" xref="S4.SS2.p4.2.m2.1.1"><times id="S4.SS2.p4.2.m2.1.1.1.cmml" xref="S4.SS2.p4.2.m2.1.1.1"></times><ci id="S4.SS2.p4.2.m2.1.1.2.cmml" xref="S4.SS2.p4.2.m2.1.1.2">𝒞</ci><ci id="S4.SS2.p4.2.m2.1.1.3.cmml" xref="S4.SS2.p4.2.m2.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.2.m2.1c">\mathcal{CW}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.2.m2.1d">caligraphic_C caligraphic_W</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S4.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.1.1.1">Proposition 4.3</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem3.p1"> <p class="ltx_p" id="S4.Thmtheorem3.p1.3"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.p1.3.3">Consider an ASHG <math alttext="(N,v)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.1.1.m1.2"><semantics id="S4.Thmtheorem3.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem3.p1.1.1.m1.2.3.2" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.1.cmml"><mo id="S4.Thmtheorem3.p1.1.1.m1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.1.cmml">(</mo><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml">N</mi><mo id="S4.Thmtheorem3.p1.1.1.m1.2.3.2.2" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem3.p1.1.1.m1.2.2" xref="S4.Thmtheorem3.p1.1.1.m1.2.2.cmml">v</mi><mo id="S4.Thmtheorem3.p1.1.1.m1.2.3.2.3" stretchy="false" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.1.1.m1.2b"><interval closure="open" id="S4.Thmtheorem3.p1.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.3.2"><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1">𝑁</ci><ci id="S4.Thmtheorem3.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.1.1.m1.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.1.1.m1.2d">( italic_N , italic_v )</annotation></semantics></math>. Then a partition maximizes <math alttext="\mathcal{SW}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.2.2.m2.1"><semantics id="S4.Thmtheorem3.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem3.p1.2.2.m2.1.1" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem3.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.2.cmml">𝒮</mi><mo id="S4.Thmtheorem3.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem3.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.2.2.m2.1b"><apply id="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1"><times id="S4.Thmtheorem3.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.1"></times><ci id="S4.Thmtheorem3.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.2">𝒮</ci><ci id="S4.Thmtheorem3.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.2.2.m2.1c">\mathcal{SW}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.2.2.m2.1d">caligraphic_S caligraphic_W</annotation></semantics></math> if and only if it maximizes <math alttext="\mathcal{CW}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.3.3.m3.1"><semantics id="S4.Thmtheorem3.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem3.p1.3.3.m3.1.1" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem3.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.2.cmml">𝒞</mi><mo id="S4.Thmtheorem3.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem3.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.3.3.m3.1b"><apply id="S4.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1"><times id="S4.Thmtheorem3.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.1"></times><ci id="S4.Thmtheorem3.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.2">𝒞</ci><ci id="S4.Thmtheorem3.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.3.3.m3.1c">\mathcal{CW}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.3.3.m3.1d">caligraphic_C caligraphic_W</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.SS2.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.2.p1"> <p class="ltx_p" id="S4.SS2.2.p1.2">Consider two partitions <math alttext="\pi" class="ltx_Math" display="inline" id="S4.SS2.2.p1.1.m1.1"><semantics id="S4.SS2.2.p1.1.m1.1a"><mi id="S4.SS2.2.p1.1.m1.1.1" xref="S4.SS2.2.p1.1.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.1.m1.1b"><ci id="S4.SS2.2.p1.1.m1.1.1.cmml" xref="S4.SS2.2.p1.1.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.1.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.1.m1.1d">italic_π</annotation></semantics></math> and <math alttext="\pi^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.2.p1.2.m2.1"><semantics id="S4.SS2.2.p1.2.m2.1a"><msup id="S4.SS2.2.p1.2.m2.1.1" xref="S4.SS2.2.p1.2.m2.1.1.cmml"><mi id="S4.SS2.2.p1.2.m2.1.1.2" xref="S4.SS2.2.p1.2.m2.1.1.2.cmml">π</mi><mo id="S4.SS2.2.p1.2.m2.1.1.3" xref="S4.SS2.2.p1.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.2.m2.1b"><apply id="S4.SS2.2.p1.2.m2.1.1.cmml" xref="S4.SS2.2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p1.2.m2.1.1.1.cmml" xref="S4.SS2.2.p1.2.m2.1.1">superscript</csymbol><ci id="S4.SS2.2.p1.2.m2.1.1.2.cmml" xref="S4.SS2.2.p1.2.m2.1.1.2">𝜋</ci><ci id="S4.SS2.2.p1.2.m2.1.1.3.cmml" xref="S4.SS2.2.p1.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.2.m2.1c">\pi^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.2.m2.1d">italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.Thmtheorem2" title="Proposition 4.2. ‣ 4.2 Logarithmic Approximation for Nonnegative Total Value ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">4.2</span></a>, it holds that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx2"> <tbody id="S4.Ex10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mathcal{SW}(\pi)-\mathcal{SW}(\pi^{\prime})" class="ltx_Math" display="inline" id="S4.Ex10.m1.2"><semantics id="S4.Ex10.m1.2a"><mrow id="S4.Ex10.m1.2.2" xref="S4.Ex10.m1.2.2.cmml"><mrow id="S4.Ex10.m1.2.2.3" xref="S4.Ex10.m1.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex10.m1.2.2.3.2" xref="S4.Ex10.m1.2.2.3.2.cmml">𝒮</mi><mo id="S4.Ex10.m1.2.2.3.1" xref="S4.Ex10.m1.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex10.m1.2.2.3.3" xref="S4.Ex10.m1.2.2.3.3.cmml">𝒲</mi><mo id="S4.Ex10.m1.2.2.3.1a" xref="S4.Ex10.m1.2.2.3.1.cmml">⁢</mo><mrow id="S4.Ex10.m1.2.2.3.4.2" xref="S4.Ex10.m1.2.2.3.cmml"><mo id="S4.Ex10.m1.2.2.3.4.2.1" stretchy="false" xref="S4.Ex10.m1.2.2.3.cmml">(</mo><mi id="S4.Ex10.m1.1.1" xref="S4.Ex10.m1.1.1.cmml">π</mi><mo id="S4.Ex10.m1.2.2.3.4.2.2" stretchy="false" xref="S4.Ex10.m1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Ex10.m1.2.2.2" xref="S4.Ex10.m1.2.2.2.cmml">−</mo><mrow id="S4.Ex10.m1.2.2.1" xref="S4.Ex10.m1.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex10.m1.2.2.1.3" xref="S4.Ex10.m1.2.2.1.3.cmml">𝒮</mi><mo id="S4.Ex10.m1.2.2.1.2" xref="S4.Ex10.m1.2.2.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex10.m1.2.2.1.4" xref="S4.Ex10.m1.2.2.1.4.cmml">𝒲</mi><mo id="S4.Ex10.m1.2.2.1.2a" xref="S4.Ex10.m1.2.2.1.2.cmml">⁢</mo><mrow id="S4.Ex10.m1.2.2.1.1.1" xref="S4.Ex10.m1.2.2.1.1.1.1.cmml"><mo id="S4.Ex10.m1.2.2.1.1.1.2" stretchy="false" xref="S4.Ex10.m1.2.2.1.1.1.1.cmml">(</mo><msup id="S4.Ex10.m1.2.2.1.1.1.1" xref="S4.Ex10.m1.2.2.1.1.1.1.cmml"><mi id="S4.Ex10.m1.2.2.1.1.1.1.2" xref="S4.Ex10.m1.2.2.1.1.1.1.2.cmml">π</mi><mo id="S4.Ex10.m1.2.2.1.1.1.1.3" xref="S4.Ex10.m1.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.Ex10.m1.2.2.1.1.1.3" stretchy="false" xref="S4.Ex10.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex10.m1.2b"><apply id="S4.Ex10.m1.2.2.cmml" xref="S4.Ex10.m1.2.2"><minus id="S4.Ex10.m1.2.2.2.cmml" xref="S4.Ex10.m1.2.2.2"></minus><apply id="S4.Ex10.m1.2.2.3.cmml" xref="S4.Ex10.m1.2.2.3"><times id="S4.Ex10.m1.2.2.3.1.cmml" xref="S4.Ex10.m1.2.2.3.1"></times><ci id="S4.Ex10.m1.2.2.3.2.cmml" xref="S4.Ex10.m1.2.2.3.2">𝒮</ci><ci id="S4.Ex10.m1.2.2.3.3.cmml" xref="S4.Ex10.m1.2.2.3.3">𝒲</ci><ci id="S4.Ex10.m1.1.1.cmml" xref="S4.Ex10.m1.1.1">𝜋</ci></apply><apply id="S4.Ex10.m1.2.2.1.cmml" xref="S4.Ex10.m1.2.2.1"><times id="S4.Ex10.m1.2.2.1.2.cmml" xref="S4.Ex10.m1.2.2.1.2"></times><ci id="S4.Ex10.m1.2.2.1.3.cmml" xref="S4.Ex10.m1.2.2.1.3">𝒮</ci><ci id="S4.Ex10.m1.2.2.1.4.cmml" xref="S4.Ex10.m1.2.2.1.4">𝒲</ci><apply id="S4.Ex10.m1.2.2.1.1.1.1.cmml" xref="S4.Ex10.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Ex10.m1.2.2.1.1.1.1.1.cmml" xref="S4.Ex10.m1.2.2.1.1.1">superscript</csymbol><ci id="S4.Ex10.m1.2.2.1.1.1.1.2.cmml" xref="S4.Ex10.m1.2.2.1.1.1.1.2">𝜋</ci><ci id="S4.Ex10.m1.2.2.1.1.1.1.3.cmml" xref="S4.Ex10.m1.2.2.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex10.m1.2c">\displaystyle\mathcal{SW}(\pi)-\mathcal{SW}(\pi^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex10.m1.2d">caligraphic_S caligraphic_W ( italic_π ) - caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathcal{CW}(\pi)+\frac{1}{2}\mathcal{V}(N,v)-\mathcal{CW}(\pi^{% \prime})-\frac{1}{2}\mathcal{V}(N,v)" class="ltx_Math" display="inline" id="S4.Ex11.m1.6"><semantics id="S4.Ex11.m1.6a"><mrow id="S4.Ex11.m1.6.6" xref="S4.Ex11.m1.6.6.cmml"><mi id="S4.Ex11.m1.6.6.3" xref="S4.Ex11.m1.6.6.3.cmml"></mi><mo id="S4.Ex11.m1.6.6.2" xref="S4.Ex11.m1.6.6.2.cmml">=</mo><mrow id="S4.Ex11.m1.6.6.1" xref="S4.Ex11.m1.6.6.1.cmml"><mrow id="S4.Ex11.m1.6.6.1.3" xref="S4.Ex11.m1.6.6.1.3.cmml"><mrow id="S4.Ex11.m1.6.6.1.3.2" xref="S4.Ex11.m1.6.6.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex11.m1.6.6.1.3.2.2" xref="S4.Ex11.m1.6.6.1.3.2.2.cmml">𝒞</mi><mo id="S4.Ex11.m1.6.6.1.3.2.1" xref="S4.Ex11.m1.6.6.1.3.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex11.m1.6.6.1.3.2.3" xref="S4.Ex11.m1.6.6.1.3.2.3.cmml">𝒲</mi><mo id="S4.Ex11.m1.6.6.1.3.2.1a" xref="S4.Ex11.m1.6.6.1.3.2.1.cmml">⁢</mo><mrow id="S4.Ex11.m1.6.6.1.3.2.4.2" xref="S4.Ex11.m1.6.6.1.3.2.cmml"><mo id="S4.Ex11.m1.6.6.1.3.2.4.2.1" stretchy="false" xref="S4.Ex11.m1.6.6.1.3.2.cmml">(</mo><mi id="S4.Ex11.m1.1.1" xref="S4.Ex11.m1.1.1.cmml">π</mi><mo id="S4.Ex11.m1.6.6.1.3.2.4.2.2" stretchy="false" xref="S4.Ex11.m1.6.6.1.3.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex11.m1.6.6.1.3.1" xref="S4.Ex11.m1.6.6.1.3.1.cmml">+</mo><mrow id="S4.Ex11.m1.6.6.1.3.3" xref="S4.Ex11.m1.6.6.1.3.3.cmml"><mstyle displaystyle="true" id="S4.Ex11.m1.6.6.1.3.3.2" xref="S4.Ex11.m1.6.6.1.3.3.2.cmml"><mfrac id="S4.Ex11.m1.6.6.1.3.3.2a" xref="S4.Ex11.m1.6.6.1.3.3.2.cmml"><mn id="S4.Ex11.m1.6.6.1.3.3.2.2" xref="S4.Ex11.m1.6.6.1.3.3.2.2.cmml">1</mn><mn id="S4.Ex11.m1.6.6.1.3.3.2.3" xref="S4.Ex11.m1.6.6.1.3.3.2.3.cmml">2</mn></mfrac></mstyle><mo id="S4.Ex11.m1.6.6.1.3.3.1" xref="S4.Ex11.m1.6.6.1.3.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex11.m1.6.6.1.3.3.3" xref="S4.Ex11.m1.6.6.1.3.3.3.cmml">𝒱</mi><mo id="S4.Ex11.m1.6.6.1.3.3.1a" xref="S4.Ex11.m1.6.6.1.3.3.1.cmml">⁢</mo><mrow id="S4.Ex11.m1.6.6.1.3.3.4.2" xref="S4.Ex11.m1.6.6.1.3.3.4.1.cmml"><mo id="S4.Ex11.m1.6.6.1.3.3.4.2.1" stretchy="false" xref="S4.Ex11.m1.6.6.1.3.3.4.1.cmml">(</mo><mi id="S4.Ex11.m1.2.2" xref="S4.Ex11.m1.2.2.cmml">N</mi><mo id="S4.Ex11.m1.6.6.1.3.3.4.2.2" xref="S4.Ex11.m1.6.6.1.3.3.4.1.cmml">,</mo><mi id="S4.Ex11.m1.3.3" xref="S4.Ex11.m1.3.3.cmml">v</mi><mo id="S4.Ex11.m1.6.6.1.3.3.4.2.3" stretchy="false" xref="S4.Ex11.m1.6.6.1.3.3.4.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex11.m1.6.6.1.2" xref="S4.Ex11.m1.6.6.1.2.cmml">−</mo><mrow id="S4.Ex11.m1.6.6.1.1" xref="S4.Ex11.m1.6.6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex11.m1.6.6.1.1.3" xref="S4.Ex11.m1.6.6.1.1.3.cmml">𝒞</mi><mo id="S4.Ex11.m1.6.6.1.1.2" xref="S4.Ex11.m1.6.6.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex11.m1.6.6.1.1.4" xref="S4.Ex11.m1.6.6.1.1.4.cmml">𝒲</mi><mo id="S4.Ex11.m1.6.6.1.1.2a" xref="S4.Ex11.m1.6.6.1.1.2.cmml">⁢</mo><mrow id="S4.Ex11.m1.6.6.1.1.1.1" xref="S4.Ex11.m1.6.6.1.1.1.1.1.cmml"><mo id="S4.Ex11.m1.6.6.1.1.1.1.2" stretchy="false" xref="S4.Ex11.m1.6.6.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex11.m1.6.6.1.1.1.1.1" xref="S4.Ex11.m1.6.6.1.1.1.1.1.cmml"><mi id="S4.Ex11.m1.6.6.1.1.1.1.1.2" xref="S4.Ex11.m1.6.6.1.1.1.1.1.2.cmml">π</mi><mo id="S4.Ex11.m1.6.6.1.1.1.1.1.3" xref="S4.Ex11.m1.6.6.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.Ex11.m1.6.6.1.1.1.1.3" stretchy="false" xref="S4.Ex11.m1.6.6.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex11.m1.6.6.1.2a" xref="S4.Ex11.m1.6.6.1.2.cmml">−</mo><mrow id="S4.Ex11.m1.6.6.1.4" xref="S4.Ex11.m1.6.6.1.4.cmml"><mstyle displaystyle="true" id="S4.Ex11.m1.6.6.1.4.2" xref="S4.Ex11.m1.6.6.1.4.2.cmml"><mfrac id="S4.Ex11.m1.6.6.1.4.2a" xref="S4.Ex11.m1.6.6.1.4.2.cmml"><mn id="S4.Ex11.m1.6.6.1.4.2.2" xref="S4.Ex11.m1.6.6.1.4.2.2.cmml">1</mn><mn id="S4.Ex11.m1.6.6.1.4.2.3" xref="S4.Ex11.m1.6.6.1.4.2.3.cmml">2</mn></mfrac></mstyle><mo id="S4.Ex11.m1.6.6.1.4.1" xref="S4.Ex11.m1.6.6.1.4.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex11.m1.6.6.1.4.3" xref="S4.Ex11.m1.6.6.1.4.3.cmml">𝒱</mi><mo id="S4.Ex11.m1.6.6.1.4.1a" xref="S4.Ex11.m1.6.6.1.4.1.cmml">⁢</mo><mrow id="S4.Ex11.m1.6.6.1.4.4.2" xref="S4.Ex11.m1.6.6.1.4.4.1.cmml"><mo id="S4.Ex11.m1.6.6.1.4.4.2.1" stretchy="false" xref="S4.Ex11.m1.6.6.1.4.4.1.cmml">(</mo><mi id="S4.Ex11.m1.4.4" xref="S4.Ex11.m1.4.4.cmml">N</mi><mo id="S4.Ex11.m1.6.6.1.4.4.2.2" xref="S4.Ex11.m1.6.6.1.4.4.1.cmml">,</mo><mi id="S4.Ex11.m1.5.5" xref="S4.Ex11.m1.5.5.cmml">v</mi><mo id="S4.Ex11.m1.6.6.1.4.4.2.3" stretchy="false" xref="S4.Ex11.m1.6.6.1.4.4.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex11.m1.6b"><apply id="S4.Ex11.m1.6.6.cmml" xref="S4.Ex11.m1.6.6"><eq id="S4.Ex11.m1.6.6.2.cmml" xref="S4.Ex11.m1.6.6.2"></eq><csymbol cd="latexml" id="S4.Ex11.m1.6.6.3.cmml" xref="S4.Ex11.m1.6.6.3">absent</csymbol><apply id="S4.Ex11.m1.6.6.1.cmml" xref="S4.Ex11.m1.6.6.1"><minus id="S4.Ex11.m1.6.6.1.2.cmml" xref="S4.Ex11.m1.6.6.1.2"></minus><apply id="S4.Ex11.m1.6.6.1.3.cmml" xref="S4.Ex11.m1.6.6.1.3"><plus id="S4.Ex11.m1.6.6.1.3.1.cmml" xref="S4.Ex11.m1.6.6.1.3.1"></plus><apply id="S4.Ex11.m1.6.6.1.3.2.cmml" xref="S4.Ex11.m1.6.6.1.3.2"><times id="S4.Ex11.m1.6.6.1.3.2.1.cmml" xref="S4.Ex11.m1.6.6.1.3.2.1"></times><ci id="S4.Ex11.m1.6.6.1.3.2.2.cmml" xref="S4.Ex11.m1.6.6.1.3.2.2">𝒞</ci><ci id="S4.Ex11.m1.6.6.1.3.2.3.cmml" xref="S4.Ex11.m1.6.6.1.3.2.3">𝒲</ci><ci id="S4.Ex11.m1.1.1.cmml" xref="S4.Ex11.m1.1.1">𝜋</ci></apply><apply id="S4.Ex11.m1.6.6.1.3.3.cmml" xref="S4.Ex11.m1.6.6.1.3.3"><times id="S4.Ex11.m1.6.6.1.3.3.1.cmml" xref="S4.Ex11.m1.6.6.1.3.3.1"></times><apply id="S4.Ex11.m1.6.6.1.3.3.2.cmml" xref="S4.Ex11.m1.6.6.1.3.3.2"><divide id="S4.Ex11.m1.6.6.1.3.3.2.1.cmml" xref="S4.Ex11.m1.6.6.1.3.3.2"></divide><cn id="S4.Ex11.m1.6.6.1.3.3.2.2.cmml" type="integer" xref="S4.Ex11.m1.6.6.1.3.3.2.2">1</cn><cn id="S4.Ex11.m1.6.6.1.3.3.2.3.cmml" type="integer" xref="S4.Ex11.m1.6.6.1.3.3.2.3">2</cn></apply><ci id="S4.Ex11.m1.6.6.1.3.3.3.cmml" xref="S4.Ex11.m1.6.6.1.3.3.3">𝒱</ci><interval closure="open" id="S4.Ex11.m1.6.6.1.3.3.4.1.cmml" xref="S4.Ex11.m1.6.6.1.3.3.4.2"><ci id="S4.Ex11.m1.2.2.cmml" xref="S4.Ex11.m1.2.2">𝑁</ci><ci id="S4.Ex11.m1.3.3.cmml" xref="S4.Ex11.m1.3.3">𝑣</ci></interval></apply></apply><apply id="S4.Ex11.m1.6.6.1.1.cmml" xref="S4.Ex11.m1.6.6.1.1"><times id="S4.Ex11.m1.6.6.1.1.2.cmml" xref="S4.Ex11.m1.6.6.1.1.2"></times><ci id="S4.Ex11.m1.6.6.1.1.3.cmml" xref="S4.Ex11.m1.6.6.1.1.3">𝒞</ci><ci id="S4.Ex11.m1.6.6.1.1.4.cmml" xref="S4.Ex11.m1.6.6.1.1.4">𝒲</ci><apply id="S4.Ex11.m1.6.6.1.1.1.1.1.cmml" xref="S4.Ex11.m1.6.6.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex11.m1.6.6.1.1.1.1.1.1.cmml" xref="S4.Ex11.m1.6.6.1.1.1.1">superscript</csymbol><ci id="S4.Ex11.m1.6.6.1.1.1.1.1.2.cmml" xref="S4.Ex11.m1.6.6.1.1.1.1.1.2">𝜋</ci><ci id="S4.Ex11.m1.6.6.1.1.1.1.1.3.cmml" xref="S4.Ex11.m1.6.6.1.1.1.1.1.3">′</ci></apply></apply><apply id="S4.Ex11.m1.6.6.1.4.cmml" xref="S4.Ex11.m1.6.6.1.4"><times id="S4.Ex11.m1.6.6.1.4.1.cmml" xref="S4.Ex11.m1.6.6.1.4.1"></times><apply id="S4.Ex11.m1.6.6.1.4.2.cmml" xref="S4.Ex11.m1.6.6.1.4.2"><divide id="S4.Ex11.m1.6.6.1.4.2.1.cmml" xref="S4.Ex11.m1.6.6.1.4.2"></divide><cn id="S4.Ex11.m1.6.6.1.4.2.2.cmml" type="integer" xref="S4.Ex11.m1.6.6.1.4.2.2">1</cn><cn id="S4.Ex11.m1.6.6.1.4.2.3.cmml" type="integer" xref="S4.Ex11.m1.6.6.1.4.2.3">2</cn></apply><ci id="S4.Ex11.m1.6.6.1.4.3.cmml" xref="S4.Ex11.m1.6.6.1.4.3">𝒱</ci><interval closure="open" id="S4.Ex11.m1.6.6.1.4.4.1.cmml" xref="S4.Ex11.m1.6.6.1.4.4.2"><ci id="S4.Ex11.m1.4.4.cmml" xref="S4.Ex11.m1.4.4">𝑁</ci><ci id="S4.Ex11.m1.5.5.cmml" xref="S4.Ex11.m1.5.5">𝑣</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex11.m1.6c">\displaystyle=\mathcal{CW}(\pi)+\frac{1}{2}\mathcal{V}(N,v)-\mathcal{CW}(\pi^{% \prime})-\frac{1}{2}\mathcal{V}(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex11.m1.6d">= caligraphic_C caligraphic_W ( italic_π ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG caligraphic_V ( italic_N , italic_v ) - caligraphic_C caligraphic_W ( italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG caligraphic_V ( italic_N , italic_v )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathcal{CW}(\pi)-\mathcal{CW}(\pi^{\prime})\text{.}" class="ltx_Math" display="inline" id="S4.Ex12.m1.2"><semantics id="S4.Ex12.m1.2a"><mrow id="S4.Ex12.m1.2.2" xref="S4.Ex12.m1.2.2.cmml"><mi id="S4.Ex12.m1.2.2.3" xref="S4.Ex12.m1.2.2.3.cmml"></mi><mo id="S4.Ex12.m1.2.2.2" xref="S4.Ex12.m1.2.2.2.cmml">=</mo><mrow id="S4.Ex12.m1.2.2.1" xref="S4.Ex12.m1.2.2.1.cmml"><mrow id="S4.Ex12.m1.2.2.1.3" xref="S4.Ex12.m1.2.2.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex12.m1.2.2.1.3.2" xref="S4.Ex12.m1.2.2.1.3.2.cmml">𝒞</mi><mo id="S4.Ex12.m1.2.2.1.3.1" xref="S4.Ex12.m1.2.2.1.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex12.m1.2.2.1.3.3" xref="S4.Ex12.m1.2.2.1.3.3.cmml">𝒲</mi><mo id="S4.Ex12.m1.2.2.1.3.1a" xref="S4.Ex12.m1.2.2.1.3.1.cmml">⁢</mo><mrow id="S4.Ex12.m1.2.2.1.3.4.2" xref="S4.Ex12.m1.2.2.1.3.cmml"><mo id="S4.Ex12.m1.2.2.1.3.4.2.1" stretchy="false" xref="S4.Ex12.m1.2.2.1.3.cmml">(</mo><mi id="S4.Ex12.m1.1.1" xref="S4.Ex12.m1.1.1.cmml">π</mi><mo id="S4.Ex12.m1.2.2.1.3.4.2.2" stretchy="false" xref="S4.Ex12.m1.2.2.1.3.cmml">)</mo></mrow></mrow><mo id="S4.Ex12.m1.2.2.1.2" xref="S4.Ex12.m1.2.2.1.2.cmml">−</mo><mrow id="S4.Ex12.m1.2.2.1.1" xref="S4.Ex12.m1.2.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex12.m1.2.2.1.1.3" xref="S4.Ex12.m1.2.2.1.1.3.cmml">𝒞</mi><mo id="S4.Ex12.m1.2.2.1.1.2" xref="S4.Ex12.m1.2.2.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex12.m1.2.2.1.1.4" xref="S4.Ex12.m1.2.2.1.1.4.cmml">𝒲</mi><mo id="S4.Ex12.m1.2.2.1.1.2a" xref="S4.Ex12.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="S4.Ex12.m1.2.2.1.1.1.1" xref="S4.Ex12.m1.2.2.1.1.1.1.1.cmml"><mo id="S4.Ex12.m1.2.2.1.1.1.1.2" stretchy="false" xref="S4.Ex12.m1.2.2.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex12.m1.2.2.1.1.1.1.1" xref="S4.Ex12.m1.2.2.1.1.1.1.1.cmml"><mi id="S4.Ex12.m1.2.2.1.1.1.1.1.2" xref="S4.Ex12.m1.2.2.1.1.1.1.1.2.cmml">π</mi><mo id="S4.Ex12.m1.2.2.1.1.1.1.1.3" xref="S4.Ex12.m1.2.2.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.Ex12.m1.2.2.1.1.1.1.3" stretchy="false" xref="S4.Ex12.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Ex12.m1.2.2.1.1.2b" xref="S4.Ex12.m1.2.2.1.1.2.cmml">⁢</mo><mtext id="S4.Ex12.m1.2.2.1.1.5" xref="S4.Ex12.m1.2.2.1.1.5a.cmml">.</mtext></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex12.m1.2b"><apply id="S4.Ex12.m1.2.2.cmml" xref="S4.Ex12.m1.2.2"><eq id="S4.Ex12.m1.2.2.2.cmml" xref="S4.Ex12.m1.2.2.2"></eq><csymbol cd="latexml" id="S4.Ex12.m1.2.2.3.cmml" xref="S4.Ex12.m1.2.2.3">absent</csymbol><apply id="S4.Ex12.m1.2.2.1.cmml" xref="S4.Ex12.m1.2.2.1"><minus id="S4.Ex12.m1.2.2.1.2.cmml" xref="S4.Ex12.m1.2.2.1.2"></minus><apply id="S4.Ex12.m1.2.2.1.3.cmml" xref="S4.Ex12.m1.2.2.1.3"><times id="S4.Ex12.m1.2.2.1.3.1.cmml" xref="S4.Ex12.m1.2.2.1.3.1"></times><ci id="S4.Ex12.m1.2.2.1.3.2.cmml" xref="S4.Ex12.m1.2.2.1.3.2">𝒞</ci><ci id="S4.Ex12.m1.2.2.1.3.3.cmml" xref="S4.Ex12.m1.2.2.1.3.3">𝒲</ci><ci id="S4.Ex12.m1.1.1.cmml" xref="S4.Ex12.m1.1.1">𝜋</ci></apply><apply id="S4.Ex12.m1.2.2.1.1.cmml" xref="S4.Ex12.m1.2.2.1.1"><times id="S4.Ex12.m1.2.2.1.1.2.cmml" xref="S4.Ex12.m1.2.2.1.1.2"></times><ci id="S4.Ex12.m1.2.2.1.1.3.cmml" xref="S4.Ex12.m1.2.2.1.1.3">𝒞</ci><ci id="S4.Ex12.m1.2.2.1.1.4.cmml" xref="S4.Ex12.m1.2.2.1.1.4">𝒲</ci><apply id="S4.Ex12.m1.2.2.1.1.1.1.1.cmml" xref="S4.Ex12.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex12.m1.2.2.1.1.1.1.1.1.cmml" xref="S4.Ex12.m1.2.2.1.1.1.1">superscript</csymbol><ci id="S4.Ex12.m1.2.2.1.1.1.1.1.2.cmml" xref="S4.Ex12.m1.2.2.1.1.1.1.1.2">𝜋</ci><ci id="S4.Ex12.m1.2.2.1.1.1.1.1.3.cmml" xref="S4.Ex12.m1.2.2.1.1.1.1.1.3">′</ci></apply><ci id="S4.Ex12.m1.2.2.1.1.5a.cmml" xref="S4.Ex12.m1.2.2.1.1.5"><mtext id="S4.Ex12.m1.2.2.1.1.5.cmml" xref="S4.Ex12.m1.2.2.1.1.5">.</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex12.m1.2c">\displaystyle=\mathcal{CW}(\pi)-\mathcal{CW}(\pi^{\prime})\text{.}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex12.m1.2d">= caligraphic_C caligraphic_W ( italic_π ) - caligraphic_C caligraphic_W ( italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.2.p1.4">Hence, it holds that <math alttext="\pi\in\operatorname*{arg\,max}_{\pi^{\prime}\in\Pi_{N}}\mathcal{SW}(\pi^{% \prime})" class="ltx_Math" display="inline" id="S4.SS2.2.p1.3.m1.1"><semantics id="S4.SS2.2.p1.3.m1.1a"><mrow id="S4.SS2.2.p1.3.m1.1.1" xref="S4.SS2.2.p1.3.m1.1.1.cmml"><mi id="S4.SS2.2.p1.3.m1.1.1.3" xref="S4.SS2.2.p1.3.m1.1.1.3.cmml">π</mi><mo id="S4.SS2.2.p1.3.m1.1.1.2" xref="S4.SS2.2.p1.3.m1.1.1.2.cmml">∈</mo><mrow id="S4.SS2.2.p1.3.m1.1.1.1" xref="S4.SS2.2.p1.3.m1.1.1.1.cmml"><mrow id="S4.SS2.2.p1.3.m1.1.1.1.3" xref="S4.SS2.2.p1.3.m1.1.1.1.3.cmml"><msub id="S4.SS2.2.p1.3.m1.1.1.1.3.1" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.cmml"><mrow id="S4.SS2.2.p1.3.m1.1.1.1.3.1.2" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.2.cmml"><mi id="S4.SS2.2.p1.3.m1.1.1.1.3.1.2.2" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.2.2.cmml">arg</mi><mo id="S4.SS2.2.p1.3.m1.1.1.1.3.1.2.1" lspace="0.170em" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.2.1.cmml">⁢</mo><mi id="S4.SS2.2.p1.3.m1.1.1.1.3.1.2.3" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.2.3.cmml">max</mi></mrow><mrow id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.cmml"><msup id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.2" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.2.cmml"><mi id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.2.2" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.2.2.cmml">π</mi><mo id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.2.3" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.2.3.cmml">′</mo></msup><mo id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.1" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.1.cmml">∈</mo><msub id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.3" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.3.cmml"><mi id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.3.2" mathvariant="normal" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.3.2.cmml">Π</mi><mi id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.3.3" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.3.3.cmml">N</mi></msub></mrow></msub><mo id="S4.SS2.2.p1.3.m1.1.1.1.3a" xref="S4.SS2.2.p1.3.m1.1.1.1.3.cmml">⁡</mo><mrow id="S4.SS2.2.p1.3.m1.1.1.1.3.2" xref="S4.SS2.2.p1.3.m1.1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.2.p1.3.m1.1.1.1.3.2.2" xref="S4.SS2.2.p1.3.m1.1.1.1.3.2.2.cmml">𝒮</mi><mo id="S4.SS2.2.p1.3.m1.1.1.1.3.2.1" xref="S4.SS2.2.p1.3.m1.1.1.1.3.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.2.p1.3.m1.1.1.1.3.2.3" xref="S4.SS2.2.p1.3.m1.1.1.1.3.2.3.cmml">𝒲</mi></mrow></mrow><mo id="S4.SS2.2.p1.3.m1.1.1.1.2" xref="S4.SS2.2.p1.3.m1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.2.p1.3.m1.1.1.1.1.1" xref="S4.SS2.2.p1.3.m1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.2.p1.3.m1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.2.p1.3.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS2.2.p1.3.m1.1.1.1.1.1.1" xref="S4.SS2.2.p1.3.m1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.2.p1.3.m1.1.1.1.1.1.1.2" xref="S4.SS2.2.p1.3.m1.1.1.1.1.1.1.2.cmml">π</mi><mo id="S4.SS2.2.p1.3.m1.1.1.1.1.1.1.3" xref="S4.SS2.2.p1.3.m1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.SS2.2.p1.3.m1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.2.p1.3.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.3.m1.1b"><apply id="S4.SS2.2.p1.3.m1.1.1.cmml" xref="S4.SS2.2.p1.3.m1.1.1"><in id="S4.SS2.2.p1.3.m1.1.1.2.cmml" xref="S4.SS2.2.p1.3.m1.1.1.2"></in><ci id="S4.SS2.2.p1.3.m1.1.1.3.cmml" xref="S4.SS2.2.p1.3.m1.1.1.3">𝜋</ci><apply id="S4.SS2.2.p1.3.m1.1.1.1.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1"><times id="S4.SS2.2.p1.3.m1.1.1.1.2.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.2"></times><apply id="S4.SS2.2.p1.3.m1.1.1.1.3.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3"><apply id="S4.SS2.2.p1.3.m1.1.1.1.3.1.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1"><csymbol cd="ambiguous" id="S4.SS2.2.p1.3.m1.1.1.1.3.1.1.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1">subscript</csymbol><apply id="S4.SS2.2.p1.3.m1.1.1.1.3.1.2.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.2"><times id="S4.SS2.2.p1.3.m1.1.1.1.3.1.2.1.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.2.1"></times><ci id="S4.SS2.2.p1.3.m1.1.1.1.3.1.2.2.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.2.2">arg</ci><ci id="S4.SS2.2.p1.3.m1.1.1.1.3.1.2.3.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.2.3">max</ci></apply><apply id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3"><in id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.1.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.1"></in><apply id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.2.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.2.1.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.2">superscript</csymbol><ci id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.2.2.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.2.2">𝜋</ci><ci id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.2.3.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.2.3">′</ci></apply><apply id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.3.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.3"><csymbol cd="ambiguous" id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.3.1.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.3">subscript</csymbol><ci id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.3.2.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.3.2">Π</ci><ci id="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.3.3.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.1.3.3.3">𝑁</ci></apply></apply></apply><apply id="S4.SS2.2.p1.3.m1.1.1.1.3.2.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.2"><times id="S4.SS2.2.p1.3.m1.1.1.1.3.2.1.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.2.1"></times><ci id="S4.SS2.2.p1.3.m1.1.1.1.3.2.2.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.2.2">𝒮</ci><ci id="S4.SS2.2.p1.3.m1.1.1.1.3.2.3.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.3.2.3">𝒲</ci></apply></apply><apply id="S4.SS2.2.p1.3.m1.1.1.1.1.1.1.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p1.3.m1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.1.1">superscript</csymbol><ci id="S4.SS2.2.p1.3.m1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.1.1.1.2">𝜋</ci><ci id="S4.SS2.2.p1.3.m1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.2.p1.3.m1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.3.m1.1c">\pi\in\operatorname*{arg\,max}_{\pi^{\prime}\in\Pi_{N}}\mathcal{SW}(\pi^{% \prime})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.3.m1.1d">italic_π ∈ start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_Π start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> if and only if <math alttext="\pi\in\operatorname*{arg\,max}_{\pi^{\prime}\in\Pi_{N}}\mathcal{CW}(\pi^{% \prime})" class="ltx_Math" display="inline" id="S4.SS2.2.p1.4.m2.1"><semantics id="S4.SS2.2.p1.4.m2.1a"><mrow id="S4.SS2.2.p1.4.m2.1.1" xref="S4.SS2.2.p1.4.m2.1.1.cmml"><mi id="S4.SS2.2.p1.4.m2.1.1.3" xref="S4.SS2.2.p1.4.m2.1.1.3.cmml">π</mi><mo id="S4.SS2.2.p1.4.m2.1.1.2" xref="S4.SS2.2.p1.4.m2.1.1.2.cmml">∈</mo><mrow id="S4.SS2.2.p1.4.m2.1.1.1" xref="S4.SS2.2.p1.4.m2.1.1.1.cmml"><mrow id="S4.SS2.2.p1.4.m2.1.1.1.3" xref="S4.SS2.2.p1.4.m2.1.1.1.3.cmml"><msub id="S4.SS2.2.p1.4.m2.1.1.1.3.1" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.cmml"><mrow id="S4.SS2.2.p1.4.m2.1.1.1.3.1.2" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.2.cmml"><mi id="S4.SS2.2.p1.4.m2.1.1.1.3.1.2.2" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.2.2.cmml">arg</mi><mo id="S4.SS2.2.p1.4.m2.1.1.1.3.1.2.1" lspace="0.170em" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.2.1.cmml">⁢</mo><mi id="S4.SS2.2.p1.4.m2.1.1.1.3.1.2.3" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.2.3.cmml">max</mi></mrow><mrow id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.cmml"><msup id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.2" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.2.cmml"><mi id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.2.2" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.2.2.cmml">π</mi><mo id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.2.3" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.2.3.cmml">′</mo></msup><mo id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.1" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.1.cmml">∈</mo><msub id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.3" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.3.cmml"><mi id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.3.2" mathvariant="normal" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.3.2.cmml">Π</mi><mi id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.3.3" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.3.3.cmml">N</mi></msub></mrow></msub><mo id="S4.SS2.2.p1.4.m2.1.1.1.3a" xref="S4.SS2.2.p1.4.m2.1.1.1.3.cmml">⁡</mo><mrow id="S4.SS2.2.p1.4.m2.1.1.1.3.2" xref="S4.SS2.2.p1.4.m2.1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.2.p1.4.m2.1.1.1.3.2.2" xref="S4.SS2.2.p1.4.m2.1.1.1.3.2.2.cmml">𝒞</mi><mo id="S4.SS2.2.p1.4.m2.1.1.1.3.2.1" xref="S4.SS2.2.p1.4.m2.1.1.1.3.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.2.p1.4.m2.1.1.1.3.2.3" xref="S4.SS2.2.p1.4.m2.1.1.1.3.2.3.cmml">𝒲</mi></mrow></mrow><mo id="S4.SS2.2.p1.4.m2.1.1.1.2" xref="S4.SS2.2.p1.4.m2.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.2.p1.4.m2.1.1.1.1.1" xref="S4.SS2.2.p1.4.m2.1.1.1.1.1.1.cmml"><mo id="S4.SS2.2.p1.4.m2.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.2.p1.4.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS2.2.p1.4.m2.1.1.1.1.1.1" xref="S4.SS2.2.p1.4.m2.1.1.1.1.1.1.cmml"><mi id="S4.SS2.2.p1.4.m2.1.1.1.1.1.1.2" xref="S4.SS2.2.p1.4.m2.1.1.1.1.1.1.2.cmml">π</mi><mo id="S4.SS2.2.p1.4.m2.1.1.1.1.1.1.3" xref="S4.SS2.2.p1.4.m2.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.SS2.2.p1.4.m2.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.2.p1.4.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.4.m2.1b"><apply id="S4.SS2.2.p1.4.m2.1.1.cmml" xref="S4.SS2.2.p1.4.m2.1.1"><in id="S4.SS2.2.p1.4.m2.1.1.2.cmml" xref="S4.SS2.2.p1.4.m2.1.1.2"></in><ci id="S4.SS2.2.p1.4.m2.1.1.3.cmml" xref="S4.SS2.2.p1.4.m2.1.1.3">𝜋</ci><apply id="S4.SS2.2.p1.4.m2.1.1.1.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1"><times id="S4.SS2.2.p1.4.m2.1.1.1.2.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.2"></times><apply id="S4.SS2.2.p1.4.m2.1.1.1.3.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3"><apply id="S4.SS2.2.p1.4.m2.1.1.1.3.1.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1"><csymbol cd="ambiguous" id="S4.SS2.2.p1.4.m2.1.1.1.3.1.1.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1">subscript</csymbol><apply id="S4.SS2.2.p1.4.m2.1.1.1.3.1.2.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.2"><times id="S4.SS2.2.p1.4.m2.1.1.1.3.1.2.1.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.2.1"></times><ci id="S4.SS2.2.p1.4.m2.1.1.1.3.1.2.2.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.2.2">arg</ci><ci id="S4.SS2.2.p1.4.m2.1.1.1.3.1.2.3.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.2.3">max</ci></apply><apply id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3"><in id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.1.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.1"></in><apply id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.2.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.2.1.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.2">superscript</csymbol><ci id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.2.2.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.2.2">𝜋</ci><ci id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.2.3.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.2.3">′</ci></apply><apply id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.3.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.3"><csymbol cd="ambiguous" id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.3.1.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.3">subscript</csymbol><ci id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.3.2.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.3.2">Π</ci><ci id="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.3.3.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.1.3.3.3">𝑁</ci></apply></apply></apply><apply id="S4.SS2.2.p1.4.m2.1.1.1.3.2.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.2"><times id="S4.SS2.2.p1.4.m2.1.1.1.3.2.1.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.2.1"></times><ci id="S4.SS2.2.p1.4.m2.1.1.1.3.2.2.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.2.2">𝒞</ci><ci id="S4.SS2.2.p1.4.m2.1.1.1.3.2.3.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.3.2.3">𝒲</ci></apply></apply><apply id="S4.SS2.2.p1.4.m2.1.1.1.1.1.1.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p1.4.m2.1.1.1.1.1.1.1.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.1.1">superscript</csymbol><ci id="S4.SS2.2.p1.4.m2.1.1.1.1.1.1.2.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.1.1.1.2">𝜋</ci><ci id="S4.SS2.2.p1.4.m2.1.1.1.1.1.1.3.cmml" xref="S4.SS2.2.p1.4.m2.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.4.m2.1c">\pi\in\operatorname*{arg\,max}_{\pi^{\prime}\in\Pi_{N}}\mathcal{CW}(\pi^{% \prime})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.4.m2.1d">italic_π ∈ start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_Π start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT caligraphic_C caligraphic_W ( italic_π start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS2.p5"> <p class="ltx_p" id="S4.SS2.p5.1">Hence, solving for social welfare maximization and correlation welfare maximization is exactly equivalent. However, this does not have any implications on approximation guarantees, as we illustrate in the next example.</p> </div> <div class="ltx_theorem ltx_theorem_example" id="S4.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem4.1.1.1">Example 4.4</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem4.p1"> <p class="ltx_p" id="S4.Thmtheorem4.p1.13">Let <math alttext="x>0" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.1.m1.1"><semantics id="S4.Thmtheorem4.p1.1.m1.1a"><mrow id="S4.Thmtheorem4.p1.1.m1.1.1" xref="S4.Thmtheorem4.p1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem4.p1.1.m1.1.1.2" xref="S4.Thmtheorem4.p1.1.m1.1.1.2.cmml">x</mi><mo id="S4.Thmtheorem4.p1.1.m1.1.1.1" xref="S4.Thmtheorem4.p1.1.m1.1.1.1.cmml">></mo><mn id="S4.Thmtheorem4.p1.1.m1.1.1.3" xref="S4.Thmtheorem4.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.1.m1.1b"><apply id="S4.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem4.p1.1.m1.1.1"><gt id="S4.Thmtheorem4.p1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.1.m1.1.1.1"></gt><ci id="S4.Thmtheorem4.p1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.1.m1.1.1.2">𝑥</ci><cn id="S4.Thmtheorem4.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.1.m1.1c">x>0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.1.m1.1d">italic_x > 0</annotation></semantics></math> be an arbitrary positive number. Consider the symmetric ASHG <math alttext="(N,v)" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.2.m2.2"><semantics id="S4.Thmtheorem4.p1.2.m2.2a"><mrow id="S4.Thmtheorem4.p1.2.m2.2.3.2" xref="S4.Thmtheorem4.p1.2.m2.2.3.1.cmml"><mo id="S4.Thmtheorem4.p1.2.m2.2.3.2.1" stretchy="false" xref="S4.Thmtheorem4.p1.2.m2.2.3.1.cmml">(</mo><mi id="S4.Thmtheorem4.p1.2.m2.1.1" xref="S4.Thmtheorem4.p1.2.m2.1.1.cmml">N</mi><mo id="S4.Thmtheorem4.p1.2.m2.2.3.2.2" xref="S4.Thmtheorem4.p1.2.m2.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.2.m2.2.2" xref="S4.Thmtheorem4.p1.2.m2.2.2.cmml">v</mi><mo id="S4.Thmtheorem4.p1.2.m2.2.3.2.3" stretchy="false" xref="S4.Thmtheorem4.p1.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.2.m2.2b"><interval closure="open" id="S4.Thmtheorem4.p1.2.m2.2.3.1.cmml" xref="S4.Thmtheorem4.p1.2.m2.2.3.2"><ci id="S4.Thmtheorem4.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem4.p1.2.m2.1.1">𝑁</ci><ci id="S4.Thmtheorem4.p1.2.m2.2.2.cmml" xref="S4.Thmtheorem4.p1.2.m2.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.2.m2.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.2.m2.2d">( italic_N , italic_v )</annotation></semantics></math> with <math alttext="N=\{a_{1},a_{2},a_{3}\}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.3.m3.3"><semantics id="S4.Thmtheorem4.p1.3.m3.3a"><mrow id="S4.Thmtheorem4.p1.3.m3.3.3" xref="S4.Thmtheorem4.p1.3.m3.3.3.cmml"><mi id="S4.Thmtheorem4.p1.3.m3.3.3.5" xref="S4.Thmtheorem4.p1.3.m3.3.3.5.cmml">N</mi><mo id="S4.Thmtheorem4.p1.3.m3.3.3.4" xref="S4.Thmtheorem4.p1.3.m3.3.3.4.cmml">=</mo><mrow id="S4.Thmtheorem4.p1.3.m3.3.3.3.3" xref="S4.Thmtheorem4.p1.3.m3.3.3.3.4.cmml"><mo id="S4.Thmtheorem4.p1.3.m3.3.3.3.3.4" stretchy="false" xref="S4.Thmtheorem4.p1.3.m3.3.3.3.4.cmml">{</mo><msub id="S4.Thmtheorem4.p1.3.m3.1.1.1.1.1" xref="S4.Thmtheorem4.p1.3.m3.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem4.p1.3.m3.1.1.1.1.1.2" xref="S4.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.cmml">a</mi><mn id="S4.Thmtheorem4.p1.3.m3.1.1.1.1.1.3" xref="S4.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem4.p1.3.m3.3.3.3.3.5" xref="S4.Thmtheorem4.p1.3.m3.3.3.3.4.cmml">,</mo><msub id="S4.Thmtheorem4.p1.3.m3.2.2.2.2.2" xref="S4.Thmtheorem4.p1.3.m3.2.2.2.2.2.cmml"><mi id="S4.Thmtheorem4.p1.3.m3.2.2.2.2.2.2" xref="S4.Thmtheorem4.p1.3.m3.2.2.2.2.2.2.cmml">a</mi><mn id="S4.Thmtheorem4.p1.3.m3.2.2.2.2.2.3" xref="S4.Thmtheorem4.p1.3.m3.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.Thmtheorem4.p1.3.m3.3.3.3.3.6" xref="S4.Thmtheorem4.p1.3.m3.3.3.3.4.cmml">,</mo><msub id="S4.Thmtheorem4.p1.3.m3.3.3.3.3.3" xref="S4.Thmtheorem4.p1.3.m3.3.3.3.3.3.cmml"><mi id="S4.Thmtheorem4.p1.3.m3.3.3.3.3.3.2" xref="S4.Thmtheorem4.p1.3.m3.3.3.3.3.3.2.cmml">a</mi><mn id="S4.Thmtheorem4.p1.3.m3.3.3.3.3.3.3" xref="S4.Thmtheorem4.p1.3.m3.3.3.3.3.3.3.cmml">3</mn></msub><mo id="S4.Thmtheorem4.p1.3.m3.3.3.3.3.7" stretchy="false" xref="S4.Thmtheorem4.p1.3.m3.3.3.3.4.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.3.m3.3b"><apply id="S4.Thmtheorem4.p1.3.m3.3.3.cmml" xref="S4.Thmtheorem4.p1.3.m3.3.3"><eq id="S4.Thmtheorem4.p1.3.m3.3.3.4.cmml" xref="S4.Thmtheorem4.p1.3.m3.3.3.4"></eq><ci id="S4.Thmtheorem4.p1.3.m3.3.3.5.cmml" xref="S4.Thmtheorem4.p1.3.m3.3.3.5">𝑁</ci><set id="S4.Thmtheorem4.p1.3.m3.3.3.3.4.cmml" xref="S4.Thmtheorem4.p1.3.m3.3.3.3.3"><apply id="S4.Thmtheorem4.p1.3.m3.1.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.3.m3.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.3.m3.1.1.1.1.1.2">𝑎</ci><cn id="S4.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.3.m3.1.1.1.1.1.3">1</cn></apply><apply id="S4.Thmtheorem4.p1.3.m3.2.2.2.2.2.cmml" xref="S4.Thmtheorem4.p1.3.m3.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.3.m3.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem4.p1.3.m3.2.2.2.2.2">subscript</csymbol><ci id="S4.Thmtheorem4.p1.3.m3.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem4.p1.3.m3.2.2.2.2.2.2">𝑎</ci><cn id="S4.Thmtheorem4.p1.3.m3.2.2.2.2.2.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.3.m3.2.2.2.2.2.3">2</cn></apply><apply id="S4.Thmtheorem4.p1.3.m3.3.3.3.3.3.cmml" xref="S4.Thmtheorem4.p1.3.m3.3.3.3.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.3.m3.3.3.3.3.3.1.cmml" xref="S4.Thmtheorem4.p1.3.m3.3.3.3.3.3">subscript</csymbol><ci id="S4.Thmtheorem4.p1.3.m3.3.3.3.3.3.2.cmml" xref="S4.Thmtheorem4.p1.3.m3.3.3.3.3.3.2">𝑎</ci><cn id="S4.Thmtheorem4.p1.3.m3.3.3.3.3.3.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.3.m3.3.3.3.3.3.3">3</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.3.m3.3c">N=\{a_{1},a_{2},a_{3}\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.3.m3.3d">italic_N = { italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT }</annotation></semantics></math> and symmetric valuation functions given by <math alttext="v(a_{1},a_{2})=1" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.4.m4.2"><semantics id="S4.Thmtheorem4.p1.4.m4.2a"><mrow id="S4.Thmtheorem4.p1.4.m4.2.2" xref="S4.Thmtheorem4.p1.4.m4.2.2.cmml"><mrow id="S4.Thmtheorem4.p1.4.m4.2.2.2" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.cmml"><mi id="S4.Thmtheorem4.p1.4.m4.2.2.2.4" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.4.cmml">v</mi><mo id="S4.Thmtheorem4.p1.4.m4.2.2.2.3" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.3.cmml">⁢</mo><mrow id="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.2.3.cmml"><mo id="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.3" stretchy="false" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.2.3.cmml">(</mo><msub id="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.1" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.1.2" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.1.2.cmml">a</mi><mn id="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.1.3" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.4" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.2.3.cmml">,</mo><msub id="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.2" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.2.cmml"><mi id="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.2.2" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.2.2.cmml">a</mi><mn id="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.2.3" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.5" stretchy="false" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem4.p1.4.m4.2.2.3" xref="S4.Thmtheorem4.p1.4.m4.2.2.3.cmml">=</mo><mn id="S4.Thmtheorem4.p1.4.m4.2.2.4" xref="S4.Thmtheorem4.p1.4.m4.2.2.4.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.4.m4.2b"><apply id="S4.Thmtheorem4.p1.4.m4.2.2.cmml" xref="S4.Thmtheorem4.p1.4.m4.2.2"><eq id="S4.Thmtheorem4.p1.4.m4.2.2.3.cmml" xref="S4.Thmtheorem4.p1.4.m4.2.2.3"></eq><apply id="S4.Thmtheorem4.p1.4.m4.2.2.2.cmml" xref="S4.Thmtheorem4.p1.4.m4.2.2.2"><times id="S4.Thmtheorem4.p1.4.m4.2.2.2.3.cmml" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.3"></times><ci id="S4.Thmtheorem4.p1.4.m4.2.2.2.4.cmml" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.4">𝑣</ci><interval closure="open" id="S4.Thmtheorem4.p1.4.m4.2.2.2.2.3.cmml" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2"><apply id="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.1.2">𝑎</ci><cn id="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.1.3">1</cn></apply><apply id="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.2">subscript</csymbol><ci id="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.2.2">𝑎</ci><cn id="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.2.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.4.m4.2.2.2.2.2.2.3">2</cn></apply></interval></apply><cn id="S4.Thmtheorem4.p1.4.m4.2.2.4.cmml" type="integer" xref="S4.Thmtheorem4.p1.4.m4.2.2.4">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.4.m4.2c">v(a_{1},a_{2})=1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.4.m4.2d">italic_v ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 1</annotation></semantics></math>, <math alttext="v(a_{1},a_{3})=-x" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.5.m5.2"><semantics id="S4.Thmtheorem4.p1.5.m5.2a"><mrow id="S4.Thmtheorem4.p1.5.m5.2.2" xref="S4.Thmtheorem4.p1.5.m5.2.2.cmml"><mrow id="S4.Thmtheorem4.p1.5.m5.2.2.2" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.cmml"><mi id="S4.Thmtheorem4.p1.5.m5.2.2.2.4" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.4.cmml">v</mi><mo id="S4.Thmtheorem4.p1.5.m5.2.2.2.3" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.3.cmml">⁢</mo><mrow id="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.2.3.cmml"><mo id="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.3" stretchy="false" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.2.3.cmml">(</mo><msub id="S4.Thmtheorem4.p1.5.m5.1.1.1.1.1.1" xref="S4.Thmtheorem4.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem4.p1.5.m5.1.1.1.1.1.1.2" xref="S4.Thmtheorem4.p1.5.m5.1.1.1.1.1.1.2.cmml">a</mi><mn id="S4.Thmtheorem4.p1.5.m5.1.1.1.1.1.1.3" xref="S4.Thmtheorem4.p1.5.m5.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.4" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.2.3.cmml">,</mo><msub id="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.2" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.2.cmml"><mi id="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.2.2" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.2.2.cmml">a</mi><mn id="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.2.3" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.2.3.cmml">3</mn></msub><mo id="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.5" stretchy="false" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem4.p1.5.m5.2.2.3" xref="S4.Thmtheorem4.p1.5.m5.2.2.3.cmml">=</mo><mrow id="S4.Thmtheorem4.p1.5.m5.2.2.4" xref="S4.Thmtheorem4.p1.5.m5.2.2.4.cmml"><mo id="S4.Thmtheorem4.p1.5.m5.2.2.4a" xref="S4.Thmtheorem4.p1.5.m5.2.2.4.cmml">−</mo><mi id="S4.Thmtheorem4.p1.5.m5.2.2.4.2" xref="S4.Thmtheorem4.p1.5.m5.2.2.4.2.cmml">x</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.5.m5.2b"><apply id="S4.Thmtheorem4.p1.5.m5.2.2.cmml" xref="S4.Thmtheorem4.p1.5.m5.2.2"><eq id="S4.Thmtheorem4.p1.5.m5.2.2.3.cmml" xref="S4.Thmtheorem4.p1.5.m5.2.2.3"></eq><apply id="S4.Thmtheorem4.p1.5.m5.2.2.2.cmml" xref="S4.Thmtheorem4.p1.5.m5.2.2.2"><times id="S4.Thmtheorem4.p1.5.m5.2.2.2.3.cmml" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.3"></times><ci id="S4.Thmtheorem4.p1.5.m5.2.2.2.4.cmml" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.4">𝑣</ci><interval closure="open" id="S4.Thmtheorem4.p1.5.m5.2.2.2.2.3.cmml" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2"><apply id="S4.Thmtheorem4.p1.5.m5.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.5.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.5.m5.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem4.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.5.m5.1.1.1.1.1.1.2">𝑎</ci><cn id="S4.Thmtheorem4.p1.5.m5.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.5.m5.1.1.1.1.1.1.3">1</cn></apply><apply id="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.2">subscript</csymbol><ci id="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.2.2">𝑎</ci><cn id="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.2.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.5.m5.2.2.2.2.2.2.3">3</cn></apply></interval></apply><apply id="S4.Thmtheorem4.p1.5.m5.2.2.4.cmml" xref="S4.Thmtheorem4.p1.5.m5.2.2.4"><minus id="S4.Thmtheorem4.p1.5.m5.2.2.4.1.cmml" xref="S4.Thmtheorem4.p1.5.m5.2.2.4"></minus><ci id="S4.Thmtheorem4.p1.5.m5.2.2.4.2.cmml" xref="S4.Thmtheorem4.p1.5.m5.2.2.4.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.5.m5.2c">v(a_{1},a_{3})=-x</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.5.m5.2d">italic_v ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = - italic_x</annotation></semantics></math>, and <math alttext="v(a_{2},a_{3})=0" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.6.m6.2"><semantics id="S4.Thmtheorem4.p1.6.m6.2a"><mrow id="S4.Thmtheorem4.p1.6.m6.2.2" xref="S4.Thmtheorem4.p1.6.m6.2.2.cmml"><mrow id="S4.Thmtheorem4.p1.6.m6.2.2.2" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.cmml"><mi id="S4.Thmtheorem4.p1.6.m6.2.2.2.4" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.4.cmml">v</mi><mo id="S4.Thmtheorem4.p1.6.m6.2.2.2.3" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.3.cmml">⁢</mo><mrow id="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.2.3.cmml"><mo id="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.3" stretchy="false" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.2.3.cmml">(</mo><msub id="S4.Thmtheorem4.p1.6.m6.1.1.1.1.1.1" xref="S4.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.2" xref="S4.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.2.cmml">a</mi><mn id="S4.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.3" xref="S4.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.4" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.2.3.cmml">,</mo><msub id="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.2" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.2.cmml"><mi id="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.2.2" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.2.2.cmml">a</mi><mn id="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.2.3" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.2.3.cmml">3</mn></msub><mo id="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.5" stretchy="false" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem4.p1.6.m6.2.2.3" xref="S4.Thmtheorem4.p1.6.m6.2.2.3.cmml">=</mo><mn id="S4.Thmtheorem4.p1.6.m6.2.2.4" xref="S4.Thmtheorem4.p1.6.m6.2.2.4.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.6.m6.2b"><apply id="S4.Thmtheorem4.p1.6.m6.2.2.cmml" xref="S4.Thmtheorem4.p1.6.m6.2.2"><eq id="S4.Thmtheorem4.p1.6.m6.2.2.3.cmml" xref="S4.Thmtheorem4.p1.6.m6.2.2.3"></eq><apply id="S4.Thmtheorem4.p1.6.m6.2.2.2.cmml" xref="S4.Thmtheorem4.p1.6.m6.2.2.2"><times id="S4.Thmtheorem4.p1.6.m6.2.2.2.3.cmml" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.3"></times><ci id="S4.Thmtheorem4.p1.6.m6.2.2.2.4.cmml" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.4">𝑣</ci><interval closure="open" id="S4.Thmtheorem4.p1.6.m6.2.2.2.2.3.cmml" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2"><apply id="S4.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.6.m6.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.6.m6.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.2">𝑎</ci><cn id="S4.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.6.m6.1.1.1.1.1.1.3">2</cn></apply><apply id="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.2">subscript</csymbol><ci id="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.2.2">𝑎</ci><cn id="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.2.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.6.m6.2.2.2.2.2.2.3">3</cn></apply></interval></apply><cn id="S4.Thmtheorem4.p1.6.m6.2.2.4.cmml" type="integer" xref="S4.Thmtheorem4.p1.6.m6.2.2.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.6.m6.2c">v(a_{2},a_{3})=0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.6.m6.2d">italic_v ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = 0</annotation></semantics></math>. Let <math alttext="\pi" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.7.m7.1"><semantics id="S4.Thmtheorem4.p1.7.m7.1a"><mi id="S4.Thmtheorem4.p1.7.m7.1.1" xref="S4.Thmtheorem4.p1.7.m7.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.7.m7.1b"><ci id="S4.Thmtheorem4.p1.7.m7.1.1.cmml" xref="S4.Thmtheorem4.p1.7.m7.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.7.m7.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.7.m7.1d">italic_π</annotation></semantics></math> denote the singleton partition and <math alttext="\pi^{*}=\{\{a_{1},a_{2}\},\{a_{3}\}\}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.8.m8.2"><semantics id="S4.Thmtheorem4.p1.8.m8.2a"><mrow id="S4.Thmtheorem4.p1.8.m8.2.2" xref="S4.Thmtheorem4.p1.8.m8.2.2.cmml"><msup id="S4.Thmtheorem4.p1.8.m8.2.2.4" xref="S4.Thmtheorem4.p1.8.m8.2.2.4.cmml"><mi id="S4.Thmtheorem4.p1.8.m8.2.2.4.2" xref="S4.Thmtheorem4.p1.8.m8.2.2.4.2.cmml">π</mi><mo id="S4.Thmtheorem4.p1.8.m8.2.2.4.3" xref="S4.Thmtheorem4.p1.8.m8.2.2.4.3.cmml">∗</mo></msup><mo id="S4.Thmtheorem4.p1.8.m8.2.2.3" xref="S4.Thmtheorem4.p1.8.m8.2.2.3.cmml">=</mo><mrow id="S4.Thmtheorem4.p1.8.m8.2.2.2.2" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.3.cmml"><mo id="S4.Thmtheorem4.p1.8.m8.2.2.2.2.3" stretchy="false" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.3.cmml">{</mo><mrow id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.3.cmml"><mo id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.3" stretchy="false" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.3.cmml">{</mo><msub id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.1.1" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.1.1.2" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.1.1.2.cmml">a</mi><mn id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.1.1.3" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.4" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.3.cmml">,</mo><msub id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.2" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.2.cmml"><mi id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.2.2" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.2.2.cmml">a</mi><mn id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.2.3" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.2.3.cmml">2</mn></msub><mo id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.5" stretchy="false" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.3.cmml">}</mo></mrow><mo id="S4.Thmtheorem4.p1.8.m8.2.2.2.2.4" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.3.cmml">,</mo><mrow id="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.2.cmml"><mo id="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.2" stretchy="false" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.2.cmml">{</mo><msub id="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.1" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.1.cmml"><mi id="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.1.2" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.1.2.cmml">a</mi><mn id="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.1.3" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.1.3.cmml">3</mn></msub><mo id="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.3" stretchy="false" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.2.cmml">}</mo></mrow><mo id="S4.Thmtheorem4.p1.8.m8.2.2.2.2.5" stretchy="false" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.8.m8.2b"><apply id="S4.Thmtheorem4.p1.8.m8.2.2.cmml" xref="S4.Thmtheorem4.p1.8.m8.2.2"><eq id="S4.Thmtheorem4.p1.8.m8.2.2.3.cmml" xref="S4.Thmtheorem4.p1.8.m8.2.2.3"></eq><apply id="S4.Thmtheorem4.p1.8.m8.2.2.4.cmml" xref="S4.Thmtheorem4.p1.8.m8.2.2.4"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.8.m8.2.2.4.1.cmml" xref="S4.Thmtheorem4.p1.8.m8.2.2.4">superscript</csymbol><ci id="S4.Thmtheorem4.p1.8.m8.2.2.4.2.cmml" xref="S4.Thmtheorem4.p1.8.m8.2.2.4.2">𝜋</ci><times id="S4.Thmtheorem4.p1.8.m8.2.2.4.3.cmml" xref="S4.Thmtheorem4.p1.8.m8.2.2.4.3"></times></apply><set id="S4.Thmtheorem4.p1.8.m8.2.2.2.3.cmml" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.2"><set id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2"><apply id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.1.1.2">𝑎</ci><cn id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.1.1.3">1</cn></apply><apply id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.2.cmml" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.2.1.cmml" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.2">subscript</csymbol><ci id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.2.2.cmml" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.2.2">𝑎</ci><cn id="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.2.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.8.m8.1.1.1.1.1.2.2.3">2</cn></apply></set><set id="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1"><apply id="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.1.cmml" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.1.1.cmml" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.1">subscript</csymbol><ci id="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.1.2.cmml" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.1.2">𝑎</ci><cn id="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.8.m8.2.2.2.2.2.1.1.3">3</cn></apply></set></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.8.m8.2c">\pi^{*}=\{\{a_{1},a_{2}\},\{a_{3}\}\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.8.m8.2d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = { { italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT } , { italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT } }</annotation></semantics></math>. Clearly, <math alttext="\pi^{*}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.9.m9.1"><semantics id="S4.Thmtheorem4.p1.9.m9.1a"><msup id="S4.Thmtheorem4.p1.9.m9.1.1" xref="S4.Thmtheorem4.p1.9.m9.1.1.cmml"><mi id="S4.Thmtheorem4.p1.9.m9.1.1.2" xref="S4.Thmtheorem4.p1.9.m9.1.1.2.cmml">π</mi><mo id="S4.Thmtheorem4.p1.9.m9.1.1.3" xref="S4.Thmtheorem4.p1.9.m9.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.9.m9.1b"><apply id="S4.Thmtheorem4.p1.9.m9.1.1.cmml" xref="S4.Thmtheorem4.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.9.m9.1.1.1.cmml" xref="S4.Thmtheorem4.p1.9.m9.1.1">superscript</csymbol><ci id="S4.Thmtheorem4.p1.9.m9.1.1.2.cmml" xref="S4.Thmtheorem4.p1.9.m9.1.1.2">𝜋</ci><times id="S4.Thmtheorem4.p1.9.m9.1.1.3.cmml" xref="S4.Thmtheorem4.p1.9.m9.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.9.m9.1c">\pi^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.9.m9.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is the unique partition maximizing <math alttext="\mathcal{SW}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.10.m10.1"><semantics id="S4.Thmtheorem4.p1.10.m10.1a"><mrow id="S4.Thmtheorem4.p1.10.m10.1.1" xref="S4.Thmtheorem4.p1.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p1.10.m10.1.1.2" xref="S4.Thmtheorem4.p1.10.m10.1.1.2.cmml">𝒮</mi><mo id="S4.Thmtheorem4.p1.10.m10.1.1.1" xref="S4.Thmtheorem4.p1.10.m10.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p1.10.m10.1.1.3" xref="S4.Thmtheorem4.p1.10.m10.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.10.m10.1b"><apply id="S4.Thmtheorem4.p1.10.m10.1.1.cmml" xref="S4.Thmtheorem4.p1.10.m10.1.1"><times id="S4.Thmtheorem4.p1.10.m10.1.1.1.cmml" xref="S4.Thmtheorem4.p1.10.m10.1.1.1"></times><ci id="S4.Thmtheorem4.p1.10.m10.1.1.2.cmml" xref="S4.Thmtheorem4.p1.10.m10.1.1.2">𝒮</ci><ci id="S4.Thmtheorem4.p1.10.m10.1.1.3.cmml" xref="S4.Thmtheorem4.p1.10.m10.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.10.m10.1c">\mathcal{SW}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.10.m10.1d">caligraphic_S caligraphic_W</annotation></semantics></math> and <math alttext="\mathcal{CW}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.11.m11.1"><semantics id="S4.Thmtheorem4.p1.11.m11.1a"><mrow id="S4.Thmtheorem4.p1.11.m11.1.1" xref="S4.Thmtheorem4.p1.11.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p1.11.m11.1.1.2" xref="S4.Thmtheorem4.p1.11.m11.1.1.2.cmml">𝒞</mi><mo id="S4.Thmtheorem4.p1.11.m11.1.1.1" xref="S4.Thmtheorem4.p1.11.m11.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p1.11.m11.1.1.3" xref="S4.Thmtheorem4.p1.11.m11.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.11.m11.1b"><apply id="S4.Thmtheorem4.p1.11.m11.1.1.cmml" xref="S4.Thmtheorem4.p1.11.m11.1.1"><times id="S4.Thmtheorem4.p1.11.m11.1.1.1.cmml" xref="S4.Thmtheorem4.p1.11.m11.1.1.1"></times><ci id="S4.Thmtheorem4.p1.11.m11.1.1.2.cmml" xref="S4.Thmtheorem4.p1.11.m11.1.1.2">𝒞</ci><ci id="S4.Thmtheorem4.p1.11.m11.1.1.3.cmml" xref="S4.Thmtheorem4.p1.11.m11.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.11.m11.1c">\mathcal{CW}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.11.m11.1d">caligraphic_C caligraphic_W</annotation></semantics></math>. In addition, it holds that <math alttext="\frac{1+x}{x-1}\mathcal{CW}(\pi)=\mathcal{CW}(\pi^{*})" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.12.m12.2"><semantics id="S4.Thmtheorem4.p1.12.m12.2a"><mrow id="S4.Thmtheorem4.p1.12.m12.2.2" xref="S4.Thmtheorem4.p1.12.m12.2.2.cmml"><mrow id="S4.Thmtheorem4.p1.12.m12.2.2.3" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.cmml"><mfrac id="S4.Thmtheorem4.p1.12.m12.2.2.3.2" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.cmml"><mrow id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2.cmml"><mn id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2.2" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2.2.cmml">1</mn><mo id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2.1" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2.1.cmml">+</mo><mi id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2.3" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2.3.cmml">x</mi></mrow><mrow id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3.cmml"><mi id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3.2" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3.2.cmml">x</mi><mo id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3.1" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3.1.cmml">−</mo><mn id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3.3" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3.3.cmml">1</mn></mrow></mfrac><mo id="S4.Thmtheorem4.p1.12.m12.2.2.3.1" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p1.12.m12.2.2.3.3" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.3.cmml">𝒞</mi><mo id="S4.Thmtheorem4.p1.12.m12.2.2.3.1a" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p1.12.m12.2.2.3.4" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.4.cmml">𝒲</mi><mo id="S4.Thmtheorem4.p1.12.m12.2.2.3.1b" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem4.p1.12.m12.2.2.3.5.2" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.cmml"><mo id="S4.Thmtheorem4.p1.12.m12.2.2.3.5.2.1" stretchy="false" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.cmml">(</mo><mi id="S4.Thmtheorem4.p1.12.m12.1.1" xref="S4.Thmtheorem4.p1.12.m12.1.1.cmml">π</mi><mo id="S4.Thmtheorem4.p1.12.m12.2.2.3.5.2.2" stretchy="false" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem4.p1.12.m12.2.2.2" xref="S4.Thmtheorem4.p1.12.m12.2.2.2.cmml">=</mo><mrow id="S4.Thmtheorem4.p1.12.m12.2.2.1" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p1.12.m12.2.2.1.3" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.3.cmml">𝒞</mi><mo id="S4.Thmtheorem4.p1.12.m12.2.2.1.2" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p1.12.m12.2.2.1.4" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.4.cmml">𝒲</mi><mo id="S4.Thmtheorem4.p1.12.m12.2.2.1.2a" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.1.cmml"><mo id="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.2" stretchy="false" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.1.cmml">(</mo><msup id="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.1" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.1.cmml"><mi id="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.1.2" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.1.2.cmml">π</mi><mo id="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.1.3" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.3" stretchy="false" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.12.m12.2b"><apply id="S4.Thmtheorem4.p1.12.m12.2.2.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2"><eq id="S4.Thmtheorem4.p1.12.m12.2.2.2.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.2"></eq><apply id="S4.Thmtheorem4.p1.12.m12.2.2.3.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.3"><times id="S4.Thmtheorem4.p1.12.m12.2.2.3.1.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.1"></times><apply id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2"><divide id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.1.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2"></divide><apply id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2"><plus id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2.1.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2.1"></plus><cn id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2.2.cmml" type="integer" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2.2">1</cn><ci id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2.3.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.2.3">𝑥</ci></apply><apply id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3"><minus id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3.1.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3.1"></minus><ci id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3.2.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3.2">𝑥</ci><cn id="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.2.3.3">1</cn></apply></apply><ci id="S4.Thmtheorem4.p1.12.m12.2.2.3.3.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.3">𝒞</ci><ci id="S4.Thmtheorem4.p1.12.m12.2.2.3.4.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.3.4">𝒲</ci><ci id="S4.Thmtheorem4.p1.12.m12.1.1.cmml" xref="S4.Thmtheorem4.p1.12.m12.1.1">𝜋</ci></apply><apply id="S4.Thmtheorem4.p1.12.m12.2.2.1.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.1"><times id="S4.Thmtheorem4.p1.12.m12.2.2.1.2.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.2"></times><ci id="S4.Thmtheorem4.p1.12.m12.2.2.1.3.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.3">𝒞</ci><ci id="S4.Thmtheorem4.p1.12.m12.2.2.1.4.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.4">𝒲</ci><apply id="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.1.2">𝜋</ci><times id="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.1.3.cmml" xref="S4.Thmtheorem4.p1.12.m12.2.2.1.1.1.1.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.12.m12.2c">\frac{1+x}{x-1}\mathcal{CW}(\pi)=\mathcal{CW}(\pi^{*})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.12.m12.2d">divide start_ARG 1 + italic_x end_ARG start_ARG italic_x - 1 end_ARG caligraphic_C caligraphic_W ( italic_π ) = caligraphic_C caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math>, whereas <math alttext="\frac{\mathcal{SW}(\pi)}{\mathcal{SW}(\pi^{*})}=0" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.13.m13.2"><semantics id="S4.Thmtheorem4.p1.13.m13.2a"><mrow id="S4.Thmtheorem4.p1.13.m13.2.3" xref="S4.Thmtheorem4.p1.13.m13.2.3.cmml"><mfrac id="S4.Thmtheorem4.p1.13.m13.2.2" xref="S4.Thmtheorem4.p1.13.m13.2.2.cmml"><mrow id="S4.Thmtheorem4.p1.13.m13.1.1.1" xref="S4.Thmtheorem4.p1.13.m13.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p1.13.m13.1.1.1.3" xref="S4.Thmtheorem4.p1.13.m13.1.1.1.3.cmml">𝒮</mi><mo id="S4.Thmtheorem4.p1.13.m13.1.1.1.2" xref="S4.Thmtheorem4.p1.13.m13.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p1.13.m13.1.1.1.4" xref="S4.Thmtheorem4.p1.13.m13.1.1.1.4.cmml">𝒲</mi><mo id="S4.Thmtheorem4.p1.13.m13.1.1.1.2a" xref="S4.Thmtheorem4.p1.13.m13.1.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem4.p1.13.m13.1.1.1.5.2" xref="S4.Thmtheorem4.p1.13.m13.1.1.1.cmml"><mo id="S4.Thmtheorem4.p1.13.m13.1.1.1.5.2.1" stretchy="false" xref="S4.Thmtheorem4.p1.13.m13.1.1.1.cmml">(</mo><mi id="S4.Thmtheorem4.p1.13.m13.1.1.1.1" xref="S4.Thmtheorem4.p1.13.m13.1.1.1.1.cmml">π</mi><mo id="S4.Thmtheorem4.p1.13.m13.1.1.1.5.2.2" stretchy="false" xref="S4.Thmtheorem4.p1.13.m13.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S4.Thmtheorem4.p1.13.m13.2.2.2" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p1.13.m13.2.2.2.3" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.3.cmml">𝒮</mi><mo id="S4.Thmtheorem4.p1.13.m13.2.2.2.2" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p1.13.m13.2.2.2.4" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.4.cmml">𝒲</mi><mo id="S4.Thmtheorem4.p1.13.m13.2.2.2.2a" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.2.cmml">⁢</mo><mrow id="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.1.cmml"><mo id="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.1.cmml">(</mo><msup id="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.1" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.1.2" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.1.2.cmml">π</mi><mo id="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.1.3" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.1.3.cmml">∗</mo></msup><mo id="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.3" stretchy="false" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mfrac><mo id="S4.Thmtheorem4.p1.13.m13.2.3.1" xref="S4.Thmtheorem4.p1.13.m13.2.3.1.cmml">=</mo><mn id="S4.Thmtheorem4.p1.13.m13.2.3.2" xref="S4.Thmtheorem4.p1.13.m13.2.3.2.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.13.m13.2b"><apply id="S4.Thmtheorem4.p1.13.m13.2.3.cmml" xref="S4.Thmtheorem4.p1.13.m13.2.3"><eq id="S4.Thmtheorem4.p1.13.m13.2.3.1.cmml" xref="S4.Thmtheorem4.p1.13.m13.2.3.1"></eq><apply id="S4.Thmtheorem4.p1.13.m13.2.2.cmml" xref="S4.Thmtheorem4.p1.13.m13.2.2"><divide id="S4.Thmtheorem4.p1.13.m13.2.2.3.cmml" xref="S4.Thmtheorem4.p1.13.m13.2.2"></divide><apply id="S4.Thmtheorem4.p1.13.m13.1.1.1.cmml" xref="S4.Thmtheorem4.p1.13.m13.1.1.1"><times id="S4.Thmtheorem4.p1.13.m13.1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.13.m13.1.1.1.2"></times><ci id="S4.Thmtheorem4.p1.13.m13.1.1.1.3.cmml" xref="S4.Thmtheorem4.p1.13.m13.1.1.1.3">𝒮</ci><ci id="S4.Thmtheorem4.p1.13.m13.1.1.1.4.cmml" xref="S4.Thmtheorem4.p1.13.m13.1.1.1.4">𝒲</ci><ci id="S4.Thmtheorem4.p1.13.m13.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.13.m13.1.1.1.1">𝜋</ci></apply><apply id="S4.Thmtheorem4.p1.13.m13.2.2.2.cmml" xref="S4.Thmtheorem4.p1.13.m13.2.2.2"><times id="S4.Thmtheorem4.p1.13.m13.2.2.2.2.cmml" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.2"></times><ci id="S4.Thmtheorem4.p1.13.m13.2.2.2.3.cmml" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.3">𝒮</ci><ci id="S4.Thmtheorem4.p1.13.m13.2.2.2.4.cmml" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.4">𝒲</ci><apply id="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.1.cmml" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1">superscript</csymbol><ci id="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.1.2">𝜋</ci><times id="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem4.p1.13.m13.2.2.2.1.1.1.3"></times></apply></apply></apply><cn id="S4.Thmtheorem4.p1.13.m13.2.3.2.cmml" type="integer" xref="S4.Thmtheorem4.p1.13.m13.2.3.2">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.13.m13.2c">\frac{\mathcal{SW}(\pi)}{\mathcal{SW}(\pi^{*})}=0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.13.m13.2d">divide start_ARG caligraphic_S caligraphic_W ( italic_π ) end_ARG start_ARG caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) end_ARG = 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.Thmtheorem4.p2"> <p class="ltx_p" id="S4.Thmtheorem4.p2.9">Now, consider any approximation guarantee <math alttext="c>1" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p2.1.m1.1"><semantics id="S4.Thmtheorem4.p2.1.m1.1a"><mrow id="S4.Thmtheorem4.p2.1.m1.1.1" xref="S4.Thmtheorem4.p2.1.m1.1.1.cmml"><mi id="S4.Thmtheorem4.p2.1.m1.1.1.2" xref="S4.Thmtheorem4.p2.1.m1.1.1.2.cmml">c</mi><mo id="S4.Thmtheorem4.p2.1.m1.1.1.1" xref="S4.Thmtheorem4.p2.1.m1.1.1.1.cmml">></mo><mn id="S4.Thmtheorem4.p2.1.m1.1.1.3" xref="S4.Thmtheorem4.p2.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p2.1.m1.1b"><apply id="S4.Thmtheorem4.p2.1.m1.1.1.cmml" xref="S4.Thmtheorem4.p2.1.m1.1.1"><gt id="S4.Thmtheorem4.p2.1.m1.1.1.1.cmml" xref="S4.Thmtheorem4.p2.1.m1.1.1.1"></gt><ci id="S4.Thmtheorem4.p2.1.m1.1.1.2.cmml" xref="S4.Thmtheorem4.p2.1.m1.1.1.2">𝑐</ci><cn id="S4.Thmtheorem4.p2.1.m1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem4.p2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p2.1.m1.1c">c>1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p2.1.m1.1d">italic_c > 1</annotation></semantics></math>. Then, <math alttext="\pi" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p2.2.m2.1"><semantics id="S4.Thmtheorem4.p2.2.m2.1a"><mi id="S4.Thmtheorem4.p2.2.m2.1.1" xref="S4.Thmtheorem4.p2.2.m2.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p2.2.m2.1b"><ci id="S4.Thmtheorem4.p2.2.m2.1.1.cmml" xref="S4.Thmtheorem4.p2.2.m2.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p2.2.m2.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p2.2.m2.1d">italic_π</annotation></semantics></math> provides a <math alttext="c" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p2.3.m3.1"><semantics id="S4.Thmtheorem4.p2.3.m3.1a"><mi id="S4.Thmtheorem4.p2.3.m3.1.1" xref="S4.Thmtheorem4.p2.3.m3.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p2.3.m3.1b"><ci id="S4.Thmtheorem4.p2.3.m3.1.1.cmml" xref="S4.Thmtheorem4.p2.3.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p2.3.m3.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p2.3.m3.1d">italic_c</annotation></semantics></math>-approximation of <math alttext="\mathcal{CW}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p2.4.m4.1"><semantics id="S4.Thmtheorem4.p2.4.m4.1a"><mrow id="S4.Thmtheorem4.p2.4.m4.1.1" xref="S4.Thmtheorem4.p2.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p2.4.m4.1.1.2" xref="S4.Thmtheorem4.p2.4.m4.1.1.2.cmml">𝒞</mi><mo id="S4.Thmtheorem4.p2.4.m4.1.1.1" xref="S4.Thmtheorem4.p2.4.m4.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p2.4.m4.1.1.3" xref="S4.Thmtheorem4.p2.4.m4.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p2.4.m4.1b"><apply id="S4.Thmtheorem4.p2.4.m4.1.1.cmml" xref="S4.Thmtheorem4.p2.4.m4.1.1"><times id="S4.Thmtheorem4.p2.4.m4.1.1.1.cmml" xref="S4.Thmtheorem4.p2.4.m4.1.1.1"></times><ci id="S4.Thmtheorem4.p2.4.m4.1.1.2.cmml" xref="S4.Thmtheorem4.p2.4.m4.1.1.2">𝒞</ci><ci id="S4.Thmtheorem4.p2.4.m4.1.1.3.cmml" xref="S4.Thmtheorem4.p2.4.m4.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p2.4.m4.1c">\mathcal{CW}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p2.4.m4.1d">caligraphic_C caligraphic_W</annotation></semantics></math> for <math alttext="x=\frac{c-1}{1+c}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p2.5.m5.1"><semantics id="S4.Thmtheorem4.p2.5.m5.1a"><mrow id="S4.Thmtheorem4.p2.5.m5.1.1" xref="S4.Thmtheorem4.p2.5.m5.1.1.cmml"><mi id="S4.Thmtheorem4.p2.5.m5.1.1.2" xref="S4.Thmtheorem4.p2.5.m5.1.1.2.cmml">x</mi><mo id="S4.Thmtheorem4.p2.5.m5.1.1.1" xref="S4.Thmtheorem4.p2.5.m5.1.1.1.cmml">=</mo><mfrac id="S4.Thmtheorem4.p2.5.m5.1.1.3" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.cmml"><mrow id="S4.Thmtheorem4.p2.5.m5.1.1.3.2" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.2.cmml"><mi id="S4.Thmtheorem4.p2.5.m5.1.1.3.2.2" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.2.2.cmml">c</mi><mo id="S4.Thmtheorem4.p2.5.m5.1.1.3.2.1" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.2.1.cmml">−</mo><mn id="S4.Thmtheorem4.p2.5.m5.1.1.3.2.3" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.2.3.cmml">1</mn></mrow><mrow id="S4.Thmtheorem4.p2.5.m5.1.1.3.3" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.3.cmml"><mn id="S4.Thmtheorem4.p2.5.m5.1.1.3.3.2" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.3.2.cmml">1</mn><mo id="S4.Thmtheorem4.p2.5.m5.1.1.3.3.1" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.3.1.cmml">+</mo><mi id="S4.Thmtheorem4.p2.5.m5.1.1.3.3.3" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.3.3.cmml">c</mi></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p2.5.m5.1b"><apply id="S4.Thmtheorem4.p2.5.m5.1.1.cmml" xref="S4.Thmtheorem4.p2.5.m5.1.1"><eq id="S4.Thmtheorem4.p2.5.m5.1.1.1.cmml" xref="S4.Thmtheorem4.p2.5.m5.1.1.1"></eq><ci id="S4.Thmtheorem4.p2.5.m5.1.1.2.cmml" xref="S4.Thmtheorem4.p2.5.m5.1.1.2">𝑥</ci><apply id="S4.Thmtheorem4.p2.5.m5.1.1.3.cmml" xref="S4.Thmtheorem4.p2.5.m5.1.1.3"><divide id="S4.Thmtheorem4.p2.5.m5.1.1.3.1.cmml" xref="S4.Thmtheorem4.p2.5.m5.1.1.3"></divide><apply id="S4.Thmtheorem4.p2.5.m5.1.1.3.2.cmml" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.2"><minus id="S4.Thmtheorem4.p2.5.m5.1.1.3.2.1.cmml" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.2.1"></minus><ci id="S4.Thmtheorem4.p2.5.m5.1.1.3.2.2.cmml" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.2.2">𝑐</ci><cn id="S4.Thmtheorem4.p2.5.m5.1.1.3.2.3.cmml" type="integer" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.2.3">1</cn></apply><apply id="S4.Thmtheorem4.p2.5.m5.1.1.3.3.cmml" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.3"><plus id="S4.Thmtheorem4.p2.5.m5.1.1.3.3.1.cmml" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.3.1"></plus><cn id="S4.Thmtheorem4.p2.5.m5.1.1.3.3.2.cmml" type="integer" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.3.2">1</cn><ci id="S4.Thmtheorem4.p2.5.m5.1.1.3.3.3.cmml" xref="S4.Thmtheorem4.p2.5.m5.1.1.3.3.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p2.5.m5.1c">x=\frac{c-1}{1+c}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p2.5.m5.1d">italic_x = divide start_ARG italic_c - 1 end_ARG start_ARG 1 + italic_c end_ARG</annotation></semantics></math>, whereas <math alttext="\pi" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p2.6.m6.1"><semantics id="S4.Thmtheorem4.p2.6.m6.1a"><mi id="S4.Thmtheorem4.p2.6.m6.1.1" xref="S4.Thmtheorem4.p2.6.m6.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p2.6.m6.1b"><ci id="S4.Thmtheorem4.p2.6.m6.1.1.cmml" xref="S4.Thmtheorem4.p2.6.m6.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p2.6.m6.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p2.6.m6.1d">italic_π</annotation></semantics></math> yields no <math alttext="c" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p2.7.m7.1"><semantics id="S4.Thmtheorem4.p2.7.m7.1a"><mi id="S4.Thmtheorem4.p2.7.m7.1.1" xref="S4.Thmtheorem4.p2.7.m7.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p2.7.m7.1b"><ci id="S4.Thmtheorem4.p2.7.m7.1.1.cmml" xref="S4.Thmtheorem4.p2.7.m7.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p2.7.m7.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p2.7.m7.1d">italic_c</annotation></semantics></math>-approximation of <math alttext="\mathcal{SW}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p2.8.m8.1"><semantics id="S4.Thmtheorem4.p2.8.m8.1a"><mrow id="S4.Thmtheorem4.p2.8.m8.1.1" xref="S4.Thmtheorem4.p2.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p2.8.m8.1.1.2" xref="S4.Thmtheorem4.p2.8.m8.1.1.2.cmml">𝒮</mi><mo id="S4.Thmtheorem4.p2.8.m8.1.1.1" xref="S4.Thmtheorem4.p2.8.m8.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem4.p2.8.m8.1.1.3" xref="S4.Thmtheorem4.p2.8.m8.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p2.8.m8.1b"><apply id="S4.Thmtheorem4.p2.8.m8.1.1.cmml" xref="S4.Thmtheorem4.p2.8.m8.1.1"><times id="S4.Thmtheorem4.p2.8.m8.1.1.1.cmml" xref="S4.Thmtheorem4.p2.8.m8.1.1.1"></times><ci id="S4.Thmtheorem4.p2.8.m8.1.1.2.cmml" xref="S4.Thmtheorem4.p2.8.m8.1.1.2">𝒮</ci><ci id="S4.Thmtheorem4.p2.8.m8.1.1.3.cmml" xref="S4.Thmtheorem4.p2.8.m8.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p2.8.m8.1c">\mathcal{SW}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p2.8.m8.1d">caligraphic_S caligraphic_W</annotation></semantics></math>. <math alttext="\lhd" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p2.9.m9.1"><semantics id="S4.Thmtheorem4.p2.9.m9.1a"><mo id="S4.Thmtheorem4.p2.9.m9.1.1" xref="S4.Thmtheorem4.p2.9.m9.1.1.cmml">⊲</mo><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p2.9.m9.1b"><csymbol cd="latexml" id="S4.Thmtheorem4.p2.9.m9.1.1.cmml" xref="S4.Thmtheorem4.p2.9.m9.1.1">subgroup-of</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p2.9.m9.1c">\lhd</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p2.9.m9.1d">⊲</annotation></semantics></math></p> </div> </div> <div class="ltx_para" id="S4.SS2.p6"> <p class="ltx_p" id="S4.SS2.p6.1">The reason why approximate outcomes in terms of correlation welfare do not yield any guarantee on the social welfare in <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.Thmtheorem4" title="Example 4.4. ‣ 4.2 Logarithmic Approximation for Nonnegative Total Value ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Example</span> <span class="ltx_text ltx_ref_tag">4.4</span></a> is that there is a single valuation that is very negative. It is enough that the two corresponding agents are in different coalitions to obtain a good approximation of the correlation welfare. However, the picture changes if such a situation does not occur. If the total value of an instance is nonnegative, then social welfare inherits approximation guarantees from correlation welfare.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem5.1.1.1">Lemma 4.5</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem5.p1"> <p class="ltx_p" id="S4.Thmtheorem5.p1.8"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p1.8.8">Let <math alttext="(N,v)" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.1.1.m1.2"><semantics id="S4.Thmtheorem5.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem5.p1.1.1.m1.2.3.2" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.1.cmml"><mo id="S4.Thmtheorem5.p1.1.1.m1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.1.cmml">(</mo><mi id="S4.Thmtheorem5.p1.1.1.m1.1.1" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.cmml">N</mi><mo id="S4.Thmtheorem5.p1.1.1.m1.2.3.2.2" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem5.p1.1.1.m1.2.2" xref="S4.Thmtheorem5.p1.1.1.m1.2.2.cmml">v</mi><mo id="S4.Thmtheorem5.p1.1.1.m1.2.3.2.3" stretchy="false" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.1.1.m1.2b"><interval closure="open" id="S4.Thmtheorem5.p1.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.2"><ci id="S4.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.1.1">𝑁</ci><ci id="S4.Thmtheorem5.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.1.1.m1.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.1.1.m1.2d">( italic_N , italic_v )</annotation></semantics></math> be an ASHG such that <math alttext="\mathcal{V}(N,v)\geq 0" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.2.2.m2.2"><semantics id="S4.Thmtheorem5.p1.2.2.m2.2a"><mrow id="S4.Thmtheorem5.p1.2.2.m2.2.3" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.cmml"><mrow id="S4.Thmtheorem5.p1.2.2.m2.2.3.2" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.2" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2.2.cmml">𝒱</mi><mo id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.1" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.3.2" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2.3.1.cmml"><mo id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2.3.1.cmml">(</mo><mi id="S4.Thmtheorem5.p1.2.2.m2.1.1" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.cmml">N</mi><mo id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.3.2.2" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem5.p1.2.2.m2.2.2" xref="S4.Thmtheorem5.p1.2.2.m2.2.2.cmml">v</mi><mo id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.3.2.3" stretchy="false" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem5.p1.2.2.m2.2.3.1" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.1.cmml">≥</mo><mn id="S4.Thmtheorem5.p1.2.2.m2.2.3.3" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.2.2.m2.2b"><apply id="S4.Thmtheorem5.p1.2.2.m2.2.3.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3"><geq id="S4.Thmtheorem5.p1.2.2.m2.2.3.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.1"></geq><apply id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2"><times id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2.1"></times><ci id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2.2">𝒱</ci><interval closure="open" id="S4.Thmtheorem5.p1.2.2.m2.2.3.2.3.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.2.3.2"><ci id="S4.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1">𝑁</ci><ci id="S4.Thmtheorem5.p1.2.2.m2.2.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.2.2">𝑣</ci></interval></apply><cn id="S4.Thmtheorem5.p1.2.2.m2.2.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p1.2.2.m2.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.2.2.m2.2c">\mathcal{V}(N,v)\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.2.2.m2.2d">caligraphic_V ( italic_N , italic_v ) ≥ 0</annotation></semantics></math>. Let <math alttext="c\geq 1" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.3.3.m3.1"><semantics id="S4.Thmtheorem5.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem5.p1.3.3.m3.1.1" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem5.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.2.cmml">c</mi><mo id="S4.Thmtheorem5.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.1.cmml">≥</mo><mn id="S4.Thmtheorem5.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.3.3.m3.1b"><apply id="S4.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1"><geq id="S4.Thmtheorem5.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.1"></geq><ci id="S4.Thmtheorem5.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.2">𝑐</ci><cn id="S4.Thmtheorem5.p1.3.3.m3.1.1.3.cmml" type="integer" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.3.3.m3.1c">c\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.3.3.m3.1d">italic_c ≥ 1</annotation></semantics></math> and let <math alttext="\pi^{*}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.4.4.m4.1"><semantics id="S4.Thmtheorem5.p1.4.4.m4.1a"><msup id="S4.Thmtheorem5.p1.4.4.m4.1.1" xref="S4.Thmtheorem5.p1.4.4.m4.1.1.cmml"><mi id="S4.Thmtheorem5.p1.4.4.m4.1.1.2" xref="S4.Thmtheorem5.p1.4.4.m4.1.1.2.cmml">π</mi><mo id="S4.Thmtheorem5.p1.4.4.m4.1.1.3" xref="S4.Thmtheorem5.p1.4.4.m4.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.4.4.m4.1b"><apply id="S4.Thmtheorem5.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.1.1">superscript</csymbol><ci id="S4.Thmtheorem5.p1.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.1.1.2">𝜋</ci><times id="S4.Thmtheorem5.p1.4.4.m4.1.1.3.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.4.4.m4.1c">\pi^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.4.4.m4.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be a partition maximizing <math alttext="\mathcal{CW}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.5.5.m5.1"><semantics id="S4.Thmtheorem5.p1.5.5.m5.1a"><mrow id="S4.Thmtheorem5.p1.5.5.m5.1.1" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem5.p1.5.5.m5.1.1.2" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.2.cmml">𝒞</mi><mo id="S4.Thmtheorem5.p1.5.5.m5.1.1.1" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem5.p1.5.5.m5.1.1.3" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.5.5.m5.1b"><apply id="S4.Thmtheorem5.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1"><times id="S4.Thmtheorem5.p1.5.5.m5.1.1.1.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.1"></times><ci id="S4.Thmtheorem5.p1.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.2">𝒞</ci><ci id="S4.Thmtheorem5.p1.5.5.m5.1.1.3.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.5.5.m5.1c">\mathcal{CW}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.5.5.m5.1d">caligraphic_C caligraphic_W</annotation></semantics></math>. Let <math alttext="\pi" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.6.6.m6.1"><semantics id="S4.Thmtheorem5.p1.6.6.m6.1a"><mi id="S4.Thmtheorem5.p1.6.6.m6.1.1" xref="S4.Thmtheorem5.p1.6.6.m6.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.6.6.m6.1b"><ci id="S4.Thmtheorem5.p1.6.6.m6.1.1.cmml" xref="S4.Thmtheorem5.p1.6.6.m6.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.6.6.m6.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.6.6.m6.1d">italic_π</annotation></semantics></math> be a partition with <math alttext="c\cdot\mathcal{CW}(\pi)\geq\mathcal{CW}(\pi^{*})" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.7.7.m7.2"><semantics id="S4.Thmtheorem5.p1.7.7.m7.2a"><mrow id="S4.Thmtheorem5.p1.7.7.m7.2.2" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.cmml"><mrow id="S4.Thmtheorem5.p1.7.7.m7.2.2.3" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.cmml"><mrow id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2.cmml"><mi id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2.2" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2.2.cmml">c</mi><mo id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2.1" lspace="0.222em" rspace="0.222em" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2.1.cmml">⋅</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2.3" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2.3.cmml">𝒞</mi></mrow><mo id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.1" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.3" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.3.cmml">𝒲</mi><mo id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.1a" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.4.2" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.cmml"><mo id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.4.2.1" stretchy="false" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.cmml">(</mo><mi id="S4.Thmtheorem5.p1.7.7.m7.1.1" xref="S4.Thmtheorem5.p1.7.7.m7.1.1.cmml">π</mi><mo id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.4.2.2" stretchy="false" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem5.p1.7.7.m7.2.2.2" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.2.cmml">≥</mo><mrow id="S4.Thmtheorem5.p1.7.7.m7.2.2.1" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.3" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.3.cmml">𝒞</mi><mo id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.2" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.4" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.4.cmml">𝒲</mi><mo id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.2a" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.1.cmml"><mo id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.2" stretchy="false" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.1.cmml">(</mo><msup id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.1" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.1.cmml"><mi id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.1.2" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.1.2.cmml">π</mi><mo id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.1.3" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.3" stretchy="false" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.7.7.m7.2b"><apply id="S4.Thmtheorem5.p1.7.7.m7.2.2.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2"><geq id="S4.Thmtheorem5.p1.7.7.m7.2.2.2.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.2"></geq><apply id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3"><times id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.1.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.1"></times><apply id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2"><ci id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2.1.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2.1">⋅</ci><ci id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2.2.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2.2">𝑐</ci><ci id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2.3.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.2.3">𝒞</ci></apply><ci id="S4.Thmtheorem5.p1.7.7.m7.2.2.3.3.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.3.3">𝒲</ci><ci id="S4.Thmtheorem5.p1.7.7.m7.1.1.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.1.1">𝜋</ci></apply><apply id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1"><times id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.2.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.2"></times><ci id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.3.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.3">𝒞</ci><ci id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.4.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.4">𝒲</ci><apply id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.1.1.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.1.2.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.1.2">𝜋</ci><times id="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.1.3.cmml" xref="S4.Thmtheorem5.p1.7.7.m7.2.2.1.1.1.1.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.7.7.m7.2c">c\cdot\mathcal{CW}(\pi)\geq\mathcal{CW}(\pi^{*})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.7.7.m7.2d">italic_c ⋅ caligraphic_C caligraphic_W ( italic_π ) ≥ caligraphic_C caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math>. Then it holds that <math alttext="c\cdot\mathcal{SW}(\pi)\geq\mathcal{SW}(\pi^{*})" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.8.8.m8.2"><semantics id="S4.Thmtheorem5.p1.8.8.m8.2a"><mrow id="S4.Thmtheorem5.p1.8.8.m8.2.2" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.cmml"><mrow id="S4.Thmtheorem5.p1.8.8.m8.2.2.3" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.cmml"><mrow id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2.cmml"><mi id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2.2" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2.2.cmml">c</mi><mo id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2.1" lspace="0.222em" rspace="0.222em" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2.1.cmml">⋅</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2.3" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2.3.cmml">𝒮</mi></mrow><mo id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.1" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.3" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.3.cmml">𝒲</mi><mo id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.1a" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.4.2" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.cmml"><mo id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.4.2.1" stretchy="false" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.cmml">(</mo><mi id="S4.Thmtheorem5.p1.8.8.m8.1.1" xref="S4.Thmtheorem5.p1.8.8.m8.1.1.cmml">π</mi><mo id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.4.2.2" stretchy="false" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem5.p1.8.8.m8.2.2.2" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.2.cmml">≥</mo><mrow id="S4.Thmtheorem5.p1.8.8.m8.2.2.1" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.3" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.3.cmml">𝒮</mi><mo id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.2" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.4" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.4.cmml">𝒲</mi><mo id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.2a" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.1.cmml"><mo id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.2" stretchy="false" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.1.cmml">(</mo><msup id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.1" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.1.cmml"><mi id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.1.2" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.1.2.cmml">π</mi><mo id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.1.3" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.3" stretchy="false" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.8.8.m8.2b"><apply id="S4.Thmtheorem5.p1.8.8.m8.2.2.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2"><geq id="S4.Thmtheorem5.p1.8.8.m8.2.2.2.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.2"></geq><apply id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3"><times id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.1.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.1"></times><apply id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2"><ci id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2.1.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2.1">⋅</ci><ci id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2.2.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2.2">𝑐</ci><ci id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2.3.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.2.3">𝒮</ci></apply><ci id="S4.Thmtheorem5.p1.8.8.m8.2.2.3.3.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.3.3">𝒲</ci><ci id="S4.Thmtheorem5.p1.8.8.m8.1.1.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.1.1">𝜋</ci></apply><apply id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1"><times id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.2.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.2"></times><ci id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.3.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.3">𝒮</ci><ci id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.4.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.4">𝒲</ci><apply id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.1.1.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.1.2.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.1.2">𝜋</ci><times id="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.1.3.cmml" xref="S4.Thmtheorem5.p1.8.8.m8.2.2.1.1.1.1.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.8.8.m8.2c">c\cdot\mathcal{SW}(\pi)\geq\mathcal{SW}(\pi^{*})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.8.8.m8.2d">italic_c ⋅ caligraphic_S caligraphic_W ( italic_π ) ≥ caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.SS2.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.3.p1"> <p class="ltx_p" id="S4.SS2.3.p1.7">Consider an ASHG <math alttext="(N,v)" class="ltx_Math" display="inline" id="S4.SS2.3.p1.1.m1.2"><semantics id="S4.SS2.3.p1.1.m1.2a"><mrow id="S4.SS2.3.p1.1.m1.2.3.2" xref="S4.SS2.3.p1.1.m1.2.3.1.cmml"><mo id="S4.SS2.3.p1.1.m1.2.3.2.1" stretchy="false" xref="S4.SS2.3.p1.1.m1.2.3.1.cmml">(</mo><mi id="S4.SS2.3.p1.1.m1.1.1" xref="S4.SS2.3.p1.1.m1.1.1.cmml">N</mi><mo id="S4.SS2.3.p1.1.m1.2.3.2.2" xref="S4.SS2.3.p1.1.m1.2.3.1.cmml">,</mo><mi id="S4.SS2.3.p1.1.m1.2.2" xref="S4.SS2.3.p1.1.m1.2.2.cmml">v</mi><mo id="S4.SS2.3.p1.1.m1.2.3.2.3" stretchy="false" xref="S4.SS2.3.p1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p1.1.m1.2b"><interval closure="open" id="S4.SS2.3.p1.1.m1.2.3.1.cmml" xref="S4.SS2.3.p1.1.m1.2.3.2"><ci id="S4.SS2.3.p1.1.m1.1.1.cmml" xref="S4.SS2.3.p1.1.m1.1.1">𝑁</ci><ci id="S4.SS2.3.p1.1.m1.2.2.cmml" xref="S4.SS2.3.p1.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p1.1.m1.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p1.1.m1.2d">( italic_N , italic_v )</annotation></semantics></math> such that <math alttext="\mathcal{V}(N,v)\geq 0" class="ltx_Math" display="inline" id="S4.SS2.3.p1.2.m2.2"><semantics id="S4.SS2.3.p1.2.m2.2a"><mrow id="S4.SS2.3.p1.2.m2.2.3" xref="S4.SS2.3.p1.2.m2.2.3.cmml"><mrow id="S4.SS2.3.p1.2.m2.2.3.2" xref="S4.SS2.3.p1.2.m2.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.3.p1.2.m2.2.3.2.2" xref="S4.SS2.3.p1.2.m2.2.3.2.2.cmml">𝒱</mi><mo id="S4.SS2.3.p1.2.m2.2.3.2.1" xref="S4.SS2.3.p1.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS2.3.p1.2.m2.2.3.2.3.2" xref="S4.SS2.3.p1.2.m2.2.3.2.3.1.cmml"><mo id="S4.SS2.3.p1.2.m2.2.3.2.3.2.1" stretchy="false" xref="S4.SS2.3.p1.2.m2.2.3.2.3.1.cmml">(</mo><mi id="S4.SS2.3.p1.2.m2.1.1" xref="S4.SS2.3.p1.2.m2.1.1.cmml">N</mi><mo id="S4.SS2.3.p1.2.m2.2.3.2.3.2.2" xref="S4.SS2.3.p1.2.m2.2.3.2.3.1.cmml">,</mo><mi id="S4.SS2.3.p1.2.m2.2.2" xref="S4.SS2.3.p1.2.m2.2.2.cmml">v</mi><mo id="S4.SS2.3.p1.2.m2.2.3.2.3.2.3" stretchy="false" xref="S4.SS2.3.p1.2.m2.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.3.p1.2.m2.2.3.1" xref="S4.SS2.3.p1.2.m2.2.3.1.cmml">≥</mo><mn id="S4.SS2.3.p1.2.m2.2.3.3" xref="S4.SS2.3.p1.2.m2.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p1.2.m2.2b"><apply id="S4.SS2.3.p1.2.m2.2.3.cmml" xref="S4.SS2.3.p1.2.m2.2.3"><geq id="S4.SS2.3.p1.2.m2.2.3.1.cmml" xref="S4.SS2.3.p1.2.m2.2.3.1"></geq><apply id="S4.SS2.3.p1.2.m2.2.3.2.cmml" xref="S4.SS2.3.p1.2.m2.2.3.2"><times id="S4.SS2.3.p1.2.m2.2.3.2.1.cmml" xref="S4.SS2.3.p1.2.m2.2.3.2.1"></times><ci id="S4.SS2.3.p1.2.m2.2.3.2.2.cmml" xref="S4.SS2.3.p1.2.m2.2.3.2.2">𝒱</ci><interval closure="open" id="S4.SS2.3.p1.2.m2.2.3.2.3.1.cmml" xref="S4.SS2.3.p1.2.m2.2.3.2.3.2"><ci id="S4.SS2.3.p1.2.m2.1.1.cmml" xref="S4.SS2.3.p1.2.m2.1.1">𝑁</ci><ci id="S4.SS2.3.p1.2.m2.2.2.cmml" xref="S4.SS2.3.p1.2.m2.2.2">𝑣</ci></interval></apply><cn id="S4.SS2.3.p1.2.m2.2.3.3.cmml" type="integer" xref="S4.SS2.3.p1.2.m2.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p1.2.m2.2c">\mathcal{V}(N,v)\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p1.2.m2.2d">caligraphic_V ( italic_N , italic_v ) ≥ 0</annotation></semantics></math>. Let <math alttext="c\leq 1" class="ltx_Math" display="inline" id="S4.SS2.3.p1.3.m3.1"><semantics id="S4.SS2.3.p1.3.m3.1a"><mrow id="S4.SS2.3.p1.3.m3.1.1" xref="S4.SS2.3.p1.3.m3.1.1.cmml"><mi id="S4.SS2.3.p1.3.m3.1.1.2" xref="S4.SS2.3.p1.3.m3.1.1.2.cmml">c</mi><mo id="S4.SS2.3.p1.3.m3.1.1.1" xref="S4.SS2.3.p1.3.m3.1.1.1.cmml">≤</mo><mn id="S4.SS2.3.p1.3.m3.1.1.3" xref="S4.SS2.3.p1.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p1.3.m3.1b"><apply id="S4.SS2.3.p1.3.m3.1.1.cmml" xref="S4.SS2.3.p1.3.m3.1.1"><leq id="S4.SS2.3.p1.3.m3.1.1.1.cmml" xref="S4.SS2.3.p1.3.m3.1.1.1"></leq><ci id="S4.SS2.3.p1.3.m3.1.1.2.cmml" xref="S4.SS2.3.p1.3.m3.1.1.2">𝑐</ci><cn id="S4.SS2.3.p1.3.m3.1.1.3.cmml" type="integer" xref="S4.SS2.3.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p1.3.m3.1c">c\leq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p1.3.m3.1d">italic_c ≤ 1</annotation></semantics></math> and let <math alttext="\pi^{*}" class="ltx_Math" display="inline" id="S4.SS2.3.p1.4.m4.1"><semantics id="S4.SS2.3.p1.4.m4.1a"><msup id="S4.SS2.3.p1.4.m4.1.1" xref="S4.SS2.3.p1.4.m4.1.1.cmml"><mi id="S4.SS2.3.p1.4.m4.1.1.2" xref="S4.SS2.3.p1.4.m4.1.1.2.cmml">π</mi><mo id="S4.SS2.3.p1.4.m4.1.1.3" xref="S4.SS2.3.p1.4.m4.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p1.4.m4.1b"><apply id="S4.SS2.3.p1.4.m4.1.1.cmml" xref="S4.SS2.3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.3.p1.4.m4.1.1.1.cmml" xref="S4.SS2.3.p1.4.m4.1.1">superscript</csymbol><ci id="S4.SS2.3.p1.4.m4.1.1.2.cmml" xref="S4.SS2.3.p1.4.m4.1.1.2">𝜋</ci><times id="S4.SS2.3.p1.4.m4.1.1.3.cmml" xref="S4.SS2.3.p1.4.m4.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p1.4.m4.1c">\pi^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p1.4.m4.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be a partition maximizing <math alttext="\mathcal{CW}" class="ltx_Math" display="inline" id="S4.SS2.3.p1.5.m5.1"><semantics id="S4.SS2.3.p1.5.m5.1a"><mrow id="S4.SS2.3.p1.5.m5.1.1" xref="S4.SS2.3.p1.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.3.p1.5.m5.1.1.2" xref="S4.SS2.3.p1.5.m5.1.1.2.cmml">𝒞</mi><mo id="S4.SS2.3.p1.5.m5.1.1.1" xref="S4.SS2.3.p1.5.m5.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.3.p1.5.m5.1.1.3" xref="S4.SS2.3.p1.5.m5.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p1.5.m5.1b"><apply id="S4.SS2.3.p1.5.m5.1.1.cmml" xref="S4.SS2.3.p1.5.m5.1.1"><times id="S4.SS2.3.p1.5.m5.1.1.1.cmml" xref="S4.SS2.3.p1.5.m5.1.1.1"></times><ci id="S4.SS2.3.p1.5.m5.1.1.2.cmml" xref="S4.SS2.3.p1.5.m5.1.1.2">𝒞</ci><ci id="S4.SS2.3.p1.5.m5.1.1.3.cmml" xref="S4.SS2.3.p1.5.m5.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p1.5.m5.1c">\mathcal{CW}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p1.5.m5.1d">caligraphic_C caligraphic_W</annotation></semantics></math>. Consider a partition <math alttext="\pi" class="ltx_Math" display="inline" id="S4.SS2.3.p1.6.m6.1"><semantics id="S4.SS2.3.p1.6.m6.1a"><mi id="S4.SS2.3.p1.6.m6.1.1" xref="S4.SS2.3.p1.6.m6.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p1.6.m6.1b"><ci id="S4.SS2.3.p1.6.m6.1.1.cmml" xref="S4.SS2.3.p1.6.m6.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p1.6.m6.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p1.6.m6.1d">italic_π</annotation></semantics></math> with <math alttext="\mathcal{CW}(\pi)\geq c\cdot\mathcal{CW}(\pi^{*})" class="ltx_Math" display="inline" id="S4.SS2.3.p1.7.m7.2"><semantics id="S4.SS2.3.p1.7.m7.2a"><mrow id="S4.SS2.3.p1.7.m7.2.2" xref="S4.SS2.3.p1.7.m7.2.2.cmml"><mrow id="S4.SS2.3.p1.7.m7.2.2.3" xref="S4.SS2.3.p1.7.m7.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.3.p1.7.m7.2.2.3.2" xref="S4.SS2.3.p1.7.m7.2.2.3.2.cmml">𝒞</mi><mo id="S4.SS2.3.p1.7.m7.2.2.3.1" xref="S4.SS2.3.p1.7.m7.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.3.p1.7.m7.2.2.3.3" xref="S4.SS2.3.p1.7.m7.2.2.3.3.cmml">𝒲</mi><mo id="S4.SS2.3.p1.7.m7.2.2.3.1a" xref="S4.SS2.3.p1.7.m7.2.2.3.1.cmml">⁢</mo><mrow id="S4.SS2.3.p1.7.m7.2.2.3.4.2" xref="S4.SS2.3.p1.7.m7.2.2.3.cmml"><mo id="S4.SS2.3.p1.7.m7.2.2.3.4.2.1" stretchy="false" xref="S4.SS2.3.p1.7.m7.2.2.3.cmml">(</mo><mi id="S4.SS2.3.p1.7.m7.1.1" xref="S4.SS2.3.p1.7.m7.1.1.cmml">π</mi><mo id="S4.SS2.3.p1.7.m7.2.2.3.4.2.2" stretchy="false" xref="S4.SS2.3.p1.7.m7.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.SS2.3.p1.7.m7.2.2.2" xref="S4.SS2.3.p1.7.m7.2.2.2.cmml">≥</mo><mrow id="S4.SS2.3.p1.7.m7.2.2.1" xref="S4.SS2.3.p1.7.m7.2.2.1.cmml"><mrow id="S4.SS2.3.p1.7.m7.2.2.1.3" xref="S4.SS2.3.p1.7.m7.2.2.1.3.cmml"><mi id="S4.SS2.3.p1.7.m7.2.2.1.3.2" xref="S4.SS2.3.p1.7.m7.2.2.1.3.2.cmml">c</mi><mo id="S4.SS2.3.p1.7.m7.2.2.1.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS2.3.p1.7.m7.2.2.1.3.1.cmml">⋅</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.3.p1.7.m7.2.2.1.3.3" xref="S4.SS2.3.p1.7.m7.2.2.1.3.3.cmml">𝒞</mi></mrow><mo id="S4.SS2.3.p1.7.m7.2.2.1.2" xref="S4.SS2.3.p1.7.m7.2.2.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.3.p1.7.m7.2.2.1.4" xref="S4.SS2.3.p1.7.m7.2.2.1.4.cmml">𝒲</mi><mo id="S4.SS2.3.p1.7.m7.2.2.1.2a" xref="S4.SS2.3.p1.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS2.3.p1.7.m7.2.2.1.1.1" xref="S4.SS2.3.p1.7.m7.2.2.1.1.1.1.cmml"><mo id="S4.SS2.3.p1.7.m7.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.3.p1.7.m7.2.2.1.1.1.1.cmml">(</mo><msup id="S4.SS2.3.p1.7.m7.2.2.1.1.1.1" xref="S4.SS2.3.p1.7.m7.2.2.1.1.1.1.cmml"><mi id="S4.SS2.3.p1.7.m7.2.2.1.1.1.1.2" xref="S4.SS2.3.p1.7.m7.2.2.1.1.1.1.2.cmml">π</mi><mo id="S4.SS2.3.p1.7.m7.2.2.1.1.1.1.3" xref="S4.SS2.3.p1.7.m7.2.2.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.SS2.3.p1.7.m7.2.2.1.1.1.3" stretchy="false" xref="S4.SS2.3.p1.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p1.7.m7.2b"><apply id="S4.SS2.3.p1.7.m7.2.2.cmml" xref="S4.SS2.3.p1.7.m7.2.2"><geq id="S4.SS2.3.p1.7.m7.2.2.2.cmml" xref="S4.SS2.3.p1.7.m7.2.2.2"></geq><apply id="S4.SS2.3.p1.7.m7.2.2.3.cmml" xref="S4.SS2.3.p1.7.m7.2.2.3"><times id="S4.SS2.3.p1.7.m7.2.2.3.1.cmml" xref="S4.SS2.3.p1.7.m7.2.2.3.1"></times><ci id="S4.SS2.3.p1.7.m7.2.2.3.2.cmml" xref="S4.SS2.3.p1.7.m7.2.2.3.2">𝒞</ci><ci id="S4.SS2.3.p1.7.m7.2.2.3.3.cmml" xref="S4.SS2.3.p1.7.m7.2.2.3.3">𝒲</ci><ci id="S4.SS2.3.p1.7.m7.1.1.cmml" xref="S4.SS2.3.p1.7.m7.1.1">𝜋</ci></apply><apply id="S4.SS2.3.p1.7.m7.2.2.1.cmml" xref="S4.SS2.3.p1.7.m7.2.2.1"><times id="S4.SS2.3.p1.7.m7.2.2.1.2.cmml" xref="S4.SS2.3.p1.7.m7.2.2.1.2"></times><apply id="S4.SS2.3.p1.7.m7.2.2.1.3.cmml" xref="S4.SS2.3.p1.7.m7.2.2.1.3"><ci id="S4.SS2.3.p1.7.m7.2.2.1.3.1.cmml" xref="S4.SS2.3.p1.7.m7.2.2.1.3.1">⋅</ci><ci id="S4.SS2.3.p1.7.m7.2.2.1.3.2.cmml" xref="S4.SS2.3.p1.7.m7.2.2.1.3.2">𝑐</ci><ci id="S4.SS2.3.p1.7.m7.2.2.1.3.3.cmml" xref="S4.SS2.3.p1.7.m7.2.2.1.3.3">𝒞</ci></apply><ci id="S4.SS2.3.p1.7.m7.2.2.1.4.cmml" xref="S4.SS2.3.p1.7.m7.2.2.1.4">𝒲</ci><apply id="S4.SS2.3.p1.7.m7.2.2.1.1.1.1.cmml" xref="S4.SS2.3.p1.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.3.p1.7.m7.2.2.1.1.1.1.1.cmml" xref="S4.SS2.3.p1.7.m7.2.2.1.1.1">superscript</csymbol><ci id="S4.SS2.3.p1.7.m7.2.2.1.1.1.1.2.cmml" xref="S4.SS2.3.p1.7.m7.2.2.1.1.1.1.2">𝜋</ci><times id="S4.SS2.3.p1.7.m7.2.2.1.1.1.1.3.cmml" xref="S4.SS2.3.p1.7.m7.2.2.1.1.1.1.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p1.7.m7.2c">\mathcal{CW}(\pi)\geq c\cdot\mathcal{CW}(\pi^{*})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p1.7.m7.2d">caligraphic_C caligraphic_W ( italic_π ) ≥ italic_c ⋅ caligraphic_C caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.4.p2"> <p class="ltx_p" id="S4.SS2.4.p2.3">Then it holds that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx3"> <tbody id="S4.Ex13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle c\cdot\mathcal{SW}(\pi)" class="ltx_Math" display="inline" id="S4.Ex13.m1.1"><semantics id="S4.Ex13.m1.1a"><mrow id="S4.Ex13.m1.1.2" xref="S4.Ex13.m1.1.2.cmml"><mrow id="S4.Ex13.m1.1.2.2" xref="S4.Ex13.m1.1.2.2.cmml"><mi id="S4.Ex13.m1.1.2.2.2" xref="S4.Ex13.m1.1.2.2.2.cmml">c</mi><mo id="S4.Ex13.m1.1.2.2.1" lspace="0.222em" rspace="0.222em" xref="S4.Ex13.m1.1.2.2.1.cmml">⋅</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex13.m1.1.2.2.3" xref="S4.Ex13.m1.1.2.2.3.cmml">𝒮</mi></mrow><mo id="S4.Ex13.m1.1.2.1" xref="S4.Ex13.m1.1.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex13.m1.1.2.3" xref="S4.Ex13.m1.1.2.3.cmml">𝒲</mi><mo id="S4.Ex13.m1.1.2.1a" xref="S4.Ex13.m1.1.2.1.cmml">⁢</mo><mrow id="S4.Ex13.m1.1.2.4.2" xref="S4.Ex13.m1.1.2.cmml"><mo id="S4.Ex13.m1.1.2.4.2.1" stretchy="false" xref="S4.Ex13.m1.1.2.cmml">(</mo><mi id="S4.Ex13.m1.1.1" xref="S4.Ex13.m1.1.1.cmml">π</mi><mo id="S4.Ex13.m1.1.2.4.2.2" stretchy="false" xref="S4.Ex13.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex13.m1.1b"><apply id="S4.Ex13.m1.1.2.cmml" xref="S4.Ex13.m1.1.2"><times id="S4.Ex13.m1.1.2.1.cmml" xref="S4.Ex13.m1.1.2.1"></times><apply id="S4.Ex13.m1.1.2.2.cmml" xref="S4.Ex13.m1.1.2.2"><ci id="S4.Ex13.m1.1.2.2.1.cmml" xref="S4.Ex13.m1.1.2.2.1">⋅</ci><ci id="S4.Ex13.m1.1.2.2.2.cmml" xref="S4.Ex13.m1.1.2.2.2">𝑐</ci><ci id="S4.Ex13.m1.1.2.2.3.cmml" xref="S4.Ex13.m1.1.2.2.3">𝒮</ci></apply><ci id="S4.Ex13.m1.1.2.3.cmml" xref="S4.Ex13.m1.1.2.3">𝒲</ci><ci id="S4.Ex13.m1.1.1.cmml" xref="S4.Ex13.m1.1.1">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex13.m1.1c">\displaystyle c\cdot\mathcal{SW}(\pi)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex13.m1.1d">italic_c ⋅ caligraphic_S caligraphic_W ( italic_π )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\overset{\textnormal{\lx@cref{creftypecap~refnum}{prop:wf-relatio% nship}}}{=}c\cdot\mathcal{CW}(\pi)+\frac{c}{2}\cdot\mathcal{V}(N,v)" class="ltx_Math" display="inline" id="S4.Ex13.m2.3"><semantics id="S4.Ex13.m2.3a"><mrow id="S4.Ex13.m2.3.4" xref="S4.Ex13.m2.3.4.cmml"><mrow id="S4.Ex13.m2.3.4.2" xref="S4.Ex13.m2.3.4.2.cmml"><mrow id="S4.Ex13.m2.3.4.2.2" xref="S4.Ex13.m2.3.4.2.2.cmml"><mrow id="S4.Ex13.m2.3.4.2.2.2" xref="S4.Ex13.m2.3.4.2.2.2.cmml"><mover accent="true" id="S4.Ex13.m2.3.4.2.2.2.2" xref="S4.Ex13.m2.3.4.2.2.2.2.cmml"><mo id="S4.Ex13.m2.3.4.2.2.2.2.2" xref="S4.Ex13.m2.3.4.2.2.2.2.2.cmml">=</mo><mtext id="S4.Ex13.m2.3.4.2.2.2.2.1" xref="S4.Ex13.m2.3.4.2.2.2.2.1d.cmml"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.Thmtheorem2" title="Proposition 4.2. ‣ 4.2 Logarithmic Approximation for Nonnegative Total Value ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">4.2</span></a></mtext></mover><mo id="S4.Ex13.m2.3.4.2.2.2.1" xref="S4.Ex13.m2.3.4.2.2.2.1.cmml">⁢</mo><mi id="S4.Ex13.m2.3.4.2.2.2.3" xref="S4.Ex13.m2.3.4.2.2.2.3.cmml">c</mi></mrow><mo id="S4.Ex13.m2.3.4.2.2.1" lspace="0.222em" rspace="0.222em" xref="S4.Ex13.m2.3.4.2.2.1.cmml">⋅</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex13.m2.3.4.2.2.3" xref="S4.Ex13.m2.3.4.2.2.3.cmml">𝒞</mi></mrow><mo id="S4.Ex13.m2.3.4.2.1" xref="S4.Ex13.m2.3.4.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex13.m2.3.4.2.3" xref="S4.Ex13.m2.3.4.2.3.cmml">𝒲</mi><mo id="S4.Ex13.m2.3.4.2.1a" xref="S4.Ex13.m2.3.4.2.1.cmml">⁢</mo><mrow id="S4.Ex13.m2.3.4.2.4.2" xref="S4.Ex13.m2.3.4.2.cmml"><mo id="S4.Ex13.m2.3.4.2.4.2.1" stretchy="false" xref="S4.Ex13.m2.3.4.2.cmml">(</mo><mi id="S4.Ex13.m2.1.1" xref="S4.Ex13.m2.1.1.cmml">π</mi><mo id="S4.Ex13.m2.3.4.2.4.2.2" stretchy="false" xref="S4.Ex13.m2.3.4.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex13.m2.3.4.1" xref="S4.Ex13.m2.3.4.1.cmml">+</mo><mrow id="S4.Ex13.m2.3.4.3" xref="S4.Ex13.m2.3.4.3.cmml"><mrow id="S4.Ex13.m2.3.4.3.2" xref="S4.Ex13.m2.3.4.3.2.cmml"><mstyle displaystyle="true" id="S4.Ex13.m2.3.4.3.2.2" xref="S4.Ex13.m2.3.4.3.2.2.cmml"><mfrac id="S4.Ex13.m2.3.4.3.2.2a" xref="S4.Ex13.m2.3.4.3.2.2.cmml"><mi id="S4.Ex13.m2.3.4.3.2.2.2" xref="S4.Ex13.m2.3.4.3.2.2.2.cmml">c</mi><mn id="S4.Ex13.m2.3.4.3.2.2.3" xref="S4.Ex13.m2.3.4.3.2.2.3.cmml">2</mn></mfrac></mstyle><mo id="S4.Ex13.m2.3.4.3.2.1" lspace="0.222em" rspace="0.222em" xref="S4.Ex13.m2.3.4.3.2.1.cmml">⋅</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex13.m2.3.4.3.2.3" xref="S4.Ex13.m2.3.4.3.2.3.cmml">𝒱</mi></mrow><mo id="S4.Ex13.m2.3.4.3.1" xref="S4.Ex13.m2.3.4.3.1.cmml">⁢</mo><mrow id="S4.Ex13.m2.3.4.3.3.2" xref="S4.Ex13.m2.3.4.3.3.1.cmml"><mo id="S4.Ex13.m2.3.4.3.3.2.1" stretchy="false" xref="S4.Ex13.m2.3.4.3.3.1.cmml">(</mo><mi id="S4.Ex13.m2.2.2" xref="S4.Ex13.m2.2.2.cmml">N</mi><mo id="S4.Ex13.m2.3.4.3.3.2.2" xref="S4.Ex13.m2.3.4.3.3.1.cmml">,</mo><mi id="S4.Ex13.m2.3.3" xref="S4.Ex13.m2.3.3.cmml">v</mi><mo id="S4.Ex13.m2.3.4.3.3.2.3" stretchy="false" xref="S4.Ex13.m2.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex13.m2.3b"><apply id="S4.Ex13.m2.3.4.cmml" xref="S4.Ex13.m2.3.4"><plus id="S4.Ex13.m2.3.4.1.cmml" xref="S4.Ex13.m2.3.4.1"></plus><apply id="S4.Ex13.m2.3.4.2.cmml" xref="S4.Ex13.m2.3.4.2"><times id="S4.Ex13.m2.3.4.2.1.cmml" xref="S4.Ex13.m2.3.4.2.1"></times><apply id="S4.Ex13.m2.3.4.2.2.cmml" xref="S4.Ex13.m2.3.4.2.2"><ci id="S4.Ex13.m2.3.4.2.2.1.cmml" xref="S4.Ex13.m2.3.4.2.2.1">⋅</ci><apply id="S4.Ex13.m2.3.4.2.2.2.cmml" xref="S4.Ex13.m2.3.4.2.2.2"><times id="S4.Ex13.m2.3.4.2.2.2.1.cmml" xref="S4.Ex13.m2.3.4.2.2.2.1"></times><apply id="S4.Ex13.m2.3.4.2.2.2.2.cmml" xref="S4.Ex13.m2.3.4.2.2.2.2"><ci id="S4.Ex13.m2.3.4.2.2.2.2.1d.cmml" xref="S4.Ex13.m2.3.4.2.2.2.2.1"><mtext id="S4.Ex13.m2.3.4.2.2.2.2.1.cmml" xref="S4.Ex13.m2.3.4.2.2.2.2.1"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.Thmtheorem2" title="Proposition 4.2. ‣ 4.2 Logarithmic Approximation for Nonnegative Total Value ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">4.2</span></a></mtext></ci><eq id="S4.Ex13.m2.3.4.2.2.2.2.2.cmml" xref="S4.Ex13.m2.3.4.2.2.2.2.2"></eq></apply><ci id="S4.Ex13.m2.3.4.2.2.2.3.cmml" xref="S4.Ex13.m2.3.4.2.2.2.3">𝑐</ci></apply><ci id="S4.Ex13.m2.3.4.2.2.3.cmml" xref="S4.Ex13.m2.3.4.2.2.3">𝒞</ci></apply><ci id="S4.Ex13.m2.3.4.2.3.cmml" xref="S4.Ex13.m2.3.4.2.3">𝒲</ci><ci id="S4.Ex13.m2.1.1.cmml" xref="S4.Ex13.m2.1.1">𝜋</ci></apply><apply id="S4.Ex13.m2.3.4.3.cmml" xref="S4.Ex13.m2.3.4.3"><times id="S4.Ex13.m2.3.4.3.1.cmml" xref="S4.Ex13.m2.3.4.3.1"></times><apply id="S4.Ex13.m2.3.4.3.2.cmml" xref="S4.Ex13.m2.3.4.3.2"><ci id="S4.Ex13.m2.3.4.3.2.1.cmml" xref="S4.Ex13.m2.3.4.3.2.1">⋅</ci><apply id="S4.Ex13.m2.3.4.3.2.2.cmml" xref="S4.Ex13.m2.3.4.3.2.2"><divide id="S4.Ex13.m2.3.4.3.2.2.1.cmml" xref="S4.Ex13.m2.3.4.3.2.2"></divide><ci id="S4.Ex13.m2.3.4.3.2.2.2.cmml" xref="S4.Ex13.m2.3.4.3.2.2.2">𝑐</ci><cn id="S4.Ex13.m2.3.4.3.2.2.3.cmml" type="integer" xref="S4.Ex13.m2.3.4.3.2.2.3">2</cn></apply><ci id="S4.Ex13.m2.3.4.3.2.3.cmml" xref="S4.Ex13.m2.3.4.3.2.3">𝒱</ci></apply><interval closure="open" id="S4.Ex13.m2.3.4.3.3.1.cmml" xref="S4.Ex13.m2.3.4.3.3.2"><ci id="S4.Ex13.m2.2.2.cmml" xref="S4.Ex13.m2.2.2">𝑁</ci><ci id="S4.Ex13.m2.3.3.cmml" xref="S4.Ex13.m2.3.3">𝑣</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex13.m2.3c">\displaystyle\overset{\textnormal{\lx@cref{creftypecap~refnum}{prop:wf-relatio% nship}}}{=}c\cdot\mathcal{CW}(\pi)+\frac{c}{2}\cdot\mathcal{V}(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex13.m2.3d">overOVERACCENT start_ARG = end_ARG italic_c ⋅ caligraphic_C caligraphic_W ( italic_π ) + divide start_ARG italic_c end_ARG start_ARG 2 end_ARG ⋅ caligraphic_V ( italic_N , italic_v )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq\mathcal{CW}(\pi^{*})+\frac{c}{2}\cdot\mathcal{V}(N,v)" class="ltx_Math" display="inline" id="S4.Ex14.m1.3"><semantics id="S4.Ex14.m1.3a"><mrow id="S4.Ex14.m1.3.3" xref="S4.Ex14.m1.3.3.cmml"><mi id="S4.Ex14.m1.3.3.3" xref="S4.Ex14.m1.3.3.3.cmml"></mi><mo id="S4.Ex14.m1.3.3.2" xref="S4.Ex14.m1.3.3.2.cmml">≥</mo><mrow id="S4.Ex14.m1.3.3.1" xref="S4.Ex14.m1.3.3.1.cmml"><mrow id="S4.Ex14.m1.3.3.1.1" xref="S4.Ex14.m1.3.3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex14.m1.3.3.1.1.3" xref="S4.Ex14.m1.3.3.1.1.3.cmml">𝒞</mi><mo id="S4.Ex14.m1.3.3.1.1.2" xref="S4.Ex14.m1.3.3.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex14.m1.3.3.1.1.4" xref="S4.Ex14.m1.3.3.1.1.4.cmml">𝒲</mi><mo id="S4.Ex14.m1.3.3.1.1.2a" xref="S4.Ex14.m1.3.3.1.1.2.cmml">⁢</mo><mrow id="S4.Ex14.m1.3.3.1.1.1.1" xref="S4.Ex14.m1.3.3.1.1.1.1.1.cmml"><mo id="S4.Ex14.m1.3.3.1.1.1.1.2" stretchy="false" xref="S4.Ex14.m1.3.3.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex14.m1.3.3.1.1.1.1.1" xref="S4.Ex14.m1.3.3.1.1.1.1.1.cmml"><mi id="S4.Ex14.m1.3.3.1.1.1.1.1.2" xref="S4.Ex14.m1.3.3.1.1.1.1.1.2.cmml">π</mi><mo id="S4.Ex14.m1.3.3.1.1.1.1.1.3" xref="S4.Ex14.m1.3.3.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.Ex14.m1.3.3.1.1.1.1.3" stretchy="false" xref="S4.Ex14.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex14.m1.3.3.1.2" xref="S4.Ex14.m1.3.3.1.2.cmml">+</mo><mrow id="S4.Ex14.m1.3.3.1.3" xref="S4.Ex14.m1.3.3.1.3.cmml"><mrow id="S4.Ex14.m1.3.3.1.3.2" xref="S4.Ex14.m1.3.3.1.3.2.cmml"><mstyle displaystyle="true" id="S4.Ex14.m1.3.3.1.3.2.2" xref="S4.Ex14.m1.3.3.1.3.2.2.cmml"><mfrac id="S4.Ex14.m1.3.3.1.3.2.2a" xref="S4.Ex14.m1.3.3.1.3.2.2.cmml"><mi id="S4.Ex14.m1.3.3.1.3.2.2.2" xref="S4.Ex14.m1.3.3.1.3.2.2.2.cmml">c</mi><mn id="S4.Ex14.m1.3.3.1.3.2.2.3" xref="S4.Ex14.m1.3.3.1.3.2.2.3.cmml">2</mn></mfrac></mstyle><mo id="S4.Ex14.m1.3.3.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S4.Ex14.m1.3.3.1.3.2.1.cmml">⋅</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex14.m1.3.3.1.3.2.3" xref="S4.Ex14.m1.3.3.1.3.2.3.cmml">𝒱</mi></mrow><mo id="S4.Ex14.m1.3.3.1.3.1" xref="S4.Ex14.m1.3.3.1.3.1.cmml">⁢</mo><mrow id="S4.Ex14.m1.3.3.1.3.3.2" xref="S4.Ex14.m1.3.3.1.3.3.1.cmml"><mo id="S4.Ex14.m1.3.3.1.3.3.2.1" stretchy="false" xref="S4.Ex14.m1.3.3.1.3.3.1.cmml">(</mo><mi id="S4.Ex14.m1.1.1" xref="S4.Ex14.m1.1.1.cmml">N</mi><mo id="S4.Ex14.m1.3.3.1.3.3.2.2" xref="S4.Ex14.m1.3.3.1.3.3.1.cmml">,</mo><mi id="S4.Ex14.m1.2.2" xref="S4.Ex14.m1.2.2.cmml">v</mi><mo id="S4.Ex14.m1.3.3.1.3.3.2.3" stretchy="false" xref="S4.Ex14.m1.3.3.1.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex14.m1.3b"><apply id="S4.Ex14.m1.3.3.cmml" xref="S4.Ex14.m1.3.3"><geq id="S4.Ex14.m1.3.3.2.cmml" xref="S4.Ex14.m1.3.3.2"></geq><csymbol cd="latexml" id="S4.Ex14.m1.3.3.3.cmml" xref="S4.Ex14.m1.3.3.3">absent</csymbol><apply id="S4.Ex14.m1.3.3.1.cmml" xref="S4.Ex14.m1.3.3.1"><plus id="S4.Ex14.m1.3.3.1.2.cmml" xref="S4.Ex14.m1.3.3.1.2"></plus><apply id="S4.Ex14.m1.3.3.1.1.cmml" xref="S4.Ex14.m1.3.3.1.1"><times id="S4.Ex14.m1.3.3.1.1.2.cmml" xref="S4.Ex14.m1.3.3.1.1.2"></times><ci id="S4.Ex14.m1.3.3.1.1.3.cmml" xref="S4.Ex14.m1.3.3.1.1.3">𝒞</ci><ci id="S4.Ex14.m1.3.3.1.1.4.cmml" xref="S4.Ex14.m1.3.3.1.1.4">𝒲</ci><apply id="S4.Ex14.m1.3.3.1.1.1.1.1.cmml" xref="S4.Ex14.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex14.m1.3.3.1.1.1.1.1.1.cmml" xref="S4.Ex14.m1.3.3.1.1.1.1">superscript</csymbol><ci id="S4.Ex14.m1.3.3.1.1.1.1.1.2.cmml" xref="S4.Ex14.m1.3.3.1.1.1.1.1.2">𝜋</ci><times id="S4.Ex14.m1.3.3.1.1.1.1.1.3.cmml" xref="S4.Ex14.m1.3.3.1.1.1.1.1.3"></times></apply></apply><apply id="S4.Ex14.m1.3.3.1.3.cmml" xref="S4.Ex14.m1.3.3.1.3"><times id="S4.Ex14.m1.3.3.1.3.1.cmml" xref="S4.Ex14.m1.3.3.1.3.1"></times><apply id="S4.Ex14.m1.3.3.1.3.2.cmml" xref="S4.Ex14.m1.3.3.1.3.2"><ci id="S4.Ex14.m1.3.3.1.3.2.1.cmml" xref="S4.Ex14.m1.3.3.1.3.2.1">⋅</ci><apply id="S4.Ex14.m1.3.3.1.3.2.2.cmml" xref="S4.Ex14.m1.3.3.1.3.2.2"><divide id="S4.Ex14.m1.3.3.1.3.2.2.1.cmml" xref="S4.Ex14.m1.3.3.1.3.2.2"></divide><ci id="S4.Ex14.m1.3.3.1.3.2.2.2.cmml" xref="S4.Ex14.m1.3.3.1.3.2.2.2">𝑐</ci><cn id="S4.Ex14.m1.3.3.1.3.2.2.3.cmml" type="integer" xref="S4.Ex14.m1.3.3.1.3.2.2.3">2</cn></apply><ci id="S4.Ex14.m1.3.3.1.3.2.3.cmml" xref="S4.Ex14.m1.3.3.1.3.2.3">𝒱</ci></apply><interval closure="open" id="S4.Ex14.m1.3.3.1.3.3.1.cmml" xref="S4.Ex14.m1.3.3.1.3.3.2"><ci id="S4.Ex14.m1.1.1.cmml" xref="S4.Ex14.m1.1.1">𝑁</ci><ci id="S4.Ex14.m1.2.2.cmml" xref="S4.Ex14.m1.2.2">𝑣</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex14.m1.3c">\displaystyle\geq\mathcal{CW}(\pi^{*})+\frac{c}{2}\cdot\mathcal{V}(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex14.m1.3d">≥ caligraphic_C caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) + divide start_ARG italic_c end_ARG start_ARG 2 end_ARG ⋅ caligraphic_V ( italic_N , italic_v )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\overset{\textnormal{\lx@cref{creftypecap~refnum}{prop:wf-relatio% nship}}}{=}\mathcal{SW}(\pi^{*})-\frac{1}{2}\mathcal{V}(N,v)+\frac{c}{2}\cdot% \mathcal{V}(N,v)" class="ltx_Math" display="inline" id="S4.Ex15.m1.5"><semantics id="S4.Ex15.m1.5a"><mrow id="S4.Ex15.m1.5.5" xref="S4.Ex15.m1.5.5.cmml"><mrow id="S4.Ex15.m1.5.5.1" xref="S4.Ex15.m1.5.5.1.cmml"><mrow id="S4.Ex15.m1.5.5.1.1" xref="S4.Ex15.m1.5.5.1.1.cmml"><mover accent="true" id="S4.Ex15.m1.5.5.1.1.3" xref="S4.Ex15.m1.5.5.1.1.3.cmml"><mo id="S4.Ex15.m1.5.5.1.1.3.2" xref="S4.Ex15.m1.5.5.1.1.3.2.cmml">=</mo><mtext id="S4.Ex15.m1.5.5.1.1.3.1" xref="S4.Ex15.m1.5.5.1.1.3.1d.cmml"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.Thmtheorem2" title="Proposition 4.2. ‣ 4.2 Logarithmic Approximation for Nonnegative Total Value ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">4.2</span></a></mtext></mover><mo id="S4.Ex15.m1.5.5.1.1.2" xref="S4.Ex15.m1.5.5.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex15.m1.5.5.1.1.4" xref="S4.Ex15.m1.5.5.1.1.4.cmml">𝒮</mi><mo id="S4.Ex15.m1.5.5.1.1.2a" xref="S4.Ex15.m1.5.5.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex15.m1.5.5.1.1.5" xref="S4.Ex15.m1.5.5.1.1.5.cmml">𝒲</mi><mo id="S4.Ex15.m1.5.5.1.1.2b" xref="S4.Ex15.m1.5.5.1.1.2.cmml">⁢</mo><mrow id="S4.Ex15.m1.5.5.1.1.1.1" xref="S4.Ex15.m1.5.5.1.1.1.1.1.cmml"><mo id="S4.Ex15.m1.5.5.1.1.1.1.2" stretchy="false" xref="S4.Ex15.m1.5.5.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex15.m1.5.5.1.1.1.1.1" xref="S4.Ex15.m1.5.5.1.1.1.1.1.cmml"><mi id="S4.Ex15.m1.5.5.1.1.1.1.1.2" xref="S4.Ex15.m1.5.5.1.1.1.1.1.2.cmml">π</mi><mo id="S4.Ex15.m1.5.5.1.1.1.1.1.3" xref="S4.Ex15.m1.5.5.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.Ex15.m1.5.5.1.1.1.1.3" stretchy="false" xref="S4.Ex15.m1.5.5.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex15.m1.5.5.1.2" xref="S4.Ex15.m1.5.5.1.2.cmml">−</mo><mrow id="S4.Ex15.m1.5.5.1.3" xref="S4.Ex15.m1.5.5.1.3.cmml"><mstyle displaystyle="true" id="S4.Ex15.m1.5.5.1.3.2" xref="S4.Ex15.m1.5.5.1.3.2.cmml"><mfrac id="S4.Ex15.m1.5.5.1.3.2a" xref="S4.Ex15.m1.5.5.1.3.2.cmml"><mn id="S4.Ex15.m1.5.5.1.3.2.2" xref="S4.Ex15.m1.5.5.1.3.2.2.cmml">1</mn><mn id="S4.Ex15.m1.5.5.1.3.2.3" xref="S4.Ex15.m1.5.5.1.3.2.3.cmml">2</mn></mfrac></mstyle><mo id="S4.Ex15.m1.5.5.1.3.1" xref="S4.Ex15.m1.5.5.1.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex15.m1.5.5.1.3.3" xref="S4.Ex15.m1.5.5.1.3.3.cmml">𝒱</mi><mo id="S4.Ex15.m1.5.5.1.3.1a" xref="S4.Ex15.m1.5.5.1.3.1.cmml">⁢</mo><mrow id="S4.Ex15.m1.5.5.1.3.4.2" xref="S4.Ex15.m1.5.5.1.3.4.1.cmml"><mo id="S4.Ex15.m1.5.5.1.3.4.2.1" stretchy="false" xref="S4.Ex15.m1.5.5.1.3.4.1.cmml">(</mo><mi id="S4.Ex15.m1.1.1" xref="S4.Ex15.m1.1.1.cmml">N</mi><mo id="S4.Ex15.m1.5.5.1.3.4.2.2" xref="S4.Ex15.m1.5.5.1.3.4.1.cmml">,</mo><mi id="S4.Ex15.m1.2.2" xref="S4.Ex15.m1.2.2.cmml">v</mi><mo id="S4.Ex15.m1.5.5.1.3.4.2.3" stretchy="false" xref="S4.Ex15.m1.5.5.1.3.4.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex15.m1.5.5.2" xref="S4.Ex15.m1.5.5.2.cmml">+</mo><mrow id="S4.Ex15.m1.5.5.3" xref="S4.Ex15.m1.5.5.3.cmml"><mrow id="S4.Ex15.m1.5.5.3.2" xref="S4.Ex15.m1.5.5.3.2.cmml"><mstyle displaystyle="true" id="S4.Ex15.m1.5.5.3.2.2" xref="S4.Ex15.m1.5.5.3.2.2.cmml"><mfrac id="S4.Ex15.m1.5.5.3.2.2a" xref="S4.Ex15.m1.5.5.3.2.2.cmml"><mi id="S4.Ex15.m1.5.5.3.2.2.2" xref="S4.Ex15.m1.5.5.3.2.2.2.cmml">c</mi><mn id="S4.Ex15.m1.5.5.3.2.2.3" xref="S4.Ex15.m1.5.5.3.2.2.3.cmml">2</mn></mfrac></mstyle><mo id="S4.Ex15.m1.5.5.3.2.1" lspace="0.222em" rspace="0.222em" xref="S4.Ex15.m1.5.5.3.2.1.cmml">⋅</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex15.m1.5.5.3.2.3" xref="S4.Ex15.m1.5.5.3.2.3.cmml">𝒱</mi></mrow><mo id="S4.Ex15.m1.5.5.3.1" xref="S4.Ex15.m1.5.5.3.1.cmml">⁢</mo><mrow id="S4.Ex15.m1.5.5.3.3.2" xref="S4.Ex15.m1.5.5.3.3.1.cmml"><mo id="S4.Ex15.m1.5.5.3.3.2.1" stretchy="false" xref="S4.Ex15.m1.5.5.3.3.1.cmml">(</mo><mi id="S4.Ex15.m1.3.3" xref="S4.Ex15.m1.3.3.cmml">N</mi><mo id="S4.Ex15.m1.5.5.3.3.2.2" xref="S4.Ex15.m1.5.5.3.3.1.cmml">,</mo><mi id="S4.Ex15.m1.4.4" xref="S4.Ex15.m1.4.4.cmml">v</mi><mo id="S4.Ex15.m1.5.5.3.3.2.3" stretchy="false" xref="S4.Ex15.m1.5.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex15.m1.5b"><apply id="S4.Ex15.m1.5.5.cmml" xref="S4.Ex15.m1.5.5"><plus id="S4.Ex15.m1.5.5.2.cmml" xref="S4.Ex15.m1.5.5.2"></plus><apply id="S4.Ex15.m1.5.5.1.cmml" xref="S4.Ex15.m1.5.5.1"><minus id="S4.Ex15.m1.5.5.1.2.cmml" xref="S4.Ex15.m1.5.5.1.2"></minus><apply id="S4.Ex15.m1.5.5.1.1.cmml" xref="S4.Ex15.m1.5.5.1.1"><times id="S4.Ex15.m1.5.5.1.1.2.cmml" xref="S4.Ex15.m1.5.5.1.1.2"></times><apply id="S4.Ex15.m1.5.5.1.1.3.cmml" xref="S4.Ex15.m1.5.5.1.1.3"><ci id="S4.Ex15.m1.5.5.1.1.3.1d.cmml" xref="S4.Ex15.m1.5.5.1.1.3.1"><mtext id="S4.Ex15.m1.5.5.1.1.3.1.cmml" xref="S4.Ex15.m1.5.5.1.1.3.1"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.Thmtheorem2" title="Proposition 4.2. ‣ 4.2 Logarithmic Approximation for Nonnegative Total Value ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">4.2</span></a></mtext></ci><eq id="S4.Ex15.m1.5.5.1.1.3.2.cmml" xref="S4.Ex15.m1.5.5.1.1.3.2"></eq></apply><ci id="S4.Ex15.m1.5.5.1.1.4.cmml" xref="S4.Ex15.m1.5.5.1.1.4">𝒮</ci><ci id="S4.Ex15.m1.5.5.1.1.5.cmml" xref="S4.Ex15.m1.5.5.1.1.5">𝒲</ci><apply id="S4.Ex15.m1.5.5.1.1.1.1.1.cmml" xref="S4.Ex15.m1.5.5.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex15.m1.5.5.1.1.1.1.1.1.cmml" xref="S4.Ex15.m1.5.5.1.1.1.1">superscript</csymbol><ci id="S4.Ex15.m1.5.5.1.1.1.1.1.2.cmml" xref="S4.Ex15.m1.5.5.1.1.1.1.1.2">𝜋</ci><times id="S4.Ex15.m1.5.5.1.1.1.1.1.3.cmml" xref="S4.Ex15.m1.5.5.1.1.1.1.1.3"></times></apply></apply><apply id="S4.Ex15.m1.5.5.1.3.cmml" xref="S4.Ex15.m1.5.5.1.3"><times id="S4.Ex15.m1.5.5.1.3.1.cmml" xref="S4.Ex15.m1.5.5.1.3.1"></times><apply id="S4.Ex15.m1.5.5.1.3.2.cmml" xref="S4.Ex15.m1.5.5.1.3.2"><divide id="S4.Ex15.m1.5.5.1.3.2.1.cmml" xref="S4.Ex15.m1.5.5.1.3.2"></divide><cn id="S4.Ex15.m1.5.5.1.3.2.2.cmml" type="integer" xref="S4.Ex15.m1.5.5.1.3.2.2">1</cn><cn id="S4.Ex15.m1.5.5.1.3.2.3.cmml" type="integer" xref="S4.Ex15.m1.5.5.1.3.2.3">2</cn></apply><ci id="S4.Ex15.m1.5.5.1.3.3.cmml" xref="S4.Ex15.m1.5.5.1.3.3">𝒱</ci><interval closure="open" id="S4.Ex15.m1.5.5.1.3.4.1.cmml" xref="S4.Ex15.m1.5.5.1.3.4.2"><ci id="S4.Ex15.m1.1.1.cmml" xref="S4.Ex15.m1.1.1">𝑁</ci><ci id="S4.Ex15.m1.2.2.cmml" xref="S4.Ex15.m1.2.2">𝑣</ci></interval></apply></apply><apply id="S4.Ex15.m1.5.5.3.cmml" xref="S4.Ex15.m1.5.5.3"><times id="S4.Ex15.m1.5.5.3.1.cmml" xref="S4.Ex15.m1.5.5.3.1"></times><apply id="S4.Ex15.m1.5.5.3.2.cmml" xref="S4.Ex15.m1.5.5.3.2"><ci id="S4.Ex15.m1.5.5.3.2.1.cmml" xref="S4.Ex15.m1.5.5.3.2.1">⋅</ci><apply id="S4.Ex15.m1.5.5.3.2.2.cmml" xref="S4.Ex15.m1.5.5.3.2.2"><divide id="S4.Ex15.m1.5.5.3.2.2.1.cmml" xref="S4.Ex15.m1.5.5.3.2.2"></divide><ci id="S4.Ex15.m1.5.5.3.2.2.2.cmml" xref="S4.Ex15.m1.5.5.3.2.2.2">𝑐</ci><cn id="S4.Ex15.m1.5.5.3.2.2.3.cmml" type="integer" xref="S4.Ex15.m1.5.5.3.2.2.3">2</cn></apply><ci id="S4.Ex15.m1.5.5.3.2.3.cmml" xref="S4.Ex15.m1.5.5.3.2.3">𝒱</ci></apply><interval closure="open" id="S4.Ex15.m1.5.5.3.3.1.cmml" xref="S4.Ex15.m1.5.5.3.3.2"><ci id="S4.Ex15.m1.3.3.cmml" xref="S4.Ex15.m1.3.3">𝑁</ci><ci id="S4.Ex15.m1.4.4.cmml" xref="S4.Ex15.m1.4.4">𝑣</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex15.m1.5c">\displaystyle\overset{\textnormal{\lx@cref{creftypecap~refnum}{prop:wf-relatio% nship}}}{=}\mathcal{SW}(\pi^{*})-\frac{1}{2}\mathcal{V}(N,v)+\frac{c}{2}\cdot% \mathcal{V}(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex15.m1.5d">overOVERACCENT start_ARG = end_ARG caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG caligraphic_V ( italic_N , italic_v ) + divide start_ARG italic_c end_ARG start_ARG 2 end_ARG ⋅ caligraphic_V ( italic_N , italic_v )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathcal{SW}(\pi^{*})+\frac{1}{2}\left(c-1\right)\mathcal{V}(N,v)" class="ltx_Math" display="inline" id="S4.Ex16.m1.4"><semantics id="S4.Ex16.m1.4a"><mrow id="S4.Ex16.m1.4.4" xref="S4.Ex16.m1.4.4.cmml"><mi id="S4.Ex16.m1.4.4.4" xref="S4.Ex16.m1.4.4.4.cmml"></mi><mo id="S4.Ex16.m1.4.4.3" xref="S4.Ex16.m1.4.4.3.cmml">=</mo><mrow id="S4.Ex16.m1.4.4.2" xref="S4.Ex16.m1.4.4.2.cmml"><mrow id="S4.Ex16.m1.3.3.1.1" xref="S4.Ex16.m1.3.3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex16.m1.3.3.1.1.3" xref="S4.Ex16.m1.3.3.1.1.3.cmml">𝒮</mi><mo id="S4.Ex16.m1.3.3.1.1.2" xref="S4.Ex16.m1.3.3.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex16.m1.3.3.1.1.4" xref="S4.Ex16.m1.3.3.1.1.4.cmml">𝒲</mi><mo id="S4.Ex16.m1.3.3.1.1.2a" xref="S4.Ex16.m1.3.3.1.1.2.cmml">⁢</mo><mrow id="S4.Ex16.m1.3.3.1.1.1.1" xref="S4.Ex16.m1.3.3.1.1.1.1.1.cmml"><mo id="S4.Ex16.m1.3.3.1.1.1.1.2" stretchy="false" xref="S4.Ex16.m1.3.3.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex16.m1.3.3.1.1.1.1.1" xref="S4.Ex16.m1.3.3.1.1.1.1.1.cmml"><mi id="S4.Ex16.m1.3.3.1.1.1.1.1.2" xref="S4.Ex16.m1.3.3.1.1.1.1.1.2.cmml">π</mi><mo id="S4.Ex16.m1.3.3.1.1.1.1.1.3" xref="S4.Ex16.m1.3.3.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.Ex16.m1.3.3.1.1.1.1.3" stretchy="false" xref="S4.Ex16.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex16.m1.4.4.2.3" xref="S4.Ex16.m1.4.4.2.3.cmml">+</mo><mrow id="S4.Ex16.m1.4.4.2.2" xref="S4.Ex16.m1.4.4.2.2.cmml"><mstyle displaystyle="true" id="S4.Ex16.m1.4.4.2.2.3" xref="S4.Ex16.m1.4.4.2.2.3.cmml"><mfrac id="S4.Ex16.m1.4.4.2.2.3a" xref="S4.Ex16.m1.4.4.2.2.3.cmml"><mn id="S4.Ex16.m1.4.4.2.2.3.2" xref="S4.Ex16.m1.4.4.2.2.3.2.cmml">1</mn><mn id="S4.Ex16.m1.4.4.2.2.3.3" xref="S4.Ex16.m1.4.4.2.2.3.3.cmml">2</mn></mfrac></mstyle><mo id="S4.Ex16.m1.4.4.2.2.2" xref="S4.Ex16.m1.4.4.2.2.2.cmml">⁢</mo><mrow id="S4.Ex16.m1.4.4.2.2.1.1" xref="S4.Ex16.m1.4.4.2.2.1.1.1.cmml"><mo id="S4.Ex16.m1.4.4.2.2.1.1.2" xref="S4.Ex16.m1.4.4.2.2.1.1.1.cmml">(</mo><mrow id="S4.Ex16.m1.4.4.2.2.1.1.1" xref="S4.Ex16.m1.4.4.2.2.1.1.1.cmml"><mi id="S4.Ex16.m1.4.4.2.2.1.1.1.2" xref="S4.Ex16.m1.4.4.2.2.1.1.1.2.cmml">c</mi><mo id="S4.Ex16.m1.4.4.2.2.1.1.1.1" xref="S4.Ex16.m1.4.4.2.2.1.1.1.1.cmml">−</mo><mn id="S4.Ex16.m1.4.4.2.2.1.1.1.3" xref="S4.Ex16.m1.4.4.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S4.Ex16.m1.4.4.2.2.1.1.3" xref="S4.Ex16.m1.4.4.2.2.1.1.1.cmml">)</mo></mrow><mo id="S4.Ex16.m1.4.4.2.2.2a" xref="S4.Ex16.m1.4.4.2.2.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex16.m1.4.4.2.2.4" xref="S4.Ex16.m1.4.4.2.2.4.cmml">𝒱</mi><mo id="S4.Ex16.m1.4.4.2.2.2b" xref="S4.Ex16.m1.4.4.2.2.2.cmml">⁢</mo><mrow id="S4.Ex16.m1.4.4.2.2.5.2" xref="S4.Ex16.m1.4.4.2.2.5.1.cmml"><mo id="S4.Ex16.m1.4.4.2.2.5.2.1" stretchy="false" xref="S4.Ex16.m1.4.4.2.2.5.1.cmml">(</mo><mi id="S4.Ex16.m1.1.1" xref="S4.Ex16.m1.1.1.cmml">N</mi><mo id="S4.Ex16.m1.4.4.2.2.5.2.2" xref="S4.Ex16.m1.4.4.2.2.5.1.cmml">,</mo><mi id="S4.Ex16.m1.2.2" xref="S4.Ex16.m1.2.2.cmml">v</mi><mo id="S4.Ex16.m1.4.4.2.2.5.2.3" stretchy="false" xref="S4.Ex16.m1.4.4.2.2.5.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex16.m1.4b"><apply id="S4.Ex16.m1.4.4.cmml" xref="S4.Ex16.m1.4.4"><eq id="S4.Ex16.m1.4.4.3.cmml" xref="S4.Ex16.m1.4.4.3"></eq><csymbol cd="latexml" id="S4.Ex16.m1.4.4.4.cmml" xref="S4.Ex16.m1.4.4.4">absent</csymbol><apply id="S4.Ex16.m1.4.4.2.cmml" xref="S4.Ex16.m1.4.4.2"><plus id="S4.Ex16.m1.4.4.2.3.cmml" xref="S4.Ex16.m1.4.4.2.3"></plus><apply id="S4.Ex16.m1.3.3.1.1.cmml" xref="S4.Ex16.m1.3.3.1.1"><times id="S4.Ex16.m1.3.3.1.1.2.cmml" xref="S4.Ex16.m1.3.3.1.1.2"></times><ci id="S4.Ex16.m1.3.3.1.1.3.cmml" xref="S4.Ex16.m1.3.3.1.1.3">𝒮</ci><ci id="S4.Ex16.m1.3.3.1.1.4.cmml" xref="S4.Ex16.m1.3.3.1.1.4">𝒲</ci><apply id="S4.Ex16.m1.3.3.1.1.1.1.1.cmml" xref="S4.Ex16.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex16.m1.3.3.1.1.1.1.1.1.cmml" xref="S4.Ex16.m1.3.3.1.1.1.1">superscript</csymbol><ci id="S4.Ex16.m1.3.3.1.1.1.1.1.2.cmml" xref="S4.Ex16.m1.3.3.1.1.1.1.1.2">𝜋</ci><times id="S4.Ex16.m1.3.3.1.1.1.1.1.3.cmml" xref="S4.Ex16.m1.3.3.1.1.1.1.1.3"></times></apply></apply><apply id="S4.Ex16.m1.4.4.2.2.cmml" xref="S4.Ex16.m1.4.4.2.2"><times id="S4.Ex16.m1.4.4.2.2.2.cmml" xref="S4.Ex16.m1.4.4.2.2.2"></times><apply id="S4.Ex16.m1.4.4.2.2.3.cmml" xref="S4.Ex16.m1.4.4.2.2.3"><divide id="S4.Ex16.m1.4.4.2.2.3.1.cmml" xref="S4.Ex16.m1.4.4.2.2.3"></divide><cn id="S4.Ex16.m1.4.4.2.2.3.2.cmml" type="integer" xref="S4.Ex16.m1.4.4.2.2.3.2">1</cn><cn id="S4.Ex16.m1.4.4.2.2.3.3.cmml" type="integer" xref="S4.Ex16.m1.4.4.2.2.3.3">2</cn></apply><apply id="S4.Ex16.m1.4.4.2.2.1.1.1.cmml" xref="S4.Ex16.m1.4.4.2.2.1.1"><minus id="S4.Ex16.m1.4.4.2.2.1.1.1.1.cmml" xref="S4.Ex16.m1.4.4.2.2.1.1.1.1"></minus><ci id="S4.Ex16.m1.4.4.2.2.1.1.1.2.cmml" xref="S4.Ex16.m1.4.4.2.2.1.1.1.2">𝑐</ci><cn id="S4.Ex16.m1.4.4.2.2.1.1.1.3.cmml" type="integer" xref="S4.Ex16.m1.4.4.2.2.1.1.1.3">1</cn></apply><ci id="S4.Ex16.m1.4.4.2.2.4.cmml" xref="S4.Ex16.m1.4.4.2.2.4">𝒱</ci><interval closure="open" id="S4.Ex16.m1.4.4.2.2.5.1.cmml" xref="S4.Ex16.m1.4.4.2.2.5.2"><ci id="S4.Ex16.m1.1.1.cmml" xref="S4.Ex16.m1.1.1">𝑁</ci><ci id="S4.Ex16.m1.2.2.cmml" xref="S4.Ex16.m1.2.2">𝑣</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex16.m1.4c">\displaystyle=\mathcal{SW}(\pi^{*})+\frac{1}{2}\left(c-1\right)\mathcal{V}(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex16.m1.4d">= caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_c - 1 ) caligraphic_V ( italic_N , italic_v )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq\mathcal{SW}(\pi^{*})\text{.}" class="ltx_Math" display="inline" id="S4.Ex17.m1.1"><semantics id="S4.Ex17.m1.1a"><mrow id="S4.Ex17.m1.1.1" xref="S4.Ex17.m1.1.1.cmml"><mi id="S4.Ex17.m1.1.1.3" xref="S4.Ex17.m1.1.1.3.cmml"></mi><mo id="S4.Ex17.m1.1.1.2" xref="S4.Ex17.m1.1.1.2.cmml">≥</mo><mrow id="S4.Ex17.m1.1.1.1" xref="S4.Ex17.m1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex17.m1.1.1.1.3" xref="S4.Ex17.m1.1.1.1.3.cmml">𝒮</mi><mo id="S4.Ex17.m1.1.1.1.2" xref="S4.Ex17.m1.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex17.m1.1.1.1.4" xref="S4.Ex17.m1.1.1.1.4.cmml">𝒲</mi><mo id="S4.Ex17.m1.1.1.1.2a" xref="S4.Ex17.m1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex17.m1.1.1.1.1.1" xref="S4.Ex17.m1.1.1.1.1.1.1.cmml"><mo id="S4.Ex17.m1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex17.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex17.m1.1.1.1.1.1.1" xref="S4.Ex17.m1.1.1.1.1.1.1.cmml"><mi id="S4.Ex17.m1.1.1.1.1.1.1.2" xref="S4.Ex17.m1.1.1.1.1.1.1.2.cmml">π</mi><mo id="S4.Ex17.m1.1.1.1.1.1.1.3" xref="S4.Ex17.m1.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.Ex17.m1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex17.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Ex17.m1.1.1.1.2b" xref="S4.Ex17.m1.1.1.1.2.cmml">⁢</mo><mtext id="S4.Ex17.m1.1.1.1.5" xref="S4.Ex17.m1.1.1.1.5a.cmml">.</mtext></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex17.m1.1b"><apply id="S4.Ex17.m1.1.1.cmml" xref="S4.Ex17.m1.1.1"><geq id="S4.Ex17.m1.1.1.2.cmml" xref="S4.Ex17.m1.1.1.2"></geq><csymbol cd="latexml" id="S4.Ex17.m1.1.1.3.cmml" xref="S4.Ex17.m1.1.1.3">absent</csymbol><apply id="S4.Ex17.m1.1.1.1.cmml" xref="S4.Ex17.m1.1.1.1"><times id="S4.Ex17.m1.1.1.1.2.cmml" xref="S4.Ex17.m1.1.1.1.2"></times><ci id="S4.Ex17.m1.1.1.1.3.cmml" xref="S4.Ex17.m1.1.1.1.3">𝒮</ci><ci id="S4.Ex17.m1.1.1.1.4.cmml" xref="S4.Ex17.m1.1.1.1.4">𝒲</ci><apply id="S4.Ex17.m1.1.1.1.1.1.1.cmml" xref="S4.Ex17.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex17.m1.1.1.1.1.1.1.1.cmml" xref="S4.Ex17.m1.1.1.1.1.1">superscript</csymbol><ci id="S4.Ex17.m1.1.1.1.1.1.1.2.cmml" xref="S4.Ex17.m1.1.1.1.1.1.1.2">𝜋</ci><times id="S4.Ex17.m1.1.1.1.1.1.1.3.cmml" xref="S4.Ex17.m1.1.1.1.1.1.1.3"></times></apply><ci id="S4.Ex17.m1.1.1.1.5a.cmml" xref="S4.Ex17.m1.1.1.1.5"><mtext id="S4.Ex17.m1.1.1.1.5.cmml" xref="S4.Ex17.m1.1.1.1.5">.</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex17.m1.1c">\displaystyle\geq\mathcal{SW}(\pi^{*})\text{.}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex17.m1.1d">≥ caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.4.p2.2">In the last inequality, we used that <math alttext="\mathcal{V}(N,v)\geq 0" class="ltx_Math" display="inline" id="S4.SS2.4.p2.1.m1.2"><semantics id="S4.SS2.4.p2.1.m1.2a"><mrow id="S4.SS2.4.p2.1.m1.2.3" xref="S4.SS2.4.p2.1.m1.2.3.cmml"><mrow id="S4.SS2.4.p2.1.m1.2.3.2" xref="S4.SS2.4.p2.1.m1.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.4.p2.1.m1.2.3.2.2" xref="S4.SS2.4.p2.1.m1.2.3.2.2.cmml">𝒱</mi><mo id="S4.SS2.4.p2.1.m1.2.3.2.1" xref="S4.SS2.4.p2.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS2.4.p2.1.m1.2.3.2.3.2" xref="S4.SS2.4.p2.1.m1.2.3.2.3.1.cmml"><mo id="S4.SS2.4.p2.1.m1.2.3.2.3.2.1" stretchy="false" xref="S4.SS2.4.p2.1.m1.2.3.2.3.1.cmml">(</mo><mi id="S4.SS2.4.p2.1.m1.1.1" xref="S4.SS2.4.p2.1.m1.1.1.cmml">N</mi><mo id="S4.SS2.4.p2.1.m1.2.3.2.3.2.2" xref="S4.SS2.4.p2.1.m1.2.3.2.3.1.cmml">,</mo><mi id="S4.SS2.4.p2.1.m1.2.2" xref="S4.SS2.4.p2.1.m1.2.2.cmml">v</mi><mo id="S4.SS2.4.p2.1.m1.2.3.2.3.2.3" stretchy="false" xref="S4.SS2.4.p2.1.m1.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.4.p2.1.m1.2.3.1" xref="S4.SS2.4.p2.1.m1.2.3.1.cmml">≥</mo><mn id="S4.SS2.4.p2.1.m1.2.3.3" xref="S4.SS2.4.p2.1.m1.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.4.p2.1.m1.2b"><apply id="S4.SS2.4.p2.1.m1.2.3.cmml" xref="S4.SS2.4.p2.1.m1.2.3"><geq id="S4.SS2.4.p2.1.m1.2.3.1.cmml" xref="S4.SS2.4.p2.1.m1.2.3.1"></geq><apply id="S4.SS2.4.p2.1.m1.2.3.2.cmml" xref="S4.SS2.4.p2.1.m1.2.3.2"><times id="S4.SS2.4.p2.1.m1.2.3.2.1.cmml" xref="S4.SS2.4.p2.1.m1.2.3.2.1"></times><ci id="S4.SS2.4.p2.1.m1.2.3.2.2.cmml" xref="S4.SS2.4.p2.1.m1.2.3.2.2">𝒱</ci><interval closure="open" id="S4.SS2.4.p2.1.m1.2.3.2.3.1.cmml" xref="S4.SS2.4.p2.1.m1.2.3.2.3.2"><ci id="S4.SS2.4.p2.1.m1.1.1.cmml" xref="S4.SS2.4.p2.1.m1.1.1">𝑁</ci><ci id="S4.SS2.4.p2.1.m1.2.2.cmml" xref="S4.SS2.4.p2.1.m1.2.2">𝑣</ci></interval></apply><cn id="S4.SS2.4.p2.1.m1.2.3.3.cmml" type="integer" xref="S4.SS2.4.p2.1.m1.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.4.p2.1.m1.2c">\mathcal{V}(N,v)\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.4.p2.1.m1.2d">caligraphic_V ( italic_N , italic_v ) ≥ 0</annotation></semantics></math> and <math alttext="c\geq 1" class="ltx_Math" display="inline" id="S4.SS2.4.p2.2.m2.1"><semantics id="S4.SS2.4.p2.2.m2.1a"><mrow id="S4.SS2.4.p2.2.m2.1.1" xref="S4.SS2.4.p2.2.m2.1.1.cmml"><mi id="S4.SS2.4.p2.2.m2.1.1.2" xref="S4.SS2.4.p2.2.m2.1.1.2.cmml">c</mi><mo id="S4.SS2.4.p2.2.m2.1.1.1" xref="S4.SS2.4.p2.2.m2.1.1.1.cmml">≥</mo><mn id="S4.SS2.4.p2.2.m2.1.1.3" xref="S4.SS2.4.p2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.4.p2.2.m2.1b"><apply id="S4.SS2.4.p2.2.m2.1.1.cmml" xref="S4.SS2.4.p2.2.m2.1.1"><geq id="S4.SS2.4.p2.2.m2.1.1.1.cmml" xref="S4.SS2.4.p2.2.m2.1.1.1"></geq><ci id="S4.SS2.4.p2.2.m2.1.1.2.cmml" xref="S4.SS2.4.p2.2.m2.1.1.2">𝑐</ci><cn id="S4.SS2.4.p2.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.4.p2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.4.p2.2.m2.1c">c\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.4.p2.2.m2.1d">italic_c ≥ 1</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS2.p7"> <p class="ltx_p" id="S4.SS2.p7.2">As a second lemma, we establish a relationship between maximizing welfare when partitions can only contain two coalitions and when partitions are unconstrained. Its proof is an adaptation of a similar result by Charikar and Wirth <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx18" title="">CW04</a>]</cite> concerning <math alttext="\mathcal{CW}" class="ltx_Math" display="inline" id="S4.SS2.p7.1.m1.1"><semantics id="S4.SS2.p7.1.m1.1a"><mrow id="S4.SS2.p7.1.m1.1.1" xref="S4.SS2.p7.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p7.1.m1.1.1.2" xref="S4.SS2.p7.1.m1.1.1.2.cmml">𝒞</mi><mo id="S4.SS2.p7.1.m1.1.1.1" xref="S4.SS2.p7.1.m1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p7.1.m1.1.1.3" xref="S4.SS2.p7.1.m1.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.1.m1.1b"><apply id="S4.SS2.p7.1.m1.1.1.cmml" xref="S4.SS2.p7.1.m1.1.1"><times id="S4.SS2.p7.1.m1.1.1.1.cmml" xref="S4.SS2.p7.1.m1.1.1.1"></times><ci id="S4.SS2.p7.1.m1.1.1.2.cmml" xref="S4.SS2.p7.1.m1.1.1.2">𝒞</ci><ci id="S4.SS2.p7.1.m1.1.1.3.cmml" xref="S4.SS2.p7.1.m1.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.1.m1.1c">\mathcal{CW}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.1.m1.1d">caligraphic_C caligraphic_W</annotation></semantics></math> for valuations in <math alttext="\{-1,1\}" class="ltx_Math" display="inline" id="S4.SS2.p7.2.m2.2"><semantics id="S4.SS2.p7.2.m2.2a"><mrow id="S4.SS2.p7.2.m2.2.2.1" xref="S4.SS2.p7.2.m2.2.2.2.cmml"><mo id="S4.SS2.p7.2.m2.2.2.1.2" stretchy="false" xref="S4.SS2.p7.2.m2.2.2.2.cmml">{</mo><mrow id="S4.SS2.p7.2.m2.2.2.1.1" xref="S4.SS2.p7.2.m2.2.2.1.1.cmml"><mo id="S4.SS2.p7.2.m2.2.2.1.1a" xref="S4.SS2.p7.2.m2.2.2.1.1.cmml">−</mo><mn id="S4.SS2.p7.2.m2.2.2.1.1.2" xref="S4.SS2.p7.2.m2.2.2.1.1.2.cmml">1</mn></mrow><mo id="S4.SS2.p7.2.m2.2.2.1.3" xref="S4.SS2.p7.2.m2.2.2.2.cmml">,</mo><mn id="S4.SS2.p7.2.m2.1.1" xref="S4.SS2.p7.2.m2.1.1.cmml">1</mn><mo id="S4.SS2.p7.2.m2.2.2.1.4" stretchy="false" xref="S4.SS2.p7.2.m2.2.2.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.2.m2.2b"><set id="S4.SS2.p7.2.m2.2.2.2.cmml" xref="S4.SS2.p7.2.m2.2.2.1"><apply id="S4.SS2.p7.2.m2.2.2.1.1.cmml" xref="S4.SS2.p7.2.m2.2.2.1.1"><minus id="S4.SS2.p7.2.m2.2.2.1.1.1.cmml" xref="S4.SS2.p7.2.m2.2.2.1.1"></minus><cn id="S4.SS2.p7.2.m2.2.2.1.1.2.cmml" type="integer" xref="S4.SS2.p7.2.m2.2.2.1.1.2">1</cn></apply><cn id="S4.SS2.p7.2.m2.1.1.cmml" type="integer" xref="S4.SS2.p7.2.m2.1.1">1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.2.m2.2c">\{-1,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.2.m2.2d">{ - 1 , 1 }</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem6.1.1.1">Lemma 4.6</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem6.p1"> <p class="ltx_p" id="S4.Thmtheorem6.p1.3"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem6.p1.3.3">Let <math alttext="(N,v)" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.1.1.m1.2"><semantics id="S4.Thmtheorem6.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem6.p1.1.1.m1.2.3.2" xref="S4.Thmtheorem6.p1.1.1.m1.2.3.1.cmml"><mo id="S4.Thmtheorem6.p1.1.1.m1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem6.p1.1.1.m1.2.3.1.cmml">(</mo><mi id="S4.Thmtheorem6.p1.1.1.m1.1.1" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.cmml">N</mi><mo id="S4.Thmtheorem6.p1.1.1.m1.2.3.2.2" xref="S4.Thmtheorem6.p1.1.1.m1.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem6.p1.1.1.m1.2.2" xref="S4.Thmtheorem6.p1.1.1.m1.2.2.cmml">v</mi><mo id="S4.Thmtheorem6.p1.1.1.m1.2.3.2.3" stretchy="false" xref="S4.Thmtheorem6.p1.1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.1.1.m1.2b"><interval closure="open" id="S4.Thmtheorem6.p1.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.2.3.2"><ci id="S4.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1">𝑁</ci><ci id="S4.Thmtheorem6.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.1.1.m1.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.1.1.m1.2d">( italic_N , italic_v )</annotation></semantics></math> be an ASHG with <math alttext="\mathcal{V}(N,v)\geq 0" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.2.2.m2.2"><semantics id="S4.Thmtheorem6.p1.2.2.m2.2a"><mrow id="S4.Thmtheorem6.p1.2.2.m2.2.3" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.cmml"><mrow id="S4.Thmtheorem6.p1.2.2.m2.2.3.2" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem6.p1.2.2.m2.2.3.2.2" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.2.2.cmml">𝒱</mi><mo id="S4.Thmtheorem6.p1.2.2.m2.2.3.2.1" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem6.p1.2.2.m2.2.3.2.3.2" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.2.3.1.cmml"><mo id="S4.Thmtheorem6.p1.2.2.m2.2.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.2.3.1.cmml">(</mo><mi id="S4.Thmtheorem6.p1.2.2.m2.1.1" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.cmml">N</mi><mo id="S4.Thmtheorem6.p1.2.2.m2.2.3.2.3.2.2" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem6.p1.2.2.m2.2.2" xref="S4.Thmtheorem6.p1.2.2.m2.2.2.cmml">v</mi><mo id="S4.Thmtheorem6.p1.2.2.m2.2.3.2.3.2.3" stretchy="false" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem6.p1.2.2.m2.2.3.1" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.1.cmml">≥</mo><mn id="S4.Thmtheorem6.p1.2.2.m2.2.3.3" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.2.2.m2.2b"><apply id="S4.Thmtheorem6.p1.2.2.m2.2.3.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.2.3"><geq id="S4.Thmtheorem6.p1.2.2.m2.2.3.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.1"></geq><apply id="S4.Thmtheorem6.p1.2.2.m2.2.3.2.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.2"><times id="S4.Thmtheorem6.p1.2.2.m2.2.3.2.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.2.1"></times><ci id="S4.Thmtheorem6.p1.2.2.m2.2.3.2.2.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.2.2">𝒱</ci><interval closure="open" id="S4.Thmtheorem6.p1.2.2.m2.2.3.2.3.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.2.3.2"><ci id="S4.Thmtheorem6.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1">𝑁</ci><ci id="S4.Thmtheorem6.p1.2.2.m2.2.2.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.2.2">𝑣</ci></interval></apply><cn id="S4.Thmtheorem6.p1.2.2.m2.2.3.3.cmml" type="integer" xref="S4.Thmtheorem6.p1.2.2.m2.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.2.2.m2.2c">\mathcal{V}(N,v)\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.2.2.m2.2d">caligraphic_V ( italic_N , italic_v ) ≥ 0</annotation></semantics></math>. Then it holds that <math alttext="\max_{\pi\in\Pi_{N}}\mathcal{SW}(\pi)\leq 2\cdot\max_{\pi\in\Pi_{N}^{(2)}}% \mathcal{SW}(\pi)" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.3.3.m3.3"><semantics id="S4.Thmtheorem6.p1.3.3.m3.3a"><mrow id="S4.Thmtheorem6.p1.3.3.m3.3.4" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.cmml"><mrow id="S4.Thmtheorem6.p1.3.3.m3.3.4.2" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.cmml"><mrow id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.cmml"><msub id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.cmml"><mi id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.2" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.2.cmml">max</mi><mrow id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.cmml"><mi id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.2" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.2.cmml">π</mi><mo id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.1" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.1.cmml">∈</mo><msub id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.3" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.3.cmml"><mi id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.3.2" mathvariant="normal" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.3.2.cmml">Π</mi><mi id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.3.3" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.3.3.cmml">N</mi></msub></mrow></msub><mo id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2a" lspace="0.167em" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.cmml">⁡</mo><mrow id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2.2" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2.2.cmml">𝒮</mi><mo id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2.1" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2.3" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2.3.cmml">𝒲</mi></mrow></mrow><mo id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.1" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.3.2" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.cmml"><mo id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.3.2.1" stretchy="false" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.cmml">(</mo><mi id="S4.Thmtheorem6.p1.3.3.m3.2.2" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.cmml">π</mi><mo id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.3.2.2" stretchy="false" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem6.p1.3.3.m3.3.4.1" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.1.cmml">≤</mo><mrow id="S4.Thmtheorem6.p1.3.3.m3.3.4.3" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.cmml"><mrow id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.cmml"><mn id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.2" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.2.cmml">2</mn><mo id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.1" lspace="0.222em" rspace="0.222em" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.1.cmml">⋅</mo><mrow id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.cmml"><msub id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.1" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.1.cmml"><mi id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.1.2" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.1.2.cmml">max</mi><mrow id="S4.Thmtheorem6.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.cmml"><mi id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.3" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.3.cmml">π</mi><mo id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.2" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.2.cmml">∈</mo><msubsup id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.cmml"><mi id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.2.2" mathvariant="normal" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.2.2.cmml">Π</mi><mi id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.2.3" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.2.3.cmml">N</mi><mrow id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1.3" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.cmml"><mo id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1.3.1" stretchy="false" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.cmml">(</mo><mn id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1.1" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1.1.cmml">2</mn><mo id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1.3.2" stretchy="false" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.cmml">)</mo></mrow></msubsup></mrow></msub><mo id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3a" lspace="0.167em" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.cmml">⁡</mo><mrow id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2.2" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2.2.cmml">𝒮</mi><mo id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2.1" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2.3" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2.3.cmml">𝒲</mi></mrow></mrow></mrow><mo id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.1" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.3.2" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.cmml"><mo id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.3.2.1" stretchy="false" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.cmml">(</mo><mi id="S4.Thmtheorem6.p1.3.3.m3.3.3" xref="S4.Thmtheorem6.p1.3.3.m3.3.3.cmml">π</mi><mo id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.3.2.2" stretchy="false" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.3.3.m3.3b"><apply id="S4.Thmtheorem6.p1.3.3.m3.3.4.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4"><leq id="S4.Thmtheorem6.p1.3.3.m3.3.4.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.1"></leq><apply id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2"><times id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.1"></times><apply id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2"><apply id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1">subscript</csymbol><max id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.2"></max><apply id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3"><in id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.1"></in><ci id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.2">𝜋</ci><apply id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.3.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.3">subscript</csymbol><ci id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.3.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.3.2">Π</ci><ci id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.3.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.1.3.3.3">𝑁</ci></apply></apply></apply><apply id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2"><times id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2.1"></times><ci id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2.2">𝒮</ci><ci id="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.2.2.2.3">𝒲</ci></apply></apply><ci id="S4.Thmtheorem6.p1.3.3.m3.2.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.2.2">𝜋</ci></apply><apply id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3"><times id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.1"></times><apply id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2"><ci id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.1">⋅</ci><cn id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.2.cmml" type="integer" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.2">2</cn><apply id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3"><apply id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.1"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.1.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.1">subscript</csymbol><max id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.1.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.1.2"></max><apply id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1"><in id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.2"></in><ci id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.3">𝜋</ci><apply id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4">superscript</csymbol><apply id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.2.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4">subscript</csymbol><ci id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.2.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.2.2">Π</ci><ci id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.2.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.4.2.3">𝑁</ci></apply><cn id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1.1.cmml" type="integer" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1.1">2</cn></apply></apply></apply><apply id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2"><times id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2.1"></times><ci id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2.2">𝒮</ci><ci id="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.4.3.2.3.2.3">𝒲</ci></apply></apply></apply><ci id="S4.Thmtheorem6.p1.3.3.m3.3.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.3.3">𝜋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.3.3.m3.3c">\max_{\pi\in\Pi_{N}}\mathcal{SW}(\pi)\leq 2\cdot\max_{\pi\in\Pi_{N}^{(2)}}% \mathcal{SW}(\pi)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.3.3.m3.3d">roman_max start_POSTSUBSCRIPT italic_π ∈ roman_Π start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT caligraphic_S caligraphic_W ( italic_π ) ≤ 2 ⋅ roman_max start_POSTSUBSCRIPT italic_π ∈ roman_Π start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT caligraphic_S caligraphic_W ( italic_π )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.SS2.7"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.5.p1"> <p class="ltx_p" id="S4.SS2.5.p1.4">Consider an ASHG <math alttext="(N,v)" class="ltx_Math" display="inline" id="S4.SS2.5.p1.1.m1.2"><semantics id="S4.SS2.5.p1.1.m1.2a"><mrow id="S4.SS2.5.p1.1.m1.2.3.2" xref="S4.SS2.5.p1.1.m1.2.3.1.cmml"><mo id="S4.SS2.5.p1.1.m1.2.3.2.1" stretchy="false" xref="S4.SS2.5.p1.1.m1.2.3.1.cmml">(</mo><mi id="S4.SS2.5.p1.1.m1.1.1" xref="S4.SS2.5.p1.1.m1.1.1.cmml">N</mi><mo id="S4.SS2.5.p1.1.m1.2.3.2.2" xref="S4.SS2.5.p1.1.m1.2.3.1.cmml">,</mo><mi id="S4.SS2.5.p1.1.m1.2.2" xref="S4.SS2.5.p1.1.m1.2.2.cmml">v</mi><mo id="S4.SS2.5.p1.1.m1.2.3.2.3" stretchy="false" xref="S4.SS2.5.p1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.5.p1.1.m1.2b"><interval closure="open" id="S4.SS2.5.p1.1.m1.2.3.1.cmml" xref="S4.SS2.5.p1.1.m1.2.3.2"><ci id="S4.SS2.5.p1.1.m1.1.1.cmml" xref="S4.SS2.5.p1.1.m1.1.1">𝑁</ci><ci id="S4.SS2.5.p1.1.m1.2.2.cmml" xref="S4.SS2.5.p1.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.5.p1.1.m1.2c">(N,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.5.p1.1.m1.2d">( italic_N , italic_v )</annotation></semantics></math> with <math alttext="\mathcal{V}(N,v)\geq 0" class="ltx_Math" display="inline" id="S4.SS2.5.p1.2.m2.2"><semantics id="S4.SS2.5.p1.2.m2.2a"><mrow id="S4.SS2.5.p1.2.m2.2.3" xref="S4.SS2.5.p1.2.m2.2.3.cmml"><mrow id="S4.SS2.5.p1.2.m2.2.3.2" xref="S4.SS2.5.p1.2.m2.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.5.p1.2.m2.2.3.2.2" xref="S4.SS2.5.p1.2.m2.2.3.2.2.cmml">𝒱</mi><mo id="S4.SS2.5.p1.2.m2.2.3.2.1" xref="S4.SS2.5.p1.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS2.5.p1.2.m2.2.3.2.3.2" xref="S4.SS2.5.p1.2.m2.2.3.2.3.1.cmml"><mo id="S4.SS2.5.p1.2.m2.2.3.2.3.2.1" stretchy="false" xref="S4.SS2.5.p1.2.m2.2.3.2.3.1.cmml">(</mo><mi id="S4.SS2.5.p1.2.m2.1.1" xref="S4.SS2.5.p1.2.m2.1.1.cmml">N</mi><mo id="S4.SS2.5.p1.2.m2.2.3.2.3.2.2" xref="S4.SS2.5.p1.2.m2.2.3.2.3.1.cmml">,</mo><mi id="S4.SS2.5.p1.2.m2.2.2" xref="S4.SS2.5.p1.2.m2.2.2.cmml">v</mi><mo id="S4.SS2.5.p1.2.m2.2.3.2.3.2.3" stretchy="false" xref="S4.SS2.5.p1.2.m2.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.5.p1.2.m2.2.3.1" xref="S4.SS2.5.p1.2.m2.2.3.1.cmml">≥</mo><mn id="S4.SS2.5.p1.2.m2.2.3.3" xref="S4.SS2.5.p1.2.m2.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.5.p1.2.m2.2b"><apply id="S4.SS2.5.p1.2.m2.2.3.cmml" xref="S4.SS2.5.p1.2.m2.2.3"><geq id="S4.SS2.5.p1.2.m2.2.3.1.cmml" xref="S4.SS2.5.p1.2.m2.2.3.1"></geq><apply id="S4.SS2.5.p1.2.m2.2.3.2.cmml" xref="S4.SS2.5.p1.2.m2.2.3.2"><times id="S4.SS2.5.p1.2.m2.2.3.2.1.cmml" xref="S4.SS2.5.p1.2.m2.2.3.2.1"></times><ci id="S4.SS2.5.p1.2.m2.2.3.2.2.cmml" xref="S4.SS2.5.p1.2.m2.2.3.2.2">𝒱</ci><interval closure="open" id="S4.SS2.5.p1.2.m2.2.3.2.3.1.cmml" xref="S4.SS2.5.p1.2.m2.2.3.2.3.2"><ci id="S4.SS2.5.p1.2.m2.1.1.cmml" xref="S4.SS2.5.p1.2.m2.1.1">𝑁</ci><ci id="S4.SS2.5.p1.2.m2.2.2.cmml" xref="S4.SS2.5.p1.2.m2.2.2">𝑣</ci></interval></apply><cn id="S4.SS2.5.p1.2.m2.2.3.3.cmml" type="integer" xref="S4.SS2.5.p1.2.m2.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.5.p1.2.m2.2c">\mathcal{V}(N,v)\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.5.p1.2.m2.2d">caligraphic_V ( italic_N , italic_v ) ≥ 0</annotation></semantics></math>. Let <math alttext="\pi^{*}" class="ltx_Math" display="inline" id="S4.SS2.5.p1.3.m3.1"><semantics id="S4.SS2.5.p1.3.m3.1a"><msup id="S4.SS2.5.p1.3.m3.1.1" xref="S4.SS2.5.p1.3.m3.1.1.cmml"><mi id="S4.SS2.5.p1.3.m3.1.1.2" xref="S4.SS2.5.p1.3.m3.1.1.2.cmml">π</mi><mo id="S4.SS2.5.p1.3.m3.1.1.3" xref="S4.SS2.5.p1.3.m3.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.5.p1.3.m3.1b"><apply id="S4.SS2.5.p1.3.m3.1.1.cmml" xref="S4.SS2.5.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.5.p1.3.m3.1.1.1.cmml" xref="S4.SS2.5.p1.3.m3.1.1">superscript</csymbol><ci id="S4.SS2.5.p1.3.m3.1.1.2.cmml" xref="S4.SS2.5.p1.3.m3.1.1.2">𝜋</ci><times id="S4.SS2.5.p1.3.m3.1.1.3.cmml" xref="S4.SS2.5.p1.3.m3.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.5.p1.3.m3.1c">\pi^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.5.p1.3.m3.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be a partition maximizing <math alttext="\mathcal{SW}" class="ltx_Math" display="inline" id="S4.SS2.5.p1.4.m4.1"><semantics id="S4.SS2.5.p1.4.m4.1a"><mrow id="S4.SS2.5.p1.4.m4.1.1" xref="S4.SS2.5.p1.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.5.p1.4.m4.1.1.2" xref="S4.SS2.5.p1.4.m4.1.1.2.cmml">𝒮</mi><mo id="S4.SS2.5.p1.4.m4.1.1.1" xref="S4.SS2.5.p1.4.m4.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.5.p1.4.m4.1.1.3" xref="S4.SS2.5.p1.4.m4.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.5.p1.4.m4.1b"><apply id="S4.SS2.5.p1.4.m4.1.1.cmml" xref="S4.SS2.5.p1.4.m4.1.1"><times id="S4.SS2.5.p1.4.m4.1.1.1.cmml" xref="S4.SS2.5.p1.4.m4.1.1.1"></times><ci id="S4.SS2.5.p1.4.m4.1.1.2.cmml" xref="S4.SS2.5.p1.4.m4.1.1.2">𝒮</ci><ci id="S4.SS2.5.p1.4.m4.1.1.3.cmml" xref="S4.SS2.5.p1.4.m4.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.5.p1.4.m4.1c">\mathcal{SW}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.5.p1.4.m4.1d">caligraphic_S caligraphic_W</annotation></semantics></math>. Define the quantities</p> <ul class="ltx_itemize" id="S4.I1"> <li class="ltx_item" id="S4.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i1.p1"> <p class="ltx_p" id="S4.I1.i1.p1.1"><math alttext="W:=\sum_{i\in N,j\in\pi^{*}(i)}v_{i}(j)" class="ltx_Math" display="inline" id="S4.I1.i1.p1.1.m1.4"><semantics id="S4.I1.i1.p1.1.m1.4a"><mrow id="S4.I1.i1.p1.1.m1.4.5" xref="S4.I1.i1.p1.1.m1.4.5.cmml"><mi id="S4.I1.i1.p1.1.m1.4.5.2" xref="S4.I1.i1.p1.1.m1.4.5.2.cmml">W</mi><mo id="S4.I1.i1.p1.1.m1.4.5.1" lspace="0.278em" rspace="0.111em" xref="S4.I1.i1.p1.1.m1.4.5.1.cmml">:=</mo><mrow id="S4.I1.i1.p1.1.m1.4.5.3" xref="S4.I1.i1.p1.1.m1.4.5.3.cmml"><msub id="S4.I1.i1.p1.1.m1.4.5.3.1" xref="S4.I1.i1.p1.1.m1.4.5.3.1.cmml"><mo id="S4.I1.i1.p1.1.m1.4.5.3.1.2" xref="S4.I1.i1.p1.1.m1.4.5.3.1.2.cmml">∑</mo><mrow id="S4.I1.i1.p1.1.m1.3.3.3.3" xref="S4.I1.i1.p1.1.m1.3.3.3.4.cmml"><mrow id="S4.I1.i1.p1.1.m1.2.2.2.2.1" xref="S4.I1.i1.p1.1.m1.2.2.2.2.1.cmml"><mi id="S4.I1.i1.p1.1.m1.2.2.2.2.1.2" xref="S4.I1.i1.p1.1.m1.2.2.2.2.1.2.cmml">i</mi><mo id="S4.I1.i1.p1.1.m1.2.2.2.2.1.1" xref="S4.I1.i1.p1.1.m1.2.2.2.2.1.1.cmml">∈</mo><mi id="S4.I1.i1.p1.1.m1.2.2.2.2.1.3" xref="S4.I1.i1.p1.1.m1.2.2.2.2.1.3.cmml">N</mi></mrow><mo id="S4.I1.i1.p1.1.m1.3.3.3.3.3" xref="S4.I1.i1.p1.1.m1.3.3.3.4a.cmml">,</mo><mrow id="S4.I1.i1.p1.1.m1.3.3.3.3.2" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.cmml"><mi id="S4.I1.i1.p1.1.m1.3.3.3.3.2.2" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.2.cmml">j</mi><mo id="S4.I1.i1.p1.1.m1.3.3.3.3.2.1" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.1.cmml">∈</mo><mrow id="S4.I1.i1.p1.1.m1.3.3.3.3.2.3" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.cmml"><msup id="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.2" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.2.cmml"><mi id="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.2.2" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.2.2.cmml">π</mi><mo id="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.2.3" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.2.3.cmml">∗</mo></msup><mo id="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.1" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.1.cmml">⁢</mo><mrow id="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.3.2" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.cmml"><mo id="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.3.2.1" stretchy="false" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.cmml">(</mo><mi id="S4.I1.i1.p1.1.m1.1.1.1.1" xref="S4.I1.i1.p1.1.m1.1.1.1.1.cmml">i</mi><mo id="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.3.2.2" stretchy="false" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.cmml">)</mo></mrow></mrow></mrow></mrow></msub><mrow id="S4.I1.i1.p1.1.m1.4.5.3.2" xref="S4.I1.i1.p1.1.m1.4.5.3.2.cmml"><msub id="S4.I1.i1.p1.1.m1.4.5.3.2.2" xref="S4.I1.i1.p1.1.m1.4.5.3.2.2.cmml"><mi id="S4.I1.i1.p1.1.m1.4.5.3.2.2.2" xref="S4.I1.i1.p1.1.m1.4.5.3.2.2.2.cmml">v</mi><mi id="S4.I1.i1.p1.1.m1.4.5.3.2.2.3" xref="S4.I1.i1.p1.1.m1.4.5.3.2.2.3.cmml">i</mi></msub><mo id="S4.I1.i1.p1.1.m1.4.5.3.2.1" xref="S4.I1.i1.p1.1.m1.4.5.3.2.1.cmml">⁢</mo><mrow id="S4.I1.i1.p1.1.m1.4.5.3.2.3.2" xref="S4.I1.i1.p1.1.m1.4.5.3.2.cmml"><mo id="S4.I1.i1.p1.1.m1.4.5.3.2.3.2.1" stretchy="false" xref="S4.I1.i1.p1.1.m1.4.5.3.2.cmml">(</mo><mi id="S4.I1.i1.p1.1.m1.4.4" xref="S4.I1.i1.p1.1.m1.4.4.cmml">j</mi><mo id="S4.I1.i1.p1.1.m1.4.5.3.2.3.2.2" stretchy="false" xref="S4.I1.i1.p1.1.m1.4.5.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.1.m1.4b"><apply id="S4.I1.i1.p1.1.m1.4.5.cmml" xref="S4.I1.i1.p1.1.m1.4.5"><csymbol cd="latexml" id="S4.I1.i1.p1.1.m1.4.5.1.cmml" xref="S4.I1.i1.p1.1.m1.4.5.1">assign</csymbol><ci id="S4.I1.i1.p1.1.m1.4.5.2.cmml" xref="S4.I1.i1.p1.1.m1.4.5.2">𝑊</ci><apply id="S4.I1.i1.p1.1.m1.4.5.3.cmml" xref="S4.I1.i1.p1.1.m1.4.5.3"><apply id="S4.I1.i1.p1.1.m1.4.5.3.1.cmml" xref="S4.I1.i1.p1.1.m1.4.5.3.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.1.m1.4.5.3.1.1.cmml" xref="S4.I1.i1.p1.1.m1.4.5.3.1">subscript</csymbol><sum id="S4.I1.i1.p1.1.m1.4.5.3.1.2.cmml" xref="S4.I1.i1.p1.1.m1.4.5.3.1.2"></sum><apply id="S4.I1.i1.p1.1.m1.3.3.3.4.cmml" xref="S4.I1.i1.p1.1.m1.3.3.3.3"><csymbol cd="ambiguous" id="S4.I1.i1.p1.1.m1.3.3.3.4a.cmml" xref="S4.I1.i1.p1.1.m1.3.3.3.3.3">formulae-sequence</csymbol><apply id="S4.I1.i1.p1.1.m1.2.2.2.2.1.cmml" xref="S4.I1.i1.p1.1.m1.2.2.2.2.1"><in id="S4.I1.i1.p1.1.m1.2.2.2.2.1.1.cmml" xref="S4.I1.i1.p1.1.m1.2.2.2.2.1.1"></in><ci id="S4.I1.i1.p1.1.m1.2.2.2.2.1.2.cmml" xref="S4.I1.i1.p1.1.m1.2.2.2.2.1.2">𝑖</ci><ci id="S4.I1.i1.p1.1.m1.2.2.2.2.1.3.cmml" xref="S4.I1.i1.p1.1.m1.2.2.2.2.1.3">𝑁</ci></apply><apply id="S4.I1.i1.p1.1.m1.3.3.3.3.2.cmml" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2"><in id="S4.I1.i1.p1.1.m1.3.3.3.3.2.1.cmml" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.1"></in><ci id="S4.I1.i1.p1.1.m1.3.3.3.3.2.2.cmml" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.2">𝑗</ci><apply id="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.cmml" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.3"><times id="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.1.cmml" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.1"></times><apply id="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.2.cmml" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.2"><csymbol cd="ambiguous" id="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.2.1.cmml" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.2">superscript</csymbol><ci id="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.2.2.cmml" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.2.2">𝜋</ci><times id="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.2.3.cmml" xref="S4.I1.i1.p1.1.m1.3.3.3.3.2.3.2.3"></times></apply><ci id="S4.I1.i1.p1.1.m1.1.1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1.1.1">𝑖</ci></apply></apply></apply></apply><apply id="S4.I1.i1.p1.1.m1.4.5.3.2.cmml" xref="S4.I1.i1.p1.1.m1.4.5.3.2"><times id="S4.I1.i1.p1.1.m1.4.5.3.2.1.cmml" xref="S4.I1.i1.p1.1.m1.4.5.3.2.1"></times><apply id="S4.I1.i1.p1.1.m1.4.5.3.2.2.cmml" xref="S4.I1.i1.p1.1.m1.4.5.3.2.2"><csymbol cd="ambiguous" id="S4.I1.i1.p1.1.m1.4.5.3.2.2.1.cmml" xref="S4.I1.i1.p1.1.m1.4.5.3.2.2">subscript</csymbol><ci id="S4.I1.i1.p1.1.m1.4.5.3.2.2.2.cmml" xref="S4.I1.i1.p1.1.m1.4.5.3.2.2.2">𝑣</ci><ci id="S4.I1.i1.p1.1.m1.4.5.3.2.2.3.cmml" xref="S4.I1.i1.p1.1.m1.4.5.3.2.2.3">𝑖</ci></apply><ci id="S4.I1.i1.p1.1.m1.4.4.cmml" xref="S4.I1.i1.p1.1.m1.4.4">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.1.m1.4c">W:=\sum_{i\in N,j\in\pi^{*}(i)}v_{i}(j)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.1.m1.4d">italic_W := ∑ start_POSTSUBSCRIPT italic_i ∈ italic_N , italic_j ∈ italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_i ) end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math>,</p> </div> </li> <li class="ltx_item" id="S4.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i2.p1"> <p class="ltx_p" id="S4.I1.i2.p1.1"><math alttext="A:=\sum_{i\in N,j\notin\pi^{*}(i)}v_{i}(j)" class="ltx_Math" display="inline" id="S4.I1.i2.p1.1.m1.4"><semantics id="S4.I1.i2.p1.1.m1.4a"><mrow id="S4.I1.i2.p1.1.m1.4.5" xref="S4.I1.i2.p1.1.m1.4.5.cmml"><mi id="S4.I1.i2.p1.1.m1.4.5.2" xref="S4.I1.i2.p1.1.m1.4.5.2.cmml">A</mi><mo id="S4.I1.i2.p1.1.m1.4.5.1" lspace="0.278em" rspace="0.111em" xref="S4.I1.i2.p1.1.m1.4.5.1.cmml">:=</mo><mrow id="S4.I1.i2.p1.1.m1.4.5.3" xref="S4.I1.i2.p1.1.m1.4.5.3.cmml"><msub id="S4.I1.i2.p1.1.m1.4.5.3.1" xref="S4.I1.i2.p1.1.m1.4.5.3.1.cmml"><mo id="S4.I1.i2.p1.1.m1.4.5.3.1.2" xref="S4.I1.i2.p1.1.m1.4.5.3.1.2.cmml">∑</mo><mrow id="S4.I1.i2.p1.1.m1.3.3.3.3" xref="S4.I1.i2.p1.1.m1.3.3.3.4.cmml"><mrow id="S4.I1.i2.p1.1.m1.2.2.2.2.1" xref="S4.I1.i2.p1.1.m1.2.2.2.2.1.cmml"><mi id="S4.I1.i2.p1.1.m1.2.2.2.2.1.2" xref="S4.I1.i2.p1.1.m1.2.2.2.2.1.2.cmml">i</mi><mo id="S4.I1.i2.p1.1.m1.2.2.2.2.1.1" xref="S4.I1.i2.p1.1.m1.2.2.2.2.1.1.cmml">∈</mo><mi id="S4.I1.i2.p1.1.m1.2.2.2.2.1.3" xref="S4.I1.i2.p1.1.m1.2.2.2.2.1.3.cmml">N</mi></mrow><mo id="S4.I1.i2.p1.1.m1.3.3.3.3.3" xref="S4.I1.i2.p1.1.m1.3.3.3.4a.cmml">,</mo><mrow id="S4.I1.i2.p1.1.m1.3.3.3.3.2" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.cmml"><mi id="S4.I1.i2.p1.1.m1.3.3.3.3.2.2" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.2.cmml">j</mi><mo id="S4.I1.i2.p1.1.m1.3.3.3.3.2.1" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.1.cmml">∉</mo><mrow id="S4.I1.i2.p1.1.m1.3.3.3.3.2.3" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.cmml"><msup id="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.2" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.2.cmml"><mi id="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.2.2" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.2.2.cmml">π</mi><mo id="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.2.3" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.2.3.cmml">∗</mo></msup><mo id="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.1" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.1.cmml">⁢</mo><mrow id="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.3.2" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.cmml"><mo id="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.3.2.1" stretchy="false" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.cmml">(</mo><mi id="S4.I1.i2.p1.1.m1.1.1.1.1" xref="S4.I1.i2.p1.1.m1.1.1.1.1.cmml">i</mi><mo id="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.3.2.2" stretchy="false" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.cmml">)</mo></mrow></mrow></mrow></mrow></msub><mrow id="S4.I1.i2.p1.1.m1.4.5.3.2" xref="S4.I1.i2.p1.1.m1.4.5.3.2.cmml"><msub id="S4.I1.i2.p1.1.m1.4.5.3.2.2" xref="S4.I1.i2.p1.1.m1.4.5.3.2.2.cmml"><mi id="S4.I1.i2.p1.1.m1.4.5.3.2.2.2" xref="S4.I1.i2.p1.1.m1.4.5.3.2.2.2.cmml">v</mi><mi id="S4.I1.i2.p1.1.m1.4.5.3.2.2.3" xref="S4.I1.i2.p1.1.m1.4.5.3.2.2.3.cmml">i</mi></msub><mo id="S4.I1.i2.p1.1.m1.4.5.3.2.1" xref="S4.I1.i2.p1.1.m1.4.5.3.2.1.cmml">⁢</mo><mrow id="S4.I1.i2.p1.1.m1.4.5.3.2.3.2" xref="S4.I1.i2.p1.1.m1.4.5.3.2.cmml"><mo id="S4.I1.i2.p1.1.m1.4.5.3.2.3.2.1" stretchy="false" xref="S4.I1.i2.p1.1.m1.4.5.3.2.cmml">(</mo><mi id="S4.I1.i2.p1.1.m1.4.4" xref="S4.I1.i2.p1.1.m1.4.4.cmml">j</mi><mo id="S4.I1.i2.p1.1.m1.4.5.3.2.3.2.2" stretchy="false" xref="S4.I1.i2.p1.1.m1.4.5.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.1.m1.4b"><apply id="S4.I1.i2.p1.1.m1.4.5.cmml" xref="S4.I1.i2.p1.1.m1.4.5"><csymbol cd="latexml" id="S4.I1.i2.p1.1.m1.4.5.1.cmml" xref="S4.I1.i2.p1.1.m1.4.5.1">assign</csymbol><ci id="S4.I1.i2.p1.1.m1.4.5.2.cmml" xref="S4.I1.i2.p1.1.m1.4.5.2">𝐴</ci><apply id="S4.I1.i2.p1.1.m1.4.5.3.cmml" xref="S4.I1.i2.p1.1.m1.4.5.3"><apply id="S4.I1.i2.p1.1.m1.4.5.3.1.cmml" xref="S4.I1.i2.p1.1.m1.4.5.3.1"><csymbol cd="ambiguous" id="S4.I1.i2.p1.1.m1.4.5.3.1.1.cmml" xref="S4.I1.i2.p1.1.m1.4.5.3.1">subscript</csymbol><sum id="S4.I1.i2.p1.1.m1.4.5.3.1.2.cmml" xref="S4.I1.i2.p1.1.m1.4.5.3.1.2"></sum><apply id="S4.I1.i2.p1.1.m1.3.3.3.4.cmml" xref="S4.I1.i2.p1.1.m1.3.3.3.3"><csymbol cd="ambiguous" id="S4.I1.i2.p1.1.m1.3.3.3.4a.cmml" xref="S4.I1.i2.p1.1.m1.3.3.3.3.3">formulae-sequence</csymbol><apply id="S4.I1.i2.p1.1.m1.2.2.2.2.1.cmml" xref="S4.I1.i2.p1.1.m1.2.2.2.2.1"><in id="S4.I1.i2.p1.1.m1.2.2.2.2.1.1.cmml" xref="S4.I1.i2.p1.1.m1.2.2.2.2.1.1"></in><ci id="S4.I1.i2.p1.1.m1.2.2.2.2.1.2.cmml" xref="S4.I1.i2.p1.1.m1.2.2.2.2.1.2">𝑖</ci><ci id="S4.I1.i2.p1.1.m1.2.2.2.2.1.3.cmml" xref="S4.I1.i2.p1.1.m1.2.2.2.2.1.3">𝑁</ci></apply><apply id="S4.I1.i2.p1.1.m1.3.3.3.3.2.cmml" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2"><notin id="S4.I1.i2.p1.1.m1.3.3.3.3.2.1.cmml" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.1"></notin><ci id="S4.I1.i2.p1.1.m1.3.3.3.3.2.2.cmml" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.2">𝑗</ci><apply id="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.cmml" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.3"><times id="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.1.cmml" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.1"></times><apply id="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.2.cmml" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.2"><csymbol cd="ambiguous" id="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.2.1.cmml" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.2">superscript</csymbol><ci id="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.2.2.cmml" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.2.2">𝜋</ci><times id="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.2.3.cmml" xref="S4.I1.i2.p1.1.m1.3.3.3.3.2.3.2.3"></times></apply><ci id="S4.I1.i2.p1.1.m1.1.1.1.1.cmml" xref="S4.I1.i2.p1.1.m1.1.1.1.1">𝑖</ci></apply></apply></apply></apply><apply id="S4.I1.i2.p1.1.m1.4.5.3.2.cmml" xref="S4.I1.i2.p1.1.m1.4.5.3.2"><times id="S4.I1.i2.p1.1.m1.4.5.3.2.1.cmml" xref="S4.I1.i2.p1.1.m1.4.5.3.2.1"></times><apply id="S4.I1.i2.p1.1.m1.4.5.3.2.2.cmml" xref="S4.I1.i2.p1.1.m1.4.5.3.2.2"><csymbol cd="ambiguous" id="S4.I1.i2.p1.1.m1.4.5.3.2.2.1.cmml" xref="S4.I1.i2.p1.1.m1.4.5.3.2.2">subscript</csymbol><ci id="S4.I1.i2.p1.1.m1.4.5.3.2.2.2.cmml" xref="S4.I1.i2.p1.1.m1.4.5.3.2.2.2">𝑣</ci><ci id="S4.I1.i2.p1.1.m1.4.5.3.2.2.3.cmml" xref="S4.I1.i2.p1.1.m1.4.5.3.2.2.3">𝑖</ci></apply><ci id="S4.I1.i2.p1.1.m1.4.4.cmml" xref="S4.I1.i2.p1.1.m1.4.4">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.1.m1.4c">A:=\sum_{i\in N,j\notin\pi^{*}(i)}v_{i}(j)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.1.m1.4d">italic_A := ∑ start_POSTSUBSCRIPT italic_i ∈ italic_N , italic_j ∉ italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_i ) end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math>,</p> </div> </li> </ul> </div> <div class="ltx_para" id="S4.SS2.6.p2"> <p class="ltx_p" id="S4.SS2.6.p2.1">These measure the total value of within and across coalitions in the optimal partition <math alttext="\pi^{*}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.1.m1.1"><semantics id="S4.SS2.6.p2.1.m1.1a"><msup id="S4.SS2.6.p2.1.m1.1.1" xref="S4.SS2.6.p2.1.m1.1.1.cmml"><mi id="S4.SS2.6.p2.1.m1.1.1.2" xref="S4.SS2.6.p2.1.m1.1.1.2.cmml">π</mi><mo id="S4.SS2.6.p2.1.m1.1.1.3" xref="S4.SS2.6.p2.1.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.1.m1.1b"><apply id="S4.SS2.6.p2.1.m1.1.1.cmml" xref="S4.SS2.6.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.1.m1.1.1.1.cmml" xref="S4.SS2.6.p2.1.m1.1.1">superscript</csymbol><ci id="S4.SS2.6.p2.1.m1.1.1.2.cmml" xref="S4.SS2.6.p2.1.m1.1.1.2">𝜋</ci><times id="S4.SS2.6.p2.1.m1.1.1.3.cmml" xref="S4.SS2.6.p2.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.1.m1.1c">\pi^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.1.m1.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. By definition, it holds that</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex18"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{SW}(\pi^{*})=W\quad\text{and}\quad\mathcal{V}(N,v)=W+A\text{.}" class="ltx_Math" display="block" id="S4.Ex18.m1.6"><semantics id="S4.Ex18.m1.6a"><mrow id="S4.Ex18.m1.6.6.2" xref="S4.Ex18.m1.6.6.3.cmml"><mrow id="S4.Ex18.m1.5.5.1.1" xref="S4.Ex18.m1.5.5.1.1.cmml"><mrow id="S4.Ex18.m1.5.5.1.1.1" xref="S4.Ex18.m1.5.5.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex18.m1.5.5.1.1.1.3" xref="S4.Ex18.m1.5.5.1.1.1.3.cmml">𝒮</mi><mo id="S4.Ex18.m1.5.5.1.1.1.2" xref="S4.Ex18.m1.5.5.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex18.m1.5.5.1.1.1.4" xref="S4.Ex18.m1.5.5.1.1.1.4.cmml">𝒲</mi><mo id="S4.Ex18.m1.5.5.1.1.1.2a" xref="S4.Ex18.m1.5.5.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex18.m1.5.5.1.1.1.1.1" xref="S4.Ex18.m1.5.5.1.1.1.1.1.1.cmml"><mo id="S4.Ex18.m1.5.5.1.1.1.1.1.2" stretchy="false" xref="S4.Ex18.m1.5.5.1.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex18.m1.5.5.1.1.1.1.1.1" xref="S4.Ex18.m1.5.5.1.1.1.1.1.1.cmml"><mi id="S4.Ex18.m1.5.5.1.1.1.1.1.1.2" xref="S4.Ex18.m1.5.5.1.1.1.1.1.1.2.cmml">π</mi><mo id="S4.Ex18.m1.5.5.1.1.1.1.1.1.3" xref="S4.Ex18.m1.5.5.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.Ex18.m1.5.5.1.1.1.1.1.3" stretchy="false" xref="S4.Ex18.m1.5.5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex18.m1.5.5.1.1.2" xref="S4.Ex18.m1.5.5.1.1.2.cmml">=</mo><mrow id="S4.Ex18.m1.5.5.1.1.3.2" xref="S4.Ex18.m1.5.5.1.1.3.1.cmml"><mi id="S4.Ex18.m1.3.3" xref="S4.Ex18.m1.3.3.cmml">W</mi><mspace id="S4.Ex18.m1.5.5.1.1.3.2.1" width="1em" xref="S4.Ex18.m1.5.5.1.1.3.1.cmml"></mspace><mtext id="S4.Ex18.m1.4.4" xref="S4.Ex18.m1.4.4a.cmml">and</mtext></mrow></mrow><mspace id="S4.Ex18.m1.6.6.2.3" width="1em" xref="S4.Ex18.m1.6.6.3a.cmml"></mspace><mrow id="S4.Ex18.m1.6.6.2.2" xref="S4.Ex18.m1.6.6.2.2.cmml"><mrow id="S4.Ex18.m1.6.6.2.2.2" xref="S4.Ex18.m1.6.6.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex18.m1.6.6.2.2.2.2" xref="S4.Ex18.m1.6.6.2.2.2.2.cmml">𝒱</mi><mo id="S4.Ex18.m1.6.6.2.2.2.1" xref="S4.Ex18.m1.6.6.2.2.2.1.cmml">⁢</mo><mrow id="S4.Ex18.m1.6.6.2.2.2.3.2" xref="S4.Ex18.m1.6.6.2.2.2.3.1.cmml"><mo id="S4.Ex18.m1.6.6.2.2.2.3.2.1" stretchy="false" xref="S4.Ex18.m1.6.6.2.2.2.3.1.cmml">(</mo><mi id="S4.Ex18.m1.1.1" xref="S4.Ex18.m1.1.1.cmml">N</mi><mo id="S4.Ex18.m1.6.6.2.2.2.3.2.2" xref="S4.Ex18.m1.6.6.2.2.2.3.1.cmml">,</mo><mi id="S4.Ex18.m1.2.2" xref="S4.Ex18.m1.2.2.cmml">v</mi><mo id="S4.Ex18.m1.6.6.2.2.2.3.2.3" stretchy="false" xref="S4.Ex18.m1.6.6.2.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex18.m1.6.6.2.2.1" xref="S4.Ex18.m1.6.6.2.2.1.cmml">=</mo><mrow id="S4.Ex18.m1.6.6.2.2.3" xref="S4.Ex18.m1.6.6.2.2.3.cmml"><mi id="S4.Ex18.m1.6.6.2.2.3.2" xref="S4.Ex18.m1.6.6.2.2.3.2.cmml">W</mi><mo id="S4.Ex18.m1.6.6.2.2.3.1" xref="S4.Ex18.m1.6.6.2.2.3.1.cmml">+</mo><mrow id="S4.Ex18.m1.6.6.2.2.3.3" xref="S4.Ex18.m1.6.6.2.2.3.3.cmml"><mi id="S4.Ex18.m1.6.6.2.2.3.3.2" xref="S4.Ex18.m1.6.6.2.2.3.3.2.cmml">A</mi><mo id="S4.Ex18.m1.6.6.2.2.3.3.1" xref="S4.Ex18.m1.6.6.2.2.3.3.1.cmml">⁢</mo><mtext id="S4.Ex18.m1.6.6.2.2.3.3.3" xref="S4.Ex18.m1.6.6.2.2.3.3.3a.cmml">.</mtext></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex18.m1.6b"><apply id="S4.Ex18.m1.6.6.3.cmml" xref="S4.Ex18.m1.6.6.2"><csymbol cd="ambiguous" id="S4.Ex18.m1.6.6.3a.cmml" xref="S4.Ex18.m1.6.6.2.3">formulae-sequence</csymbol><apply id="S4.Ex18.m1.5.5.1.1.cmml" xref="S4.Ex18.m1.5.5.1.1"><eq id="S4.Ex18.m1.5.5.1.1.2.cmml" xref="S4.Ex18.m1.5.5.1.1.2"></eq><apply id="S4.Ex18.m1.5.5.1.1.1.cmml" xref="S4.Ex18.m1.5.5.1.1.1"><times id="S4.Ex18.m1.5.5.1.1.1.2.cmml" xref="S4.Ex18.m1.5.5.1.1.1.2"></times><ci id="S4.Ex18.m1.5.5.1.1.1.3.cmml" xref="S4.Ex18.m1.5.5.1.1.1.3">𝒮</ci><ci id="S4.Ex18.m1.5.5.1.1.1.4.cmml" xref="S4.Ex18.m1.5.5.1.1.1.4">𝒲</ci><apply id="S4.Ex18.m1.5.5.1.1.1.1.1.1.cmml" xref="S4.Ex18.m1.5.5.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex18.m1.5.5.1.1.1.1.1.1.1.cmml" xref="S4.Ex18.m1.5.5.1.1.1.1.1">superscript</csymbol><ci id="S4.Ex18.m1.5.5.1.1.1.1.1.1.2.cmml" xref="S4.Ex18.m1.5.5.1.1.1.1.1.1.2">𝜋</ci><times id="S4.Ex18.m1.5.5.1.1.1.1.1.1.3.cmml" xref="S4.Ex18.m1.5.5.1.1.1.1.1.1.3"></times></apply></apply><list id="S4.Ex18.m1.5.5.1.1.3.1.cmml" xref="S4.Ex18.m1.5.5.1.1.3.2"><ci id="S4.Ex18.m1.3.3.cmml" xref="S4.Ex18.m1.3.3">𝑊</ci><ci id="S4.Ex18.m1.4.4a.cmml" xref="S4.Ex18.m1.4.4"><mtext id="S4.Ex18.m1.4.4.cmml" xref="S4.Ex18.m1.4.4">and</mtext></ci></list></apply><apply id="S4.Ex18.m1.6.6.2.2.cmml" xref="S4.Ex18.m1.6.6.2.2"><eq id="S4.Ex18.m1.6.6.2.2.1.cmml" xref="S4.Ex18.m1.6.6.2.2.1"></eq><apply id="S4.Ex18.m1.6.6.2.2.2.cmml" xref="S4.Ex18.m1.6.6.2.2.2"><times id="S4.Ex18.m1.6.6.2.2.2.1.cmml" xref="S4.Ex18.m1.6.6.2.2.2.1"></times><ci id="S4.Ex18.m1.6.6.2.2.2.2.cmml" xref="S4.Ex18.m1.6.6.2.2.2.2">𝒱</ci><interval closure="open" id="S4.Ex18.m1.6.6.2.2.2.3.1.cmml" xref="S4.Ex18.m1.6.6.2.2.2.3.2"><ci id="S4.Ex18.m1.1.1.cmml" xref="S4.Ex18.m1.1.1">𝑁</ci><ci id="S4.Ex18.m1.2.2.cmml" xref="S4.Ex18.m1.2.2">𝑣</ci></interval></apply><apply id="S4.Ex18.m1.6.6.2.2.3.cmml" xref="S4.Ex18.m1.6.6.2.2.3"><plus id="S4.Ex18.m1.6.6.2.2.3.1.cmml" xref="S4.Ex18.m1.6.6.2.2.3.1"></plus><ci id="S4.Ex18.m1.6.6.2.2.3.2.cmml" xref="S4.Ex18.m1.6.6.2.2.3.2">𝑊</ci><apply id="S4.Ex18.m1.6.6.2.2.3.3.cmml" xref="S4.Ex18.m1.6.6.2.2.3.3"><times id="S4.Ex18.m1.6.6.2.2.3.3.1.cmml" xref="S4.Ex18.m1.6.6.2.2.3.3.1"></times><ci id="S4.Ex18.m1.6.6.2.2.3.3.2.cmml" xref="S4.Ex18.m1.6.6.2.2.3.3.2">𝐴</ci><ci id="S4.Ex18.m1.6.6.2.2.3.3.3a.cmml" xref="S4.Ex18.m1.6.6.2.2.3.3.3"><mtext id="S4.Ex18.m1.6.6.2.2.3.3.3.cmml" xref="S4.Ex18.m1.6.6.2.2.3.3.3">.</mtext></ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex18.m1.6c">\mathcal{SW}(\pi^{*})=W\quad\text{and}\quad\mathcal{V}(N,v)=W+A\text{.}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex18.m1.6d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = italic_W and caligraphic_V ( italic_N , italic_v ) = italic_W + italic_A .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.6.p2.14">We now establish a lower bound for the optimal solution welfare of partitions with two coalitions. Therefore, define <math alttext="\pi^{*}_{2}:=\operatorname*{arg\,max}_{\pi\in\Pi_{N}^{(2)}}\mathcal{SW}(\pi)" class="ltx_Math" display="inline" id="S4.SS2.6.p2.2.m1.2"><semantics id="S4.SS2.6.p2.2.m1.2a"><mrow id="S4.SS2.6.p2.2.m1.2.3" xref="S4.SS2.6.p2.2.m1.2.3.cmml"><msubsup id="S4.SS2.6.p2.2.m1.2.3.2" xref="S4.SS2.6.p2.2.m1.2.3.2.cmml"><mi id="S4.SS2.6.p2.2.m1.2.3.2.2.2" xref="S4.SS2.6.p2.2.m1.2.3.2.2.2.cmml">π</mi><mn id="S4.SS2.6.p2.2.m1.2.3.2.3" xref="S4.SS2.6.p2.2.m1.2.3.2.3.cmml">2</mn><mo id="S4.SS2.6.p2.2.m1.2.3.2.2.3" xref="S4.SS2.6.p2.2.m1.2.3.2.2.3.cmml">∗</mo></msubsup><mo id="S4.SS2.6.p2.2.m1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S4.SS2.6.p2.2.m1.2.3.1.cmml">:=</mo><mrow id="S4.SS2.6.p2.2.m1.2.3.3" xref="S4.SS2.6.p2.2.m1.2.3.3.cmml"><mrow id="S4.SS2.6.p2.2.m1.2.3.3.2" xref="S4.SS2.6.p2.2.m1.2.3.3.2.cmml"><msub id="S4.SS2.6.p2.2.m1.2.3.3.2.1" xref="S4.SS2.6.p2.2.m1.2.3.3.2.1.cmml"><mrow id="S4.SS2.6.p2.2.m1.2.3.3.2.1.2" xref="S4.SS2.6.p2.2.m1.2.3.3.2.1.2.cmml"><mi id="S4.SS2.6.p2.2.m1.2.3.3.2.1.2.2" xref="S4.SS2.6.p2.2.m1.2.3.3.2.1.2.2.cmml">arg</mi><mo id="S4.SS2.6.p2.2.m1.2.3.3.2.1.2.1" lspace="0.170em" xref="S4.SS2.6.p2.2.m1.2.3.3.2.1.2.1.cmml">⁢</mo><mi id="S4.SS2.6.p2.2.m1.2.3.3.2.1.2.3" xref="S4.SS2.6.p2.2.m1.2.3.3.2.1.2.3.cmml">max</mi></mrow><mrow id="S4.SS2.6.p2.2.m1.1.1.1" xref="S4.SS2.6.p2.2.m1.1.1.1.cmml"><mi id="S4.SS2.6.p2.2.m1.1.1.1.3" xref="S4.SS2.6.p2.2.m1.1.1.1.3.cmml">π</mi><mo id="S4.SS2.6.p2.2.m1.1.1.1.2" xref="S4.SS2.6.p2.2.m1.1.1.1.2.cmml">∈</mo><msubsup id="S4.SS2.6.p2.2.m1.1.1.1.4" xref="S4.SS2.6.p2.2.m1.1.1.1.4.cmml"><mi id="S4.SS2.6.p2.2.m1.1.1.1.4.2.2" mathvariant="normal" xref="S4.SS2.6.p2.2.m1.1.1.1.4.2.2.cmml">Π</mi><mi id="S4.SS2.6.p2.2.m1.1.1.1.4.2.3" xref="S4.SS2.6.p2.2.m1.1.1.1.4.2.3.cmml">N</mi><mrow id="S4.SS2.6.p2.2.m1.1.1.1.1.1.3" xref="S4.SS2.6.p2.2.m1.1.1.1.4.cmml"><mo id="S4.SS2.6.p2.2.m1.1.1.1.1.1.3.1" stretchy="false" xref="S4.SS2.6.p2.2.m1.1.1.1.4.cmml">(</mo><mn id="S4.SS2.6.p2.2.m1.1.1.1.1.1.1" xref="S4.SS2.6.p2.2.m1.1.1.1.1.1.1.cmml">2</mn><mo id="S4.SS2.6.p2.2.m1.1.1.1.1.1.3.2" stretchy="false" xref="S4.SS2.6.p2.2.m1.1.1.1.4.cmml">)</mo></mrow></msubsup></mrow></msub><mo id="S4.SS2.6.p2.2.m1.2.3.3.2a" xref="S4.SS2.6.p2.2.m1.2.3.3.2.cmml">⁡</mo><mrow id="S4.SS2.6.p2.2.m1.2.3.3.2.2" xref="S4.SS2.6.p2.2.m1.2.3.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.6.p2.2.m1.2.3.3.2.2.2" xref="S4.SS2.6.p2.2.m1.2.3.3.2.2.2.cmml">𝒮</mi><mo id="S4.SS2.6.p2.2.m1.2.3.3.2.2.1" xref="S4.SS2.6.p2.2.m1.2.3.3.2.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.6.p2.2.m1.2.3.3.2.2.3" xref="S4.SS2.6.p2.2.m1.2.3.3.2.2.3.cmml">𝒲</mi></mrow></mrow><mo id="S4.SS2.6.p2.2.m1.2.3.3.1" xref="S4.SS2.6.p2.2.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.SS2.6.p2.2.m1.2.3.3.3.2" xref="S4.SS2.6.p2.2.m1.2.3.3.cmml"><mo id="S4.SS2.6.p2.2.m1.2.3.3.3.2.1" stretchy="false" xref="S4.SS2.6.p2.2.m1.2.3.3.cmml">(</mo><mi id="S4.SS2.6.p2.2.m1.2.2" xref="S4.SS2.6.p2.2.m1.2.2.cmml">π</mi><mo id="S4.SS2.6.p2.2.m1.2.3.3.3.2.2" stretchy="false" xref="S4.SS2.6.p2.2.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.2.m1.2b"><apply id="S4.SS2.6.p2.2.m1.2.3.cmml" xref="S4.SS2.6.p2.2.m1.2.3"><csymbol cd="latexml" id="S4.SS2.6.p2.2.m1.2.3.1.cmml" xref="S4.SS2.6.p2.2.m1.2.3.1">assign</csymbol><apply id="S4.SS2.6.p2.2.m1.2.3.2.cmml" xref="S4.SS2.6.p2.2.m1.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.6.p2.2.m1.2.3.2.1.cmml" xref="S4.SS2.6.p2.2.m1.2.3.2">subscript</csymbol><apply id="S4.SS2.6.p2.2.m1.2.3.2.2.cmml" xref="S4.SS2.6.p2.2.m1.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.6.p2.2.m1.2.3.2.2.1.cmml" xref="S4.SS2.6.p2.2.m1.2.3.2">superscript</csymbol><ci id="S4.SS2.6.p2.2.m1.2.3.2.2.2.cmml" xref="S4.SS2.6.p2.2.m1.2.3.2.2.2">𝜋</ci><times id="S4.SS2.6.p2.2.m1.2.3.2.2.3.cmml" xref="S4.SS2.6.p2.2.m1.2.3.2.2.3"></times></apply><cn id="S4.SS2.6.p2.2.m1.2.3.2.3.cmml" type="integer" xref="S4.SS2.6.p2.2.m1.2.3.2.3">2</cn></apply><apply id="S4.SS2.6.p2.2.m1.2.3.3.cmml" xref="S4.SS2.6.p2.2.m1.2.3.3"><times id="S4.SS2.6.p2.2.m1.2.3.3.1.cmml" xref="S4.SS2.6.p2.2.m1.2.3.3.1"></times><apply id="S4.SS2.6.p2.2.m1.2.3.3.2.cmml" xref="S4.SS2.6.p2.2.m1.2.3.3.2"><apply id="S4.SS2.6.p2.2.m1.2.3.3.2.1.cmml" xref="S4.SS2.6.p2.2.m1.2.3.3.2.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.2.m1.2.3.3.2.1.1.cmml" xref="S4.SS2.6.p2.2.m1.2.3.3.2.1">subscript</csymbol><apply id="S4.SS2.6.p2.2.m1.2.3.3.2.1.2.cmml" xref="S4.SS2.6.p2.2.m1.2.3.3.2.1.2"><times id="S4.SS2.6.p2.2.m1.2.3.3.2.1.2.1.cmml" xref="S4.SS2.6.p2.2.m1.2.3.3.2.1.2.1"></times><ci id="S4.SS2.6.p2.2.m1.2.3.3.2.1.2.2.cmml" xref="S4.SS2.6.p2.2.m1.2.3.3.2.1.2.2">arg</ci><ci id="S4.SS2.6.p2.2.m1.2.3.3.2.1.2.3.cmml" xref="S4.SS2.6.p2.2.m1.2.3.3.2.1.2.3">max</ci></apply><apply id="S4.SS2.6.p2.2.m1.1.1.1.cmml" xref="S4.SS2.6.p2.2.m1.1.1.1"><in id="S4.SS2.6.p2.2.m1.1.1.1.2.cmml" xref="S4.SS2.6.p2.2.m1.1.1.1.2"></in><ci id="S4.SS2.6.p2.2.m1.1.1.1.3.cmml" xref="S4.SS2.6.p2.2.m1.1.1.1.3">𝜋</ci><apply id="S4.SS2.6.p2.2.m1.1.1.1.4.cmml" xref="S4.SS2.6.p2.2.m1.1.1.1.4"><csymbol cd="ambiguous" id="S4.SS2.6.p2.2.m1.1.1.1.4.1.cmml" xref="S4.SS2.6.p2.2.m1.1.1.1.4">superscript</csymbol><apply id="S4.SS2.6.p2.2.m1.1.1.1.4.2.cmml" xref="S4.SS2.6.p2.2.m1.1.1.1.4"><csymbol cd="ambiguous" id="S4.SS2.6.p2.2.m1.1.1.1.4.2.1.cmml" xref="S4.SS2.6.p2.2.m1.1.1.1.4">subscript</csymbol><ci id="S4.SS2.6.p2.2.m1.1.1.1.4.2.2.cmml" xref="S4.SS2.6.p2.2.m1.1.1.1.4.2.2">Π</ci><ci id="S4.SS2.6.p2.2.m1.1.1.1.4.2.3.cmml" xref="S4.SS2.6.p2.2.m1.1.1.1.4.2.3">𝑁</ci></apply><cn id="S4.SS2.6.p2.2.m1.1.1.1.1.1.1.cmml" type="integer" xref="S4.SS2.6.p2.2.m1.1.1.1.1.1.1">2</cn></apply></apply></apply><apply id="S4.SS2.6.p2.2.m1.2.3.3.2.2.cmml" xref="S4.SS2.6.p2.2.m1.2.3.3.2.2"><times id="S4.SS2.6.p2.2.m1.2.3.3.2.2.1.cmml" xref="S4.SS2.6.p2.2.m1.2.3.3.2.2.1"></times><ci id="S4.SS2.6.p2.2.m1.2.3.3.2.2.2.cmml" xref="S4.SS2.6.p2.2.m1.2.3.3.2.2.2">𝒮</ci><ci id="S4.SS2.6.p2.2.m1.2.3.3.2.2.3.cmml" xref="S4.SS2.6.p2.2.m1.2.3.3.2.2.3">𝒲</ci></apply></apply><ci id="S4.SS2.6.p2.2.m1.2.2.cmml" xref="S4.SS2.6.p2.2.m1.2.2">𝜋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.2.m1.2c">\pi^{*}_{2}:=\operatorname*{arg\,max}_{\pi\in\Pi_{N}^{(2)}}\mathcal{SW}(\pi)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.2.m1.2d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT := start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ roman_Π start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT caligraphic_S caligraphic_W ( italic_π )</annotation></semantics></math>. While the exact social welfare of <math alttext="\pi^{*}_{2}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.3.m2.1"><semantics id="S4.SS2.6.p2.3.m2.1a"><msubsup id="S4.SS2.6.p2.3.m2.1.1" xref="S4.SS2.6.p2.3.m2.1.1.cmml"><mi id="S4.SS2.6.p2.3.m2.1.1.2.2" xref="S4.SS2.6.p2.3.m2.1.1.2.2.cmml">π</mi><mn id="S4.SS2.6.p2.3.m2.1.1.3" xref="S4.SS2.6.p2.3.m2.1.1.3.cmml">2</mn><mo id="S4.SS2.6.p2.3.m2.1.1.2.3" xref="S4.SS2.6.p2.3.m2.1.1.2.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.3.m2.1b"><apply id="S4.SS2.6.p2.3.m2.1.1.cmml" xref="S4.SS2.6.p2.3.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.3.m2.1.1.1.cmml" xref="S4.SS2.6.p2.3.m2.1.1">subscript</csymbol><apply id="S4.SS2.6.p2.3.m2.1.1.2.cmml" xref="S4.SS2.6.p2.3.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.3.m2.1.1.2.1.cmml" xref="S4.SS2.6.p2.3.m2.1.1">superscript</csymbol><ci id="S4.SS2.6.p2.3.m2.1.1.2.2.cmml" xref="S4.SS2.6.p2.3.m2.1.1.2.2">𝜋</ci><times id="S4.SS2.6.p2.3.m2.1.1.2.3.cmml" xref="S4.SS2.6.p2.3.m2.1.1.2.3"></times></apply><cn id="S4.SS2.6.p2.3.m2.1.1.3.cmml" type="integer" xref="S4.SS2.6.p2.3.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.3.m2.1c">\pi^{*}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.3.m2.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is not computable, we can establish a lower bound for it. Consider the random partition where each coalition in <math alttext="\pi^{*}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.4.m3.1"><semantics id="S4.SS2.6.p2.4.m3.1a"><msup id="S4.SS2.6.p2.4.m3.1.1" xref="S4.SS2.6.p2.4.m3.1.1.cmml"><mi id="S4.SS2.6.p2.4.m3.1.1.2" xref="S4.SS2.6.p2.4.m3.1.1.2.cmml">π</mi><mo id="S4.SS2.6.p2.4.m3.1.1.3" xref="S4.SS2.6.p2.4.m3.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.4.m3.1b"><apply id="S4.SS2.6.p2.4.m3.1.1.cmml" xref="S4.SS2.6.p2.4.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.4.m3.1.1.1.cmml" xref="S4.SS2.6.p2.4.m3.1.1">superscript</csymbol><ci id="S4.SS2.6.p2.4.m3.1.1.2.cmml" xref="S4.SS2.6.p2.4.m3.1.1.2">𝜋</ci><times id="S4.SS2.6.p2.4.m3.1.1.3.cmml" xref="S4.SS2.6.p2.4.m3.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.4.m3.1c">\pi^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.4.m3.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is assigned to one of two new coalitions, <math alttext="A" class="ltx_Math" display="inline" id="S4.SS2.6.p2.5.m4.1"><semantics id="S4.SS2.6.p2.5.m4.1a"><mi id="S4.SS2.6.p2.5.m4.1.1" xref="S4.SS2.6.p2.5.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.5.m4.1b"><ci id="S4.SS2.6.p2.5.m4.1.1.cmml" xref="S4.SS2.6.p2.5.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.5.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.5.m4.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S4.SS2.6.p2.6.m5.1"><semantics id="S4.SS2.6.p2.6.m5.1a"><mi id="S4.SS2.6.p2.6.m5.1.1" xref="S4.SS2.6.p2.6.m5.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.6.m5.1b"><ci id="S4.SS2.6.p2.6.m5.1.1.cmml" xref="S4.SS2.6.p2.6.m5.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.6.m5.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.6.m5.1d">italic_B</annotation></semantics></math>, uniformly at random. Then the expected social welfare of the random partition <math alttext="\pi^{(2)}:=\{A,B\}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.7.m6.3"><semantics id="S4.SS2.6.p2.7.m6.3a"><mrow id="S4.SS2.6.p2.7.m6.3.4" xref="S4.SS2.6.p2.7.m6.3.4.cmml"><msup id="S4.SS2.6.p2.7.m6.3.4.2" xref="S4.SS2.6.p2.7.m6.3.4.2.cmml"><mi id="S4.SS2.6.p2.7.m6.3.4.2.2" xref="S4.SS2.6.p2.7.m6.3.4.2.2.cmml">π</mi><mrow id="S4.SS2.6.p2.7.m6.1.1.1.3" xref="S4.SS2.6.p2.7.m6.3.4.2.cmml"><mo id="S4.SS2.6.p2.7.m6.1.1.1.3.1" stretchy="false" xref="S4.SS2.6.p2.7.m6.3.4.2.cmml">(</mo><mn id="S4.SS2.6.p2.7.m6.1.1.1.1" xref="S4.SS2.6.p2.7.m6.1.1.1.1.cmml">2</mn><mo id="S4.SS2.6.p2.7.m6.1.1.1.3.2" stretchy="false" xref="S4.SS2.6.p2.7.m6.3.4.2.cmml">)</mo></mrow></msup><mo id="S4.SS2.6.p2.7.m6.3.4.1" lspace="0.278em" rspace="0.278em" xref="S4.SS2.6.p2.7.m6.3.4.1.cmml">:=</mo><mrow id="S4.SS2.6.p2.7.m6.3.4.3.2" xref="S4.SS2.6.p2.7.m6.3.4.3.1.cmml"><mo id="S4.SS2.6.p2.7.m6.3.4.3.2.1" stretchy="false" xref="S4.SS2.6.p2.7.m6.3.4.3.1.cmml">{</mo><mi id="S4.SS2.6.p2.7.m6.2.2" xref="S4.SS2.6.p2.7.m6.2.2.cmml">A</mi><mo id="S4.SS2.6.p2.7.m6.3.4.3.2.2" xref="S4.SS2.6.p2.7.m6.3.4.3.1.cmml">,</mo><mi id="S4.SS2.6.p2.7.m6.3.3" xref="S4.SS2.6.p2.7.m6.3.3.cmml">B</mi><mo id="S4.SS2.6.p2.7.m6.3.4.3.2.3" stretchy="false" xref="S4.SS2.6.p2.7.m6.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.7.m6.3b"><apply id="S4.SS2.6.p2.7.m6.3.4.cmml" xref="S4.SS2.6.p2.7.m6.3.4"><csymbol cd="latexml" id="S4.SS2.6.p2.7.m6.3.4.1.cmml" xref="S4.SS2.6.p2.7.m6.3.4.1">assign</csymbol><apply id="S4.SS2.6.p2.7.m6.3.4.2.cmml" xref="S4.SS2.6.p2.7.m6.3.4.2"><csymbol cd="ambiguous" id="S4.SS2.6.p2.7.m6.3.4.2.1.cmml" xref="S4.SS2.6.p2.7.m6.3.4.2">superscript</csymbol><ci id="S4.SS2.6.p2.7.m6.3.4.2.2.cmml" xref="S4.SS2.6.p2.7.m6.3.4.2.2">𝜋</ci><cn id="S4.SS2.6.p2.7.m6.1.1.1.1.cmml" type="integer" xref="S4.SS2.6.p2.7.m6.1.1.1.1">2</cn></apply><set id="S4.SS2.6.p2.7.m6.3.4.3.1.cmml" xref="S4.SS2.6.p2.7.m6.3.4.3.2"><ci id="S4.SS2.6.p2.7.m6.2.2.cmml" xref="S4.SS2.6.p2.7.m6.2.2">𝐴</ci><ci id="S4.SS2.6.p2.7.m6.3.3.cmml" xref="S4.SS2.6.p2.7.m6.3.3">𝐵</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.7.m6.3c">\pi^{(2)}:=\{A,B\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.7.m6.3d">italic_π start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT := { italic_A , italic_B }</annotation></semantics></math> is a lower bound for the social welfare of <math alttext="\pi^{*}_{2}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.8.m7.1"><semantics id="S4.SS2.6.p2.8.m7.1a"><msubsup id="S4.SS2.6.p2.8.m7.1.1" xref="S4.SS2.6.p2.8.m7.1.1.cmml"><mi id="S4.SS2.6.p2.8.m7.1.1.2.2" xref="S4.SS2.6.p2.8.m7.1.1.2.2.cmml">π</mi><mn id="S4.SS2.6.p2.8.m7.1.1.3" xref="S4.SS2.6.p2.8.m7.1.1.3.cmml">2</mn><mo id="S4.SS2.6.p2.8.m7.1.1.2.3" xref="S4.SS2.6.p2.8.m7.1.1.2.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.8.m7.1b"><apply id="S4.SS2.6.p2.8.m7.1.1.cmml" xref="S4.SS2.6.p2.8.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.8.m7.1.1.1.cmml" xref="S4.SS2.6.p2.8.m7.1.1">subscript</csymbol><apply id="S4.SS2.6.p2.8.m7.1.1.2.cmml" xref="S4.SS2.6.p2.8.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.8.m7.1.1.2.1.cmml" xref="S4.SS2.6.p2.8.m7.1.1">superscript</csymbol><ci id="S4.SS2.6.p2.8.m7.1.1.2.2.cmml" xref="S4.SS2.6.p2.8.m7.1.1.2.2">𝜋</ci><times id="S4.SS2.6.p2.8.m7.1.1.2.3.cmml" xref="S4.SS2.6.p2.8.m7.1.1.2.3"></times></apply><cn id="S4.SS2.6.p2.8.m7.1.1.3.cmml" type="integer" xref="S4.SS2.6.p2.8.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.8.m7.1c">\pi^{*}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.8.m7.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. Every pair of agents in a common coalition in <math alttext="\pi^{*}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.9.m8.1"><semantics id="S4.SS2.6.p2.9.m8.1a"><msup id="S4.SS2.6.p2.9.m8.1.1" xref="S4.SS2.6.p2.9.m8.1.1.cmml"><mi id="S4.SS2.6.p2.9.m8.1.1.2" xref="S4.SS2.6.p2.9.m8.1.1.2.cmml">π</mi><mo id="S4.SS2.6.p2.9.m8.1.1.3" xref="S4.SS2.6.p2.9.m8.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.9.m8.1b"><apply id="S4.SS2.6.p2.9.m8.1.1.cmml" xref="S4.SS2.6.p2.9.m8.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.9.m8.1.1.1.cmml" xref="S4.SS2.6.p2.9.m8.1.1">superscript</csymbol><ci id="S4.SS2.6.p2.9.m8.1.1.2.cmml" xref="S4.SS2.6.p2.9.m8.1.1.2">𝜋</ci><times id="S4.SS2.6.p2.9.m8.1.1.3.cmml" xref="S4.SS2.6.p2.9.m8.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.9.m8.1c">\pi^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.9.m8.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> will remain in a common coalition in <math alttext="\pi^{(2)}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.10.m9.1"><semantics id="S4.SS2.6.p2.10.m9.1a"><msup id="S4.SS2.6.p2.10.m9.1.2" xref="S4.SS2.6.p2.10.m9.1.2.cmml"><mi id="S4.SS2.6.p2.10.m9.1.2.2" xref="S4.SS2.6.p2.10.m9.1.2.2.cmml">π</mi><mrow id="S4.SS2.6.p2.10.m9.1.1.1.3" xref="S4.SS2.6.p2.10.m9.1.2.cmml"><mo id="S4.SS2.6.p2.10.m9.1.1.1.3.1" stretchy="false" xref="S4.SS2.6.p2.10.m9.1.2.cmml">(</mo><mn id="S4.SS2.6.p2.10.m9.1.1.1.1" xref="S4.SS2.6.p2.10.m9.1.1.1.1.cmml">2</mn><mo id="S4.SS2.6.p2.10.m9.1.1.1.3.2" stretchy="false" xref="S4.SS2.6.p2.10.m9.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.10.m9.1b"><apply id="S4.SS2.6.p2.10.m9.1.2.cmml" xref="S4.SS2.6.p2.10.m9.1.2"><csymbol cd="ambiguous" id="S4.SS2.6.p2.10.m9.1.2.1.cmml" xref="S4.SS2.6.p2.10.m9.1.2">superscript</csymbol><ci id="S4.SS2.6.p2.10.m9.1.2.2.cmml" xref="S4.SS2.6.p2.10.m9.1.2.2">𝜋</ci><cn id="S4.SS2.6.p2.10.m9.1.1.1.1.cmml" type="integer" xref="S4.SS2.6.p2.10.m9.1.1.1.1">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.10.m9.1c">\pi^{(2)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.10.m9.1d">italic_π start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT</annotation></semantics></math>. Moreover, agents that have been in different coalitions in <math alttext="\pi^{*}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.11.m10.1"><semantics id="S4.SS2.6.p2.11.m10.1a"><msup id="S4.SS2.6.p2.11.m10.1.1" xref="S4.SS2.6.p2.11.m10.1.1.cmml"><mi id="S4.SS2.6.p2.11.m10.1.1.2" xref="S4.SS2.6.p2.11.m10.1.1.2.cmml">π</mi><mo id="S4.SS2.6.p2.11.m10.1.1.3" xref="S4.SS2.6.p2.11.m10.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.11.m10.1b"><apply id="S4.SS2.6.p2.11.m10.1.1.cmml" xref="S4.SS2.6.p2.11.m10.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.11.m10.1.1.1.cmml" xref="S4.SS2.6.p2.11.m10.1.1">superscript</csymbol><ci id="S4.SS2.6.p2.11.m10.1.1.2.cmml" xref="S4.SS2.6.p2.11.m10.1.1.2">𝜋</ci><times id="S4.SS2.6.p2.11.m10.1.1.3.cmml" xref="S4.SS2.6.p2.11.m10.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.11.m10.1c">\pi^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.11.m10.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> are in the same coalition in <math alttext="\pi^{(2)}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.12.m11.1"><semantics id="S4.SS2.6.p2.12.m11.1a"><msup id="S4.SS2.6.p2.12.m11.1.2" xref="S4.SS2.6.p2.12.m11.1.2.cmml"><mi id="S4.SS2.6.p2.12.m11.1.2.2" xref="S4.SS2.6.p2.12.m11.1.2.2.cmml">π</mi><mrow id="S4.SS2.6.p2.12.m11.1.1.1.3" xref="S4.SS2.6.p2.12.m11.1.2.cmml"><mo id="S4.SS2.6.p2.12.m11.1.1.1.3.1" stretchy="false" xref="S4.SS2.6.p2.12.m11.1.2.cmml">(</mo><mn id="S4.SS2.6.p2.12.m11.1.1.1.1" xref="S4.SS2.6.p2.12.m11.1.1.1.1.cmml">2</mn><mo id="S4.SS2.6.p2.12.m11.1.1.1.3.2" stretchy="false" xref="S4.SS2.6.p2.12.m11.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.12.m11.1b"><apply id="S4.SS2.6.p2.12.m11.1.2.cmml" xref="S4.SS2.6.p2.12.m11.1.2"><csymbol cd="ambiguous" id="S4.SS2.6.p2.12.m11.1.2.1.cmml" xref="S4.SS2.6.p2.12.m11.1.2">superscript</csymbol><ci id="S4.SS2.6.p2.12.m11.1.2.2.cmml" xref="S4.SS2.6.p2.12.m11.1.2.2">𝜋</ci><cn id="S4.SS2.6.p2.12.m11.1.1.1.1.cmml" type="integer" xref="S4.SS2.6.p2.12.m11.1.1.1.1">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.12.m11.1c">\pi^{(2)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.12.m11.1d">italic_π start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT</annotation></semantics></math> with probability <math alttext="\frac{1}{2}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.13.m12.1"><semantics id="S4.SS2.6.p2.13.m12.1a"><mfrac id="S4.SS2.6.p2.13.m12.1.1" xref="S4.SS2.6.p2.13.m12.1.1.cmml"><mn id="S4.SS2.6.p2.13.m12.1.1.2" xref="S4.SS2.6.p2.13.m12.1.1.2.cmml">1</mn><mn id="S4.SS2.6.p2.13.m12.1.1.3" xref="S4.SS2.6.p2.13.m12.1.1.3.cmml">2</mn></mfrac><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.13.m12.1b"><apply id="S4.SS2.6.p2.13.m12.1.1.cmml" xref="S4.SS2.6.p2.13.m12.1.1"><divide id="S4.SS2.6.p2.13.m12.1.1.1.cmml" xref="S4.SS2.6.p2.13.m12.1.1"></divide><cn id="S4.SS2.6.p2.13.m12.1.1.2.cmml" type="integer" xref="S4.SS2.6.p2.13.m12.1.1.2">1</cn><cn id="S4.SS2.6.p2.13.m12.1.1.3.cmml" type="integer" xref="S4.SS2.6.p2.13.m12.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.13.m12.1c">\frac{1}{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.13.m12.1d">divide start_ARG 1 end_ARG start_ARG 2 end_ARG</annotation></semantics></math>. Hence, <math alttext="\mathbb{E}\left[\mathcal{SW}\left(\pi^{(2)}\right)\right]=W+\frac{1}{2}{A}\leq% \pi^{*}_{2}" class="ltx_Math" display="inline" id="S4.SS2.6.p2.14.m13.2"><semantics id="S4.SS2.6.p2.14.m13.2a"><mrow id="S4.SS2.6.p2.14.m13.2.2" xref="S4.SS2.6.p2.14.m13.2.2.cmml"><mrow id="S4.SS2.6.p2.14.m13.2.2.1" xref="S4.SS2.6.p2.14.m13.2.2.1.cmml"><mi id="S4.SS2.6.p2.14.m13.2.2.1.3" xref="S4.SS2.6.p2.14.m13.2.2.1.3.cmml">𝔼</mi><mo id="S4.SS2.6.p2.14.m13.2.2.1.2" xref="S4.SS2.6.p2.14.m13.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS2.6.p2.14.m13.2.2.1.1.1" xref="S4.SS2.6.p2.14.m13.2.2.1.1.2.cmml"><mo id="S4.SS2.6.p2.14.m13.2.2.1.1.1.2" xref="S4.SS2.6.p2.14.m13.2.2.1.1.2.1.cmml">[</mo><mrow id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.3" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.3.cmml">𝒮</mi><mo id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.2" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.4" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.4.cmml">𝒲</mi><mo id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.2a" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.2" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.1" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.1.2" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.1.2.cmml">π</mi><mrow id="S4.SS2.6.p2.14.m13.1.1.1.3" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.6.p2.14.m13.1.1.1.3.1" stretchy="false" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.1.cmml">(</mo><mn id="S4.SS2.6.p2.14.m13.1.1.1.1" xref="S4.SS2.6.p2.14.m13.1.1.1.1.cmml">2</mn><mo id="S4.SS2.6.p2.14.m13.1.1.1.3.2" stretchy="false" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.3" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.6.p2.14.m13.2.2.1.1.1.3" xref="S4.SS2.6.p2.14.m13.2.2.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S4.SS2.6.p2.14.m13.2.2.3" xref="S4.SS2.6.p2.14.m13.2.2.3.cmml">=</mo><mrow id="S4.SS2.6.p2.14.m13.2.2.4" xref="S4.SS2.6.p2.14.m13.2.2.4.cmml"><mi id="S4.SS2.6.p2.14.m13.2.2.4.2" xref="S4.SS2.6.p2.14.m13.2.2.4.2.cmml">W</mi><mo id="S4.SS2.6.p2.14.m13.2.2.4.1" xref="S4.SS2.6.p2.14.m13.2.2.4.1.cmml">+</mo><mrow id="S4.SS2.6.p2.14.m13.2.2.4.3" xref="S4.SS2.6.p2.14.m13.2.2.4.3.cmml"><mfrac id="S4.SS2.6.p2.14.m13.2.2.4.3.2" xref="S4.SS2.6.p2.14.m13.2.2.4.3.2.cmml"><mn id="S4.SS2.6.p2.14.m13.2.2.4.3.2.2" xref="S4.SS2.6.p2.14.m13.2.2.4.3.2.2.cmml">1</mn><mn id="S4.SS2.6.p2.14.m13.2.2.4.3.2.3" xref="S4.SS2.6.p2.14.m13.2.2.4.3.2.3.cmml">2</mn></mfrac><mo id="S4.SS2.6.p2.14.m13.2.2.4.3.1" xref="S4.SS2.6.p2.14.m13.2.2.4.3.1.cmml">⁢</mo><mi id="S4.SS2.6.p2.14.m13.2.2.4.3.3" xref="S4.SS2.6.p2.14.m13.2.2.4.3.3.cmml">A</mi></mrow></mrow><mo id="S4.SS2.6.p2.14.m13.2.2.5" xref="S4.SS2.6.p2.14.m13.2.2.5.cmml">≤</mo><msubsup id="S4.SS2.6.p2.14.m13.2.2.6" xref="S4.SS2.6.p2.14.m13.2.2.6.cmml"><mi id="S4.SS2.6.p2.14.m13.2.2.6.2.2" xref="S4.SS2.6.p2.14.m13.2.2.6.2.2.cmml">π</mi><mn id="S4.SS2.6.p2.14.m13.2.2.6.3" xref="S4.SS2.6.p2.14.m13.2.2.6.3.cmml">2</mn><mo id="S4.SS2.6.p2.14.m13.2.2.6.2.3" xref="S4.SS2.6.p2.14.m13.2.2.6.2.3.cmml">∗</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.6.p2.14.m13.2b"><apply id="S4.SS2.6.p2.14.m13.2.2.cmml" xref="S4.SS2.6.p2.14.m13.2.2"><and id="S4.SS2.6.p2.14.m13.2.2a.cmml" xref="S4.SS2.6.p2.14.m13.2.2"></and><apply id="S4.SS2.6.p2.14.m13.2.2b.cmml" xref="S4.SS2.6.p2.14.m13.2.2"><eq id="S4.SS2.6.p2.14.m13.2.2.3.cmml" xref="S4.SS2.6.p2.14.m13.2.2.3"></eq><apply id="S4.SS2.6.p2.14.m13.2.2.1.cmml" xref="S4.SS2.6.p2.14.m13.2.2.1"><times id="S4.SS2.6.p2.14.m13.2.2.1.2.cmml" xref="S4.SS2.6.p2.14.m13.2.2.1.2"></times><ci id="S4.SS2.6.p2.14.m13.2.2.1.3.cmml" xref="S4.SS2.6.p2.14.m13.2.2.1.3">𝔼</ci><apply id="S4.SS2.6.p2.14.m13.2.2.1.1.2.cmml" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1"><csymbol cd="latexml" id="S4.SS2.6.p2.14.m13.2.2.1.1.2.1.cmml" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.2">delimited-[]</csymbol><apply id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.cmml" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1"><times id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.2.cmml" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.2"></times><ci id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.3.cmml" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.3">𝒮</ci><ci id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.4.cmml" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.4">𝒲</ci><apply id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.1.cmml" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1">superscript</csymbol><ci id="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.6.p2.14.m13.2.2.1.1.1.1.1.1.1.2">𝜋</ci><cn id="S4.SS2.6.p2.14.m13.1.1.1.1.cmml" type="integer" xref="S4.SS2.6.p2.14.m13.1.1.1.1">2</cn></apply></apply></apply></apply><apply id="S4.SS2.6.p2.14.m13.2.2.4.cmml" xref="S4.SS2.6.p2.14.m13.2.2.4"><plus id="S4.SS2.6.p2.14.m13.2.2.4.1.cmml" xref="S4.SS2.6.p2.14.m13.2.2.4.1"></plus><ci id="S4.SS2.6.p2.14.m13.2.2.4.2.cmml" xref="S4.SS2.6.p2.14.m13.2.2.4.2">𝑊</ci><apply id="S4.SS2.6.p2.14.m13.2.2.4.3.cmml" xref="S4.SS2.6.p2.14.m13.2.2.4.3"><times id="S4.SS2.6.p2.14.m13.2.2.4.3.1.cmml" xref="S4.SS2.6.p2.14.m13.2.2.4.3.1"></times><apply id="S4.SS2.6.p2.14.m13.2.2.4.3.2.cmml" xref="S4.SS2.6.p2.14.m13.2.2.4.3.2"><divide id="S4.SS2.6.p2.14.m13.2.2.4.3.2.1.cmml" xref="S4.SS2.6.p2.14.m13.2.2.4.3.2"></divide><cn id="S4.SS2.6.p2.14.m13.2.2.4.3.2.2.cmml" type="integer" xref="S4.SS2.6.p2.14.m13.2.2.4.3.2.2">1</cn><cn id="S4.SS2.6.p2.14.m13.2.2.4.3.2.3.cmml" type="integer" xref="S4.SS2.6.p2.14.m13.2.2.4.3.2.3">2</cn></apply><ci id="S4.SS2.6.p2.14.m13.2.2.4.3.3.cmml" xref="S4.SS2.6.p2.14.m13.2.2.4.3.3">𝐴</ci></apply></apply></apply><apply id="S4.SS2.6.p2.14.m13.2.2c.cmml" xref="S4.SS2.6.p2.14.m13.2.2"><leq id="S4.SS2.6.p2.14.m13.2.2.5.cmml" xref="S4.SS2.6.p2.14.m13.2.2.5"></leq><share href="https://arxiv.org/html/2503.06017v1#S4.SS2.6.p2.14.m13.2.2.4.cmml" id="S4.SS2.6.p2.14.m13.2.2d.cmml" xref="S4.SS2.6.p2.14.m13.2.2"></share><apply id="S4.SS2.6.p2.14.m13.2.2.6.cmml" xref="S4.SS2.6.p2.14.m13.2.2.6"><csymbol cd="ambiguous" id="S4.SS2.6.p2.14.m13.2.2.6.1.cmml" xref="S4.SS2.6.p2.14.m13.2.2.6">subscript</csymbol><apply id="S4.SS2.6.p2.14.m13.2.2.6.2.cmml" xref="S4.SS2.6.p2.14.m13.2.2.6"><csymbol cd="ambiguous" id="S4.SS2.6.p2.14.m13.2.2.6.2.1.cmml" xref="S4.SS2.6.p2.14.m13.2.2.6">superscript</csymbol><ci id="S4.SS2.6.p2.14.m13.2.2.6.2.2.cmml" xref="S4.SS2.6.p2.14.m13.2.2.6.2.2">𝜋</ci><times id="S4.SS2.6.p2.14.m13.2.2.6.2.3.cmml" xref="S4.SS2.6.p2.14.m13.2.2.6.2.3"></times></apply><cn id="S4.SS2.6.p2.14.m13.2.2.6.3.cmml" type="integer" xref="S4.SS2.6.p2.14.m13.2.2.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.6.p2.14.m13.2c">\mathbb{E}\left[\mathcal{SW}\left(\pi^{(2)}\right)\right]=W+\frac{1}{2}{A}\leq% \pi^{*}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.6.p2.14.m13.2d">blackboard_E [ caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ) ] = italic_W + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_A ≤ italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.7.p3"> <p class="ltx_p" id="S4.SS2.7.p3.2">We conclude that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx4"> <tbody id="S4.Ex19"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{SW}(\pi^{*})" class="ltx_Math" display="inline" id="S4.Ex19.m1.1"><semantics id="S4.Ex19.m1.1a"><mrow id="S4.Ex19.m1.1.1" xref="S4.Ex19.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex19.m1.1.1.3" xref="S4.Ex19.m1.1.1.3.cmml">𝒮</mi><mo id="S4.Ex19.m1.1.1.2" xref="S4.Ex19.m1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex19.m1.1.1.4" xref="S4.Ex19.m1.1.1.4.cmml">𝒲</mi><mo id="S4.Ex19.m1.1.1.2a" xref="S4.Ex19.m1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex19.m1.1.1.1.1" xref="S4.Ex19.m1.1.1.1.1.1.cmml"><mo id="S4.Ex19.m1.1.1.1.1.2" stretchy="false" xref="S4.Ex19.m1.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex19.m1.1.1.1.1.1" xref="S4.Ex19.m1.1.1.1.1.1.cmml"><mi id="S4.Ex19.m1.1.1.1.1.1.2" xref="S4.Ex19.m1.1.1.1.1.1.2.cmml">π</mi><mo id="S4.Ex19.m1.1.1.1.1.1.3" xref="S4.Ex19.m1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.Ex19.m1.1.1.1.1.3" stretchy="false" xref="S4.Ex19.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex19.m1.1b"><apply id="S4.Ex19.m1.1.1.cmml" xref="S4.Ex19.m1.1.1"><times id="S4.Ex19.m1.1.1.2.cmml" xref="S4.Ex19.m1.1.1.2"></times><ci id="S4.Ex19.m1.1.1.3.cmml" xref="S4.Ex19.m1.1.1.3">𝒮</ci><ci id="S4.Ex19.m1.1.1.4.cmml" xref="S4.Ex19.m1.1.1.4">𝒲</ci><apply id="S4.Ex19.m1.1.1.1.1.1.cmml" xref="S4.Ex19.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex19.m1.1.1.1.1.1.1.cmml" xref="S4.Ex19.m1.1.1.1.1">superscript</csymbol><ci id="S4.Ex19.m1.1.1.1.1.1.2.cmml" xref="S4.Ex19.m1.1.1.1.1.1.2">𝜋</ci><times id="S4.Ex19.m1.1.1.1.1.1.3.cmml" xref="S4.Ex19.m1.1.1.1.1.1.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex19.m1.1c">\displaystyle\mathcal{SW}(\pi^{*})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex19.m1.1d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=W\leq W+\mathcal{V}(N,v)=2W+A" class="ltx_Math" display="inline" id="S4.Ex19.m2.2"><semantics id="S4.Ex19.m2.2a"><mrow id="S4.Ex19.m2.2.3" xref="S4.Ex19.m2.2.3.cmml"><mi id="S4.Ex19.m2.2.3.2" xref="S4.Ex19.m2.2.3.2.cmml"></mi><mo id="S4.Ex19.m2.2.3.3" xref="S4.Ex19.m2.2.3.3.cmml">=</mo><mi id="S4.Ex19.m2.2.3.4" xref="S4.Ex19.m2.2.3.4.cmml">W</mi><mo id="S4.Ex19.m2.2.3.5" xref="S4.Ex19.m2.2.3.5.cmml">≤</mo><mrow id="S4.Ex19.m2.2.3.6" xref="S4.Ex19.m2.2.3.6.cmml"><mi id="S4.Ex19.m2.2.3.6.2" xref="S4.Ex19.m2.2.3.6.2.cmml">W</mi><mo id="S4.Ex19.m2.2.3.6.1" xref="S4.Ex19.m2.2.3.6.1.cmml">+</mo><mrow id="S4.Ex19.m2.2.3.6.3" xref="S4.Ex19.m2.2.3.6.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex19.m2.2.3.6.3.2" xref="S4.Ex19.m2.2.3.6.3.2.cmml">𝒱</mi><mo id="S4.Ex19.m2.2.3.6.3.1" xref="S4.Ex19.m2.2.3.6.3.1.cmml">⁢</mo><mrow id="S4.Ex19.m2.2.3.6.3.3.2" xref="S4.Ex19.m2.2.3.6.3.3.1.cmml"><mo id="S4.Ex19.m2.2.3.6.3.3.2.1" stretchy="false" xref="S4.Ex19.m2.2.3.6.3.3.1.cmml">(</mo><mi id="S4.Ex19.m2.1.1" xref="S4.Ex19.m2.1.1.cmml">N</mi><mo id="S4.Ex19.m2.2.3.6.3.3.2.2" xref="S4.Ex19.m2.2.3.6.3.3.1.cmml">,</mo><mi id="S4.Ex19.m2.2.2" xref="S4.Ex19.m2.2.2.cmml">v</mi><mo id="S4.Ex19.m2.2.3.6.3.3.2.3" stretchy="false" xref="S4.Ex19.m2.2.3.6.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex19.m2.2.3.7" xref="S4.Ex19.m2.2.3.7.cmml">=</mo><mrow id="S4.Ex19.m2.2.3.8" xref="S4.Ex19.m2.2.3.8.cmml"><mrow id="S4.Ex19.m2.2.3.8.2" xref="S4.Ex19.m2.2.3.8.2.cmml"><mn id="S4.Ex19.m2.2.3.8.2.2" xref="S4.Ex19.m2.2.3.8.2.2.cmml">2</mn><mo id="S4.Ex19.m2.2.3.8.2.1" xref="S4.Ex19.m2.2.3.8.2.1.cmml">⁢</mo><mi id="S4.Ex19.m2.2.3.8.2.3" xref="S4.Ex19.m2.2.3.8.2.3.cmml">W</mi></mrow><mo id="S4.Ex19.m2.2.3.8.1" xref="S4.Ex19.m2.2.3.8.1.cmml">+</mo><mi id="S4.Ex19.m2.2.3.8.3" xref="S4.Ex19.m2.2.3.8.3.cmml">A</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex19.m2.2b"><apply id="S4.Ex19.m2.2.3.cmml" xref="S4.Ex19.m2.2.3"><and id="S4.Ex19.m2.2.3a.cmml" xref="S4.Ex19.m2.2.3"></and><apply id="S4.Ex19.m2.2.3b.cmml" xref="S4.Ex19.m2.2.3"><eq id="S4.Ex19.m2.2.3.3.cmml" xref="S4.Ex19.m2.2.3.3"></eq><csymbol cd="latexml" id="S4.Ex19.m2.2.3.2.cmml" xref="S4.Ex19.m2.2.3.2">absent</csymbol><ci id="S4.Ex19.m2.2.3.4.cmml" xref="S4.Ex19.m2.2.3.4">𝑊</ci></apply><apply id="S4.Ex19.m2.2.3c.cmml" xref="S4.Ex19.m2.2.3"><leq id="S4.Ex19.m2.2.3.5.cmml" xref="S4.Ex19.m2.2.3.5"></leq><share href="https://arxiv.org/html/2503.06017v1#S4.Ex19.m2.2.3.4.cmml" id="S4.Ex19.m2.2.3d.cmml" xref="S4.Ex19.m2.2.3"></share><apply id="S4.Ex19.m2.2.3.6.cmml" xref="S4.Ex19.m2.2.3.6"><plus id="S4.Ex19.m2.2.3.6.1.cmml" xref="S4.Ex19.m2.2.3.6.1"></plus><ci id="S4.Ex19.m2.2.3.6.2.cmml" xref="S4.Ex19.m2.2.3.6.2">𝑊</ci><apply id="S4.Ex19.m2.2.3.6.3.cmml" xref="S4.Ex19.m2.2.3.6.3"><times id="S4.Ex19.m2.2.3.6.3.1.cmml" xref="S4.Ex19.m2.2.3.6.3.1"></times><ci id="S4.Ex19.m2.2.3.6.3.2.cmml" xref="S4.Ex19.m2.2.3.6.3.2">𝒱</ci><interval closure="open" id="S4.Ex19.m2.2.3.6.3.3.1.cmml" xref="S4.Ex19.m2.2.3.6.3.3.2"><ci id="S4.Ex19.m2.1.1.cmml" xref="S4.Ex19.m2.1.1">𝑁</ci><ci id="S4.Ex19.m2.2.2.cmml" xref="S4.Ex19.m2.2.2">𝑣</ci></interval></apply></apply></apply><apply id="S4.Ex19.m2.2.3e.cmml" xref="S4.Ex19.m2.2.3"><eq id="S4.Ex19.m2.2.3.7.cmml" xref="S4.Ex19.m2.2.3.7"></eq><share href="https://arxiv.org/html/2503.06017v1#S4.Ex19.m2.2.3.6.cmml" id="S4.Ex19.m2.2.3f.cmml" xref="S4.Ex19.m2.2.3"></share><apply id="S4.Ex19.m2.2.3.8.cmml" xref="S4.Ex19.m2.2.3.8"><plus id="S4.Ex19.m2.2.3.8.1.cmml" xref="S4.Ex19.m2.2.3.8.1"></plus><apply id="S4.Ex19.m2.2.3.8.2.cmml" xref="S4.Ex19.m2.2.3.8.2"><times id="S4.Ex19.m2.2.3.8.2.1.cmml" xref="S4.Ex19.m2.2.3.8.2.1"></times><cn id="S4.Ex19.m2.2.3.8.2.2.cmml" type="integer" xref="S4.Ex19.m2.2.3.8.2.2">2</cn><ci id="S4.Ex19.m2.2.3.8.2.3.cmml" xref="S4.Ex19.m2.2.3.8.2.3">𝑊</ci></apply><ci id="S4.Ex19.m2.2.3.8.3.cmml" xref="S4.Ex19.m2.2.3.8.3">𝐴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex19.m2.2c">\displaystyle=W\leq W+\mathcal{V}(N,v)=2W+A</annotation><annotation encoding="application/x-llamapun" id="S4.Ex19.m2.2d">= italic_W ≤ italic_W + caligraphic_V ( italic_N , italic_v ) = 2 italic_W + italic_A</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=2\cdot\mathbb{E}\left[\pi^{(2)}\right]\leq 2\cdot\mathcal{SW}(% \pi^{*}_{2})\text{.}" class="ltx_Math" display="inline" id="S4.Ex20.m1.3"><semantics id="S4.Ex20.m1.3a"><mrow id="S4.Ex20.m1.3.3" xref="S4.Ex20.m1.3.3.cmml"><mi id="S4.Ex20.m1.3.3.4" xref="S4.Ex20.m1.3.3.4.cmml"></mi><mo id="S4.Ex20.m1.3.3.5" xref="S4.Ex20.m1.3.3.5.cmml">=</mo><mrow id="S4.Ex20.m1.2.2.1" xref="S4.Ex20.m1.2.2.1.cmml"><mrow id="S4.Ex20.m1.2.2.1.3" xref="S4.Ex20.m1.2.2.1.3.cmml"><mn id="S4.Ex20.m1.2.2.1.3.2" xref="S4.Ex20.m1.2.2.1.3.2.cmml">2</mn><mo id="S4.Ex20.m1.2.2.1.3.1" lspace="0.222em" rspace="0.222em" xref="S4.Ex20.m1.2.2.1.3.1.cmml">⋅</mo><mi id="S4.Ex20.m1.2.2.1.3.3" xref="S4.Ex20.m1.2.2.1.3.3.cmml">𝔼</mi></mrow><mo id="S4.Ex20.m1.2.2.1.2" xref="S4.Ex20.m1.2.2.1.2.cmml">⁢</mo><mrow id="S4.Ex20.m1.2.2.1.1.1" xref="S4.Ex20.m1.2.2.1.1.2.cmml"><mo id="S4.Ex20.m1.2.2.1.1.1.2" xref="S4.Ex20.m1.2.2.1.1.2.1.cmml">[</mo><msup id="S4.Ex20.m1.2.2.1.1.1.1" xref="S4.Ex20.m1.2.2.1.1.1.1.cmml"><mi id="S4.Ex20.m1.2.2.1.1.1.1.2" xref="S4.Ex20.m1.2.2.1.1.1.1.2.cmml">π</mi><mrow id="S4.Ex20.m1.1.1.1.3" xref="S4.Ex20.m1.2.2.1.1.1.1.cmml"><mo id="S4.Ex20.m1.1.1.1.3.1" stretchy="false" xref="S4.Ex20.m1.2.2.1.1.1.1.cmml">(</mo><mn id="S4.Ex20.m1.1.1.1.1" xref="S4.Ex20.m1.1.1.1.1.cmml">2</mn><mo id="S4.Ex20.m1.1.1.1.3.2" stretchy="false" xref="S4.Ex20.m1.2.2.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S4.Ex20.m1.2.2.1.1.1.3" xref="S4.Ex20.m1.2.2.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S4.Ex20.m1.3.3.6" xref="S4.Ex20.m1.3.3.6.cmml">≤</mo><mrow id="S4.Ex20.m1.3.3.2" xref="S4.Ex20.m1.3.3.2.cmml"><mrow id="S4.Ex20.m1.3.3.2.3" xref="S4.Ex20.m1.3.3.2.3.cmml"><mn id="S4.Ex20.m1.3.3.2.3.2" xref="S4.Ex20.m1.3.3.2.3.2.cmml">2</mn><mo id="S4.Ex20.m1.3.3.2.3.1" lspace="0.222em" rspace="0.222em" xref="S4.Ex20.m1.3.3.2.3.1.cmml">⋅</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex20.m1.3.3.2.3.3" xref="S4.Ex20.m1.3.3.2.3.3.cmml">𝒮</mi></mrow><mo id="S4.Ex20.m1.3.3.2.2" xref="S4.Ex20.m1.3.3.2.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex20.m1.3.3.2.4" xref="S4.Ex20.m1.3.3.2.4.cmml">𝒲</mi><mo id="S4.Ex20.m1.3.3.2.2a" xref="S4.Ex20.m1.3.3.2.2.cmml">⁢</mo><mrow id="S4.Ex20.m1.3.3.2.1.1" xref="S4.Ex20.m1.3.3.2.1.1.1.cmml"><mo id="S4.Ex20.m1.3.3.2.1.1.2" stretchy="false" xref="S4.Ex20.m1.3.3.2.1.1.1.cmml">(</mo><msubsup id="S4.Ex20.m1.3.3.2.1.1.1" xref="S4.Ex20.m1.3.3.2.1.1.1.cmml"><mi id="S4.Ex20.m1.3.3.2.1.1.1.2.2" xref="S4.Ex20.m1.3.3.2.1.1.1.2.2.cmml">π</mi><mn id="S4.Ex20.m1.3.3.2.1.1.1.3" xref="S4.Ex20.m1.3.3.2.1.1.1.3.cmml">2</mn><mo id="S4.Ex20.m1.3.3.2.1.1.1.2.3" xref="S4.Ex20.m1.3.3.2.1.1.1.2.3.cmml">∗</mo></msubsup><mo id="S4.Ex20.m1.3.3.2.1.1.3" stretchy="false" xref="S4.Ex20.m1.3.3.2.1.1.1.cmml">)</mo></mrow><mo id="S4.Ex20.m1.3.3.2.2b" xref="S4.Ex20.m1.3.3.2.2.cmml">⁢</mo><mtext id="S4.Ex20.m1.3.3.2.5" xref="S4.Ex20.m1.3.3.2.5a.cmml">.</mtext></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex20.m1.3b"><apply id="S4.Ex20.m1.3.3.cmml" xref="S4.Ex20.m1.3.3"><and id="S4.Ex20.m1.3.3a.cmml" xref="S4.Ex20.m1.3.3"></and><apply id="S4.Ex20.m1.3.3b.cmml" xref="S4.Ex20.m1.3.3"><eq id="S4.Ex20.m1.3.3.5.cmml" xref="S4.Ex20.m1.3.3.5"></eq><csymbol cd="latexml" id="S4.Ex20.m1.3.3.4.cmml" xref="S4.Ex20.m1.3.3.4">absent</csymbol><apply id="S4.Ex20.m1.2.2.1.cmml" xref="S4.Ex20.m1.2.2.1"><times id="S4.Ex20.m1.2.2.1.2.cmml" xref="S4.Ex20.m1.2.2.1.2"></times><apply id="S4.Ex20.m1.2.2.1.3.cmml" xref="S4.Ex20.m1.2.2.1.3"><ci id="S4.Ex20.m1.2.2.1.3.1.cmml" xref="S4.Ex20.m1.2.2.1.3.1">⋅</ci><cn id="S4.Ex20.m1.2.2.1.3.2.cmml" type="integer" xref="S4.Ex20.m1.2.2.1.3.2">2</cn><ci id="S4.Ex20.m1.2.2.1.3.3.cmml" xref="S4.Ex20.m1.2.2.1.3.3">𝔼</ci></apply><apply id="S4.Ex20.m1.2.2.1.1.2.cmml" xref="S4.Ex20.m1.2.2.1.1.1"><csymbol cd="latexml" id="S4.Ex20.m1.2.2.1.1.2.1.cmml" xref="S4.Ex20.m1.2.2.1.1.1.2">delimited-[]</csymbol><apply id="S4.Ex20.m1.2.2.1.1.1.1.cmml" xref="S4.Ex20.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex20.m1.2.2.1.1.1.1.1.cmml" xref="S4.Ex20.m1.2.2.1.1.1.1">superscript</csymbol><ci id="S4.Ex20.m1.2.2.1.1.1.1.2.cmml" xref="S4.Ex20.m1.2.2.1.1.1.1.2">𝜋</ci><cn id="S4.Ex20.m1.1.1.1.1.cmml" type="integer" xref="S4.Ex20.m1.1.1.1.1">2</cn></apply></apply></apply></apply><apply id="S4.Ex20.m1.3.3c.cmml" xref="S4.Ex20.m1.3.3"><leq id="S4.Ex20.m1.3.3.6.cmml" xref="S4.Ex20.m1.3.3.6"></leq><share href="https://arxiv.org/html/2503.06017v1#S4.Ex20.m1.2.2.1.cmml" id="S4.Ex20.m1.3.3d.cmml" xref="S4.Ex20.m1.3.3"></share><apply id="S4.Ex20.m1.3.3.2.cmml" xref="S4.Ex20.m1.3.3.2"><times id="S4.Ex20.m1.3.3.2.2.cmml" xref="S4.Ex20.m1.3.3.2.2"></times><apply id="S4.Ex20.m1.3.3.2.3.cmml" xref="S4.Ex20.m1.3.3.2.3"><ci id="S4.Ex20.m1.3.3.2.3.1.cmml" xref="S4.Ex20.m1.3.3.2.3.1">⋅</ci><cn id="S4.Ex20.m1.3.3.2.3.2.cmml" type="integer" xref="S4.Ex20.m1.3.3.2.3.2">2</cn><ci id="S4.Ex20.m1.3.3.2.3.3.cmml" xref="S4.Ex20.m1.3.3.2.3.3">𝒮</ci></apply><ci id="S4.Ex20.m1.3.3.2.4.cmml" xref="S4.Ex20.m1.3.3.2.4">𝒲</ci><apply id="S4.Ex20.m1.3.3.2.1.1.1.cmml" xref="S4.Ex20.m1.3.3.2.1.1"><csymbol cd="ambiguous" id="S4.Ex20.m1.3.3.2.1.1.1.1.cmml" xref="S4.Ex20.m1.3.3.2.1.1">subscript</csymbol><apply id="S4.Ex20.m1.3.3.2.1.1.1.2.cmml" xref="S4.Ex20.m1.3.3.2.1.1"><csymbol cd="ambiguous" id="S4.Ex20.m1.3.3.2.1.1.1.2.1.cmml" xref="S4.Ex20.m1.3.3.2.1.1">superscript</csymbol><ci id="S4.Ex20.m1.3.3.2.1.1.1.2.2.cmml" xref="S4.Ex20.m1.3.3.2.1.1.1.2.2">𝜋</ci><times id="S4.Ex20.m1.3.3.2.1.1.1.2.3.cmml" xref="S4.Ex20.m1.3.3.2.1.1.1.2.3"></times></apply><cn id="S4.Ex20.m1.3.3.2.1.1.1.3.cmml" type="integer" xref="S4.Ex20.m1.3.3.2.1.1.1.3">2</cn></apply><ci id="S4.Ex20.m1.3.3.2.5a.cmml" xref="S4.Ex20.m1.3.3.2.5"><mtext id="S4.Ex20.m1.3.3.2.5.cmml" xref="S4.Ex20.m1.3.3.2.5">.</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex20.m1.3c">\displaystyle=2\cdot\mathbb{E}\left[\pi^{(2)}\right]\leq 2\cdot\mathcal{SW}(% \pi^{*}_{2})\text{.}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex20.m1.3d">= 2 ⋅ blackboard_E [ italic_π start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ] ≤ 2 ⋅ caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.7.p3.1">There we use that <math alttext="\mathcal{V}(N,v)\geq 0" class="ltx_Math" display="inline" id="S4.SS2.7.p3.1.m1.2"><semantics id="S4.SS2.7.p3.1.m1.2a"><mrow id="S4.SS2.7.p3.1.m1.2.3" xref="S4.SS2.7.p3.1.m1.2.3.cmml"><mrow id="S4.SS2.7.p3.1.m1.2.3.2" xref="S4.SS2.7.p3.1.m1.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.7.p3.1.m1.2.3.2.2" xref="S4.SS2.7.p3.1.m1.2.3.2.2.cmml">𝒱</mi><mo id="S4.SS2.7.p3.1.m1.2.3.2.1" xref="S4.SS2.7.p3.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS2.7.p3.1.m1.2.3.2.3.2" xref="S4.SS2.7.p3.1.m1.2.3.2.3.1.cmml"><mo id="S4.SS2.7.p3.1.m1.2.3.2.3.2.1" stretchy="false" xref="S4.SS2.7.p3.1.m1.2.3.2.3.1.cmml">(</mo><mi id="S4.SS2.7.p3.1.m1.1.1" xref="S4.SS2.7.p3.1.m1.1.1.cmml">N</mi><mo id="S4.SS2.7.p3.1.m1.2.3.2.3.2.2" xref="S4.SS2.7.p3.1.m1.2.3.2.3.1.cmml">,</mo><mi id="S4.SS2.7.p3.1.m1.2.2" xref="S4.SS2.7.p3.1.m1.2.2.cmml">v</mi><mo id="S4.SS2.7.p3.1.m1.2.3.2.3.2.3" stretchy="false" xref="S4.SS2.7.p3.1.m1.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.7.p3.1.m1.2.3.1" xref="S4.SS2.7.p3.1.m1.2.3.1.cmml">≥</mo><mn id="S4.SS2.7.p3.1.m1.2.3.3" xref="S4.SS2.7.p3.1.m1.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.7.p3.1.m1.2b"><apply id="S4.SS2.7.p3.1.m1.2.3.cmml" xref="S4.SS2.7.p3.1.m1.2.3"><geq id="S4.SS2.7.p3.1.m1.2.3.1.cmml" xref="S4.SS2.7.p3.1.m1.2.3.1"></geq><apply id="S4.SS2.7.p3.1.m1.2.3.2.cmml" xref="S4.SS2.7.p3.1.m1.2.3.2"><times id="S4.SS2.7.p3.1.m1.2.3.2.1.cmml" xref="S4.SS2.7.p3.1.m1.2.3.2.1"></times><ci id="S4.SS2.7.p3.1.m1.2.3.2.2.cmml" xref="S4.SS2.7.p3.1.m1.2.3.2.2">𝒱</ci><interval closure="open" id="S4.SS2.7.p3.1.m1.2.3.2.3.1.cmml" xref="S4.SS2.7.p3.1.m1.2.3.2.3.2"><ci id="S4.SS2.7.p3.1.m1.1.1.cmml" xref="S4.SS2.7.p3.1.m1.1.1">𝑁</ci><ci id="S4.SS2.7.p3.1.m1.2.2.cmml" xref="S4.SS2.7.p3.1.m1.2.2">𝑣</ci></interval></apply><cn id="S4.SS2.7.p3.1.m1.2.3.3.cmml" type="integer" xref="S4.SS2.7.p3.1.m1.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.7.p3.1.m1.2c">\mathcal{V}(N,v)\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.7.p3.1.m1.2d">caligraphic_V ( italic_N , italic_v ) ≥ 0</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS2.p8"> <p class="ltx_p" id="S4.SS2.p8.1">We can combine the two last lemmas to apply the main theorem by Charikar and Wirth <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx18" title="">CW04</a>]</cite> and obtain a randomized <math alttext="\mathcal{O}(\log n)" class="ltx_Math" display="inline" id="S4.SS2.p8.1.m1.1"><semantics id="S4.SS2.p8.1.m1.1a"><mrow id="S4.SS2.p8.1.m1.1.1" xref="S4.SS2.p8.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p8.1.m1.1.1.3" xref="S4.SS2.p8.1.m1.1.1.3.cmml">𝒪</mi><mo id="S4.SS2.p8.1.m1.1.1.2" xref="S4.SS2.p8.1.m1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p8.1.m1.1.1.1.1" xref="S4.SS2.p8.1.m1.1.1.1.1.1.cmml"><mo id="S4.SS2.p8.1.m1.1.1.1.1.2" stretchy="false" xref="S4.SS2.p8.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p8.1.m1.1.1.1.1.1" xref="S4.SS2.p8.1.m1.1.1.1.1.1.cmml"><mi id="S4.SS2.p8.1.m1.1.1.1.1.1.1" xref="S4.SS2.p8.1.m1.1.1.1.1.1.1.cmml">log</mi><mo id="S4.SS2.p8.1.m1.1.1.1.1.1a" lspace="0.167em" xref="S4.SS2.p8.1.m1.1.1.1.1.1.cmml">⁡</mo><mi id="S4.SS2.p8.1.m1.1.1.1.1.1.2" xref="S4.SS2.p8.1.m1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S4.SS2.p8.1.m1.1.1.1.1.3" stretchy="false" xref="S4.SS2.p8.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.1.m1.1b"><apply id="S4.SS2.p8.1.m1.1.1.cmml" xref="S4.SS2.p8.1.m1.1.1"><times id="S4.SS2.p8.1.m1.1.1.2.cmml" xref="S4.SS2.p8.1.m1.1.1.2"></times><ci id="S4.SS2.p8.1.m1.1.1.3.cmml" xref="S4.SS2.p8.1.m1.1.1.3">𝒪</ci><apply id="S4.SS2.p8.1.m1.1.1.1.1.1.cmml" xref="S4.SS2.p8.1.m1.1.1.1.1"><log id="S4.SS2.p8.1.m1.1.1.1.1.1.1.cmml" xref="S4.SS2.p8.1.m1.1.1.1.1.1.1"></log><ci id="S4.SS2.p8.1.m1.1.1.1.1.1.2.cmml" xref="S4.SS2.p8.1.m1.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.1.m1.1c">\mathcal{O}(\log n)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.1.m1.1d">caligraphic_O ( roman_log italic_n )</annotation></semantics></math>-approximation algorithm.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem7.1.1.1">Theorem 4.7</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem7.p1"> <p class="ltx_p" id="S4.Thmtheorem7.p1.1"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem7.p1.1.1">There exists a randomized <math alttext="\mathcal{O}(\log n)" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.1.1.m1.1"><semantics id="S4.Thmtheorem7.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem7.p1.1.1.m1.1.1" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem7.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.3.cmml">𝒪</mi><mo id="S4.Thmtheorem7.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1.1" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1.1.cmml">log</mi><mo id="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1a" lspace="0.167em" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1.cmml">⁡</mo><mi id="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1.2" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.1.1.m1.1b"><apply id="S4.Thmtheorem7.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.1.1"><times id="S4.Thmtheorem7.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.2"></times><ci id="S4.Thmtheorem7.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.3">𝒪</ci><apply id="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1"><log id="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1.1"></log><ci id="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.1.1.m1.1c">\mathcal{O}(\log n)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.1.1.m1.1d">caligraphic_O ( roman_log italic_n )</annotation></semantics></math>-approximation algorithm for maximizing social welfare in ASHGs with nonnegative total value.</span></p> </div> </div> <div class="ltx_proof" id="S4.SS2.8"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.8.p1"> <p class="ltx_p" id="S4.SS2.8.p1.5">Theorem 1 by Charikar and Wirth <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx18" title="">CW04</a>]</cite> states the existence of a randomized <math alttext="\mathcal{O}(\log n)" class="ltx_Math" display="inline" id="S4.SS2.8.p1.1.m1.1"><semantics id="S4.SS2.8.p1.1.m1.1a"><mrow id="S4.SS2.8.p1.1.m1.1.1" xref="S4.SS2.8.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.8.p1.1.m1.1.1.3" xref="S4.SS2.8.p1.1.m1.1.1.3.cmml">𝒪</mi><mo id="S4.SS2.8.p1.1.m1.1.1.2" xref="S4.SS2.8.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.8.p1.1.m1.1.1.1.1" xref="S4.SS2.8.p1.1.m1.1.1.1.1.1.cmml"><mo id="S4.SS2.8.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S4.SS2.8.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.8.p1.1.m1.1.1.1.1.1" xref="S4.SS2.8.p1.1.m1.1.1.1.1.1.cmml"><mi id="S4.SS2.8.p1.1.m1.1.1.1.1.1.1" xref="S4.SS2.8.p1.1.m1.1.1.1.1.1.1.cmml">log</mi><mo id="S4.SS2.8.p1.1.m1.1.1.1.1.1a" lspace="0.167em" xref="S4.SS2.8.p1.1.m1.1.1.1.1.1.cmml">⁡</mo><mi id="S4.SS2.8.p1.1.m1.1.1.1.1.1.2" xref="S4.SS2.8.p1.1.m1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S4.SS2.8.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S4.SS2.8.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p1.1.m1.1b"><apply id="S4.SS2.8.p1.1.m1.1.1.cmml" xref="S4.SS2.8.p1.1.m1.1.1"><times id="S4.SS2.8.p1.1.m1.1.1.2.cmml" xref="S4.SS2.8.p1.1.m1.1.1.2"></times><ci id="S4.SS2.8.p1.1.m1.1.1.3.cmml" xref="S4.SS2.8.p1.1.m1.1.1.3">𝒪</ci><apply id="S4.SS2.8.p1.1.m1.1.1.1.1.1.cmml" xref="S4.SS2.8.p1.1.m1.1.1.1.1"><log id="S4.SS2.8.p1.1.m1.1.1.1.1.1.1.cmml" xref="S4.SS2.8.p1.1.m1.1.1.1.1.1.1"></log><ci id="S4.SS2.8.p1.1.m1.1.1.1.1.1.2.cmml" xref="S4.SS2.8.p1.1.m1.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p1.1.m1.1c">\mathcal{O}(\log n)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p1.1.m1.1d">caligraphic_O ( roman_log italic_n )</annotation></semantics></math>-approximation algorithm for <math alttext="\mathcal{CW}" class="ltx_Math" display="inline" id="S4.SS2.8.p1.2.m2.1"><semantics id="S4.SS2.8.p1.2.m2.1a"><mrow id="S4.SS2.8.p1.2.m2.1.1" xref="S4.SS2.8.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.8.p1.2.m2.1.1.2" xref="S4.SS2.8.p1.2.m2.1.1.2.cmml">𝒞</mi><mo id="S4.SS2.8.p1.2.m2.1.1.1" xref="S4.SS2.8.p1.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.8.p1.2.m2.1.1.3" xref="S4.SS2.8.p1.2.m2.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p1.2.m2.1b"><apply id="S4.SS2.8.p1.2.m2.1.1.cmml" xref="S4.SS2.8.p1.2.m2.1.1"><times id="S4.SS2.8.p1.2.m2.1.1.1.cmml" xref="S4.SS2.8.p1.2.m2.1.1.1"></times><ci id="S4.SS2.8.p1.2.m2.1.1.2.cmml" xref="S4.SS2.8.p1.2.m2.1.1.2">𝒞</ci><ci id="S4.SS2.8.p1.2.m2.1.1.3.cmml" xref="S4.SS2.8.p1.2.m2.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p1.2.m2.1c">\mathcal{CW}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p1.2.m2.1d">caligraphic_C caligraphic_W</annotation></semantics></math> under the constraint that partitions are in <math alttext="\Pi_{N}^{(2)}" class="ltx_Math" display="inline" id="S4.SS2.8.p1.3.m3.1"><semantics id="S4.SS2.8.p1.3.m3.1a"><msubsup id="S4.SS2.8.p1.3.m3.1.2" xref="S4.SS2.8.p1.3.m3.1.2.cmml"><mi id="S4.SS2.8.p1.3.m3.1.2.2.2" mathvariant="normal" xref="S4.SS2.8.p1.3.m3.1.2.2.2.cmml">Π</mi><mi id="S4.SS2.8.p1.3.m3.1.2.2.3" xref="S4.SS2.8.p1.3.m3.1.2.2.3.cmml">N</mi><mrow id="S4.SS2.8.p1.3.m3.1.1.1.3" xref="S4.SS2.8.p1.3.m3.1.2.cmml"><mo id="S4.SS2.8.p1.3.m3.1.1.1.3.1" stretchy="false" xref="S4.SS2.8.p1.3.m3.1.2.cmml">(</mo><mn id="S4.SS2.8.p1.3.m3.1.1.1.1" xref="S4.SS2.8.p1.3.m3.1.1.1.1.cmml">2</mn><mo id="S4.SS2.8.p1.3.m3.1.1.1.3.2" stretchy="false" xref="S4.SS2.8.p1.3.m3.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p1.3.m3.1b"><apply id="S4.SS2.8.p1.3.m3.1.2.cmml" xref="S4.SS2.8.p1.3.m3.1.2"><csymbol cd="ambiguous" id="S4.SS2.8.p1.3.m3.1.2.1.cmml" xref="S4.SS2.8.p1.3.m3.1.2">superscript</csymbol><apply id="S4.SS2.8.p1.3.m3.1.2.2.cmml" xref="S4.SS2.8.p1.3.m3.1.2"><csymbol cd="ambiguous" id="S4.SS2.8.p1.3.m3.1.2.2.1.cmml" xref="S4.SS2.8.p1.3.m3.1.2">subscript</csymbol><ci id="S4.SS2.8.p1.3.m3.1.2.2.2.cmml" xref="S4.SS2.8.p1.3.m3.1.2.2.2">Π</ci><ci id="S4.SS2.8.p1.3.m3.1.2.2.3.cmml" xref="S4.SS2.8.p1.3.m3.1.2.2.3">𝑁</ci></apply><cn id="S4.SS2.8.p1.3.m3.1.1.1.1.cmml" type="integer" xref="S4.SS2.8.p1.3.m3.1.1.1.1">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p1.3.m3.1c">\Pi_{N}^{(2)}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p1.3.m3.1d">roman_Π start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.Thmtheorem5" title="Lemma 4.5. ‣ 4.2 Logarithmic Approximation for Nonnegative Total Value ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.5</span></a>, the same approximation guarantee is obtained for <math alttext="\mathcal{SW}" class="ltx_Math" display="inline" id="S4.SS2.8.p1.4.m4.1"><semantics id="S4.SS2.8.p1.4.m4.1a"><mrow id="S4.SS2.8.p1.4.m4.1.1" xref="S4.SS2.8.p1.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.8.p1.4.m4.1.1.2" xref="S4.SS2.8.p1.4.m4.1.1.2.cmml">𝒮</mi><mo id="S4.SS2.8.p1.4.m4.1.1.1" xref="S4.SS2.8.p1.4.m4.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.8.p1.4.m4.1.1.3" xref="S4.SS2.8.p1.4.m4.1.1.3.cmml">𝒲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p1.4.m4.1b"><apply id="S4.SS2.8.p1.4.m4.1.1.cmml" xref="S4.SS2.8.p1.4.m4.1.1"><times id="S4.SS2.8.p1.4.m4.1.1.1.cmml" xref="S4.SS2.8.p1.4.m4.1.1.1"></times><ci id="S4.SS2.8.p1.4.m4.1.1.2.cmml" xref="S4.SS2.8.p1.4.m4.1.1.2">𝒮</ci><ci id="S4.SS2.8.p1.4.m4.1.1.3.cmml" xref="S4.SS2.8.p1.4.m4.1.1.3">𝒲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p1.4.m4.1c">\mathcal{SW}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p1.4.m4.1d">caligraphic_S caligraphic_W</annotation></semantics></math> under the same constraint. Finally, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.Thmtheorem6" title="Lemma 4.6. ‣ 4.2 Logarithmic Approximation for Nonnegative Total Value ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.6</span></a> guarantees that the maximum welfare of any partition is better by at most a factor of <math alttext="2" class="ltx_Math" display="inline" id="S4.SS2.8.p1.5.m5.1"><semantics id="S4.SS2.8.p1.5.m5.1a"><mn id="S4.SS2.8.p1.5.m5.1.1" xref="S4.SS2.8.p1.5.m5.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.8.p1.5.m5.1b"><cn id="S4.SS2.8.p1.5.m5.1.1.cmml" type="integer" xref="S4.SS2.8.p1.5.m5.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.8.p1.5.m5.1c">2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.8.p1.5.m5.1d">2</annotation></semantics></math>. ∎</p> </div> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Beyond Worst-Case Analysis</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">In light of the hardness result by Flammini et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx26" title="">FKV22</a>]</cite> for approximating social welfare in aversion-to-enemies games, it is natural to ask how well we can approximate welfare in such games generated by stochastic models. In this section, we introduce two such models where the valuations originate from either Erdős-Rényi or multipartite graphs. Erdős-Rényi graphs serve as a common testbed for graph optimization problems and help us set the stage for the more challenging setting of multipartite graphs. Interestingly, our main theorems demonstrate that greedy algorithms are remarkably effective in these models, yielding constant-factor and logarithmic-factor approximations of social welfare.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.8">In this section, <math alttext="G=(N,v)" class="ltx_Math" display="inline" id="S5.p2.1.m1.2"><semantics id="S5.p2.1.m1.2a"><mrow id="S5.p2.1.m1.2.3" xref="S5.p2.1.m1.2.3.cmml"><mi id="S5.p2.1.m1.2.3.2" xref="S5.p2.1.m1.2.3.2.cmml">G</mi><mo id="S5.p2.1.m1.2.3.1" xref="S5.p2.1.m1.2.3.1.cmml">=</mo><mrow id="S5.p2.1.m1.2.3.3.2" xref="S5.p2.1.m1.2.3.3.1.cmml"><mo id="S5.p2.1.m1.2.3.3.2.1" stretchy="false" xref="S5.p2.1.m1.2.3.3.1.cmml">(</mo><mi id="S5.p2.1.m1.1.1" xref="S5.p2.1.m1.1.1.cmml">N</mi><mo id="S5.p2.1.m1.2.3.3.2.2" xref="S5.p2.1.m1.2.3.3.1.cmml">,</mo><mi id="S5.p2.1.m1.2.2" xref="S5.p2.1.m1.2.2.cmml">v</mi><mo id="S5.p2.1.m1.2.3.3.2.3" stretchy="false" xref="S5.p2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.2b"><apply id="S5.p2.1.m1.2.3.cmml" xref="S5.p2.1.m1.2.3"><eq id="S5.p2.1.m1.2.3.1.cmml" xref="S5.p2.1.m1.2.3.1"></eq><ci id="S5.p2.1.m1.2.3.2.cmml" xref="S5.p2.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S5.p2.1.m1.2.3.3.1.cmml" xref="S5.p2.1.m1.2.3.3.2"><ci id="S5.p2.1.m1.1.1.cmml" xref="S5.p2.1.m1.1.1">𝑁</ci><ci id="S5.p2.1.m1.2.2.cmml" xref="S5.p2.1.m1.2.2">𝑣</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.2c">G=(N,v)</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.2d">italic_G = ( italic_N , italic_v )</annotation></semantics></math> refers to a fixed symmetric aversion-to-enemies game. In any partition of <math alttext="N" class="ltx_Math" display="inline" id="S5.p2.2.m2.1"><semantics id="S5.p2.2.m2.1a"><mi id="S5.p2.2.m2.1.1" xref="S5.p2.2.m2.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S5.p2.2.m2.1b"><ci id="S5.p2.2.m2.1.1.cmml" xref="S5.p2.2.m2.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.2.m2.1c">N</annotation><annotation encoding="application/x-llamapun" id="S5.p2.2.m2.1d">italic_N</annotation></semantics></math>, a valuation of <math alttext="-n" class="ltx_Math" display="inline" id="S5.p2.3.m3.1"><semantics id="S5.p2.3.m3.1a"><mrow id="S5.p2.3.m3.1.1" xref="S5.p2.3.m3.1.1.cmml"><mo id="S5.p2.3.m3.1.1a" xref="S5.p2.3.m3.1.1.cmml">−</mo><mi id="S5.p2.3.m3.1.1.2" xref="S5.p2.3.m3.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.3.m3.1b"><apply id="S5.p2.3.m3.1.1.cmml" xref="S5.p2.3.m3.1.1"><minus id="S5.p2.3.m3.1.1.1.cmml" xref="S5.p2.3.m3.1.1"></minus><ci id="S5.p2.3.m3.1.1.2.cmml" xref="S5.p2.3.m3.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.3.m3.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.p2.3.m3.1d">- italic_n</annotation></semantics></math> within a coalition implies a negative utility for the corresponding agents. Consequently, removing one of these agents from the coalition and forming a singleton coalition would increase the overall social welfare. This observation suggests that in an optimal partition <math alttext="\pi^{*}" class="ltx_Math" display="inline" id="S5.p2.4.m4.1"><semantics id="S5.p2.4.m4.1a"><msup id="S5.p2.4.m4.1.1" xref="S5.p2.4.m4.1.1.cmml"><mi id="S5.p2.4.m4.1.1.2" xref="S5.p2.4.m4.1.1.2.cmml">π</mi><mo id="S5.p2.4.m4.1.1.3" xref="S5.p2.4.m4.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S5.p2.4.m4.1b"><apply id="S5.p2.4.m4.1.1.cmml" xref="S5.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S5.p2.4.m4.1.1.1.cmml" xref="S5.p2.4.m4.1.1">superscript</csymbol><ci id="S5.p2.4.m4.1.1.2.cmml" xref="S5.p2.4.m4.1.1.2">𝜋</ci><times id="S5.p2.4.m4.1.1.3.cmml" xref="S5.p2.4.m4.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.4.m4.1c">\pi^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.4.m4.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, no coalition contains agents with a mutual valuation of <math alttext="-n" class="ltx_Math" display="inline" id="S5.p2.5.m5.1"><semantics id="S5.p2.5.m5.1a"><mrow id="S5.p2.5.m5.1.1" xref="S5.p2.5.m5.1.1.cmml"><mo id="S5.p2.5.m5.1.1a" xref="S5.p2.5.m5.1.1.cmml">−</mo><mi id="S5.p2.5.m5.1.1.2" xref="S5.p2.5.m5.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.5.m5.1b"><apply id="S5.p2.5.m5.1.1.cmml" xref="S5.p2.5.m5.1.1"><minus id="S5.p2.5.m5.1.1.1.cmml" xref="S5.p2.5.m5.1.1"></minus><ci id="S5.p2.5.m5.1.1.2.cmml" xref="S5.p2.5.m5.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.5.m5.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.p2.5.m5.1d">- italic_n</annotation></semantics></math>. Let <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.p2.6.m6.1"><semantics id="S5.p2.6.m6.1a"><msup id="S5.p2.6.m6.1.1" xref="S5.p2.6.m6.1.1.cmml"><mi id="S5.p2.6.m6.1.1.2" xref="S5.p2.6.m6.1.1.2.cmml">G</mi><mo id="S5.p2.6.m6.1.1.3" xref="S5.p2.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.p2.6.m6.1b"><apply id="S5.p2.6.m6.1.1.cmml" xref="S5.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S5.p2.6.m6.1.1.1.cmml" xref="S5.p2.6.m6.1.1">superscript</csymbol><ci id="S5.p2.6.m6.1.1.2.cmml" xref="S5.p2.6.m6.1.1.2">𝐺</ci><ci id="S5.p2.6.m6.1.1.3.cmml" xref="S5.p2.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.6.m6.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.6.m6.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> denote the subgraph of <math alttext="G" class="ltx_Math" display="inline" id="S5.p2.7.m7.1"><semantics id="S5.p2.7.m7.1a"><mi id="S5.p2.7.m7.1.1" xref="S5.p2.7.m7.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.p2.7.m7.1b"><ci id="S5.p2.7.m7.1.1.cmml" xref="S5.p2.7.m7.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.7.m7.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.p2.7.m7.1d">italic_G</annotation></semantics></math>, obtained by removing all edges with weight <math alttext="-n" class="ltx_Math" display="inline" id="S5.p2.8.m8.1"><semantics id="S5.p2.8.m8.1a"><mrow id="S5.p2.8.m8.1.1" xref="S5.p2.8.m8.1.1.cmml"><mo id="S5.p2.8.m8.1.1a" xref="S5.p2.8.m8.1.1.cmml">−</mo><mi id="S5.p2.8.m8.1.1.2" xref="S5.p2.8.m8.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.8.m8.1b"><apply id="S5.p2.8.m8.1.1.cmml" xref="S5.p2.8.m8.1.1"><minus id="S5.p2.8.m8.1.1.1.cmml" xref="S5.p2.8.m8.1.1"></minus><ci id="S5.p2.8.m8.1.1.2.cmml" xref="S5.p2.8.m8.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.8.m8.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.p2.8.m8.1d">- italic_n</annotation></semantics></math>. We now present a useful lemma.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S5.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.1.1.1">Lemma 5.1</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem1.p1"> <p class="ltx_p" id="S5.Thmtheorem1.p1.3"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem1.p1.3.3">If the size of the maximum clique in <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.1.1.m1.1"><semantics id="S5.Thmtheorem1.p1.1.1.m1.1a"><msup id="S5.Thmtheorem1.p1.1.1.m1.1.1" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem1.p1.1.1.m1.1.1.2" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.2.cmml">G</mi><mo id="S5.Thmtheorem1.p1.1.1.m1.1.1.3" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.1.1.m1.1b"><apply id="S5.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1">superscript</csymbol><ci id="S5.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.2">𝐺</ci><ci id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.1.1.m1.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.1.1.m1.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is <math alttext="t" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.2.2.m2.1"><semantics id="S5.Thmtheorem1.p1.2.2.m2.1a"><mi id="S5.Thmtheorem1.p1.2.2.m2.1.1" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.2.2.m2.1b"><ci id="S5.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.2.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.2.2.m2.1d">italic_t</annotation></semantics></math>, then <math alttext="\mathcal{SW}(\pi^{*})\leq n(t-1)" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.3.3.m3.2"><semantics id="S5.Thmtheorem1.p1.3.3.m3.2a"><mrow id="S5.Thmtheorem1.p1.3.3.m3.2.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.cmml"><mrow id="S5.Thmtheorem1.p1.3.3.m3.1.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3.cmml">𝒮</mi><mo id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.2" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.4" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.4.cmml">𝒲</mi><mo id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.2a" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml">(</mo><msup id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.2" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.2.cmml">π</mi><mo id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.3" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem1.p1.3.3.m3.2.2.3" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.3.cmml">≤</mo><mrow id="S5.Thmtheorem1.p1.3.3.m3.2.2.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.cmml"><mi id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3.cmml">n</mi><mo id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.cmml">⁢</mo><mrow id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.cmml"><mo id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.2" stretchy="false" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.cmml"><mi id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.2.cmml">t</mi><mo id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.1.cmml">−</mo><mn id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.3" stretchy="false" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.3.3.m3.2b"><apply id="S5.Thmtheorem1.p1.3.3.m3.2.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2"><leq id="S5.Thmtheorem1.p1.3.3.m3.2.2.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.3"></leq><apply id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1"><times id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.2"></times><ci id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3">𝒮</ci><ci id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.4.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.4">𝒲</ci><apply id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.2">𝜋</ci><times id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.3"></times></apply></apply><apply id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2"><times id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2"></times><ci id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3">𝑛</ci><apply id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1"><minus id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.1"></minus><ci id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.2">𝑡</ci><cn id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.3.3.m3.2c">\mathcal{SW}(\pi^{*})\leq n(t-1)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.3.3.m3.2d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ≤ italic_n ( italic_t - 1 )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S5.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.1.p1"> <p class="ltx_p" id="S5.1.p1.9">No coalition in the partition <math alttext="\pi^{*}" class="ltx_Math" display="inline" id="S5.1.p1.1.m1.1"><semantics id="S5.1.p1.1.m1.1a"><msup id="S5.1.p1.1.m1.1.1" xref="S5.1.p1.1.m1.1.1.cmml"><mi id="S5.1.p1.1.m1.1.1.2" xref="S5.1.p1.1.m1.1.1.2.cmml">π</mi><mo id="S5.1.p1.1.m1.1.1.3" xref="S5.1.p1.1.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S5.1.p1.1.m1.1b"><apply id="S5.1.p1.1.m1.1.1.cmml" xref="S5.1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.1.p1.1.m1.1.1.1.cmml" xref="S5.1.p1.1.m1.1.1">superscript</csymbol><ci id="S5.1.p1.1.m1.1.1.2.cmml" xref="S5.1.p1.1.m1.1.1.2">𝜋</ci><times id="S5.1.p1.1.m1.1.1.3.cmml" xref="S5.1.p1.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.1.m1.1c">\pi^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.1.m1.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> contains an edge with weight <math alttext="-n" class="ltx_Math" display="inline" id="S5.1.p1.2.m2.1"><semantics id="S5.1.p1.2.m2.1a"><mrow id="S5.1.p1.2.m2.1.1" xref="S5.1.p1.2.m2.1.1.cmml"><mo id="S5.1.p1.2.m2.1.1a" xref="S5.1.p1.2.m2.1.1.cmml">−</mo><mi id="S5.1.p1.2.m2.1.1.2" xref="S5.1.p1.2.m2.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.2.m2.1b"><apply id="S5.1.p1.2.m2.1.1.cmml" xref="S5.1.p1.2.m2.1.1"><minus id="S5.1.p1.2.m2.1.1.1.cmml" xref="S5.1.p1.2.m2.1.1"></minus><ci id="S5.1.p1.2.m2.1.1.2.cmml" xref="S5.1.p1.2.m2.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.2.m2.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.2.m2.1d">- italic_n</annotation></semantics></math>. Therefore, each coalition in <math alttext="\pi^{*}" class="ltx_Math" display="inline" id="S5.1.p1.3.m3.1"><semantics id="S5.1.p1.3.m3.1a"><msup id="S5.1.p1.3.m3.1.1" xref="S5.1.p1.3.m3.1.1.cmml"><mi id="S5.1.p1.3.m3.1.1.2" xref="S5.1.p1.3.m3.1.1.2.cmml">π</mi><mo id="S5.1.p1.3.m3.1.1.3" xref="S5.1.p1.3.m3.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S5.1.p1.3.m3.1b"><apply id="S5.1.p1.3.m3.1.1.cmml" xref="S5.1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.1.p1.3.m3.1.1.1.cmml" xref="S5.1.p1.3.m3.1.1">superscript</csymbol><ci id="S5.1.p1.3.m3.1.1.2.cmml" xref="S5.1.p1.3.m3.1.1.2">𝜋</ci><times id="S5.1.p1.3.m3.1.1.3.cmml" xref="S5.1.p1.3.m3.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.3.m3.1c">\pi^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.3.m3.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> forms a clique in <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.1.p1.4.m4.1"><semantics id="S5.1.p1.4.m4.1a"><msup id="S5.1.p1.4.m4.1.1" xref="S5.1.p1.4.m4.1.1.cmml"><mi id="S5.1.p1.4.m4.1.1.2" xref="S5.1.p1.4.m4.1.1.2.cmml">G</mi><mo id="S5.1.p1.4.m4.1.1.3" xref="S5.1.p1.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.1.p1.4.m4.1b"><apply id="S5.1.p1.4.m4.1.1.cmml" xref="S5.1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.1.p1.4.m4.1.1.1.cmml" xref="S5.1.p1.4.m4.1.1">superscript</csymbol><ci id="S5.1.p1.4.m4.1.1.2.cmml" xref="S5.1.p1.4.m4.1.1.2">𝐺</ci><ci id="S5.1.p1.4.m4.1.1.3.cmml" xref="S5.1.p1.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.4.m4.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.4.m4.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Since the size of a maximum clique in <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.1.p1.5.m5.1"><semantics id="S5.1.p1.5.m5.1a"><msup id="S5.1.p1.5.m5.1.1" xref="S5.1.p1.5.m5.1.1.cmml"><mi id="S5.1.p1.5.m5.1.1.2" xref="S5.1.p1.5.m5.1.1.2.cmml">G</mi><mo id="S5.1.p1.5.m5.1.1.3" xref="S5.1.p1.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.1.p1.5.m5.1b"><apply id="S5.1.p1.5.m5.1.1.cmml" xref="S5.1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S5.1.p1.5.m5.1.1.1.cmml" xref="S5.1.p1.5.m5.1.1">superscript</csymbol><ci id="S5.1.p1.5.m5.1.1.2.cmml" xref="S5.1.p1.5.m5.1.1.2">𝐺</ci><ci id="S5.1.p1.5.m5.1.1.3.cmml" xref="S5.1.p1.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.5.m5.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.5.m5.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is <math alttext="t" class="ltx_Math" display="inline" id="S5.1.p1.6.m6.1"><semantics id="S5.1.p1.6.m6.1a"><mi id="S5.1.p1.6.m6.1.1" xref="S5.1.p1.6.m6.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.6.m6.1b"><ci id="S5.1.p1.6.m6.1.1.cmml" xref="S5.1.p1.6.m6.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.6.m6.1c">t</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.6.m6.1d">italic_t</annotation></semantics></math>, the size of every coalition in <math alttext="\pi^{*}" class="ltx_Math" display="inline" id="S5.1.p1.7.m7.1"><semantics id="S5.1.p1.7.m7.1a"><msup id="S5.1.p1.7.m7.1.1" xref="S5.1.p1.7.m7.1.1.cmml"><mi id="S5.1.p1.7.m7.1.1.2" xref="S5.1.p1.7.m7.1.1.2.cmml">π</mi><mo id="S5.1.p1.7.m7.1.1.3" xref="S5.1.p1.7.m7.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S5.1.p1.7.m7.1b"><apply id="S5.1.p1.7.m7.1.1.cmml" xref="S5.1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S5.1.p1.7.m7.1.1.1.cmml" xref="S5.1.p1.7.m7.1.1">superscript</csymbol><ci id="S5.1.p1.7.m7.1.1.2.cmml" xref="S5.1.p1.7.m7.1.1.2">𝜋</ci><times id="S5.1.p1.7.m7.1.1.3.cmml" xref="S5.1.p1.7.m7.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.7.m7.1c">\pi^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.7.m7.1d">italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is at most <math alttext="t" class="ltx_Math" display="inline" id="S5.1.p1.8.m8.1"><semantics id="S5.1.p1.8.m8.1a"><mi id="S5.1.p1.8.m8.1.1" xref="S5.1.p1.8.m8.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.8.m8.1b"><ci id="S5.1.p1.8.m8.1.1.cmml" xref="S5.1.p1.8.m8.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.8.m8.1c">t</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.8.m8.1d">italic_t</annotation></semantics></math>. Consequently, the utility of each agent is bounded by <math alttext="t-1" class="ltx_Math" display="inline" id="S5.1.p1.9.m9.1"><semantics id="S5.1.p1.9.m9.1a"><mrow id="S5.1.p1.9.m9.1.1" xref="S5.1.p1.9.m9.1.1.cmml"><mi id="S5.1.p1.9.m9.1.1.2" xref="S5.1.p1.9.m9.1.1.2.cmml">t</mi><mo id="S5.1.p1.9.m9.1.1.1" xref="S5.1.p1.9.m9.1.1.1.cmml">−</mo><mn id="S5.1.p1.9.m9.1.1.3" xref="S5.1.p1.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.9.m9.1b"><apply id="S5.1.p1.9.m9.1.1.cmml" xref="S5.1.p1.9.m9.1.1"><minus id="S5.1.p1.9.m9.1.1.1.cmml" xref="S5.1.p1.9.m9.1.1.1"></minus><ci id="S5.1.p1.9.m9.1.1.2.cmml" xref="S5.1.p1.9.m9.1.1.2">𝑡</ci><cn id="S5.1.p1.9.m9.1.1.3.cmml" type="integer" xref="S5.1.p1.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.9.m9.1c">t-1</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.9.m9.1d">italic_t - 1</annotation></semantics></math>. We conclude that</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex21"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{SW}(\pi^{*})=\sum_{i\in N}u_{i}(\pi^{*}(i))\leq n(t-1).\qed" class="ltx_Math" display="block" id="S5.Ex21.m1.2"><semantics id="S5.Ex21.m1.2a"><mrow id="S5.Ex21.m1.2.2.1" xref="S5.Ex21.m1.2.2.1.1.cmml"><mrow id="S5.Ex21.m1.2.2.1.1" xref="S5.Ex21.m1.2.2.1.1.cmml"><mrow id="S5.Ex21.m1.2.2.1.1.1" xref="S5.Ex21.m1.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex21.m1.2.2.1.1.1.3" xref="S5.Ex21.m1.2.2.1.1.1.3.cmml">𝒮</mi><mo id="S5.Ex21.m1.2.2.1.1.1.2" xref="S5.Ex21.m1.2.2.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex21.m1.2.2.1.1.1.4" xref="S5.Ex21.m1.2.2.1.1.1.4.cmml">𝒲</mi><mo id="S5.Ex21.m1.2.2.1.1.1.2a" xref="S5.Ex21.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex21.m1.2.2.1.1.1.1.1" xref="S5.Ex21.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S5.Ex21.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S5.Ex21.m1.2.2.1.1.1.1.1.1.cmml">(</mo><msup id="S5.Ex21.m1.2.2.1.1.1.1.1.1" xref="S5.Ex21.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S5.Ex21.m1.2.2.1.1.1.1.1.1.2" xref="S5.Ex21.m1.2.2.1.1.1.1.1.1.2.cmml">π</mi><mo id="S5.Ex21.m1.2.2.1.1.1.1.1.1.3" xref="S5.Ex21.m1.2.2.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.Ex21.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S5.Ex21.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex21.m1.2.2.1.1.5" rspace="0.111em" xref="S5.Ex21.m1.2.2.1.1.5.cmml">=</mo><mrow id="S5.Ex21.m1.2.2.1.1.2" xref="S5.Ex21.m1.2.2.1.1.2.cmml"><munder id="S5.Ex21.m1.2.2.1.1.2.2" xref="S5.Ex21.m1.2.2.1.1.2.2.cmml"><mo id="S5.Ex21.m1.2.2.1.1.2.2.2" movablelimits="false" xref="S5.Ex21.m1.2.2.1.1.2.2.2.cmml">∑</mo><mrow id="S5.Ex21.m1.2.2.1.1.2.2.3" xref="S5.Ex21.m1.2.2.1.1.2.2.3.cmml"><mi id="S5.Ex21.m1.2.2.1.1.2.2.3.2" xref="S5.Ex21.m1.2.2.1.1.2.2.3.2.cmml">i</mi><mo id="S5.Ex21.m1.2.2.1.1.2.2.3.1" xref="S5.Ex21.m1.2.2.1.1.2.2.3.1.cmml">∈</mo><mi id="S5.Ex21.m1.2.2.1.1.2.2.3.3" xref="S5.Ex21.m1.2.2.1.1.2.2.3.3.cmml">N</mi></mrow></munder><mrow id="S5.Ex21.m1.2.2.1.1.2.1" xref="S5.Ex21.m1.2.2.1.1.2.1.cmml"><msub id="S5.Ex21.m1.2.2.1.1.2.1.3" xref="S5.Ex21.m1.2.2.1.1.2.1.3.cmml"><mi id="S5.Ex21.m1.2.2.1.1.2.1.3.2" xref="S5.Ex21.m1.2.2.1.1.2.1.3.2.cmml">u</mi><mi id="S5.Ex21.m1.2.2.1.1.2.1.3.3" xref="S5.Ex21.m1.2.2.1.1.2.1.3.3.cmml">i</mi></msub><mo id="S5.Ex21.m1.2.2.1.1.2.1.2" xref="S5.Ex21.m1.2.2.1.1.2.1.2.cmml">⁢</mo><mrow id="S5.Ex21.m1.2.2.1.1.2.1.1.1" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.cmml"><mo id="S5.Ex21.m1.2.2.1.1.2.1.1.1.2" stretchy="false" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.cmml">(</mo><mrow id="S5.Ex21.m1.2.2.1.1.2.1.1.1.1" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.cmml"><msup id="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.2" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.2.cmml"><mi id="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.2.2" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.2.2.cmml">π</mi><mo id="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.2.3" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.2.3.cmml">∗</mo></msup><mo id="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.1" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.3.2" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.cmml"><mo id="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.3.2.1" stretchy="false" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.cmml">(</mo><mi id="S5.Ex21.m1.1.1" xref="S5.Ex21.m1.1.1.cmml">i</mi><mo id="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.3.2.2" stretchy="false" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex21.m1.2.2.1.1.2.1.1.1.3" stretchy="false" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Ex21.m1.2.2.1.1.6" xref="S5.Ex21.m1.2.2.1.1.6.cmml">≤</mo><mrow id="S5.Ex21.m1.2.2.1.1.3" xref="S5.Ex21.m1.2.2.1.1.3.cmml"><mi id="S5.Ex21.m1.2.2.1.1.3.3" xref="S5.Ex21.m1.2.2.1.1.3.3.cmml">n</mi><mo id="S5.Ex21.m1.2.2.1.1.3.2" xref="S5.Ex21.m1.2.2.1.1.3.2.cmml">⁢</mo><mrow id="S5.Ex21.m1.2.2.1.1.3.1.1" xref="S5.Ex21.m1.2.2.1.1.3.1.1.1.cmml"><mo id="S5.Ex21.m1.2.2.1.1.3.1.1.2" stretchy="false" xref="S5.Ex21.m1.2.2.1.1.3.1.1.1.cmml">(</mo><mrow id="S5.Ex21.m1.2.2.1.1.3.1.1.1" xref="S5.Ex21.m1.2.2.1.1.3.1.1.1.cmml"><mi id="S5.Ex21.m1.2.2.1.1.3.1.1.1.2" xref="S5.Ex21.m1.2.2.1.1.3.1.1.1.2.cmml">t</mi><mo id="S5.Ex21.m1.2.2.1.1.3.1.1.1.1" xref="S5.Ex21.m1.2.2.1.1.3.1.1.1.1.cmml">−</mo><mn id="S5.Ex21.m1.2.2.1.1.3.1.1.1.3" xref="S5.Ex21.m1.2.2.1.1.3.1.1.1.3.cmml">1</mn></mrow><mo id="S5.Ex21.m1.2.2.1.1.3.1.1.3" stretchy="false" xref="S5.Ex21.m1.2.2.1.1.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Ex21.m1.2.2.1.2" lspace="0em" rspace="0.0835em" xref="S5.Ex21.m1.2.2.1.1.cmml">.</mo><mo class="ltx_mathvariant_italic" id="S5.Ex21.m1.2.2.1.3" lspace="0.0835em" mathvariant="italic" separator="true" xref="S5.Ex21.m1.2.2.1.1.cmml">∎</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex21.m1.2b"><apply id="S5.Ex21.m1.2.2.1.1.cmml" xref="S5.Ex21.m1.2.2.1"><and id="S5.Ex21.m1.2.2.1.1a.cmml" xref="S5.Ex21.m1.2.2.1"></and><apply id="S5.Ex21.m1.2.2.1.1b.cmml" xref="S5.Ex21.m1.2.2.1"><eq id="S5.Ex21.m1.2.2.1.1.5.cmml" xref="S5.Ex21.m1.2.2.1.1.5"></eq><apply id="S5.Ex21.m1.2.2.1.1.1.cmml" xref="S5.Ex21.m1.2.2.1.1.1"><times id="S5.Ex21.m1.2.2.1.1.1.2.cmml" xref="S5.Ex21.m1.2.2.1.1.1.2"></times><ci id="S5.Ex21.m1.2.2.1.1.1.3.cmml" xref="S5.Ex21.m1.2.2.1.1.1.3">𝒮</ci><ci id="S5.Ex21.m1.2.2.1.1.1.4.cmml" xref="S5.Ex21.m1.2.2.1.1.1.4">𝒲</ci><apply id="S5.Ex21.m1.2.2.1.1.1.1.1.1.cmml" xref="S5.Ex21.m1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Ex21.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S5.Ex21.m1.2.2.1.1.1.1.1">superscript</csymbol><ci id="S5.Ex21.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S5.Ex21.m1.2.2.1.1.1.1.1.1.2">𝜋</ci><times id="S5.Ex21.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S5.Ex21.m1.2.2.1.1.1.1.1.1.3"></times></apply></apply><apply id="S5.Ex21.m1.2.2.1.1.2.cmml" xref="S5.Ex21.m1.2.2.1.1.2"><apply id="S5.Ex21.m1.2.2.1.1.2.2.cmml" xref="S5.Ex21.m1.2.2.1.1.2.2"><csymbol cd="ambiguous" id="S5.Ex21.m1.2.2.1.1.2.2.1.cmml" xref="S5.Ex21.m1.2.2.1.1.2.2">subscript</csymbol><sum id="S5.Ex21.m1.2.2.1.1.2.2.2.cmml" xref="S5.Ex21.m1.2.2.1.1.2.2.2"></sum><apply id="S5.Ex21.m1.2.2.1.1.2.2.3.cmml" xref="S5.Ex21.m1.2.2.1.1.2.2.3"><in id="S5.Ex21.m1.2.2.1.1.2.2.3.1.cmml" xref="S5.Ex21.m1.2.2.1.1.2.2.3.1"></in><ci id="S5.Ex21.m1.2.2.1.1.2.2.3.2.cmml" xref="S5.Ex21.m1.2.2.1.1.2.2.3.2">𝑖</ci><ci id="S5.Ex21.m1.2.2.1.1.2.2.3.3.cmml" xref="S5.Ex21.m1.2.2.1.1.2.2.3.3">𝑁</ci></apply></apply><apply id="S5.Ex21.m1.2.2.1.1.2.1.cmml" xref="S5.Ex21.m1.2.2.1.1.2.1"><times id="S5.Ex21.m1.2.2.1.1.2.1.2.cmml" xref="S5.Ex21.m1.2.2.1.1.2.1.2"></times><apply id="S5.Ex21.m1.2.2.1.1.2.1.3.cmml" xref="S5.Ex21.m1.2.2.1.1.2.1.3"><csymbol cd="ambiguous" id="S5.Ex21.m1.2.2.1.1.2.1.3.1.cmml" xref="S5.Ex21.m1.2.2.1.1.2.1.3">subscript</csymbol><ci id="S5.Ex21.m1.2.2.1.1.2.1.3.2.cmml" xref="S5.Ex21.m1.2.2.1.1.2.1.3.2">𝑢</ci><ci id="S5.Ex21.m1.2.2.1.1.2.1.3.3.cmml" xref="S5.Ex21.m1.2.2.1.1.2.1.3.3">𝑖</ci></apply><apply id="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.cmml" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1"><times id="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.1.cmml" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.1"></times><apply id="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.2.cmml" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.2.1.cmml" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.2">superscript</csymbol><ci id="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.2.2.cmml" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.2.2">𝜋</ci><times id="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.2.3.cmml" xref="S5.Ex21.m1.2.2.1.1.2.1.1.1.1.2.3"></times></apply><ci id="S5.Ex21.m1.1.1.cmml" xref="S5.Ex21.m1.1.1">𝑖</ci></apply></apply></apply></apply><apply id="S5.Ex21.m1.2.2.1.1c.cmml" xref="S5.Ex21.m1.2.2.1"><leq id="S5.Ex21.m1.2.2.1.1.6.cmml" xref="S5.Ex21.m1.2.2.1.1.6"></leq><share href="https://arxiv.org/html/2503.06017v1#S5.Ex21.m1.2.2.1.1.2.cmml" id="S5.Ex21.m1.2.2.1.1d.cmml" xref="S5.Ex21.m1.2.2.1"></share><apply id="S5.Ex21.m1.2.2.1.1.3.cmml" xref="S5.Ex21.m1.2.2.1.1.3"><times id="S5.Ex21.m1.2.2.1.1.3.2.cmml" xref="S5.Ex21.m1.2.2.1.1.3.2"></times><ci id="S5.Ex21.m1.2.2.1.1.3.3.cmml" xref="S5.Ex21.m1.2.2.1.1.3.3">𝑛</ci><apply id="S5.Ex21.m1.2.2.1.1.3.1.1.1.cmml" xref="S5.Ex21.m1.2.2.1.1.3.1.1"><minus id="S5.Ex21.m1.2.2.1.1.3.1.1.1.1.cmml" xref="S5.Ex21.m1.2.2.1.1.3.1.1.1.1"></minus><ci id="S5.Ex21.m1.2.2.1.1.3.1.1.1.2.cmml" xref="S5.Ex21.m1.2.2.1.1.3.1.1.1.2">𝑡</ci><cn id="S5.Ex21.m1.2.2.1.1.3.1.1.1.3.cmml" type="integer" xref="S5.Ex21.m1.2.2.1.1.3.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex21.m1.2c">\mathcal{SW}(\pi^{*})=\sum_{i\in N}u_{i}(\pi^{*}(i))\leq n(t-1).\qed</annotation><annotation encoding="application/x-llamapun" id="S5.Ex21.m1.2d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_i ∈ italic_N end_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_i ) ) ≤ italic_n ( italic_t - 1 ) . italic_∎</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <section class="ltx_subsection" id="S5.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.1 </span>Erdős-Rényi Graphs</h3> <div class="ltx_para" id="S5.SS1.p1"> <p class="ltx_p" id="S5.SS1.p1.3">In our first model, we assume a set of agents, each pair of which is incompatible with probability <math alttext="1-p" class="ltx_Math" display="inline" id="S5.SS1.p1.1.m1.1"><semantics id="S5.SS1.p1.1.m1.1a"><mrow id="S5.SS1.p1.1.m1.1.1" xref="S5.SS1.p1.1.m1.1.1.cmml"><mn id="S5.SS1.p1.1.m1.1.1.2" xref="S5.SS1.p1.1.m1.1.1.2.cmml">1</mn><mo id="S5.SS1.p1.1.m1.1.1.1" xref="S5.SS1.p1.1.m1.1.1.1.cmml">−</mo><mi id="S5.SS1.p1.1.m1.1.1.3" xref="S5.SS1.p1.1.m1.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.1.m1.1b"><apply id="S5.SS1.p1.1.m1.1.1.cmml" xref="S5.SS1.p1.1.m1.1.1"><minus id="S5.SS1.p1.1.m1.1.1.1.cmml" xref="S5.SS1.p1.1.m1.1.1.1"></minus><cn id="S5.SS1.p1.1.m1.1.1.2.cmml" type="integer" xref="S5.SS1.p1.1.m1.1.1.2">1</cn><ci id="S5.SS1.p1.1.m1.1.1.3.cmml" xref="S5.SS1.p1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.1.m1.1c">1-p</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.1.m1.1d">1 - italic_p</annotation></semantics></math>. We model this as a symmetric aversion-to-enemies game by assigning a valuation of <math alttext="-n" class="ltx_Math" display="inline" id="S5.SS1.p1.2.m2.1"><semantics id="S5.SS1.p1.2.m2.1a"><mrow id="S5.SS1.p1.2.m2.1.1" xref="S5.SS1.p1.2.m2.1.1.cmml"><mo id="S5.SS1.p1.2.m2.1.1a" xref="S5.SS1.p1.2.m2.1.1.cmml">−</mo><mi id="S5.SS1.p1.2.m2.1.1.2" xref="S5.SS1.p1.2.m2.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.2.m2.1b"><apply id="S5.SS1.p1.2.m2.1.1.cmml" xref="S5.SS1.p1.2.m2.1.1"><minus id="S5.SS1.p1.2.m2.1.1.1.cmml" xref="S5.SS1.p1.2.m2.1.1"></minus><ci id="S5.SS1.p1.2.m2.1.1.2.cmml" xref="S5.SS1.p1.2.m2.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.2.m2.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.2.m2.1d">- italic_n</annotation></semantics></math> between incompatible agents and a valuation of <math alttext="1" class="ltx_Math" display="inline" id="S5.SS1.p1.3.m3.1"><semantics id="S5.SS1.p1.3.m3.1a"><mn id="S5.SS1.p1.3.m3.1.1" xref="S5.SS1.p1.3.m3.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.SS1.p1.3.m3.1b"><cn id="S5.SS1.p1.3.m3.1.1.cmml" type="integer" xref="S5.SS1.p1.3.m3.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p1.3.m3.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p1.3.m3.1d">1</annotation></semantics></math> between compatible agents. This corresponds to sampling its underlying graph as follows.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S5.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.1.1.1">Definition 5.2</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem2.p1"> <p class="ltx_p" id="S5.Thmtheorem2.p1.6">A <em class="ltx_emph ltx_font_italic" id="S5.Thmtheorem2.p1.6.1">weighted Erdős-Rényi graph</em> <math alttext="G=(n,p)" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.1.m1.2"><semantics id="S5.Thmtheorem2.p1.1.m1.2a"><mrow id="S5.Thmtheorem2.p1.1.m1.2.3" xref="S5.Thmtheorem2.p1.1.m1.2.3.cmml"><mi id="S5.Thmtheorem2.p1.1.m1.2.3.2" xref="S5.Thmtheorem2.p1.1.m1.2.3.2.cmml">G</mi><mo id="S5.Thmtheorem2.p1.1.m1.2.3.1" xref="S5.Thmtheorem2.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S5.Thmtheorem2.p1.1.m1.2.3.3.2" xref="S5.Thmtheorem2.p1.1.m1.2.3.3.1.cmml"><mo id="S5.Thmtheorem2.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem2.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem2.p1.1.m1.1.1" xref="S5.Thmtheorem2.p1.1.m1.1.1.cmml">n</mi><mo id="S5.Thmtheorem2.p1.1.m1.2.3.3.2.2" xref="S5.Thmtheorem2.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S5.Thmtheorem2.p1.1.m1.2.2" xref="S5.Thmtheorem2.p1.1.m1.2.2.cmml">p</mi><mo id="S5.Thmtheorem2.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S5.Thmtheorem2.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.1.m1.2b"><apply id="S5.Thmtheorem2.p1.1.m1.2.3.cmml" xref="S5.Thmtheorem2.p1.1.m1.2.3"><eq id="S5.Thmtheorem2.p1.1.m1.2.3.1.cmml" xref="S5.Thmtheorem2.p1.1.m1.2.3.1"></eq><ci id="S5.Thmtheorem2.p1.1.m1.2.3.2.cmml" xref="S5.Thmtheorem2.p1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S5.Thmtheorem2.p1.1.m1.2.3.3.1.cmml" xref="S5.Thmtheorem2.p1.1.m1.2.3.3.2"><ci id="S5.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S5.Thmtheorem2.p1.1.m1.1.1">𝑛</ci><ci id="S5.Thmtheorem2.p1.1.m1.2.2.cmml" xref="S5.Thmtheorem2.p1.1.m1.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.1.m1.2c">G=(n,p)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.1.m1.2d">italic_G = ( italic_n , italic_p )</annotation></semantics></math> is a random weighted graph with <math alttext="n" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.2.m2.1"><semantics id="S5.Thmtheorem2.p1.2.m2.1a"><mi id="S5.Thmtheorem2.p1.2.m2.1.1" xref="S5.Thmtheorem2.p1.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.2.m2.1b"><ci id="S5.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S5.Thmtheorem2.p1.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.2.m2.1d">italic_n</annotation></semantics></math> vertices such that, independently, each edge takes a weight of <math alttext="-n" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.3.m3.1"><semantics id="S5.Thmtheorem2.p1.3.m3.1a"><mrow id="S5.Thmtheorem2.p1.3.m3.1.1" xref="S5.Thmtheorem2.p1.3.m3.1.1.cmml"><mo id="S5.Thmtheorem2.p1.3.m3.1.1a" xref="S5.Thmtheorem2.p1.3.m3.1.1.cmml">−</mo><mi id="S5.Thmtheorem2.p1.3.m3.1.1.2" xref="S5.Thmtheorem2.p1.3.m3.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.3.m3.1b"><apply id="S5.Thmtheorem2.p1.3.m3.1.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1"><minus id="S5.Thmtheorem2.p1.3.m3.1.1.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1"></minus><ci id="S5.Thmtheorem2.p1.3.m3.1.1.2.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.3.m3.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.3.m3.1d">- italic_n</annotation></semantics></math> with probability <math alttext="p" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.4.m4.1"><semantics id="S5.Thmtheorem2.p1.4.m4.1a"><mi id="S5.Thmtheorem2.p1.4.m4.1.1" xref="S5.Thmtheorem2.p1.4.m4.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.4.m4.1b"><ci id="S5.Thmtheorem2.p1.4.m4.1.1.cmml" xref="S5.Thmtheorem2.p1.4.m4.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.4.m4.1c">p</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.4.m4.1d">italic_p</annotation></semantics></math> and a weight of <math alttext="1" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.5.m5.1"><semantics id="S5.Thmtheorem2.p1.5.m5.1a"><mn id="S5.Thmtheorem2.p1.5.m5.1.1" xref="S5.Thmtheorem2.p1.5.m5.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.5.m5.1b"><cn id="S5.Thmtheorem2.p1.5.m5.1.1.cmml" type="integer" xref="S5.Thmtheorem2.p1.5.m5.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.5.m5.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.5.m5.1d">1</annotation></semantics></math> with probability <math alttext="1-p" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.6.m6.1"><semantics id="S5.Thmtheorem2.p1.6.m6.1a"><mrow id="S5.Thmtheorem2.p1.6.m6.1.1" xref="S5.Thmtheorem2.p1.6.m6.1.1.cmml"><mn id="S5.Thmtheorem2.p1.6.m6.1.1.2" xref="S5.Thmtheorem2.p1.6.m6.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem2.p1.6.m6.1.1.1" xref="S5.Thmtheorem2.p1.6.m6.1.1.1.cmml">−</mo><mi id="S5.Thmtheorem2.p1.6.m6.1.1.3" xref="S5.Thmtheorem2.p1.6.m6.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.6.m6.1b"><apply id="S5.Thmtheorem2.p1.6.m6.1.1.cmml" xref="S5.Thmtheorem2.p1.6.m6.1.1"><minus id="S5.Thmtheorem2.p1.6.m6.1.1.1.cmml" xref="S5.Thmtheorem2.p1.6.m6.1.1.1"></minus><cn id="S5.Thmtheorem2.p1.6.m6.1.1.2.cmml" type="integer" xref="S5.Thmtheorem2.p1.6.m6.1.1.2">1</cn><ci id="S5.Thmtheorem2.p1.6.m6.1.1.3.cmml" xref="S5.Thmtheorem2.p1.6.m6.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.6.m6.1c">1-p</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.6.m6.1d">1 - italic_p</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S5.SS1.p2"> <p class="ltx_p" id="S5.SS1.p2.5">We will show that a simple and natural greedy algorithm yields a constant-factor approximation of the maximum welfare with high probability. For this, we use the <span class="ltx_text ltx_font_italic" id="S5.SS1.p2.5.1">greedy clique formation algorithm</span> in Section 5.2 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx11" title="">BK24</a>]</cite> applied to the subgraph <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS1.p2.1.m1.1"><semantics id="S5.SS1.p2.1.m1.1a"><msup id="S5.SS1.p2.1.m1.1.1" xref="S5.SS1.p2.1.m1.1.1.cmml"><mi id="S5.SS1.p2.1.m1.1.1.2" xref="S5.SS1.p2.1.m1.1.1.2.cmml">G</mi><mo id="S5.SS1.p2.1.m1.1.1.3" xref="S5.SS1.p2.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.1.m1.1b"><apply id="S5.SS1.p2.1.m1.1.1.cmml" xref="S5.SS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS1.p2.1.m1.1.1.1.cmml" xref="S5.SS1.p2.1.m1.1.1">superscript</csymbol><ci id="S5.SS1.p2.1.m1.1.1.2.cmml" xref="S5.SS1.p2.1.m1.1.1.2">𝐺</ci><ci id="S5.SS1.p2.1.m1.1.1.3.cmml" xref="S5.SS1.p2.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.1.m1.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.1.m1.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> formed by removing all negative edges from a graph <math alttext="G" class="ltx_Math" display="inline" id="S5.SS1.p2.2.m2.1"><semantics id="S5.SS1.p2.2.m2.1a"><mi id="S5.SS1.p2.2.m2.1.1" xref="S5.SS1.p2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.2.m2.1b"><ci id="S5.SS1.p2.2.m2.1.1.cmml" xref="S5.SS1.p2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.2.m2.1d">italic_G</annotation></semantics></math>. The algorithm greedily forms maximal cliques in <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS1.p2.3.m3.1"><semantics id="S5.SS1.p2.3.m3.1a"><msup id="S5.SS1.p2.3.m3.1.1" xref="S5.SS1.p2.3.m3.1.1.cmml"><mi id="S5.SS1.p2.3.m3.1.1.2" xref="S5.SS1.p2.3.m3.1.1.2.cmml">G</mi><mo id="S5.SS1.p2.3.m3.1.1.3" xref="S5.SS1.p2.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.3.m3.1b"><apply id="S5.SS1.p2.3.m3.1.1.cmml" xref="S5.SS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS1.p2.3.m3.1.1.1.cmml" xref="S5.SS1.p2.3.m3.1.1">superscript</csymbol><ci id="S5.SS1.p2.3.m3.1.1.2.cmml" xref="S5.SS1.p2.3.m3.1.1.2">𝐺</ci><ci id="S5.SS1.p2.3.m3.1.1.3.cmml" xref="S5.SS1.p2.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.3.m3.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.3.m3.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. as long as the cliques reach a certain size threshold <math alttext="t=\left\lceil\frac{\log_{1/p}n}{2}\right\rceil" class="ltx_Math" display="inline" id="S5.SS1.p2.4.m4.1"><semantics id="S5.SS1.p2.4.m4.1a"><mrow id="S5.SS1.p2.4.m4.1.2" xref="S5.SS1.p2.4.m4.1.2.cmml"><mi id="S5.SS1.p2.4.m4.1.2.2" xref="S5.SS1.p2.4.m4.1.2.2.cmml">t</mi><mo id="S5.SS1.p2.4.m4.1.2.1" xref="S5.SS1.p2.4.m4.1.2.1.cmml">=</mo><mrow id="S5.SS1.p2.4.m4.1.2.3.2" xref="S5.SS1.p2.4.m4.1.2.3.1.cmml"><mo id="S5.SS1.p2.4.m4.1.2.3.2.1" xref="S5.SS1.p2.4.m4.1.2.3.1.1.cmml">⌈</mo><mfrac id="S5.SS1.p2.4.m4.1.1" xref="S5.SS1.p2.4.m4.1.1.cmml"><mrow id="S5.SS1.p2.4.m4.1.1.2" xref="S5.SS1.p2.4.m4.1.1.2.cmml"><msub id="S5.SS1.p2.4.m4.1.1.2.1" xref="S5.SS1.p2.4.m4.1.1.2.1.cmml"><mi id="S5.SS1.p2.4.m4.1.1.2.1.2" xref="S5.SS1.p2.4.m4.1.1.2.1.2.cmml">log</mi><mrow id="S5.SS1.p2.4.m4.1.1.2.1.3" xref="S5.SS1.p2.4.m4.1.1.2.1.3.cmml"><mn id="S5.SS1.p2.4.m4.1.1.2.1.3.2" xref="S5.SS1.p2.4.m4.1.1.2.1.3.2.cmml">1</mn><mo id="S5.SS1.p2.4.m4.1.1.2.1.3.1" xref="S5.SS1.p2.4.m4.1.1.2.1.3.1.cmml">/</mo><mi id="S5.SS1.p2.4.m4.1.1.2.1.3.3" xref="S5.SS1.p2.4.m4.1.1.2.1.3.3.cmml">p</mi></mrow></msub><mo id="S5.SS1.p2.4.m4.1.1.2a" lspace="0.167em" xref="S5.SS1.p2.4.m4.1.1.2.cmml">⁡</mo><mi id="S5.SS1.p2.4.m4.1.1.2.2" xref="S5.SS1.p2.4.m4.1.1.2.2.cmml">n</mi></mrow><mn id="S5.SS1.p2.4.m4.1.1.3" xref="S5.SS1.p2.4.m4.1.1.3.cmml">2</mn></mfrac><mo id="S5.SS1.p2.4.m4.1.2.3.2.2" xref="S5.SS1.p2.4.m4.1.2.3.1.1.cmml">⌉</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.4.m4.1b"><apply id="S5.SS1.p2.4.m4.1.2.cmml" xref="S5.SS1.p2.4.m4.1.2"><eq id="S5.SS1.p2.4.m4.1.2.1.cmml" xref="S5.SS1.p2.4.m4.1.2.1"></eq><ci id="S5.SS1.p2.4.m4.1.2.2.cmml" xref="S5.SS1.p2.4.m4.1.2.2">𝑡</ci><apply id="S5.SS1.p2.4.m4.1.2.3.1.cmml" xref="S5.SS1.p2.4.m4.1.2.3.2"><ceiling id="S5.SS1.p2.4.m4.1.2.3.1.1.cmml" xref="S5.SS1.p2.4.m4.1.2.3.2.1"></ceiling><apply id="S5.SS1.p2.4.m4.1.1.cmml" xref="S5.SS1.p2.4.m4.1.1"><divide id="S5.SS1.p2.4.m4.1.1.1.cmml" xref="S5.SS1.p2.4.m4.1.1"></divide><apply id="S5.SS1.p2.4.m4.1.1.2.cmml" xref="S5.SS1.p2.4.m4.1.1.2"><apply id="S5.SS1.p2.4.m4.1.1.2.1.cmml" xref="S5.SS1.p2.4.m4.1.1.2.1"><csymbol cd="ambiguous" id="S5.SS1.p2.4.m4.1.1.2.1.1.cmml" xref="S5.SS1.p2.4.m4.1.1.2.1">subscript</csymbol><log id="S5.SS1.p2.4.m4.1.1.2.1.2.cmml" xref="S5.SS1.p2.4.m4.1.1.2.1.2"></log><apply id="S5.SS1.p2.4.m4.1.1.2.1.3.cmml" xref="S5.SS1.p2.4.m4.1.1.2.1.3"><divide id="S5.SS1.p2.4.m4.1.1.2.1.3.1.cmml" xref="S5.SS1.p2.4.m4.1.1.2.1.3.1"></divide><cn id="S5.SS1.p2.4.m4.1.1.2.1.3.2.cmml" type="integer" xref="S5.SS1.p2.4.m4.1.1.2.1.3.2">1</cn><ci id="S5.SS1.p2.4.m4.1.1.2.1.3.3.cmml" xref="S5.SS1.p2.4.m4.1.1.2.1.3.3">𝑝</ci></apply></apply><ci id="S5.SS1.p2.4.m4.1.1.2.2.cmml" xref="S5.SS1.p2.4.m4.1.1.2.2">𝑛</ci></apply><cn id="S5.SS1.p2.4.m4.1.1.3.cmml" type="integer" xref="S5.SS1.p2.4.m4.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.4.m4.1c">t=\left\lceil\frac{\log_{1/p}n}{2}\right\rceil</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.4.m4.1d">italic_t = ⌈ divide start_ARG roman_log start_POSTSUBSCRIPT 1 / italic_p end_POSTSUBSCRIPT italic_n end_ARG start_ARG 2 end_ARG ⌉</annotation></semantics></math>. If, at any point, the size of the created maximal clique is smaller than <math alttext="t" class="ltx_Math" display="inline" id="S5.SS1.p2.5.m5.1"><semantics id="S5.SS1.p2.5.m5.1a"><mi id="S5.SS1.p2.5.m5.1.1" xref="S5.SS1.p2.5.m5.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p2.5.m5.1b"><ci id="S5.SS1.p2.5.m5.1.1.cmml" xref="S5.SS1.p2.5.m5.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p2.5.m5.1c">t</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p2.5.m5.1d">italic_t</annotation></semantics></math>, the algorithm outputs the existing cliques as coalitions, and assigns singleton coalitions to the remaining agents. The following theorem measures the performance of this algorithm. It follows from the proof of Theorem 5.2 by Bullinger and Kraiczy <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx11" title="">BK24</a>]</cite>.<span class="ltx_note ltx_role_footnote" id="footnote5"><sup class="ltx_note_mark">5</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">5</sup><span class="ltx_tag ltx_tag_note">5</span>We remark that in their model the edges in the cliques occur with probability <math alttext="p" class="ltx_Math" display="inline" id="footnote5.m1.1"><semantics id="footnote5.m1.1b"><mi id="footnote5.m1.1.1" xref="footnote5.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="footnote5.m1.1c"><ci id="footnote5.m1.1.1.cmml" xref="footnote5.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote5.m1.1d">p</annotation><annotation encoding="application/x-llamapun" id="footnote5.m1.1e">italic_p</annotation></semantics></math> whereas they occur with probability <math alttext="1-p" class="ltx_Math" display="inline" id="footnote5.m2.1"><semantics id="footnote5.m2.1b"><mrow id="footnote5.m2.1.1" xref="footnote5.m2.1.1.cmml"><mn id="footnote5.m2.1.1.2" xref="footnote5.m2.1.1.2.cmml">1</mn><mo id="footnote5.m2.1.1.1" xref="footnote5.m2.1.1.1.cmml">−</mo><mi id="footnote5.m2.1.1.3" xref="footnote5.m2.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote5.m2.1c"><apply id="footnote5.m2.1.1.cmml" xref="footnote5.m2.1.1"><minus id="footnote5.m2.1.1.1.cmml" xref="footnote5.m2.1.1.1"></minus><cn id="footnote5.m2.1.1.2.cmml" type="integer" xref="footnote5.m2.1.1.2">1</cn><ci id="footnote5.m2.1.1.3.cmml" xref="footnote5.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote5.m2.1d">1-p</annotation><annotation encoding="application/x-llamapun" id="footnote5.m2.1e">1 - italic_p</annotation></semantics></math> in our model.</span></span></span></p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S5.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.1.1.1">Theorem 5.3</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.2.2"> </span>(<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx11" title="">BK24</a>]</cite>)<span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.3.3">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem3.p1"> <p class="ltx_p" id="S5.Thmtheorem3.p1.5"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p1.5.5">Consider an Erdős-Rényi graph <math alttext="G=(n,p)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.1.1.m1.2"><semantics id="S5.Thmtheorem3.p1.1.1.m1.2a"><mrow id="S5.Thmtheorem3.p1.1.1.m1.2.3" xref="S5.Thmtheorem3.p1.1.1.m1.2.3.cmml"><mi id="S5.Thmtheorem3.p1.1.1.m1.2.3.2" xref="S5.Thmtheorem3.p1.1.1.m1.2.3.2.cmml">G</mi><mo id="S5.Thmtheorem3.p1.1.1.m1.2.3.1" xref="S5.Thmtheorem3.p1.1.1.m1.2.3.1.cmml">=</mo><mrow id="S5.Thmtheorem3.p1.1.1.m1.2.3.3.2" xref="S5.Thmtheorem3.p1.1.1.m1.2.3.3.1.cmml"><mo id="S5.Thmtheorem3.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.1.1.m1.2.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.1.1.m1.1.1" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.cmml">n</mi><mo id="S5.Thmtheorem3.p1.1.1.m1.2.3.3.2.2" xref="S5.Thmtheorem3.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.1.1.m1.2.2" xref="S5.Thmtheorem3.p1.1.1.m1.2.2.cmml">p</mi><mo id="S5.Thmtheorem3.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S5.Thmtheorem3.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.1.1.m1.2b"><apply id="S5.Thmtheorem3.p1.1.1.m1.2.3.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.2.3"><eq id="S5.Thmtheorem3.p1.1.1.m1.2.3.1.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.2.3.1"></eq><ci id="S5.Thmtheorem3.p1.1.1.m1.2.3.2.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S5.Thmtheorem3.p1.1.1.m1.2.3.3.1.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.2.3.3.2"><ci id="S5.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1">𝑛</ci><ci id="S5.Thmtheorem3.p1.1.1.m1.2.2.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.1.1.m1.2c">G=(n,p)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.1.1.m1.2d">italic_G = ( italic_n , italic_p )</annotation></semantics></math> and let <math alttext="b=\frac{1}{p}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.2.2.m2.1"><semantics id="S5.Thmtheorem3.p1.2.2.m2.1a"><mrow id="S5.Thmtheorem3.p1.2.2.m2.1.1" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem3.p1.2.2.m2.1.1.2" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.2.cmml">b</mi><mo id="S5.Thmtheorem3.p1.2.2.m2.1.1.1" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.1.cmml">=</mo><mfrac id="S5.Thmtheorem3.p1.2.2.m2.1.1.3" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.3.cmml"><mn id="S5.Thmtheorem3.p1.2.2.m2.1.1.3.2" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.3.2.cmml">1</mn><mi id="S5.Thmtheorem3.p1.2.2.m2.1.1.3.3" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.3.3.cmml">p</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.2.2.m2.1b"><apply id="S5.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.1"><eq id="S5.Thmtheorem3.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.1"></eq><ci id="S5.Thmtheorem3.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.2">𝑏</ci><apply id="S5.Thmtheorem3.p1.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.3"><divide id="S5.Thmtheorem3.p1.2.2.m2.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.3"></divide><cn id="S5.Thmtheorem3.p1.2.2.m2.1.1.3.2.cmml" type="integer" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.3.2">1</cn><ci id="S5.Thmtheorem3.p1.2.2.m2.1.1.3.3.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.2.2.m2.1c">b=\frac{1}{p}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.2.2.m2.1d">italic_b = divide start_ARG 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Then, with probability at least <math alttext="1-e^{-\Omega\left(\log_{b}^{3}n\right)}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.3.3.m3.1"><semantics id="S5.Thmtheorem3.p1.3.3.m3.1a"><mrow id="S5.Thmtheorem3.p1.3.3.m3.1.2" xref="S5.Thmtheorem3.p1.3.3.m3.1.2.cmml"><mn id="S5.Thmtheorem3.p1.3.3.m3.1.2.2" xref="S5.Thmtheorem3.p1.3.3.m3.1.2.2.cmml">1</mn><mo id="S5.Thmtheorem3.p1.3.3.m3.1.2.1" xref="S5.Thmtheorem3.p1.3.3.m3.1.2.1.cmml">−</mo><msup id="S5.Thmtheorem3.p1.3.3.m3.1.2.3" xref="S5.Thmtheorem3.p1.3.3.m3.1.2.3.cmml"><mi id="S5.Thmtheorem3.p1.3.3.m3.1.2.3.2" xref="S5.Thmtheorem3.p1.3.3.m3.1.2.3.2.cmml">e</mi><mrow id="S5.Thmtheorem3.p1.3.3.m3.1.1.1" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.cmml"><mo id="S5.Thmtheorem3.p1.3.3.m3.1.1.1a" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.cmml">−</mo><mrow id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.3" mathvariant="normal" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.3.cmml">Ω</mi><mo id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.2" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.2" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.cmml"><msubsup id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.2.2" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.2.2.cmml">log</mi><mi id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.2.3" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.2.3.cmml">b</mi><mn id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.3" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.3.cmml">3</mn></msubsup><mo id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1a" lspace="0.167em" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.cmml">⁡</mo><mi id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.2" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.3.3.m3.1b"><apply id="S5.Thmtheorem3.p1.3.3.m3.1.2.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.2"><minus id="S5.Thmtheorem3.p1.3.3.m3.1.2.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.2.1"></minus><cn id="S5.Thmtheorem3.p1.3.3.m3.1.2.2.cmml" type="integer" xref="S5.Thmtheorem3.p1.3.3.m3.1.2.2">1</cn><apply id="S5.Thmtheorem3.p1.3.3.m3.1.2.3.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.3.3.m3.1.2.3.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.2.3">superscript</csymbol><ci id="S5.Thmtheorem3.p1.3.3.m3.1.2.3.2.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.2.3.2">𝑒</ci><apply id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1"><minus id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1"></minus><apply id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1"><times id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.2"></times><ci id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.3">Ω</ci><apply id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1"><apply id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1">superscript</csymbol><apply id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.2.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1">subscript</csymbol><log id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.2.2.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.2.2"></log><ci id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.2.3.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.2.3">𝑏</ci></apply><cn id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.1.3">3</cn></apply><ci id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.3.3.m3.1c">1-e^{-\Omega\left(\log_{b}^{3}n\right)}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.3.3.m3.1d">1 - italic_e start_POSTSUPERSCRIPT - roman_Ω ( roman_log start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_n ) end_POSTSUPERSCRIPT</annotation></semantics></math>, the greedy clique formation algorithm assigns all except at most <math alttext="\frac{n}{\log_{b}^{2}n}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.4.4.m4.1"><semantics id="S5.Thmtheorem3.p1.4.4.m4.1a"><mfrac id="S5.Thmtheorem3.p1.4.4.m4.1.1" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.cmml"><mi id="S5.Thmtheorem3.p1.4.4.m4.1.1.2" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.2.cmml">n</mi><mrow id="S5.Thmtheorem3.p1.4.4.m4.1.1.3" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.cmml"><msubsup id="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.cmml"><mi id="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.2.2" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.2.2.cmml">log</mi><mi id="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.2.3" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.2.3.cmml">b</mi><mn id="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.3" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.3.cmml">2</mn></msubsup><mo id="S5.Thmtheorem3.p1.4.4.m4.1.1.3a" lspace="0.167em" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.cmml">⁡</mo><mi id="S5.Thmtheorem3.p1.4.4.m4.1.1.3.2" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.2.cmml">n</mi></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.4.4.m4.1b"><apply id="S5.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1"><divide id="S5.Thmtheorem3.p1.4.4.m4.1.1.1.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1"></divide><ci id="S5.Thmtheorem3.p1.4.4.m4.1.1.2.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.2">𝑛</ci><apply id="S5.Thmtheorem3.p1.4.4.m4.1.1.3.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3"><apply id="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.1.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1">superscript</csymbol><apply id="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.2.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.2.1.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1">subscript</csymbol><log id="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.2.2.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.2.2"></log><ci id="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.2.3.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.2.3">𝑏</ci></apply><cn id="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.1.3">2</cn></apply><ci id="S5.Thmtheorem3.p1.4.4.m4.1.1.3.2.cmml" xref="S5.Thmtheorem3.p1.4.4.m4.1.1.3.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.4.4.m4.1c">\frac{n}{\log_{b}^{2}n}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.4.4.m4.1d">divide start_ARG italic_n end_ARG start_ARG roman_log start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n end_ARG</annotation></semantics></math> to cliques of size <math alttext="\left\lceil\frac{\log_{b}n}{2}\right\rceil" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.5.5.m5.1"><semantics id="S5.Thmtheorem3.p1.5.5.m5.1a"><mrow id="S5.Thmtheorem3.p1.5.5.m5.1.2.2" xref="S5.Thmtheorem3.p1.5.5.m5.1.2.1.cmml"><mo id="S5.Thmtheorem3.p1.5.5.m5.1.2.2.1" xref="S5.Thmtheorem3.p1.5.5.m5.1.2.1.1.cmml">⌈</mo><mfrac id="S5.Thmtheorem3.p1.5.5.m5.1.1" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.cmml"><mrow id="S5.Thmtheorem3.p1.5.5.m5.1.1.2" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2.cmml"><msub id="S5.Thmtheorem3.p1.5.5.m5.1.1.2.1" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2.1.cmml"><mi id="S5.Thmtheorem3.p1.5.5.m5.1.1.2.1.2" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2.1.2.cmml">log</mi><mi id="S5.Thmtheorem3.p1.5.5.m5.1.1.2.1.3" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2.1.3.cmml">b</mi></msub><mo id="S5.Thmtheorem3.p1.5.5.m5.1.1.2a" lspace="0.167em" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2.cmml">⁡</mo><mi id="S5.Thmtheorem3.p1.5.5.m5.1.1.2.2" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2.2.cmml">n</mi></mrow><mn id="S5.Thmtheorem3.p1.5.5.m5.1.1.3" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.3.cmml">2</mn></mfrac><mo id="S5.Thmtheorem3.p1.5.5.m5.1.2.2.2" xref="S5.Thmtheorem3.p1.5.5.m5.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.5.5.m5.1b"><apply id="S5.Thmtheorem3.p1.5.5.m5.1.2.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.2.2"><ceiling id="S5.Thmtheorem3.p1.5.5.m5.1.2.1.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.2.2.1"></ceiling><apply id="S5.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1"><divide id="S5.Thmtheorem3.p1.5.5.m5.1.1.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1"></divide><apply id="S5.Thmtheorem3.p1.5.5.m5.1.1.2.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2"><apply id="S5.Thmtheorem3.p1.5.5.m5.1.1.2.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.5.5.m5.1.1.2.1.1.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2.1">subscript</csymbol><log id="S5.Thmtheorem3.p1.5.5.m5.1.1.2.1.2.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2.1.2"></log><ci id="S5.Thmtheorem3.p1.5.5.m5.1.1.2.1.3.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2.1.3">𝑏</ci></apply><ci id="S5.Thmtheorem3.p1.5.5.m5.1.1.2.2.cmml" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.2.2">𝑛</ci></apply><cn id="S5.Thmtheorem3.p1.5.5.m5.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.5.5.m5.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.5.5.m5.1c">\left\lceil\frac{\log_{b}n}{2}\right\rceil</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.5.5.m5.1d">⌈ divide start_ARG roman_log start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_n end_ARG start_ARG 2 end_ARG ⌉</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S5.SS1.p3"> <p class="ltx_p" id="S5.SS1.p3.1">We apply the theorem to obtain a constant-factor approximation of maximum welfare. Essentially, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem3" title="Theorem 5.3 ([BK24]). ‣ 5.1 Erdős-Rényi Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">5.3</span></a> allows to obtain a coalition with social welfare <math alttext="\Theta(n\log n)" class="ltx_Math" display="inline" id="S5.SS1.p3.1.m1.1"><semantics id="S5.SS1.p3.1.m1.1a"><mrow id="S5.SS1.p3.1.m1.1.1" xref="S5.SS1.p3.1.m1.1.1.cmml"><mi id="S5.SS1.p3.1.m1.1.1.3" mathvariant="normal" xref="S5.SS1.p3.1.m1.1.1.3.cmml">Θ</mi><mo id="S5.SS1.p3.1.m1.1.1.2" xref="S5.SS1.p3.1.m1.1.1.2.cmml">⁢</mo><mrow id="S5.SS1.p3.1.m1.1.1.1.1" xref="S5.SS1.p3.1.m1.1.1.1.1.1.cmml"><mo id="S5.SS1.p3.1.m1.1.1.1.1.2" stretchy="false" xref="S5.SS1.p3.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS1.p3.1.m1.1.1.1.1.1" xref="S5.SS1.p3.1.m1.1.1.1.1.1.cmml"><mi id="S5.SS1.p3.1.m1.1.1.1.1.1.2" xref="S5.SS1.p3.1.m1.1.1.1.1.1.2.cmml">n</mi><mo id="S5.SS1.p3.1.m1.1.1.1.1.1.1" lspace="0.167em" xref="S5.SS1.p3.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.SS1.p3.1.m1.1.1.1.1.1.3" xref="S5.SS1.p3.1.m1.1.1.1.1.1.3.cmml"><mi id="S5.SS1.p3.1.m1.1.1.1.1.1.3.1" xref="S5.SS1.p3.1.m1.1.1.1.1.1.3.1.cmml">log</mi><mo id="S5.SS1.p3.1.m1.1.1.1.1.1.3a" lspace="0.167em" xref="S5.SS1.p3.1.m1.1.1.1.1.1.3.cmml">⁡</mo><mi id="S5.SS1.p3.1.m1.1.1.1.1.1.3.2" xref="S5.SS1.p3.1.m1.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S5.SS1.p3.1.m1.1.1.1.1.3" stretchy="false" xref="S5.SS1.p3.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p3.1.m1.1b"><apply id="S5.SS1.p3.1.m1.1.1.cmml" xref="S5.SS1.p3.1.m1.1.1"><times id="S5.SS1.p3.1.m1.1.1.2.cmml" xref="S5.SS1.p3.1.m1.1.1.2"></times><ci id="S5.SS1.p3.1.m1.1.1.3.cmml" xref="S5.SS1.p3.1.m1.1.1.3">Θ</ci><apply id="S5.SS1.p3.1.m1.1.1.1.1.1.cmml" xref="S5.SS1.p3.1.m1.1.1.1.1"><times id="S5.SS1.p3.1.m1.1.1.1.1.1.1.cmml" xref="S5.SS1.p3.1.m1.1.1.1.1.1.1"></times><ci id="S5.SS1.p3.1.m1.1.1.1.1.1.2.cmml" xref="S5.SS1.p3.1.m1.1.1.1.1.1.2">𝑛</ci><apply id="S5.SS1.p3.1.m1.1.1.1.1.1.3.cmml" xref="S5.SS1.p3.1.m1.1.1.1.1.1.3"><log id="S5.SS1.p3.1.m1.1.1.1.1.1.3.1.cmml" xref="S5.SS1.p3.1.m1.1.1.1.1.1.3.1"></log><ci id="S5.SS1.p3.1.m1.1.1.1.1.1.3.2.cmml" xref="S5.SS1.p3.1.m1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p3.1.m1.1c">\Theta(n\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p3.1.m1.1d">roman_Θ ( italic_n roman_log italic_n )</annotation></semantics></math> while we apply <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem1" title="Lemma 5.1. ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5.1</span></a> to show that the maximum welfare is also of this order.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S5.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem4.1.1.1">Theorem 5.4</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem4.p1"> <p class="ltx_p" id="S5.Thmtheorem4.p1.2"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem4.p1.2.2">Let <math alttext="p\in(0,1)" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.1.1.m1.2"><semantics id="S5.Thmtheorem4.p1.1.1.m1.2a"><mrow id="S5.Thmtheorem4.p1.1.1.m1.2.3" xref="S5.Thmtheorem4.p1.1.1.m1.2.3.cmml"><mi id="S5.Thmtheorem4.p1.1.1.m1.2.3.2" xref="S5.Thmtheorem4.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S5.Thmtheorem4.p1.1.1.m1.2.3.1" xref="S5.Thmtheorem4.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S5.Thmtheorem4.p1.1.1.m1.2.3.3.2" xref="S5.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml"><mo id="S5.Thmtheorem4.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S5.Thmtheorem4.p1.1.1.m1.1.1" xref="S5.Thmtheorem4.p1.1.1.m1.1.1.cmml">0</mn><mo id="S5.Thmtheorem4.p1.1.1.m1.2.3.3.2.2" xref="S5.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml">,</mo><mn id="S5.Thmtheorem4.p1.1.1.m1.2.2" xref="S5.Thmtheorem4.p1.1.1.m1.2.2.cmml">1</mn><mo id="S5.Thmtheorem4.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S5.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.1.1.m1.2b"><apply id="S5.Thmtheorem4.p1.1.1.m1.2.3.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.2.3"><in id="S5.Thmtheorem4.p1.1.1.m1.2.3.1.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.2.3.1"></in><ci id="S5.Thmtheorem4.p1.1.1.m1.2.3.2.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S5.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.2.3.3.2"><cn id="S5.Thmtheorem4.p1.1.1.m1.1.1.cmml" type="integer" xref="S5.Thmtheorem4.p1.1.1.m1.1.1">0</cn><cn id="S5.Thmtheorem4.p1.1.1.m1.2.2.cmml" type="integer" xref="S5.Thmtheorem4.p1.1.1.m1.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.1.1.m1.2c">p\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.1.1.m1.2d">italic_p ∈ ( 0 , 1 )</annotation></semantics></math>. Then there exists a constant-factor approximation algorithm for aversion-to-enemies games given by a weighted Erdős-Rényi graph <math alttext="G=(n,p)" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.2.2.m2.2"><semantics id="S5.Thmtheorem4.p1.2.2.m2.2a"><mrow id="S5.Thmtheorem4.p1.2.2.m2.2.3" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.cmml"><mi id="S5.Thmtheorem4.p1.2.2.m2.2.3.2" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.2.cmml">G</mi><mo id="S5.Thmtheorem4.p1.2.2.m2.2.3.1" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.1.cmml">=</mo><mrow id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.2" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.1.cmml"><mo id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem4.p1.2.2.m2.1.1" xref="S5.Thmtheorem4.p1.2.2.m2.1.1.cmml">n</mi><mo id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.2.2" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.1.cmml">,</mo><mi id="S5.Thmtheorem4.p1.2.2.m2.2.2" xref="S5.Thmtheorem4.p1.2.2.m2.2.2.cmml">p</mi><mo id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.2.3" stretchy="false" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.2.2.m2.2b"><apply id="S5.Thmtheorem4.p1.2.2.m2.2.3.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.3"><eq id="S5.Thmtheorem4.p1.2.2.m2.2.3.1.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.1"></eq><ci id="S5.Thmtheorem4.p1.2.2.m2.2.3.2.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.2">𝐺</ci><interval closure="open" id="S5.Thmtheorem4.p1.2.2.m2.2.3.3.1.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.3.3.2"><ci id="S5.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.1.1">𝑛</ci><ci id="S5.Thmtheorem4.p1.2.2.m2.2.2.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.2.2.m2.2c">G=(n,p)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.2.2.m2.2d">italic_G = ( italic_n , italic_p )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S5.SS1.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.SS1.1.p1"> <p class="ltx_p" id="S5.SS1.1.p1.7">Note that <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS1.1.p1.1.m1.1"><semantics id="S5.SS1.1.p1.1.m1.1a"><msup id="S5.SS1.1.p1.1.m1.1.1" xref="S5.SS1.1.p1.1.m1.1.1.cmml"><mi id="S5.SS1.1.p1.1.m1.1.1.2" xref="S5.SS1.1.p1.1.m1.1.1.2.cmml">G</mi><mo id="S5.SS1.1.p1.1.m1.1.1.3" xref="S5.SS1.1.p1.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.1.m1.1b"><apply id="S5.SS1.1.p1.1.m1.1.1.cmml" xref="S5.SS1.1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS1.1.p1.1.m1.1.1.1.cmml" xref="S5.SS1.1.p1.1.m1.1.1">superscript</csymbol><ci id="S5.SS1.1.p1.1.m1.1.1.2.cmml" xref="S5.SS1.1.p1.1.m1.1.1.2">𝐺</ci><ci id="S5.SS1.1.p1.1.m1.1.1.3.cmml" xref="S5.SS1.1.p1.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.1.m1.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.1.m1.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is a weighted Erdős-Rényi graph where every edge was sampled with probability <math alttext="p" class="ltx_Math" display="inline" id="S5.SS1.1.p1.2.m2.1"><semantics id="S5.SS1.1.p1.2.m2.1a"><mi id="S5.SS1.1.p1.2.m2.1.1" xref="S5.SS1.1.p1.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.2.m2.1b"><ci id="S5.SS1.1.p1.2.m2.1.1.cmml" xref="S5.SS1.1.p1.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.2.m2.1d">italic_p</annotation></semantics></math>, i.e. <math alttext="G^{\prime}=(n,p)" class="ltx_Math" display="inline" id="S5.SS1.1.p1.3.m3.2"><semantics id="S5.SS1.1.p1.3.m3.2a"><mrow id="S5.SS1.1.p1.3.m3.2.3" xref="S5.SS1.1.p1.3.m3.2.3.cmml"><msup id="S5.SS1.1.p1.3.m3.2.3.2" xref="S5.SS1.1.p1.3.m3.2.3.2.cmml"><mi id="S5.SS1.1.p1.3.m3.2.3.2.2" xref="S5.SS1.1.p1.3.m3.2.3.2.2.cmml">G</mi><mo id="S5.SS1.1.p1.3.m3.2.3.2.3" xref="S5.SS1.1.p1.3.m3.2.3.2.3.cmml">′</mo></msup><mo id="S5.SS1.1.p1.3.m3.2.3.1" xref="S5.SS1.1.p1.3.m3.2.3.1.cmml">=</mo><mrow id="S5.SS1.1.p1.3.m3.2.3.3.2" xref="S5.SS1.1.p1.3.m3.2.3.3.1.cmml"><mo id="S5.SS1.1.p1.3.m3.2.3.3.2.1" stretchy="false" xref="S5.SS1.1.p1.3.m3.2.3.3.1.cmml">(</mo><mi id="S5.SS1.1.p1.3.m3.1.1" xref="S5.SS1.1.p1.3.m3.1.1.cmml">n</mi><mo id="S5.SS1.1.p1.3.m3.2.3.3.2.2" xref="S5.SS1.1.p1.3.m3.2.3.3.1.cmml">,</mo><mi id="S5.SS1.1.p1.3.m3.2.2" xref="S5.SS1.1.p1.3.m3.2.2.cmml">p</mi><mo id="S5.SS1.1.p1.3.m3.2.3.3.2.3" stretchy="false" xref="S5.SS1.1.p1.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.3.m3.2b"><apply id="S5.SS1.1.p1.3.m3.2.3.cmml" xref="S5.SS1.1.p1.3.m3.2.3"><eq id="S5.SS1.1.p1.3.m3.2.3.1.cmml" xref="S5.SS1.1.p1.3.m3.2.3.1"></eq><apply id="S5.SS1.1.p1.3.m3.2.3.2.cmml" xref="S5.SS1.1.p1.3.m3.2.3.2"><csymbol cd="ambiguous" id="S5.SS1.1.p1.3.m3.2.3.2.1.cmml" xref="S5.SS1.1.p1.3.m3.2.3.2">superscript</csymbol><ci id="S5.SS1.1.p1.3.m3.2.3.2.2.cmml" xref="S5.SS1.1.p1.3.m3.2.3.2.2">𝐺</ci><ci id="S5.SS1.1.p1.3.m3.2.3.2.3.cmml" xref="S5.SS1.1.p1.3.m3.2.3.2.3">′</ci></apply><interval closure="open" id="S5.SS1.1.p1.3.m3.2.3.3.1.cmml" xref="S5.SS1.1.p1.3.m3.2.3.3.2"><ci id="S5.SS1.1.p1.3.m3.1.1.cmml" xref="S5.SS1.1.p1.3.m3.1.1">𝑛</ci><ci id="S5.SS1.1.p1.3.m3.2.2.cmml" xref="S5.SS1.1.p1.3.m3.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.3.m3.2c">G^{\prime}=(n,p)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.3.m3.2d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( italic_n , italic_p )</annotation></semantics></math>. Let <math alttext="b=\frac{1}{p}" class="ltx_Math" display="inline" id="S5.SS1.1.p1.4.m4.1"><semantics id="S5.SS1.1.p1.4.m4.1a"><mrow id="S5.SS1.1.p1.4.m4.1.1" xref="S5.SS1.1.p1.4.m4.1.1.cmml"><mi id="S5.SS1.1.p1.4.m4.1.1.2" xref="S5.SS1.1.p1.4.m4.1.1.2.cmml">b</mi><mo id="S5.SS1.1.p1.4.m4.1.1.1" xref="S5.SS1.1.p1.4.m4.1.1.1.cmml">=</mo><mfrac id="S5.SS1.1.p1.4.m4.1.1.3" xref="S5.SS1.1.p1.4.m4.1.1.3.cmml"><mn id="S5.SS1.1.p1.4.m4.1.1.3.2" xref="S5.SS1.1.p1.4.m4.1.1.3.2.cmml">1</mn><mi id="S5.SS1.1.p1.4.m4.1.1.3.3" xref="S5.SS1.1.p1.4.m4.1.1.3.3.cmml">p</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.4.m4.1b"><apply id="S5.SS1.1.p1.4.m4.1.1.cmml" xref="S5.SS1.1.p1.4.m4.1.1"><eq id="S5.SS1.1.p1.4.m4.1.1.1.cmml" xref="S5.SS1.1.p1.4.m4.1.1.1"></eq><ci id="S5.SS1.1.p1.4.m4.1.1.2.cmml" xref="S5.SS1.1.p1.4.m4.1.1.2">𝑏</ci><apply id="S5.SS1.1.p1.4.m4.1.1.3.cmml" xref="S5.SS1.1.p1.4.m4.1.1.3"><divide id="S5.SS1.1.p1.4.m4.1.1.3.1.cmml" xref="S5.SS1.1.p1.4.m4.1.1.3"></divide><cn id="S5.SS1.1.p1.4.m4.1.1.3.2.cmml" type="integer" xref="S5.SS1.1.p1.4.m4.1.1.3.2">1</cn><ci id="S5.SS1.1.p1.4.m4.1.1.3.3.cmml" xref="S5.SS1.1.p1.4.m4.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.4.m4.1c">b=\frac{1}{p}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.4.m4.1d">italic_b = divide start_ARG 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> and let <math alttext="\pi" class="ltx_Math" display="inline" id="S5.SS1.1.p1.5.m5.1"><semantics id="S5.SS1.1.p1.5.m5.1a"><mi id="S5.SS1.1.p1.5.m5.1.1" xref="S5.SS1.1.p1.5.m5.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.5.m5.1b"><ci id="S5.SS1.1.p1.5.m5.1.1.cmml" xref="S5.SS1.1.p1.5.m5.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.5.m5.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.5.m5.1d">italic_π</annotation></semantics></math> be the resulting partition after applying the greedy clique formation algorithm to the subgraph <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS1.1.p1.6.m6.1"><semantics id="S5.SS1.1.p1.6.m6.1a"><msup id="S5.SS1.1.p1.6.m6.1.1" xref="S5.SS1.1.p1.6.m6.1.1.cmml"><mi id="S5.SS1.1.p1.6.m6.1.1.2" xref="S5.SS1.1.p1.6.m6.1.1.2.cmml">G</mi><mo id="S5.SS1.1.p1.6.m6.1.1.3" xref="S5.SS1.1.p1.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.6.m6.1b"><apply id="S5.SS1.1.p1.6.m6.1.1.cmml" xref="S5.SS1.1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S5.SS1.1.p1.6.m6.1.1.1.cmml" xref="S5.SS1.1.p1.6.m6.1.1">superscript</csymbol><ci id="S5.SS1.1.p1.6.m6.1.1.2.cmml" xref="S5.SS1.1.p1.6.m6.1.1.2">𝐺</ci><ci id="S5.SS1.1.p1.6.m6.1.1.3.cmml" xref="S5.SS1.1.p1.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.6.m6.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.6.m6.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem3" title="Theorem 5.3 ([BK24]). ‣ 5.1 Erdős-Rényi Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">5.3</span></a>, it follows that <math alttext="\mathbb{E}[\mathcal{SW}(\pi)]=\Theta(n\log n)" class="ltx_Math" display="inline" id="S5.SS1.1.p1.7.m7.3"><semantics id="S5.SS1.1.p1.7.m7.3a"><mrow id="S5.SS1.1.p1.7.m7.3.3" xref="S5.SS1.1.p1.7.m7.3.3.cmml"><mrow id="S5.SS1.1.p1.7.m7.2.2.1" xref="S5.SS1.1.p1.7.m7.2.2.1.cmml"><mi id="S5.SS1.1.p1.7.m7.2.2.1.3" xref="S5.SS1.1.p1.7.m7.2.2.1.3.cmml">𝔼</mi><mo id="S5.SS1.1.p1.7.m7.2.2.1.2" xref="S5.SS1.1.p1.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="S5.SS1.1.p1.7.m7.2.2.1.1.1" xref="S5.SS1.1.p1.7.m7.2.2.1.1.2.cmml"><mo id="S5.SS1.1.p1.7.m7.2.2.1.1.1.2" stretchy="false" xref="S5.SS1.1.p1.7.m7.2.2.1.1.2.1.cmml">[</mo><mrow id="S5.SS1.1.p1.7.m7.2.2.1.1.1.1" xref="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.2" xref="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.2.cmml">𝒮</mi><mo id="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.1" xref="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.3" xref="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.3.cmml">𝒲</mi><mo id="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.1a" xref="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.4.2" xref="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.cmml"><mo id="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.4.2.1" stretchy="false" xref="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.cmml">(</mo><mi id="S5.SS1.1.p1.7.m7.1.1" xref="S5.SS1.1.p1.7.m7.1.1.cmml">π</mi><mo id="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.4.2.2" stretchy="false" xref="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS1.1.p1.7.m7.2.2.1.1.1.3" stretchy="false" xref="S5.SS1.1.p1.7.m7.2.2.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S5.SS1.1.p1.7.m7.3.3.3" xref="S5.SS1.1.p1.7.m7.3.3.3.cmml">=</mo><mrow id="S5.SS1.1.p1.7.m7.3.3.2" xref="S5.SS1.1.p1.7.m7.3.3.2.cmml"><mi id="S5.SS1.1.p1.7.m7.3.3.2.3" mathvariant="normal" xref="S5.SS1.1.p1.7.m7.3.3.2.3.cmml">Θ</mi><mo id="S5.SS1.1.p1.7.m7.3.3.2.2" xref="S5.SS1.1.p1.7.m7.3.3.2.2.cmml">⁢</mo><mrow id="S5.SS1.1.p1.7.m7.3.3.2.1.1" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.cmml"><mo id="S5.SS1.1.p1.7.m7.3.3.2.1.1.2" stretchy="false" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.cmml">(</mo><mrow id="S5.SS1.1.p1.7.m7.3.3.2.1.1.1" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.cmml"><mi id="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.2" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.2.cmml">n</mi><mo id="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.1" lspace="0.167em" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.1.cmml">⁢</mo><mrow id="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.3" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.3.cmml"><mi id="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.3.1" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.3.1.cmml">log</mi><mo id="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.3a" lspace="0.167em" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.3.cmml">⁡</mo><mi id="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.3.2" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S5.SS1.1.p1.7.m7.3.3.2.1.1.3" stretchy="false" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.1.p1.7.m7.3b"><apply id="S5.SS1.1.p1.7.m7.3.3.cmml" xref="S5.SS1.1.p1.7.m7.3.3"><eq id="S5.SS1.1.p1.7.m7.3.3.3.cmml" xref="S5.SS1.1.p1.7.m7.3.3.3"></eq><apply id="S5.SS1.1.p1.7.m7.2.2.1.cmml" xref="S5.SS1.1.p1.7.m7.2.2.1"><times id="S5.SS1.1.p1.7.m7.2.2.1.2.cmml" xref="S5.SS1.1.p1.7.m7.2.2.1.2"></times><ci id="S5.SS1.1.p1.7.m7.2.2.1.3.cmml" xref="S5.SS1.1.p1.7.m7.2.2.1.3">𝔼</ci><apply id="S5.SS1.1.p1.7.m7.2.2.1.1.2.cmml" xref="S5.SS1.1.p1.7.m7.2.2.1.1.1"><csymbol cd="latexml" id="S5.SS1.1.p1.7.m7.2.2.1.1.2.1.cmml" xref="S5.SS1.1.p1.7.m7.2.2.1.1.1.2">delimited-[]</csymbol><apply id="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.cmml" xref="S5.SS1.1.p1.7.m7.2.2.1.1.1.1"><times id="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.1.cmml" xref="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.1"></times><ci id="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.2.cmml" xref="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.2">𝒮</ci><ci id="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.3.cmml" xref="S5.SS1.1.p1.7.m7.2.2.1.1.1.1.3">𝒲</ci><ci id="S5.SS1.1.p1.7.m7.1.1.cmml" xref="S5.SS1.1.p1.7.m7.1.1">𝜋</ci></apply></apply></apply><apply id="S5.SS1.1.p1.7.m7.3.3.2.cmml" xref="S5.SS1.1.p1.7.m7.3.3.2"><times id="S5.SS1.1.p1.7.m7.3.3.2.2.cmml" xref="S5.SS1.1.p1.7.m7.3.3.2.2"></times><ci id="S5.SS1.1.p1.7.m7.3.3.2.3.cmml" xref="S5.SS1.1.p1.7.m7.3.3.2.3">Θ</ci><apply id="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.cmml" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1"><times id="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.1.cmml" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.1"></times><ci id="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.2.cmml" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.2">𝑛</ci><apply id="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.3.cmml" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.3"><log id="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.3.1.cmml" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.3.1"></log><ci id="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.3.2.cmml" xref="S5.SS1.1.p1.7.m7.3.3.2.1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.1.p1.7.m7.3c">\mathbb{E}[\mathcal{SW}(\pi)]=\Theta(n\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.1.p1.7.m7.3d">blackboard_E [ caligraphic_S caligraphic_W ( italic_π ) ] = roman_Θ ( italic_n roman_log italic_n )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS1.2.p2"> <p class="ltx_p" id="S5.SS1.2.p2.7">In addition, the size of the maximum clique in <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS1.2.p2.1.m1.1"><semantics id="S5.SS1.2.p2.1.m1.1a"><msup id="S5.SS1.2.p2.1.m1.1.1" xref="S5.SS1.2.p2.1.m1.1.1.cmml"><mi id="S5.SS1.2.p2.1.m1.1.1.2" xref="S5.SS1.2.p2.1.m1.1.1.2.cmml">G</mi><mo id="S5.SS1.2.p2.1.m1.1.1.3" xref="S5.SS1.2.p2.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p2.1.m1.1b"><apply id="S5.SS1.2.p2.1.m1.1.1.cmml" xref="S5.SS1.2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS1.2.p2.1.m1.1.1.1.cmml" xref="S5.SS1.2.p2.1.m1.1.1">superscript</csymbol><ci id="S5.SS1.2.p2.1.m1.1.1.2.cmml" xref="S5.SS1.2.p2.1.m1.1.1.2">𝐺</ci><ci id="S5.SS1.2.p2.1.m1.1.1.3.cmml" xref="S5.SS1.2.p2.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p2.1.m1.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p2.1.m1.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is <math alttext="\mathcal{O}(\log_{b}n)" class="ltx_Math" display="inline" id="S5.SS1.2.p2.2.m2.1"><semantics id="S5.SS1.2.p2.2.m2.1a"><mrow id="S5.SS1.2.p2.2.m2.1.1" xref="S5.SS1.2.p2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS1.2.p2.2.m2.1.1.3" xref="S5.SS1.2.p2.2.m2.1.1.3.cmml">𝒪</mi><mo id="S5.SS1.2.p2.2.m2.1.1.2" xref="S5.SS1.2.p2.2.m2.1.1.2.cmml">⁢</mo><mrow id="S5.SS1.2.p2.2.m2.1.1.1.1" xref="S5.SS1.2.p2.2.m2.1.1.1.1.1.cmml"><mo id="S5.SS1.2.p2.2.m2.1.1.1.1.2" stretchy="false" xref="S5.SS1.2.p2.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS1.2.p2.2.m2.1.1.1.1.1" xref="S5.SS1.2.p2.2.m2.1.1.1.1.1.cmml"><msub id="S5.SS1.2.p2.2.m2.1.1.1.1.1.1" xref="S5.SS1.2.p2.2.m2.1.1.1.1.1.1.cmml"><mi id="S5.SS1.2.p2.2.m2.1.1.1.1.1.1.2" xref="S5.SS1.2.p2.2.m2.1.1.1.1.1.1.2.cmml">log</mi><mi id="S5.SS1.2.p2.2.m2.1.1.1.1.1.1.3" xref="S5.SS1.2.p2.2.m2.1.1.1.1.1.1.3.cmml">b</mi></msub><mo id="S5.SS1.2.p2.2.m2.1.1.1.1.1a" lspace="0.167em" xref="S5.SS1.2.p2.2.m2.1.1.1.1.1.cmml">⁡</mo><mi id="S5.SS1.2.p2.2.m2.1.1.1.1.1.2" xref="S5.SS1.2.p2.2.m2.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S5.SS1.2.p2.2.m2.1.1.1.1.3" stretchy="false" xref="S5.SS1.2.p2.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p2.2.m2.1b"><apply id="S5.SS1.2.p2.2.m2.1.1.cmml" xref="S5.SS1.2.p2.2.m2.1.1"><times id="S5.SS1.2.p2.2.m2.1.1.2.cmml" xref="S5.SS1.2.p2.2.m2.1.1.2"></times><ci id="S5.SS1.2.p2.2.m2.1.1.3.cmml" xref="S5.SS1.2.p2.2.m2.1.1.3">𝒪</ci><apply id="S5.SS1.2.p2.2.m2.1.1.1.1.1.cmml" xref="S5.SS1.2.p2.2.m2.1.1.1.1"><apply id="S5.SS1.2.p2.2.m2.1.1.1.1.1.1.cmml" xref="S5.SS1.2.p2.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS1.2.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="S5.SS1.2.p2.2.m2.1.1.1.1.1.1">subscript</csymbol><log id="S5.SS1.2.p2.2.m2.1.1.1.1.1.1.2.cmml" xref="S5.SS1.2.p2.2.m2.1.1.1.1.1.1.2"></log><ci id="S5.SS1.2.p2.2.m2.1.1.1.1.1.1.3.cmml" xref="S5.SS1.2.p2.2.m2.1.1.1.1.1.1.3">𝑏</ci></apply><ci id="S5.SS1.2.p2.2.m2.1.1.1.1.1.2.cmml" xref="S5.SS1.2.p2.2.m2.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p2.2.m2.1c">\mathcal{O}(\log_{b}n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p2.2.m2.1d">caligraphic_O ( roman_log start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_n )</annotation></semantics></math> with probability <math alttext="1" class="ltx_Math" display="inline" id="S5.SS1.2.p2.3.m3.1"><semantics id="S5.SS1.2.p2.3.m3.1a"><mn id="S5.SS1.2.p2.3.m3.1.1" xref="S5.SS1.2.p2.3.m3.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p2.3.m3.1b"><cn id="S5.SS1.2.p2.3.m3.1.1.cmml" type="integer" xref="S5.SS1.2.p2.3.m3.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p2.3.m3.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p2.3.m3.1d">1</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx28" title="">GM75</a>]</cite>. Hence, by <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem1" title="Lemma 5.1. ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5.1</span></a>, the expected maximum welfare of a partition in <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS1.2.p2.4.m4.1"><semantics id="S5.SS1.2.p2.4.m4.1a"><msup id="S5.SS1.2.p2.4.m4.1.1" xref="S5.SS1.2.p2.4.m4.1.1.cmml"><mi id="S5.SS1.2.p2.4.m4.1.1.2" xref="S5.SS1.2.p2.4.m4.1.1.2.cmml">G</mi><mo id="S5.SS1.2.p2.4.m4.1.1.3" xref="S5.SS1.2.p2.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p2.4.m4.1b"><apply id="S5.SS1.2.p2.4.m4.1.1.cmml" xref="S5.SS1.2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S5.SS1.2.p2.4.m4.1.1.1.cmml" xref="S5.SS1.2.p2.4.m4.1.1">superscript</csymbol><ci id="S5.SS1.2.p2.4.m4.1.1.2.cmml" xref="S5.SS1.2.p2.4.m4.1.1.2">𝐺</ci><ci id="S5.SS1.2.p2.4.m4.1.1.3.cmml" xref="S5.SS1.2.p2.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p2.4.m4.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p2.4.m4.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is <math alttext="\mathcal{O}(n\log n)" class="ltx_Math" display="inline" id="S5.SS1.2.p2.5.m5.1"><semantics id="S5.SS1.2.p2.5.m5.1a"><mrow id="S5.SS1.2.p2.5.m5.1.1" xref="S5.SS1.2.p2.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS1.2.p2.5.m5.1.1.3" xref="S5.SS1.2.p2.5.m5.1.1.3.cmml">𝒪</mi><mo id="S5.SS1.2.p2.5.m5.1.1.2" xref="S5.SS1.2.p2.5.m5.1.1.2.cmml">⁢</mo><mrow id="S5.SS1.2.p2.5.m5.1.1.1.1" xref="S5.SS1.2.p2.5.m5.1.1.1.1.1.cmml"><mo id="S5.SS1.2.p2.5.m5.1.1.1.1.2" stretchy="false" xref="S5.SS1.2.p2.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS1.2.p2.5.m5.1.1.1.1.1" xref="S5.SS1.2.p2.5.m5.1.1.1.1.1.cmml"><mi id="S5.SS1.2.p2.5.m5.1.1.1.1.1.2" xref="S5.SS1.2.p2.5.m5.1.1.1.1.1.2.cmml">n</mi><mo id="S5.SS1.2.p2.5.m5.1.1.1.1.1.1" lspace="0.167em" xref="S5.SS1.2.p2.5.m5.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.SS1.2.p2.5.m5.1.1.1.1.1.3" xref="S5.SS1.2.p2.5.m5.1.1.1.1.1.3.cmml"><mi id="S5.SS1.2.p2.5.m5.1.1.1.1.1.3.1" xref="S5.SS1.2.p2.5.m5.1.1.1.1.1.3.1.cmml">log</mi><mo id="S5.SS1.2.p2.5.m5.1.1.1.1.1.3a" lspace="0.167em" xref="S5.SS1.2.p2.5.m5.1.1.1.1.1.3.cmml">⁡</mo><mi id="S5.SS1.2.p2.5.m5.1.1.1.1.1.3.2" xref="S5.SS1.2.p2.5.m5.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S5.SS1.2.p2.5.m5.1.1.1.1.3" stretchy="false" xref="S5.SS1.2.p2.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p2.5.m5.1b"><apply id="S5.SS1.2.p2.5.m5.1.1.cmml" xref="S5.SS1.2.p2.5.m5.1.1"><times id="S5.SS1.2.p2.5.m5.1.1.2.cmml" xref="S5.SS1.2.p2.5.m5.1.1.2"></times><ci id="S5.SS1.2.p2.5.m5.1.1.3.cmml" xref="S5.SS1.2.p2.5.m5.1.1.3">𝒪</ci><apply id="S5.SS1.2.p2.5.m5.1.1.1.1.1.cmml" xref="S5.SS1.2.p2.5.m5.1.1.1.1"><times id="S5.SS1.2.p2.5.m5.1.1.1.1.1.1.cmml" xref="S5.SS1.2.p2.5.m5.1.1.1.1.1.1"></times><ci id="S5.SS1.2.p2.5.m5.1.1.1.1.1.2.cmml" xref="S5.SS1.2.p2.5.m5.1.1.1.1.1.2">𝑛</ci><apply id="S5.SS1.2.p2.5.m5.1.1.1.1.1.3.cmml" xref="S5.SS1.2.p2.5.m5.1.1.1.1.1.3"><log id="S5.SS1.2.p2.5.m5.1.1.1.1.1.3.1.cmml" xref="S5.SS1.2.p2.5.m5.1.1.1.1.1.3.1"></log><ci id="S5.SS1.2.p2.5.m5.1.1.1.1.1.3.2.cmml" xref="S5.SS1.2.p2.5.m5.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p2.5.m5.1c">\mathcal{O}(n\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p2.5.m5.1d">caligraphic_O ( italic_n roman_log italic_n )</annotation></semantics></math>. Note that all constants hidden in the asymptotic behavior only depend on <math alttext="b" class="ltx_Math" display="inline" id="S5.SS1.2.p2.6.m6.1"><semantics id="S5.SS1.2.p2.6.m6.1a"><mi id="S5.SS1.2.p2.6.m6.1.1" xref="S5.SS1.2.p2.6.m6.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p2.6.m6.1b"><ci id="S5.SS1.2.p2.6.m6.1.1.cmml" xref="S5.SS1.2.p2.6.m6.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p2.6.m6.1c">b</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p2.6.m6.1d">italic_b</annotation></semantics></math> and, therefore, on <math alttext="p" class="ltx_Math" display="inline" id="S5.SS1.2.p2.7.m7.1"><semantics id="S5.SS1.2.p2.7.m7.1a"><mi id="S5.SS1.2.p2.7.m7.1.1" xref="S5.SS1.2.p2.7.m7.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.2.p2.7.m7.1b"><ci id="S5.SS1.2.p2.7.m7.1.1.cmml" xref="S5.SS1.2.p2.7.m7.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.2.p2.7.m7.1c">p</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.2.p2.7.m7.1d">italic_p</annotation></semantics></math>. It follows that the greedy clique formation algorithm yields a constant-factor approximation. ∎</p> </div> </div> <div class="ltx_para" id="S5.SS1.p4"> <p class="ltx_p" id="S5.SS1.p4.11">To get a feeling on the constant hidden in the previous theorem, one can reason as follows. For large enough <math alttext="n" class="ltx_Math" display="inline" id="S5.SS1.p4.1.m1.1"><semantics id="S5.SS1.p4.1.m1.1a"><mi id="S5.SS1.p4.1.m1.1.1" xref="S5.SS1.p4.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.1.m1.1b"><ci id="S5.SS1.p4.1.m1.1.1.cmml" xref="S5.SS1.p4.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.1.m1.1d">italic_n</annotation></semantics></math>, with probability <math alttext="1-\frac{1}{n}" class="ltx_Math" display="inline" id="S5.SS1.p4.2.m2.1"><semantics id="S5.SS1.p4.2.m2.1a"><mrow id="S5.SS1.p4.2.m2.1.1" xref="S5.SS1.p4.2.m2.1.1.cmml"><mn id="S5.SS1.p4.2.m2.1.1.2" xref="S5.SS1.p4.2.m2.1.1.2.cmml">1</mn><mo id="S5.SS1.p4.2.m2.1.1.1" xref="S5.SS1.p4.2.m2.1.1.1.cmml">−</mo><mfrac id="S5.SS1.p4.2.m2.1.1.3" xref="S5.SS1.p4.2.m2.1.1.3.cmml"><mn id="S5.SS1.p4.2.m2.1.1.3.2" xref="S5.SS1.p4.2.m2.1.1.3.2.cmml">1</mn><mi id="S5.SS1.p4.2.m2.1.1.3.3" xref="S5.SS1.p4.2.m2.1.1.3.3.cmml">n</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.2.m2.1b"><apply id="S5.SS1.p4.2.m2.1.1.cmml" xref="S5.SS1.p4.2.m2.1.1"><minus id="S5.SS1.p4.2.m2.1.1.1.cmml" xref="S5.SS1.p4.2.m2.1.1.1"></minus><cn id="S5.SS1.p4.2.m2.1.1.2.cmml" type="integer" xref="S5.SS1.p4.2.m2.1.1.2">1</cn><apply id="S5.SS1.p4.2.m2.1.1.3.cmml" xref="S5.SS1.p4.2.m2.1.1.3"><divide id="S5.SS1.p4.2.m2.1.1.3.1.cmml" xref="S5.SS1.p4.2.m2.1.1.3"></divide><cn id="S5.SS1.p4.2.m2.1.1.3.2.cmml" type="integer" xref="S5.SS1.p4.2.m2.1.1.3.2">1</cn><ci id="S5.SS1.p4.2.m2.1.1.3.3.cmml" xref="S5.SS1.p4.2.m2.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.2.m2.1c">1-\frac{1}{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.2.m2.1d">1 - divide start_ARG 1 end_ARG start_ARG italic_n end_ARG</annotation></semantics></math>, greedy clique formation will place <math alttext="\frac{2n}{3}" class="ltx_Math" display="inline" id="S5.SS1.p4.3.m3.1"><semantics id="S5.SS1.p4.3.m3.1a"><mfrac id="S5.SS1.p4.3.m3.1.1" xref="S5.SS1.p4.3.m3.1.1.cmml"><mrow id="S5.SS1.p4.3.m3.1.1.2" xref="S5.SS1.p4.3.m3.1.1.2.cmml"><mn id="S5.SS1.p4.3.m3.1.1.2.2" xref="S5.SS1.p4.3.m3.1.1.2.2.cmml">2</mn><mo id="S5.SS1.p4.3.m3.1.1.2.1" xref="S5.SS1.p4.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S5.SS1.p4.3.m3.1.1.2.3" xref="S5.SS1.p4.3.m3.1.1.2.3.cmml">n</mi></mrow><mn id="S5.SS1.p4.3.m3.1.1.3" xref="S5.SS1.p4.3.m3.1.1.3.cmml">3</mn></mfrac><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.3.m3.1b"><apply id="S5.SS1.p4.3.m3.1.1.cmml" xref="S5.SS1.p4.3.m3.1.1"><divide id="S5.SS1.p4.3.m3.1.1.1.cmml" xref="S5.SS1.p4.3.m3.1.1"></divide><apply id="S5.SS1.p4.3.m3.1.1.2.cmml" xref="S5.SS1.p4.3.m3.1.1.2"><times id="S5.SS1.p4.3.m3.1.1.2.1.cmml" xref="S5.SS1.p4.3.m3.1.1.2.1"></times><cn id="S5.SS1.p4.3.m3.1.1.2.2.cmml" type="integer" xref="S5.SS1.p4.3.m3.1.1.2.2">2</cn><ci id="S5.SS1.p4.3.m3.1.1.2.3.cmml" xref="S5.SS1.p4.3.m3.1.1.2.3">𝑛</ci></apply><cn id="S5.SS1.p4.3.m3.1.1.3.cmml" type="integer" xref="S5.SS1.p4.3.m3.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.3.m3.1c">\frac{2n}{3}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.3.m3.1d">divide start_ARG 2 italic_n end_ARG start_ARG 3 end_ARG</annotation></semantics></math> agents into coalitions of size <math alttext="\frac{\log_{b}n}{2}" class="ltx_Math" display="inline" id="S5.SS1.p4.4.m4.1"><semantics id="S5.SS1.p4.4.m4.1a"><mfrac id="S5.SS1.p4.4.m4.1.1" xref="S5.SS1.p4.4.m4.1.1.cmml"><mrow id="S5.SS1.p4.4.m4.1.1.2" xref="S5.SS1.p4.4.m4.1.1.2.cmml"><msub id="S5.SS1.p4.4.m4.1.1.2.1" xref="S5.SS1.p4.4.m4.1.1.2.1.cmml"><mi id="S5.SS1.p4.4.m4.1.1.2.1.2" xref="S5.SS1.p4.4.m4.1.1.2.1.2.cmml">log</mi><mi id="S5.SS1.p4.4.m4.1.1.2.1.3" xref="S5.SS1.p4.4.m4.1.1.2.1.3.cmml">b</mi></msub><mo id="S5.SS1.p4.4.m4.1.1.2a" lspace="0.167em" xref="S5.SS1.p4.4.m4.1.1.2.cmml">⁡</mo><mi id="S5.SS1.p4.4.m4.1.1.2.2" xref="S5.SS1.p4.4.m4.1.1.2.2.cmml">n</mi></mrow><mn id="S5.SS1.p4.4.m4.1.1.3" xref="S5.SS1.p4.4.m4.1.1.3.cmml">2</mn></mfrac><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.4.m4.1b"><apply id="S5.SS1.p4.4.m4.1.1.cmml" xref="S5.SS1.p4.4.m4.1.1"><divide id="S5.SS1.p4.4.m4.1.1.1.cmml" xref="S5.SS1.p4.4.m4.1.1"></divide><apply id="S5.SS1.p4.4.m4.1.1.2.cmml" xref="S5.SS1.p4.4.m4.1.1.2"><apply id="S5.SS1.p4.4.m4.1.1.2.1.cmml" xref="S5.SS1.p4.4.m4.1.1.2.1"><csymbol cd="ambiguous" id="S5.SS1.p4.4.m4.1.1.2.1.1.cmml" xref="S5.SS1.p4.4.m4.1.1.2.1">subscript</csymbol><log id="S5.SS1.p4.4.m4.1.1.2.1.2.cmml" xref="S5.SS1.p4.4.m4.1.1.2.1.2"></log><ci id="S5.SS1.p4.4.m4.1.1.2.1.3.cmml" xref="S5.SS1.p4.4.m4.1.1.2.1.3">𝑏</ci></apply><ci id="S5.SS1.p4.4.m4.1.1.2.2.cmml" xref="S5.SS1.p4.4.m4.1.1.2.2">𝑛</ci></apply><cn id="S5.SS1.p4.4.m4.1.1.3.cmml" type="integer" xref="S5.SS1.p4.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.4.m4.1c">\frac{\log_{b}n}{2}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.4.m4.1d">divide start_ARG roman_log start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_n end_ARG start_ARG 2 end_ARG</annotation></semantics></math>, resulting in an expected social welfare of at least <math alttext="(1-\frac{1}{n})\frac{2n}{3}\frac{\log_{b}n}{2}\geq\frac{n\log_{b}n}{4}" class="ltx_Math" display="inline" id="S5.SS1.p4.5.m5.1"><semantics id="S5.SS1.p4.5.m5.1a"><mrow id="S5.SS1.p4.5.m5.1.1" xref="S5.SS1.p4.5.m5.1.1.cmml"><mrow id="S5.SS1.p4.5.m5.1.1.1" xref="S5.SS1.p4.5.m5.1.1.1.cmml"><mrow id="S5.SS1.p4.5.m5.1.1.1.1.1" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.cmml"><mo id="S5.SS1.p4.5.m5.1.1.1.1.1.2" stretchy="false" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS1.p4.5.m5.1.1.1.1.1.1" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.cmml"><mn id="S5.SS1.p4.5.m5.1.1.1.1.1.1.2" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.SS1.p4.5.m5.1.1.1.1.1.1.1" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.1.cmml">−</mo><mfrac id="S5.SS1.p4.5.m5.1.1.1.1.1.1.3" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.3.cmml"><mn id="S5.SS1.p4.5.m5.1.1.1.1.1.1.3.2" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.3.2.cmml">1</mn><mi id="S5.SS1.p4.5.m5.1.1.1.1.1.1.3.3" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.3.3.cmml">n</mi></mfrac></mrow><mo id="S5.SS1.p4.5.m5.1.1.1.1.1.3" stretchy="false" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.SS1.p4.5.m5.1.1.1.2" xref="S5.SS1.p4.5.m5.1.1.1.2.cmml">⁢</mo><mfrac id="S5.SS1.p4.5.m5.1.1.1.3" xref="S5.SS1.p4.5.m5.1.1.1.3.cmml"><mrow id="S5.SS1.p4.5.m5.1.1.1.3.2" xref="S5.SS1.p4.5.m5.1.1.1.3.2.cmml"><mn id="S5.SS1.p4.5.m5.1.1.1.3.2.2" xref="S5.SS1.p4.5.m5.1.1.1.3.2.2.cmml">2</mn><mo id="S5.SS1.p4.5.m5.1.1.1.3.2.1" xref="S5.SS1.p4.5.m5.1.1.1.3.2.1.cmml">⁢</mo><mi id="S5.SS1.p4.5.m5.1.1.1.3.2.3" xref="S5.SS1.p4.5.m5.1.1.1.3.2.3.cmml">n</mi></mrow><mn id="S5.SS1.p4.5.m5.1.1.1.3.3" xref="S5.SS1.p4.5.m5.1.1.1.3.3.cmml">3</mn></mfrac><mo id="S5.SS1.p4.5.m5.1.1.1.2a" xref="S5.SS1.p4.5.m5.1.1.1.2.cmml">⁢</mo><mfrac id="S5.SS1.p4.5.m5.1.1.1.4" xref="S5.SS1.p4.5.m5.1.1.1.4.cmml"><mrow id="S5.SS1.p4.5.m5.1.1.1.4.2" xref="S5.SS1.p4.5.m5.1.1.1.4.2.cmml"><msub id="S5.SS1.p4.5.m5.1.1.1.4.2.1" xref="S5.SS1.p4.5.m5.1.1.1.4.2.1.cmml"><mi id="S5.SS1.p4.5.m5.1.1.1.4.2.1.2" xref="S5.SS1.p4.5.m5.1.1.1.4.2.1.2.cmml">log</mi><mi id="S5.SS1.p4.5.m5.1.1.1.4.2.1.3" xref="S5.SS1.p4.5.m5.1.1.1.4.2.1.3.cmml">b</mi></msub><mo id="S5.SS1.p4.5.m5.1.1.1.4.2a" lspace="0.167em" xref="S5.SS1.p4.5.m5.1.1.1.4.2.cmml">⁡</mo><mi id="S5.SS1.p4.5.m5.1.1.1.4.2.2" xref="S5.SS1.p4.5.m5.1.1.1.4.2.2.cmml">n</mi></mrow><mn id="S5.SS1.p4.5.m5.1.1.1.4.3" xref="S5.SS1.p4.5.m5.1.1.1.4.3.cmml">2</mn></mfrac></mrow><mo id="S5.SS1.p4.5.m5.1.1.2" xref="S5.SS1.p4.5.m5.1.1.2.cmml">≥</mo><mfrac id="S5.SS1.p4.5.m5.1.1.3" xref="S5.SS1.p4.5.m5.1.1.3.cmml"><mrow id="S5.SS1.p4.5.m5.1.1.3.2" xref="S5.SS1.p4.5.m5.1.1.3.2.cmml"><mi id="S5.SS1.p4.5.m5.1.1.3.2.2" xref="S5.SS1.p4.5.m5.1.1.3.2.2.cmml">n</mi><mo id="S5.SS1.p4.5.m5.1.1.3.2.1" lspace="0.167em" xref="S5.SS1.p4.5.m5.1.1.3.2.1.cmml">⁢</mo><mrow id="S5.SS1.p4.5.m5.1.1.3.2.3" xref="S5.SS1.p4.5.m5.1.1.3.2.3.cmml"><msub id="S5.SS1.p4.5.m5.1.1.3.2.3.1" xref="S5.SS1.p4.5.m5.1.1.3.2.3.1.cmml"><mi id="S5.SS1.p4.5.m5.1.1.3.2.3.1.2" xref="S5.SS1.p4.5.m5.1.1.3.2.3.1.2.cmml">log</mi><mi id="S5.SS1.p4.5.m5.1.1.3.2.3.1.3" xref="S5.SS1.p4.5.m5.1.1.3.2.3.1.3.cmml">b</mi></msub><mo id="S5.SS1.p4.5.m5.1.1.3.2.3a" lspace="0.167em" xref="S5.SS1.p4.5.m5.1.1.3.2.3.cmml">⁡</mo><mi id="S5.SS1.p4.5.m5.1.1.3.2.3.2" xref="S5.SS1.p4.5.m5.1.1.3.2.3.2.cmml">n</mi></mrow></mrow><mn id="S5.SS1.p4.5.m5.1.1.3.3" xref="S5.SS1.p4.5.m5.1.1.3.3.cmml">4</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.5.m5.1b"><apply id="S5.SS1.p4.5.m5.1.1.cmml" xref="S5.SS1.p4.5.m5.1.1"><geq id="S5.SS1.p4.5.m5.1.1.2.cmml" xref="S5.SS1.p4.5.m5.1.1.2"></geq><apply id="S5.SS1.p4.5.m5.1.1.1.cmml" xref="S5.SS1.p4.5.m5.1.1.1"><times id="S5.SS1.p4.5.m5.1.1.1.2.cmml" xref="S5.SS1.p4.5.m5.1.1.1.2"></times><apply id="S5.SS1.p4.5.m5.1.1.1.1.1.1.cmml" xref="S5.SS1.p4.5.m5.1.1.1.1.1"><minus id="S5.SS1.p4.5.m5.1.1.1.1.1.1.1.cmml" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.1"></minus><cn id="S5.SS1.p4.5.m5.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.2">1</cn><apply id="S5.SS1.p4.5.m5.1.1.1.1.1.1.3.cmml" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.3"><divide id="S5.SS1.p4.5.m5.1.1.1.1.1.1.3.1.cmml" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.3"></divide><cn id="S5.SS1.p4.5.m5.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.3.2">1</cn><ci id="S5.SS1.p4.5.m5.1.1.1.1.1.1.3.3.cmml" xref="S5.SS1.p4.5.m5.1.1.1.1.1.1.3.3">𝑛</ci></apply></apply><apply id="S5.SS1.p4.5.m5.1.1.1.3.cmml" xref="S5.SS1.p4.5.m5.1.1.1.3"><divide id="S5.SS1.p4.5.m5.1.1.1.3.1.cmml" xref="S5.SS1.p4.5.m5.1.1.1.3"></divide><apply id="S5.SS1.p4.5.m5.1.1.1.3.2.cmml" xref="S5.SS1.p4.5.m5.1.1.1.3.2"><times id="S5.SS1.p4.5.m5.1.1.1.3.2.1.cmml" xref="S5.SS1.p4.5.m5.1.1.1.3.2.1"></times><cn id="S5.SS1.p4.5.m5.1.1.1.3.2.2.cmml" type="integer" xref="S5.SS1.p4.5.m5.1.1.1.3.2.2">2</cn><ci id="S5.SS1.p4.5.m5.1.1.1.3.2.3.cmml" xref="S5.SS1.p4.5.m5.1.1.1.3.2.3">𝑛</ci></apply><cn id="S5.SS1.p4.5.m5.1.1.1.3.3.cmml" type="integer" xref="S5.SS1.p4.5.m5.1.1.1.3.3">3</cn></apply><apply id="S5.SS1.p4.5.m5.1.1.1.4.cmml" xref="S5.SS1.p4.5.m5.1.1.1.4"><divide id="S5.SS1.p4.5.m5.1.1.1.4.1.cmml" xref="S5.SS1.p4.5.m5.1.1.1.4"></divide><apply id="S5.SS1.p4.5.m5.1.1.1.4.2.cmml" xref="S5.SS1.p4.5.m5.1.1.1.4.2"><apply id="S5.SS1.p4.5.m5.1.1.1.4.2.1.cmml" xref="S5.SS1.p4.5.m5.1.1.1.4.2.1"><csymbol cd="ambiguous" id="S5.SS1.p4.5.m5.1.1.1.4.2.1.1.cmml" xref="S5.SS1.p4.5.m5.1.1.1.4.2.1">subscript</csymbol><log id="S5.SS1.p4.5.m5.1.1.1.4.2.1.2.cmml" xref="S5.SS1.p4.5.m5.1.1.1.4.2.1.2"></log><ci id="S5.SS1.p4.5.m5.1.1.1.4.2.1.3.cmml" xref="S5.SS1.p4.5.m5.1.1.1.4.2.1.3">𝑏</ci></apply><ci id="S5.SS1.p4.5.m5.1.1.1.4.2.2.cmml" xref="S5.SS1.p4.5.m5.1.1.1.4.2.2">𝑛</ci></apply><cn id="S5.SS1.p4.5.m5.1.1.1.4.3.cmml" type="integer" xref="S5.SS1.p4.5.m5.1.1.1.4.3">2</cn></apply></apply><apply id="S5.SS1.p4.5.m5.1.1.3.cmml" xref="S5.SS1.p4.5.m5.1.1.3"><divide id="S5.SS1.p4.5.m5.1.1.3.1.cmml" xref="S5.SS1.p4.5.m5.1.1.3"></divide><apply id="S5.SS1.p4.5.m5.1.1.3.2.cmml" xref="S5.SS1.p4.5.m5.1.1.3.2"><times id="S5.SS1.p4.5.m5.1.1.3.2.1.cmml" xref="S5.SS1.p4.5.m5.1.1.3.2.1"></times><ci id="S5.SS1.p4.5.m5.1.1.3.2.2.cmml" xref="S5.SS1.p4.5.m5.1.1.3.2.2">𝑛</ci><apply id="S5.SS1.p4.5.m5.1.1.3.2.3.cmml" xref="S5.SS1.p4.5.m5.1.1.3.2.3"><apply id="S5.SS1.p4.5.m5.1.1.3.2.3.1.cmml" xref="S5.SS1.p4.5.m5.1.1.3.2.3.1"><csymbol cd="ambiguous" id="S5.SS1.p4.5.m5.1.1.3.2.3.1.1.cmml" xref="S5.SS1.p4.5.m5.1.1.3.2.3.1">subscript</csymbol><log id="S5.SS1.p4.5.m5.1.1.3.2.3.1.2.cmml" xref="S5.SS1.p4.5.m5.1.1.3.2.3.1.2"></log><ci id="S5.SS1.p4.5.m5.1.1.3.2.3.1.3.cmml" xref="S5.SS1.p4.5.m5.1.1.3.2.3.1.3">𝑏</ci></apply><ci id="S5.SS1.p4.5.m5.1.1.3.2.3.2.cmml" xref="S5.SS1.p4.5.m5.1.1.3.2.3.2">𝑛</ci></apply></apply><cn id="S5.SS1.p4.5.m5.1.1.3.3.cmml" type="integer" xref="S5.SS1.p4.5.m5.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.5.m5.1c">(1-\frac{1}{n})\frac{2n}{3}\frac{\log_{b}n}{2}\geq\frac{n\log_{b}n}{4}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.5.m5.1d">( 1 - divide start_ARG 1 end_ARG start_ARG italic_n end_ARG ) divide start_ARG 2 italic_n end_ARG start_ARG 3 end_ARG divide start_ARG roman_log start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_n end_ARG start_ARG 2 end_ARG ≥ divide start_ARG italic_n roman_log start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_n end_ARG start_ARG 4 end_ARG</annotation></semantics></math>. Moreover, for large enough <math alttext="n" class="ltx_Math" display="inline" id="S5.SS1.p4.6.m6.1"><semantics id="S5.SS1.p4.6.m6.1a"><mi id="S5.SS1.p4.6.m6.1.1" xref="S5.SS1.p4.6.m6.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.6.m6.1b"><ci id="S5.SS1.p4.6.m6.1.1.cmml" xref="S5.SS1.p4.6.m6.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.6.m6.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.6.m6.1d">italic_n</annotation></semantics></math>, with probability <math alttext="1-\frac{1}{n^{2}}" class="ltx_Math" display="inline" id="S5.SS1.p4.7.m7.1"><semantics id="S5.SS1.p4.7.m7.1a"><mrow id="S5.SS1.p4.7.m7.1.1" xref="S5.SS1.p4.7.m7.1.1.cmml"><mn id="S5.SS1.p4.7.m7.1.1.2" xref="S5.SS1.p4.7.m7.1.1.2.cmml">1</mn><mo id="S5.SS1.p4.7.m7.1.1.1" xref="S5.SS1.p4.7.m7.1.1.1.cmml">−</mo><mfrac id="S5.SS1.p4.7.m7.1.1.3" xref="S5.SS1.p4.7.m7.1.1.3.cmml"><mn id="S5.SS1.p4.7.m7.1.1.3.2" xref="S5.SS1.p4.7.m7.1.1.3.2.cmml">1</mn><msup id="S5.SS1.p4.7.m7.1.1.3.3" xref="S5.SS1.p4.7.m7.1.1.3.3.cmml"><mi id="S5.SS1.p4.7.m7.1.1.3.3.2" xref="S5.SS1.p4.7.m7.1.1.3.3.2.cmml">n</mi><mn id="S5.SS1.p4.7.m7.1.1.3.3.3" xref="S5.SS1.p4.7.m7.1.1.3.3.3.cmml">2</mn></msup></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.7.m7.1b"><apply id="S5.SS1.p4.7.m7.1.1.cmml" xref="S5.SS1.p4.7.m7.1.1"><minus id="S5.SS1.p4.7.m7.1.1.1.cmml" xref="S5.SS1.p4.7.m7.1.1.1"></minus><cn id="S5.SS1.p4.7.m7.1.1.2.cmml" type="integer" xref="S5.SS1.p4.7.m7.1.1.2">1</cn><apply id="S5.SS1.p4.7.m7.1.1.3.cmml" xref="S5.SS1.p4.7.m7.1.1.3"><divide id="S5.SS1.p4.7.m7.1.1.3.1.cmml" xref="S5.SS1.p4.7.m7.1.1.3"></divide><cn id="S5.SS1.p4.7.m7.1.1.3.2.cmml" type="integer" xref="S5.SS1.p4.7.m7.1.1.3.2">1</cn><apply id="S5.SS1.p4.7.m7.1.1.3.3.cmml" xref="S5.SS1.p4.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S5.SS1.p4.7.m7.1.1.3.3.1.cmml" xref="S5.SS1.p4.7.m7.1.1.3.3">superscript</csymbol><ci id="S5.SS1.p4.7.m7.1.1.3.3.2.cmml" xref="S5.SS1.p4.7.m7.1.1.3.3.2">𝑛</ci><cn id="S5.SS1.p4.7.m7.1.1.3.3.3.cmml" type="integer" xref="S5.SS1.p4.7.m7.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.7.m7.1c">1-\frac{1}{n^{2}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.7.m7.1d">1 - divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math>, the maximum clique is of size at most <math alttext="4\log_{\frac{1}{p}}n" class="ltx_Math" display="inline" id="S5.SS1.p4.8.m8.1"><semantics id="S5.SS1.p4.8.m8.1a"><mrow id="S5.SS1.p4.8.m8.1.1" xref="S5.SS1.p4.8.m8.1.1.cmml"><mn id="S5.SS1.p4.8.m8.1.1.2" xref="S5.SS1.p4.8.m8.1.1.2.cmml">4</mn><mo id="S5.SS1.p4.8.m8.1.1.1" lspace="0.167em" xref="S5.SS1.p4.8.m8.1.1.1.cmml">⁢</mo><mrow id="S5.SS1.p4.8.m8.1.1.3" xref="S5.SS1.p4.8.m8.1.1.3.cmml"><msub id="S5.SS1.p4.8.m8.1.1.3.1" xref="S5.SS1.p4.8.m8.1.1.3.1.cmml"><mi id="S5.SS1.p4.8.m8.1.1.3.1.2" xref="S5.SS1.p4.8.m8.1.1.3.1.2.cmml">log</mi><mfrac id="S5.SS1.p4.8.m8.1.1.3.1.3" xref="S5.SS1.p4.8.m8.1.1.3.1.3.cmml"><mn id="S5.SS1.p4.8.m8.1.1.3.1.3.2" xref="S5.SS1.p4.8.m8.1.1.3.1.3.2.cmml">1</mn><mi id="S5.SS1.p4.8.m8.1.1.3.1.3.3" xref="S5.SS1.p4.8.m8.1.1.3.1.3.3.cmml">p</mi></mfrac></msub><mo id="S5.SS1.p4.8.m8.1.1.3a" lspace="0.167em" xref="S5.SS1.p4.8.m8.1.1.3.cmml">⁡</mo><mi id="S5.SS1.p4.8.m8.1.1.3.2" xref="S5.SS1.p4.8.m8.1.1.3.2.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.8.m8.1b"><apply id="S5.SS1.p4.8.m8.1.1.cmml" xref="S5.SS1.p4.8.m8.1.1"><times id="S5.SS1.p4.8.m8.1.1.1.cmml" xref="S5.SS1.p4.8.m8.1.1.1"></times><cn id="S5.SS1.p4.8.m8.1.1.2.cmml" type="integer" xref="S5.SS1.p4.8.m8.1.1.2">4</cn><apply id="S5.SS1.p4.8.m8.1.1.3.cmml" xref="S5.SS1.p4.8.m8.1.1.3"><apply id="S5.SS1.p4.8.m8.1.1.3.1.cmml" xref="S5.SS1.p4.8.m8.1.1.3.1"><csymbol cd="ambiguous" id="S5.SS1.p4.8.m8.1.1.3.1.1.cmml" xref="S5.SS1.p4.8.m8.1.1.3.1">subscript</csymbol><log id="S5.SS1.p4.8.m8.1.1.3.1.2.cmml" xref="S5.SS1.p4.8.m8.1.1.3.1.2"></log><apply id="S5.SS1.p4.8.m8.1.1.3.1.3.cmml" xref="S5.SS1.p4.8.m8.1.1.3.1.3"><divide id="S5.SS1.p4.8.m8.1.1.3.1.3.1.cmml" xref="S5.SS1.p4.8.m8.1.1.3.1.3"></divide><cn id="S5.SS1.p4.8.m8.1.1.3.1.3.2.cmml" type="integer" xref="S5.SS1.p4.8.m8.1.1.3.1.3.2">1</cn><ci id="S5.SS1.p4.8.m8.1.1.3.1.3.3.cmml" xref="S5.SS1.p4.8.m8.1.1.3.1.3.3">𝑝</ci></apply></apply><ci id="S5.SS1.p4.8.m8.1.1.3.2.cmml" xref="S5.SS1.p4.8.m8.1.1.3.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.8.m8.1c">4\log_{\frac{1}{p}}n</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.8.m8.1d">4 roman_log start_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p end_ARG end_POSTSUBSCRIPT italic_n</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx28" title="">GM75</a>]</cite>. Hence the expected maximum welfare is at most <math alttext="n(4\log_{\frac{1}{1-p}}n+\frac{1}{n^{2}}n^{2})=n(4\log_{\frac{1}{1-p}}n+1)" class="ltx_Math" display="inline" id="S5.SS1.p4.9.m9.2"><semantics id="S5.SS1.p4.9.m9.2a"><mrow id="S5.SS1.p4.9.m9.2.2" xref="S5.SS1.p4.9.m9.2.2.cmml"><mrow id="S5.SS1.p4.9.m9.1.1.1" xref="S5.SS1.p4.9.m9.1.1.1.cmml"><mi id="S5.SS1.p4.9.m9.1.1.1.3" xref="S5.SS1.p4.9.m9.1.1.1.3.cmml">n</mi><mo id="S5.SS1.p4.9.m9.1.1.1.2" xref="S5.SS1.p4.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS1.p4.9.m9.1.1.1.1.1" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.cmml"><mo id="S5.SS1.p4.9.m9.1.1.1.1.1.2" stretchy="false" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS1.p4.9.m9.1.1.1.1.1.1" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.cmml"><mrow id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.cmml"><mn id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.2" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.2.cmml">4</mn><mo id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.1" lspace="0.167em" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.cmml"><msub id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.cmml"><mi id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.2" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.2.cmml">log</mi><mfrac id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.cmml"><mn id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.2" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.2.cmml">1</mn><mrow id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3.cmml"><mn id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3.2" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3.2.cmml">1</mn><mo id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3.1" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3.1.cmml">−</mo><mi id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3.3" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3.3.cmml">p</mi></mrow></mfrac></msub><mo id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3a" lspace="0.167em" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.cmml">⁡</mo><mi id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.2" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.2.cmml">n</mi></mrow></mrow><mo id="S5.SS1.p4.9.m9.1.1.1.1.1.1.1" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.cmml"><mfrac id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.cmml"><mn id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.2" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.2.cmml">1</mn><msup id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.3" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.3.cmml"><mi id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.3.2" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.3.2.cmml">n</mi><mn id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.3.3" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.3.3.cmml">2</mn></msup></mfrac><mo id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.1" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.1.cmml">⁢</mo><msup id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.3" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.3.cmml"><mi id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.3.2" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.3.2.cmml">n</mi><mn id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.3.3" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.3.3.cmml">2</mn></msup></mrow></mrow><mo id="S5.SS1.p4.9.m9.1.1.1.1.1.3" stretchy="false" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS1.p4.9.m9.2.2.3" xref="S5.SS1.p4.9.m9.2.2.3.cmml">=</mo><mrow id="S5.SS1.p4.9.m9.2.2.2" xref="S5.SS1.p4.9.m9.2.2.2.cmml"><mi id="S5.SS1.p4.9.m9.2.2.2.3" xref="S5.SS1.p4.9.m9.2.2.2.3.cmml">n</mi><mo id="S5.SS1.p4.9.m9.2.2.2.2" xref="S5.SS1.p4.9.m9.2.2.2.2.cmml">⁢</mo><mrow id="S5.SS1.p4.9.m9.2.2.2.1.1" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.cmml"><mo id="S5.SS1.p4.9.m9.2.2.2.1.1.2" stretchy="false" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.SS1.p4.9.m9.2.2.2.1.1.1" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.cmml"><mrow id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.cmml"><mn id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.2" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.2.cmml">4</mn><mo id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.1" lspace="0.167em" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.1.cmml">⁢</mo><mrow id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.cmml"><msub id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.cmml"><mi id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.2" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.2.cmml">log</mi><mfrac id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.cmml"><mn id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.2" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.2.cmml">1</mn><mrow id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3.cmml"><mn id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3.2" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3.2.cmml">1</mn><mo id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3.1" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3.1.cmml">−</mo><mi id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3.3" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3.3.cmml">p</mi></mrow></mfrac></msub><mo id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3a" lspace="0.167em" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.cmml">⁡</mo><mi id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.2" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.2.cmml">n</mi></mrow></mrow><mo id="S5.SS1.p4.9.m9.2.2.2.1.1.1.1" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.1.cmml">+</mo><mn id="S5.SS1.p4.9.m9.2.2.2.1.1.1.3" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S5.SS1.p4.9.m9.2.2.2.1.1.3" stretchy="false" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.9.m9.2b"><apply id="S5.SS1.p4.9.m9.2.2.cmml" xref="S5.SS1.p4.9.m9.2.2"><eq id="S5.SS1.p4.9.m9.2.2.3.cmml" xref="S5.SS1.p4.9.m9.2.2.3"></eq><apply id="S5.SS1.p4.9.m9.1.1.1.cmml" xref="S5.SS1.p4.9.m9.1.1.1"><times id="S5.SS1.p4.9.m9.1.1.1.2.cmml" xref="S5.SS1.p4.9.m9.1.1.1.2"></times><ci id="S5.SS1.p4.9.m9.1.1.1.3.cmml" xref="S5.SS1.p4.9.m9.1.1.1.3">𝑛</ci><apply id="S5.SS1.p4.9.m9.1.1.1.1.1.1.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1"><plus id="S5.SS1.p4.9.m9.1.1.1.1.1.1.1.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.1"></plus><apply id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2"><times id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.1.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.1"></times><cn id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.2.cmml" type="integer" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.2">4</cn><apply id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3"><apply id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1"><csymbol cd="ambiguous" id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.1.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1">subscript</csymbol><log id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.2.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.2"></log><apply id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3"><divide id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.1.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3"></divide><cn id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.2.cmml" type="integer" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.2">1</cn><apply id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3"><minus id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3.1.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3.1"></minus><cn id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3.2.cmml" type="integer" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3.2">1</cn><ci id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3.3.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.1.3.3.3">𝑝</ci></apply></apply></apply><ci id="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.2.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.2.3.2">𝑛</ci></apply></apply><apply id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3"><times id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.1.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.1"></times><apply id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2"><divide id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.1.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2"></divide><cn id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.2.cmml" type="integer" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.2">1</cn><apply id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.3.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.3.1.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.3">superscript</csymbol><ci id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.3.2.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.3.2">𝑛</ci><cn id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.3.3.cmml" type="integer" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.2.3.3">2</cn></apply></apply><apply id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.3.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.3.1.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.3">superscript</csymbol><ci id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.3.2.cmml" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.3.2">𝑛</ci><cn id="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S5.SS1.p4.9.m9.1.1.1.1.1.1.3.3.3">2</cn></apply></apply></apply></apply><apply id="S5.SS1.p4.9.m9.2.2.2.cmml" xref="S5.SS1.p4.9.m9.2.2.2"><times id="S5.SS1.p4.9.m9.2.2.2.2.cmml" xref="S5.SS1.p4.9.m9.2.2.2.2"></times><ci id="S5.SS1.p4.9.m9.2.2.2.3.cmml" xref="S5.SS1.p4.9.m9.2.2.2.3">𝑛</ci><apply id="S5.SS1.p4.9.m9.2.2.2.1.1.1.cmml" xref="S5.SS1.p4.9.m9.2.2.2.1.1"><plus id="S5.SS1.p4.9.m9.2.2.2.1.1.1.1.cmml" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.1"></plus><apply id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.cmml" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2"><times id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.1.cmml" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.1"></times><cn id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.2.cmml" type="integer" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.2">4</cn><apply id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.cmml" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3"><apply id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.cmml" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1"><csymbol cd="ambiguous" id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.1.cmml" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1">subscript</csymbol><log id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.2.cmml" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.2"></log><apply id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.cmml" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3"><divide id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.1.cmml" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3"></divide><cn id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.2.cmml" type="integer" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.2">1</cn><apply id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3.cmml" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3"><minus id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3.1.cmml" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3.1"></minus><cn id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3.2.cmml" type="integer" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3.2">1</cn><ci id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3.3.cmml" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.1.3.3.3">𝑝</ci></apply></apply></apply><ci id="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.2.cmml" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.2.3.2">𝑛</ci></apply></apply><cn id="S5.SS1.p4.9.m9.2.2.2.1.1.1.3.cmml" type="integer" xref="S5.SS1.p4.9.m9.2.2.2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.9.m9.2c">n(4\log_{\frac{1}{1-p}}n+\frac{1}{n^{2}}n^{2})=n(4\log_{\frac{1}{1-p}}n+1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.9.m9.2d">italic_n ( 4 roman_log start_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG 1 - italic_p end_ARG end_POSTSUBSCRIPT italic_n + divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = italic_n ( 4 roman_log start_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG 1 - italic_p end_ARG end_POSTSUBSCRIPT italic_n + 1 )</annotation></semantics></math>, where the second term bounds the maximum welfare of a partition with the formation of a clique of all agents for the remaining cases occurring with probability at most<math alttext="\frac{1}{n^{2}}" class="ltx_Math" display="inline" id="S5.SS1.p4.10.m10.1"><semantics id="S5.SS1.p4.10.m10.1a"><mfrac id="S5.SS1.p4.10.m10.1.1" xref="S5.SS1.p4.10.m10.1.1.cmml"><mn id="S5.SS1.p4.10.m10.1.1.2" xref="S5.SS1.p4.10.m10.1.1.2.cmml">1</mn><msup id="S5.SS1.p4.10.m10.1.1.3" xref="S5.SS1.p4.10.m10.1.1.3.cmml"><mi id="S5.SS1.p4.10.m10.1.1.3.2" xref="S5.SS1.p4.10.m10.1.1.3.2.cmml">n</mi><mn id="S5.SS1.p4.10.m10.1.1.3.3" xref="S5.SS1.p4.10.m10.1.1.3.3.cmml">2</mn></msup></mfrac><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.10.m10.1b"><apply id="S5.SS1.p4.10.m10.1.1.cmml" xref="S5.SS1.p4.10.m10.1.1"><divide id="S5.SS1.p4.10.m10.1.1.1.cmml" xref="S5.SS1.p4.10.m10.1.1"></divide><cn id="S5.SS1.p4.10.m10.1.1.2.cmml" type="integer" xref="S5.SS1.p4.10.m10.1.1.2">1</cn><apply id="S5.SS1.p4.10.m10.1.1.3.cmml" xref="S5.SS1.p4.10.m10.1.1.3"><csymbol cd="ambiguous" id="S5.SS1.p4.10.m10.1.1.3.1.cmml" xref="S5.SS1.p4.10.m10.1.1.3">superscript</csymbol><ci id="S5.SS1.p4.10.m10.1.1.3.2.cmml" xref="S5.SS1.p4.10.m10.1.1.3.2">𝑛</ci><cn id="S5.SS1.p4.10.m10.1.1.3.3.cmml" type="integer" xref="S5.SS1.p4.10.m10.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.10.m10.1c">\frac{1}{n^{2}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.10.m10.1d">divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math>. This yields a ratio of about <math alttext="\frac{1}{16}\log_{\frac{1}{1-p}}b" class="ltx_Math" display="inline" id="S5.SS1.p4.11.m11.1"><semantics id="S5.SS1.p4.11.m11.1a"><mrow id="S5.SS1.p4.11.m11.1.1" xref="S5.SS1.p4.11.m11.1.1.cmml"><mfrac id="S5.SS1.p4.11.m11.1.1.2" xref="S5.SS1.p4.11.m11.1.1.2.cmml"><mn id="S5.SS1.p4.11.m11.1.1.2.2" xref="S5.SS1.p4.11.m11.1.1.2.2.cmml">1</mn><mn id="S5.SS1.p4.11.m11.1.1.2.3" xref="S5.SS1.p4.11.m11.1.1.2.3.cmml">16</mn></mfrac><mo id="S5.SS1.p4.11.m11.1.1.1" lspace="0.167em" xref="S5.SS1.p4.11.m11.1.1.1.cmml">⁢</mo><mrow id="S5.SS1.p4.11.m11.1.1.3" xref="S5.SS1.p4.11.m11.1.1.3.cmml"><msub id="S5.SS1.p4.11.m11.1.1.3.1" xref="S5.SS1.p4.11.m11.1.1.3.1.cmml"><mi id="S5.SS1.p4.11.m11.1.1.3.1.2" xref="S5.SS1.p4.11.m11.1.1.3.1.2.cmml">log</mi><mfrac id="S5.SS1.p4.11.m11.1.1.3.1.3" xref="S5.SS1.p4.11.m11.1.1.3.1.3.cmml"><mn id="S5.SS1.p4.11.m11.1.1.3.1.3.2" xref="S5.SS1.p4.11.m11.1.1.3.1.3.2.cmml">1</mn><mrow id="S5.SS1.p4.11.m11.1.1.3.1.3.3" xref="S5.SS1.p4.11.m11.1.1.3.1.3.3.cmml"><mn id="S5.SS1.p4.11.m11.1.1.3.1.3.3.2" xref="S5.SS1.p4.11.m11.1.1.3.1.3.3.2.cmml">1</mn><mo id="S5.SS1.p4.11.m11.1.1.3.1.3.3.1" xref="S5.SS1.p4.11.m11.1.1.3.1.3.3.1.cmml">−</mo><mi id="S5.SS1.p4.11.m11.1.1.3.1.3.3.3" xref="S5.SS1.p4.11.m11.1.1.3.1.3.3.3.cmml">p</mi></mrow></mfrac></msub><mo id="S5.SS1.p4.11.m11.1.1.3a" lspace="0.167em" xref="S5.SS1.p4.11.m11.1.1.3.cmml">⁡</mo><mi id="S5.SS1.p4.11.m11.1.1.3.2" xref="S5.SS1.p4.11.m11.1.1.3.2.cmml">b</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.11.m11.1b"><apply id="S5.SS1.p4.11.m11.1.1.cmml" xref="S5.SS1.p4.11.m11.1.1"><times id="S5.SS1.p4.11.m11.1.1.1.cmml" xref="S5.SS1.p4.11.m11.1.1.1"></times><apply id="S5.SS1.p4.11.m11.1.1.2.cmml" xref="S5.SS1.p4.11.m11.1.1.2"><divide id="S5.SS1.p4.11.m11.1.1.2.1.cmml" xref="S5.SS1.p4.11.m11.1.1.2"></divide><cn id="S5.SS1.p4.11.m11.1.1.2.2.cmml" type="integer" xref="S5.SS1.p4.11.m11.1.1.2.2">1</cn><cn id="S5.SS1.p4.11.m11.1.1.2.3.cmml" type="integer" xref="S5.SS1.p4.11.m11.1.1.2.3">16</cn></apply><apply id="S5.SS1.p4.11.m11.1.1.3.cmml" xref="S5.SS1.p4.11.m11.1.1.3"><apply id="S5.SS1.p4.11.m11.1.1.3.1.cmml" xref="S5.SS1.p4.11.m11.1.1.3.1"><csymbol cd="ambiguous" id="S5.SS1.p4.11.m11.1.1.3.1.1.cmml" xref="S5.SS1.p4.11.m11.1.1.3.1">subscript</csymbol><log id="S5.SS1.p4.11.m11.1.1.3.1.2.cmml" xref="S5.SS1.p4.11.m11.1.1.3.1.2"></log><apply id="S5.SS1.p4.11.m11.1.1.3.1.3.cmml" xref="S5.SS1.p4.11.m11.1.1.3.1.3"><divide id="S5.SS1.p4.11.m11.1.1.3.1.3.1.cmml" xref="S5.SS1.p4.11.m11.1.1.3.1.3"></divide><cn id="S5.SS1.p4.11.m11.1.1.3.1.3.2.cmml" type="integer" xref="S5.SS1.p4.11.m11.1.1.3.1.3.2">1</cn><apply id="S5.SS1.p4.11.m11.1.1.3.1.3.3.cmml" xref="S5.SS1.p4.11.m11.1.1.3.1.3.3"><minus id="S5.SS1.p4.11.m11.1.1.3.1.3.3.1.cmml" xref="S5.SS1.p4.11.m11.1.1.3.1.3.3.1"></minus><cn id="S5.SS1.p4.11.m11.1.1.3.1.3.3.2.cmml" type="integer" xref="S5.SS1.p4.11.m11.1.1.3.1.3.3.2">1</cn><ci id="S5.SS1.p4.11.m11.1.1.3.1.3.3.3.cmml" xref="S5.SS1.p4.11.m11.1.1.3.1.3.3.3">𝑝</ci></apply></apply></apply><ci id="S5.SS1.p4.11.m11.1.1.3.2.cmml" xref="S5.SS1.p4.11.m11.1.1.3.2">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.11.m11.1c">\frac{1}{16}\log_{\frac{1}{1-p}}b</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.11.m11.1d">divide start_ARG 1 end_ARG start_ARG 16 end_ARG roman_log start_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG 1 - italic_p end_ARG end_POSTSUBSCRIPT italic_b</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S5.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.2 </span>Random Multipartite Graphs</h3> <div class="ltx_para" id="S5.SS2.p1"> <p class="ltx_p" id="S5.SS2.p1.4">Consider <math alttext="k\geq 2" class="ltx_Math" display="inline" id="S5.SS2.p1.1.m1.1"><semantics id="S5.SS2.p1.1.m1.1a"><mrow id="S5.SS2.p1.1.m1.1.1" xref="S5.SS2.p1.1.m1.1.1.cmml"><mi id="S5.SS2.p1.1.m1.1.1.2" xref="S5.SS2.p1.1.m1.1.1.2.cmml">k</mi><mo id="S5.SS2.p1.1.m1.1.1.1" xref="S5.SS2.p1.1.m1.1.1.1.cmml">≥</mo><mn id="S5.SS2.p1.1.m1.1.1.3" xref="S5.SS2.p1.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.1.m1.1b"><apply id="S5.SS2.p1.1.m1.1.1.cmml" xref="S5.SS2.p1.1.m1.1.1"><geq id="S5.SS2.p1.1.m1.1.1.1.cmml" xref="S5.SS2.p1.1.m1.1.1.1"></geq><ci id="S5.SS2.p1.1.m1.1.1.2.cmml" xref="S5.SS2.p1.1.m1.1.1.2">𝑘</ci><cn id="S5.SS2.p1.1.m1.1.1.3.cmml" type="integer" xref="S5.SS2.p1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.1.m1.1c">k\geq 2</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.1.m1.1d">italic_k ≥ 2</annotation></semantics></math> distinct groups of agents, where the goal is to form diverse coalitions that contain at most one agent from each group. We model this by assigning a negative edge weight of <math alttext="-n" class="ltx_Math" display="inline" id="S5.SS2.p1.2.m2.1"><semantics id="S5.SS2.p1.2.m2.1a"><mrow id="S5.SS2.p1.2.m2.1.1" xref="S5.SS2.p1.2.m2.1.1.cmml"><mo id="S5.SS2.p1.2.m2.1.1a" xref="S5.SS2.p1.2.m2.1.1.cmml">−</mo><mi id="S5.SS2.p1.2.m2.1.1.2" xref="S5.SS2.p1.2.m2.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.2.m2.1b"><apply id="S5.SS2.p1.2.m2.1.1.cmml" xref="S5.SS2.p1.2.m2.1.1"><minus id="S5.SS2.p1.2.m2.1.1.1.cmml" xref="S5.SS2.p1.2.m2.1.1"></minus><ci id="S5.SS2.p1.2.m2.1.1.2.cmml" xref="S5.SS2.p1.2.m2.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.2.m2.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.2.m2.1d">- italic_n</annotation></semantics></math> to any pair of agents within the same group, rendering them incompatible. Additionally, certain pairs of agents from different groups may also be incompatible. In a random <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.p1.3.m3.1"><semantics id="S5.SS2.p1.3.m3.1a"><mi id="S5.SS2.p1.3.m3.1.1" xref="S5.SS2.p1.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.3.m3.1b"><ci id="S5.SS2.p1.3.m3.1.1.cmml" xref="S5.SS2.p1.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.3.m3.1d">italic_k</annotation></semantics></math>-partite graph, any pair of agents from different groups is incompatible with probability <math alttext="p" class="ltx_Math" display="inline" id="S5.SS2.p1.4.m4.1"><semantics id="S5.SS2.p1.4.m4.1a"><mi id="S5.SS2.p1.4.m4.1.1" xref="S5.SS2.p1.4.m4.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.4.m4.1b"><ci id="S5.SS2.p1.4.m4.1.1.cmml" xref="S5.SS2.p1.4.m4.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.4.m4.1c">p</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.4.m4.1d">italic_p</annotation></semantics></math>. This problem can be formalized as follows.</p> </div> <div class="ltx_para" id="S5.SS2.p2"> <p class="ltx_p" id="S5.SS2.p2.7">Consider a <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.p2.1.m1.1"><semantics id="S5.SS2.p2.1.m1.1a"><mi id="S5.SS2.p2.1.m1.1.1" xref="S5.SS2.p2.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.1.m1.1b"><ci id="S5.SS2.p2.1.m1.1.1.cmml" xref="S5.SS2.p2.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.1.m1.1d">italic_k</annotation></semantics></math>-partite graph where vertices represent agents. The graph consists of <math alttext="n" class="ltx_Math" display="inline" id="S5.SS2.p2.2.m2.1"><semantics id="S5.SS2.p2.2.m2.1a"><mi id="S5.SS2.p2.2.m2.1.1" xref="S5.SS2.p2.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.2.m2.1b"><ci id="S5.SS2.p2.2.m2.1.1.cmml" xref="S5.SS2.p2.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.2.m2.1d">italic_n</annotation></semantics></math> vertices partitioned into <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.p2.3.m3.1"><semantics id="S5.SS2.p2.3.m3.1a"><mi id="S5.SS2.p2.3.m3.1.1" xref="S5.SS2.p2.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.3.m3.1b"><ci id="S5.SS2.p2.3.m3.1.1.cmml" xref="S5.SS2.p2.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.3.m3.1d">italic_k</annotation></semantics></math> disjoint “color” classes <math alttext="V_{1},V_{2},\dots,V_{k}" class="ltx_Math" display="inline" id="S5.SS2.p2.4.m4.4"><semantics id="S5.SS2.p2.4.m4.4a"><mrow id="S5.SS2.p2.4.m4.4.4.3" xref="S5.SS2.p2.4.m4.4.4.4.cmml"><msub id="S5.SS2.p2.4.m4.2.2.1.1" xref="S5.SS2.p2.4.m4.2.2.1.1.cmml"><mi id="S5.SS2.p2.4.m4.2.2.1.1.2" xref="S5.SS2.p2.4.m4.2.2.1.1.2.cmml">V</mi><mn id="S5.SS2.p2.4.m4.2.2.1.1.3" xref="S5.SS2.p2.4.m4.2.2.1.1.3.cmml">1</mn></msub><mo id="S5.SS2.p2.4.m4.4.4.3.4" xref="S5.SS2.p2.4.m4.4.4.4.cmml">,</mo><msub id="S5.SS2.p2.4.m4.3.3.2.2" xref="S5.SS2.p2.4.m4.3.3.2.2.cmml"><mi id="S5.SS2.p2.4.m4.3.3.2.2.2" xref="S5.SS2.p2.4.m4.3.3.2.2.2.cmml">V</mi><mn id="S5.SS2.p2.4.m4.3.3.2.2.3" xref="S5.SS2.p2.4.m4.3.3.2.2.3.cmml">2</mn></msub><mo id="S5.SS2.p2.4.m4.4.4.3.5" xref="S5.SS2.p2.4.m4.4.4.4.cmml">,</mo><mi id="S5.SS2.p2.4.m4.1.1" mathvariant="normal" xref="S5.SS2.p2.4.m4.1.1.cmml">…</mi><mo id="S5.SS2.p2.4.m4.4.4.3.6" xref="S5.SS2.p2.4.m4.4.4.4.cmml">,</mo><msub id="S5.SS2.p2.4.m4.4.4.3.3" xref="S5.SS2.p2.4.m4.4.4.3.3.cmml"><mi id="S5.SS2.p2.4.m4.4.4.3.3.2" xref="S5.SS2.p2.4.m4.4.4.3.3.2.cmml">V</mi><mi id="S5.SS2.p2.4.m4.4.4.3.3.3" xref="S5.SS2.p2.4.m4.4.4.3.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.4.m4.4b"><list id="S5.SS2.p2.4.m4.4.4.4.cmml" xref="S5.SS2.p2.4.m4.4.4.3"><apply id="S5.SS2.p2.4.m4.2.2.1.1.cmml" xref="S5.SS2.p2.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS2.p2.4.m4.2.2.1.1.1.cmml" xref="S5.SS2.p2.4.m4.2.2.1.1">subscript</csymbol><ci id="S5.SS2.p2.4.m4.2.2.1.1.2.cmml" xref="S5.SS2.p2.4.m4.2.2.1.1.2">𝑉</ci><cn id="S5.SS2.p2.4.m4.2.2.1.1.3.cmml" type="integer" xref="S5.SS2.p2.4.m4.2.2.1.1.3">1</cn></apply><apply id="S5.SS2.p2.4.m4.3.3.2.2.cmml" xref="S5.SS2.p2.4.m4.3.3.2.2"><csymbol cd="ambiguous" id="S5.SS2.p2.4.m4.3.3.2.2.1.cmml" xref="S5.SS2.p2.4.m4.3.3.2.2">subscript</csymbol><ci id="S5.SS2.p2.4.m4.3.3.2.2.2.cmml" xref="S5.SS2.p2.4.m4.3.3.2.2.2">𝑉</ci><cn id="S5.SS2.p2.4.m4.3.3.2.2.3.cmml" type="integer" xref="S5.SS2.p2.4.m4.3.3.2.2.3">2</cn></apply><ci id="S5.SS2.p2.4.m4.1.1.cmml" xref="S5.SS2.p2.4.m4.1.1">…</ci><apply id="S5.SS2.p2.4.m4.4.4.3.3.cmml" xref="S5.SS2.p2.4.m4.4.4.3.3"><csymbol cd="ambiguous" id="S5.SS2.p2.4.m4.4.4.3.3.1.cmml" xref="S5.SS2.p2.4.m4.4.4.3.3">subscript</csymbol><ci id="S5.SS2.p2.4.m4.4.4.3.3.2.cmml" xref="S5.SS2.p2.4.m4.4.4.3.3.2">𝑉</ci><ci id="S5.SS2.p2.4.m4.4.4.3.3.3.cmml" xref="S5.SS2.p2.4.m4.4.4.3.3.3">𝑘</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.4.m4.4c">V_{1},V_{2},\dots,V_{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.4.m4.4d">italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. All our results hold if <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.p2.5.m5.1"><semantics id="S5.SS2.p2.5.m5.1a"><mi id="S5.SS2.p2.5.m5.1.1" xref="S5.SS2.p2.5.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.5.m5.1b"><ci id="S5.SS2.p2.5.m5.1.1.cmml" xref="S5.SS2.p2.5.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.5.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.5.m5.1d">italic_k</annotation></semantics></math> is either a constant or any function satisfying <math alttext="k=o\left(\frac{n}{\log n}\right)" class="ltx_Math" display="inline" id="S5.SS2.p2.6.m6.1"><semantics id="S5.SS2.p2.6.m6.1a"><mrow id="S5.SS2.p2.6.m6.1.2" xref="S5.SS2.p2.6.m6.1.2.cmml"><mi id="S5.SS2.p2.6.m6.1.2.2" xref="S5.SS2.p2.6.m6.1.2.2.cmml">k</mi><mo id="S5.SS2.p2.6.m6.1.2.1" xref="S5.SS2.p2.6.m6.1.2.1.cmml">=</mo><mrow id="S5.SS2.p2.6.m6.1.2.3" xref="S5.SS2.p2.6.m6.1.2.3.cmml"><mi id="S5.SS2.p2.6.m6.1.2.3.2" xref="S5.SS2.p2.6.m6.1.2.3.2.cmml">o</mi><mo id="S5.SS2.p2.6.m6.1.2.3.1" xref="S5.SS2.p2.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS2.p2.6.m6.1.2.3.3.2" xref="S5.SS2.p2.6.m6.1.1.cmml"><mo id="S5.SS2.p2.6.m6.1.2.3.3.2.1" xref="S5.SS2.p2.6.m6.1.1.cmml">(</mo><mfrac id="S5.SS2.p2.6.m6.1.1" xref="S5.SS2.p2.6.m6.1.1.cmml"><mi id="S5.SS2.p2.6.m6.1.1.2" xref="S5.SS2.p2.6.m6.1.1.2.cmml">n</mi><mrow id="S5.SS2.p2.6.m6.1.1.3" xref="S5.SS2.p2.6.m6.1.1.3.cmml"><mi id="S5.SS2.p2.6.m6.1.1.3.1" xref="S5.SS2.p2.6.m6.1.1.3.1.cmml">log</mi><mo id="S5.SS2.p2.6.m6.1.1.3a" lspace="0.167em" xref="S5.SS2.p2.6.m6.1.1.3.cmml">⁡</mo><mi id="S5.SS2.p2.6.m6.1.1.3.2" xref="S5.SS2.p2.6.m6.1.1.3.2.cmml">n</mi></mrow></mfrac><mo id="S5.SS2.p2.6.m6.1.2.3.3.2.2" xref="S5.SS2.p2.6.m6.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.6.m6.1b"><apply id="S5.SS2.p2.6.m6.1.2.cmml" xref="S5.SS2.p2.6.m6.1.2"><eq id="S5.SS2.p2.6.m6.1.2.1.cmml" xref="S5.SS2.p2.6.m6.1.2.1"></eq><ci id="S5.SS2.p2.6.m6.1.2.2.cmml" xref="S5.SS2.p2.6.m6.1.2.2">𝑘</ci><apply id="S5.SS2.p2.6.m6.1.2.3.cmml" xref="S5.SS2.p2.6.m6.1.2.3"><times id="S5.SS2.p2.6.m6.1.2.3.1.cmml" xref="S5.SS2.p2.6.m6.1.2.3.1"></times><ci id="S5.SS2.p2.6.m6.1.2.3.2.cmml" xref="S5.SS2.p2.6.m6.1.2.3.2">𝑜</ci><apply id="S5.SS2.p2.6.m6.1.1.cmml" xref="S5.SS2.p2.6.m6.1.2.3.3.2"><divide id="S5.SS2.p2.6.m6.1.1.1.cmml" xref="S5.SS2.p2.6.m6.1.2.3.3.2"></divide><ci id="S5.SS2.p2.6.m6.1.1.2.cmml" xref="S5.SS2.p2.6.m6.1.1.2">𝑛</ci><apply id="S5.SS2.p2.6.m6.1.1.3.cmml" xref="S5.SS2.p2.6.m6.1.1.3"><log id="S5.SS2.p2.6.m6.1.1.3.1.cmml" xref="S5.SS2.p2.6.m6.1.1.3.1"></log><ci id="S5.SS2.p2.6.m6.1.1.3.2.cmml" xref="S5.SS2.p2.6.m6.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.6.m6.1c">k=o\left(\frac{n}{\log n}\right)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.6.m6.1d">italic_k = italic_o ( divide start_ARG italic_n end_ARG start_ARG roman_log italic_n end_ARG )</annotation></semantics></math>. Without loss of generality, assume that the color classes are sorted in nonincreasing order by the number of vertices they contain, i.e., <math alttext="|V_{1}|\geq|V_{2}|\geq\cdots\geq|V_{k}|" class="ltx_Math" display="inline" id="S5.SS2.p2.7.m7.3"><semantics id="S5.SS2.p2.7.m7.3a"><mrow id="S5.SS2.p2.7.m7.3.3" xref="S5.SS2.p2.7.m7.3.3.cmml"><mrow id="S5.SS2.p2.7.m7.1.1.1.1" xref="S5.SS2.p2.7.m7.1.1.1.2.cmml"><mo id="S5.SS2.p2.7.m7.1.1.1.1.2" stretchy="false" xref="S5.SS2.p2.7.m7.1.1.1.2.1.cmml">|</mo><msub id="S5.SS2.p2.7.m7.1.1.1.1.1" xref="S5.SS2.p2.7.m7.1.1.1.1.1.cmml"><mi id="S5.SS2.p2.7.m7.1.1.1.1.1.2" xref="S5.SS2.p2.7.m7.1.1.1.1.1.2.cmml">V</mi><mn id="S5.SS2.p2.7.m7.1.1.1.1.1.3" xref="S5.SS2.p2.7.m7.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS2.p2.7.m7.1.1.1.1.3" stretchy="false" xref="S5.SS2.p2.7.m7.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.SS2.p2.7.m7.3.3.5" xref="S5.SS2.p2.7.m7.3.3.5.cmml">≥</mo><mrow id="S5.SS2.p2.7.m7.2.2.2.1" xref="S5.SS2.p2.7.m7.2.2.2.2.cmml"><mo id="S5.SS2.p2.7.m7.2.2.2.1.2" stretchy="false" xref="S5.SS2.p2.7.m7.2.2.2.2.1.cmml">|</mo><msub id="S5.SS2.p2.7.m7.2.2.2.1.1" xref="S5.SS2.p2.7.m7.2.2.2.1.1.cmml"><mi id="S5.SS2.p2.7.m7.2.2.2.1.1.2" xref="S5.SS2.p2.7.m7.2.2.2.1.1.2.cmml">V</mi><mn id="S5.SS2.p2.7.m7.2.2.2.1.1.3" xref="S5.SS2.p2.7.m7.2.2.2.1.1.3.cmml">2</mn></msub><mo id="S5.SS2.p2.7.m7.2.2.2.1.3" stretchy="false" xref="S5.SS2.p2.7.m7.2.2.2.2.1.cmml">|</mo></mrow><mo id="S5.SS2.p2.7.m7.3.3.6" xref="S5.SS2.p2.7.m7.3.3.6.cmml">≥</mo><mi id="S5.SS2.p2.7.m7.3.3.7" mathvariant="normal" xref="S5.SS2.p2.7.m7.3.3.7.cmml">⋯</mi><mo id="S5.SS2.p2.7.m7.3.3.8" xref="S5.SS2.p2.7.m7.3.3.8.cmml">≥</mo><mrow id="S5.SS2.p2.7.m7.3.3.3.1" xref="S5.SS2.p2.7.m7.3.3.3.2.cmml"><mo id="S5.SS2.p2.7.m7.3.3.3.1.2" stretchy="false" xref="S5.SS2.p2.7.m7.3.3.3.2.1.cmml">|</mo><msub id="S5.SS2.p2.7.m7.3.3.3.1.1" xref="S5.SS2.p2.7.m7.3.3.3.1.1.cmml"><mi id="S5.SS2.p2.7.m7.3.3.3.1.1.2" xref="S5.SS2.p2.7.m7.3.3.3.1.1.2.cmml">V</mi><mi id="S5.SS2.p2.7.m7.3.3.3.1.1.3" xref="S5.SS2.p2.7.m7.3.3.3.1.1.3.cmml">k</mi></msub><mo id="S5.SS2.p2.7.m7.3.3.3.1.3" stretchy="false" xref="S5.SS2.p2.7.m7.3.3.3.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.7.m7.3b"><apply id="S5.SS2.p2.7.m7.3.3.cmml" xref="S5.SS2.p2.7.m7.3.3"><and id="S5.SS2.p2.7.m7.3.3a.cmml" xref="S5.SS2.p2.7.m7.3.3"></and><apply id="S5.SS2.p2.7.m7.3.3b.cmml" xref="S5.SS2.p2.7.m7.3.3"><geq id="S5.SS2.p2.7.m7.3.3.5.cmml" xref="S5.SS2.p2.7.m7.3.3.5"></geq><apply id="S5.SS2.p2.7.m7.1.1.1.2.cmml" xref="S5.SS2.p2.7.m7.1.1.1.1"><abs id="S5.SS2.p2.7.m7.1.1.1.2.1.cmml" xref="S5.SS2.p2.7.m7.1.1.1.1.2"></abs><apply id="S5.SS2.p2.7.m7.1.1.1.1.1.cmml" xref="S5.SS2.p2.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p2.7.m7.1.1.1.1.1.1.cmml" xref="S5.SS2.p2.7.m7.1.1.1.1.1">subscript</csymbol><ci id="S5.SS2.p2.7.m7.1.1.1.1.1.2.cmml" xref="S5.SS2.p2.7.m7.1.1.1.1.1.2">𝑉</ci><cn id="S5.SS2.p2.7.m7.1.1.1.1.1.3.cmml" type="integer" xref="S5.SS2.p2.7.m7.1.1.1.1.1.3">1</cn></apply></apply><apply id="S5.SS2.p2.7.m7.2.2.2.2.cmml" xref="S5.SS2.p2.7.m7.2.2.2.1"><abs id="S5.SS2.p2.7.m7.2.2.2.2.1.cmml" xref="S5.SS2.p2.7.m7.2.2.2.1.2"></abs><apply id="S5.SS2.p2.7.m7.2.2.2.1.1.cmml" xref="S5.SS2.p2.7.m7.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS2.p2.7.m7.2.2.2.1.1.1.cmml" xref="S5.SS2.p2.7.m7.2.2.2.1.1">subscript</csymbol><ci id="S5.SS2.p2.7.m7.2.2.2.1.1.2.cmml" xref="S5.SS2.p2.7.m7.2.2.2.1.1.2">𝑉</ci><cn id="S5.SS2.p2.7.m7.2.2.2.1.1.3.cmml" type="integer" xref="S5.SS2.p2.7.m7.2.2.2.1.1.3">2</cn></apply></apply></apply><apply id="S5.SS2.p2.7.m7.3.3c.cmml" xref="S5.SS2.p2.7.m7.3.3"><geq id="S5.SS2.p2.7.m7.3.3.6.cmml" xref="S5.SS2.p2.7.m7.3.3.6"></geq><share href="https://arxiv.org/html/2503.06017v1#S5.SS2.p2.7.m7.2.2.2.cmml" id="S5.SS2.p2.7.m7.3.3d.cmml" xref="S5.SS2.p2.7.m7.3.3"></share><ci id="S5.SS2.p2.7.m7.3.3.7.cmml" xref="S5.SS2.p2.7.m7.3.3.7">⋯</ci></apply><apply id="S5.SS2.p2.7.m7.3.3e.cmml" xref="S5.SS2.p2.7.m7.3.3"><geq id="S5.SS2.p2.7.m7.3.3.8.cmml" xref="S5.SS2.p2.7.m7.3.3.8"></geq><share href="https://arxiv.org/html/2503.06017v1#S5.SS2.p2.7.m7.3.3.7.cmml" id="S5.SS2.p2.7.m7.3.3f.cmml" xref="S5.SS2.p2.7.m7.3.3"></share><apply id="S5.SS2.p2.7.m7.3.3.3.2.cmml" xref="S5.SS2.p2.7.m7.3.3.3.1"><abs id="S5.SS2.p2.7.m7.3.3.3.2.1.cmml" xref="S5.SS2.p2.7.m7.3.3.3.1.2"></abs><apply id="S5.SS2.p2.7.m7.3.3.3.1.1.cmml" xref="S5.SS2.p2.7.m7.3.3.3.1.1"><csymbol cd="ambiguous" id="S5.SS2.p2.7.m7.3.3.3.1.1.1.cmml" xref="S5.SS2.p2.7.m7.3.3.3.1.1">subscript</csymbol><ci id="S5.SS2.p2.7.m7.3.3.3.1.1.2.cmml" xref="S5.SS2.p2.7.m7.3.3.3.1.1.2">𝑉</ci><ci id="S5.SS2.p2.7.m7.3.3.3.1.1.3.cmml" xref="S5.SS2.p2.7.m7.3.3.3.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.7.m7.3c">|V_{1}|\geq|V_{2}|\geq\cdots\geq|V_{k}|</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.7.m7.3d">| italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≥ | italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | ≥ ⋯ ≥ | italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS2.p3"> <p class="ltx_p" id="S5.SS2.p3.7">A <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.p3.1.m1.1"><semantics id="S5.SS2.p3.1.m1.1a"><mi id="S5.SS2.p3.1.m1.1.1" xref="S5.SS2.p3.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.1.m1.1b"><ci id="S5.SS2.p3.1.m1.1.1.cmml" xref="S5.SS2.p3.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.1.m1.1d">italic_k</annotation></semantics></math>-partite graph is said to be <em class="ltx_emph ltx_font_italic" id="S5.SS2.p3.7.1">balanced</em> if <math alttext="|V_{k}|\geq q|V_{1}|" class="ltx_Math" display="inline" id="S5.SS2.p3.2.m2.2"><semantics id="S5.SS2.p3.2.m2.2a"><mrow id="S5.SS2.p3.2.m2.2.2" xref="S5.SS2.p3.2.m2.2.2.cmml"><mrow id="S5.SS2.p3.2.m2.1.1.1.1" xref="S5.SS2.p3.2.m2.1.1.1.2.cmml"><mo id="S5.SS2.p3.2.m2.1.1.1.1.2" stretchy="false" xref="S5.SS2.p3.2.m2.1.1.1.2.1.cmml">|</mo><msub id="S5.SS2.p3.2.m2.1.1.1.1.1" xref="S5.SS2.p3.2.m2.1.1.1.1.1.cmml"><mi id="S5.SS2.p3.2.m2.1.1.1.1.1.2" xref="S5.SS2.p3.2.m2.1.1.1.1.1.2.cmml">V</mi><mi id="S5.SS2.p3.2.m2.1.1.1.1.1.3" xref="S5.SS2.p3.2.m2.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S5.SS2.p3.2.m2.1.1.1.1.3" stretchy="false" xref="S5.SS2.p3.2.m2.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.SS2.p3.2.m2.2.2.3" xref="S5.SS2.p3.2.m2.2.2.3.cmml">≥</mo><mrow id="S5.SS2.p3.2.m2.2.2.2" xref="S5.SS2.p3.2.m2.2.2.2.cmml"><mi id="S5.SS2.p3.2.m2.2.2.2.3" xref="S5.SS2.p3.2.m2.2.2.2.3.cmml">q</mi><mo id="S5.SS2.p3.2.m2.2.2.2.2" xref="S5.SS2.p3.2.m2.2.2.2.2.cmml">⁢</mo><mrow id="S5.SS2.p3.2.m2.2.2.2.1.1" xref="S5.SS2.p3.2.m2.2.2.2.1.2.cmml"><mo id="S5.SS2.p3.2.m2.2.2.2.1.1.2" stretchy="false" xref="S5.SS2.p3.2.m2.2.2.2.1.2.1.cmml">|</mo><msub id="S5.SS2.p3.2.m2.2.2.2.1.1.1" xref="S5.SS2.p3.2.m2.2.2.2.1.1.1.cmml"><mi id="S5.SS2.p3.2.m2.2.2.2.1.1.1.2" xref="S5.SS2.p3.2.m2.2.2.2.1.1.1.2.cmml">V</mi><mn id="S5.SS2.p3.2.m2.2.2.2.1.1.1.3" xref="S5.SS2.p3.2.m2.2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS2.p3.2.m2.2.2.2.1.1.3" stretchy="false" xref="S5.SS2.p3.2.m2.2.2.2.1.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.2.m2.2b"><apply id="S5.SS2.p3.2.m2.2.2.cmml" xref="S5.SS2.p3.2.m2.2.2"><geq id="S5.SS2.p3.2.m2.2.2.3.cmml" xref="S5.SS2.p3.2.m2.2.2.3"></geq><apply id="S5.SS2.p3.2.m2.1.1.1.2.cmml" xref="S5.SS2.p3.2.m2.1.1.1.1"><abs id="S5.SS2.p3.2.m2.1.1.1.2.1.cmml" xref="S5.SS2.p3.2.m2.1.1.1.1.2"></abs><apply id="S5.SS2.p3.2.m2.1.1.1.1.1.cmml" xref="S5.SS2.p3.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p3.2.m2.1.1.1.1.1.1.cmml" xref="S5.SS2.p3.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S5.SS2.p3.2.m2.1.1.1.1.1.2.cmml" xref="S5.SS2.p3.2.m2.1.1.1.1.1.2">𝑉</ci><ci id="S5.SS2.p3.2.m2.1.1.1.1.1.3.cmml" xref="S5.SS2.p3.2.m2.1.1.1.1.1.3">𝑘</ci></apply></apply><apply id="S5.SS2.p3.2.m2.2.2.2.cmml" xref="S5.SS2.p3.2.m2.2.2.2"><times id="S5.SS2.p3.2.m2.2.2.2.2.cmml" xref="S5.SS2.p3.2.m2.2.2.2.2"></times><ci id="S5.SS2.p3.2.m2.2.2.2.3.cmml" xref="S5.SS2.p3.2.m2.2.2.2.3">𝑞</ci><apply id="S5.SS2.p3.2.m2.2.2.2.1.2.cmml" xref="S5.SS2.p3.2.m2.2.2.2.1.1"><abs id="S5.SS2.p3.2.m2.2.2.2.1.2.1.cmml" xref="S5.SS2.p3.2.m2.2.2.2.1.1.2"></abs><apply id="S5.SS2.p3.2.m2.2.2.2.1.1.1.cmml" xref="S5.SS2.p3.2.m2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p3.2.m2.2.2.2.1.1.1.1.cmml" xref="S5.SS2.p3.2.m2.2.2.2.1.1.1">subscript</csymbol><ci id="S5.SS2.p3.2.m2.2.2.2.1.1.1.2.cmml" xref="S5.SS2.p3.2.m2.2.2.2.1.1.1.2">𝑉</ci><cn id="S5.SS2.p3.2.m2.2.2.2.1.1.1.3.cmml" type="integer" xref="S5.SS2.p3.2.m2.2.2.2.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.2.m2.2c">|V_{k}|\geq q|V_{1}|</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.2.m2.2d">| italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≥ italic_q | italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT |</annotation></semantics></math> holds for some constant <math alttext="q\in(0,1)" class="ltx_Math" display="inline" id="S5.SS2.p3.3.m3.2"><semantics id="S5.SS2.p3.3.m3.2a"><mrow id="S5.SS2.p3.3.m3.2.3" xref="S5.SS2.p3.3.m3.2.3.cmml"><mi id="S5.SS2.p3.3.m3.2.3.2" xref="S5.SS2.p3.3.m3.2.3.2.cmml">q</mi><mo id="S5.SS2.p3.3.m3.2.3.1" xref="S5.SS2.p3.3.m3.2.3.1.cmml">∈</mo><mrow id="S5.SS2.p3.3.m3.2.3.3.2" xref="S5.SS2.p3.3.m3.2.3.3.1.cmml"><mo id="S5.SS2.p3.3.m3.2.3.3.2.1" stretchy="false" xref="S5.SS2.p3.3.m3.2.3.3.1.cmml">(</mo><mn id="S5.SS2.p3.3.m3.1.1" xref="S5.SS2.p3.3.m3.1.1.cmml">0</mn><mo id="S5.SS2.p3.3.m3.2.3.3.2.2" xref="S5.SS2.p3.3.m3.2.3.3.1.cmml">,</mo><mn id="S5.SS2.p3.3.m3.2.2" xref="S5.SS2.p3.3.m3.2.2.cmml">1</mn><mo id="S5.SS2.p3.3.m3.2.3.3.2.3" stretchy="false" xref="S5.SS2.p3.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.3.m3.2b"><apply id="S5.SS2.p3.3.m3.2.3.cmml" xref="S5.SS2.p3.3.m3.2.3"><in id="S5.SS2.p3.3.m3.2.3.1.cmml" xref="S5.SS2.p3.3.m3.2.3.1"></in><ci id="S5.SS2.p3.3.m3.2.3.2.cmml" xref="S5.SS2.p3.3.m3.2.3.2">𝑞</ci><interval closure="open" id="S5.SS2.p3.3.m3.2.3.3.1.cmml" xref="S5.SS2.p3.3.m3.2.3.3.2"><cn id="S5.SS2.p3.3.m3.1.1.cmml" type="integer" xref="S5.SS2.p3.3.m3.1.1">0</cn><cn id="S5.SS2.p3.3.m3.2.2.cmml" type="integer" xref="S5.SS2.p3.3.m3.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.3.m3.2c">q\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.3.m3.2d">italic_q ∈ ( 0 , 1 )</annotation></semantics></math>. A <em class="ltx_emph ltx_font_italic" id="S5.SS2.p3.7.2">Turán graph</em> is a special case of a balanced <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.p3.4.m4.1"><semantics id="S5.SS2.p3.4.m4.1a"><mi id="S5.SS2.p3.4.m4.1.1" xref="S5.SS2.p3.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.4.m4.1b"><ci id="S5.SS2.p3.4.m4.1.1.cmml" xref="S5.SS2.p3.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.4.m4.1d">italic_k</annotation></semantics></math>-partite graph with <math alttext="n" class="ltx_Math" display="inline" id="S5.SS2.p3.5.m5.1"><semantics id="S5.SS2.p3.5.m5.1a"><mi id="S5.SS2.p3.5.m5.1.1" xref="S5.SS2.p3.5.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.5.m5.1b"><ci id="S5.SS2.p3.5.m5.1.1.cmml" xref="S5.SS2.p3.5.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.5.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.5.m5.1d">italic_n</annotation></semantics></math> vertices, where each color class contains the same number of vertices, i.e., for all <math alttext="i\in[k]" class="ltx_Math" display="inline" id="S5.SS2.p3.6.m6.1"><semantics id="S5.SS2.p3.6.m6.1a"><mrow id="S5.SS2.p3.6.m6.1.2" xref="S5.SS2.p3.6.m6.1.2.cmml"><mi id="S5.SS2.p3.6.m6.1.2.2" xref="S5.SS2.p3.6.m6.1.2.2.cmml">i</mi><mo id="S5.SS2.p3.6.m6.1.2.1" xref="S5.SS2.p3.6.m6.1.2.1.cmml">∈</mo><mrow id="S5.SS2.p3.6.m6.1.2.3.2" xref="S5.SS2.p3.6.m6.1.2.3.1.cmml"><mo id="S5.SS2.p3.6.m6.1.2.3.2.1" stretchy="false" xref="S5.SS2.p3.6.m6.1.2.3.1.1.cmml">[</mo><mi id="S5.SS2.p3.6.m6.1.1" xref="S5.SS2.p3.6.m6.1.1.cmml">k</mi><mo id="S5.SS2.p3.6.m6.1.2.3.2.2" stretchy="false" xref="S5.SS2.p3.6.m6.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.6.m6.1b"><apply id="S5.SS2.p3.6.m6.1.2.cmml" xref="S5.SS2.p3.6.m6.1.2"><in id="S5.SS2.p3.6.m6.1.2.1.cmml" xref="S5.SS2.p3.6.m6.1.2.1"></in><ci id="S5.SS2.p3.6.m6.1.2.2.cmml" xref="S5.SS2.p3.6.m6.1.2.2">𝑖</ci><apply id="S5.SS2.p3.6.m6.1.2.3.1.cmml" xref="S5.SS2.p3.6.m6.1.2.3.2"><csymbol cd="latexml" id="S5.SS2.p3.6.m6.1.2.3.1.1.cmml" xref="S5.SS2.p3.6.m6.1.2.3.2.1">delimited-[]</csymbol><ci id="S5.SS2.p3.6.m6.1.1.cmml" xref="S5.SS2.p3.6.m6.1.1">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.6.m6.1c">i\in[k]</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.6.m6.1d">italic_i ∈ [ italic_k ]</annotation></semantics></math>, we require <math alttext="|V_{i}|=\frac{n}{k}" class="ltx_Math" display="inline" id="S5.SS2.p3.7.m7.1"><semantics id="S5.SS2.p3.7.m7.1a"><mrow id="S5.SS2.p3.7.m7.1.1" xref="S5.SS2.p3.7.m7.1.1.cmml"><mrow id="S5.SS2.p3.7.m7.1.1.1.1" xref="S5.SS2.p3.7.m7.1.1.1.2.cmml"><mo id="S5.SS2.p3.7.m7.1.1.1.1.2" stretchy="false" xref="S5.SS2.p3.7.m7.1.1.1.2.1.cmml">|</mo><msub id="S5.SS2.p3.7.m7.1.1.1.1.1" xref="S5.SS2.p3.7.m7.1.1.1.1.1.cmml"><mi id="S5.SS2.p3.7.m7.1.1.1.1.1.2" xref="S5.SS2.p3.7.m7.1.1.1.1.1.2.cmml">V</mi><mi id="S5.SS2.p3.7.m7.1.1.1.1.1.3" xref="S5.SS2.p3.7.m7.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.SS2.p3.7.m7.1.1.1.1.3" stretchy="false" xref="S5.SS2.p3.7.m7.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.SS2.p3.7.m7.1.1.2" xref="S5.SS2.p3.7.m7.1.1.2.cmml">=</mo><mfrac id="S5.SS2.p3.7.m7.1.1.3" xref="S5.SS2.p3.7.m7.1.1.3.cmml"><mi id="S5.SS2.p3.7.m7.1.1.3.2" xref="S5.SS2.p3.7.m7.1.1.3.2.cmml">n</mi><mi id="S5.SS2.p3.7.m7.1.1.3.3" xref="S5.SS2.p3.7.m7.1.1.3.3.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p3.7.m7.1b"><apply id="S5.SS2.p3.7.m7.1.1.cmml" xref="S5.SS2.p3.7.m7.1.1"><eq id="S5.SS2.p3.7.m7.1.1.2.cmml" xref="S5.SS2.p3.7.m7.1.1.2"></eq><apply id="S5.SS2.p3.7.m7.1.1.1.2.cmml" xref="S5.SS2.p3.7.m7.1.1.1.1"><abs id="S5.SS2.p3.7.m7.1.1.1.2.1.cmml" xref="S5.SS2.p3.7.m7.1.1.1.1.2"></abs><apply id="S5.SS2.p3.7.m7.1.1.1.1.1.cmml" xref="S5.SS2.p3.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p3.7.m7.1.1.1.1.1.1.cmml" xref="S5.SS2.p3.7.m7.1.1.1.1.1">subscript</csymbol><ci id="S5.SS2.p3.7.m7.1.1.1.1.1.2.cmml" xref="S5.SS2.p3.7.m7.1.1.1.1.1.2">𝑉</ci><ci id="S5.SS2.p3.7.m7.1.1.1.1.1.3.cmml" xref="S5.SS2.p3.7.m7.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S5.SS2.p3.7.m7.1.1.3.cmml" xref="S5.SS2.p3.7.m7.1.1.3"><divide id="S5.SS2.p3.7.m7.1.1.3.1.cmml" xref="S5.SS2.p3.7.m7.1.1.3"></divide><ci id="S5.SS2.p3.7.m7.1.1.3.2.cmml" xref="S5.SS2.p3.7.m7.1.1.3.2">𝑛</ci><ci id="S5.SS2.p3.7.m7.1.1.3.3.cmml" xref="S5.SS2.p3.7.m7.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p3.7.m7.1c">|V_{i}|=\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p3.7.m7.1d">| italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | = divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS2.p4"> <p class="ltx_p" id="S5.SS2.p4.1">We capture these in our second model of random graphs inducing aversion-to-enemies games.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S5.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem5.1.1.1">Definition 5.5</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem5.p1"> <p class="ltx_p" id="S5.Thmtheorem5.p1.10">A <em class="ltx_emph ltx_font_italic" id="S5.Thmtheorem5.p1.1.1">random balanced <math alttext="k" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.1.1.m1.1"><semantics id="S5.Thmtheorem5.p1.1.1.m1.1a"><mi id="S5.Thmtheorem5.p1.1.1.m1.1.1" xref="S5.Thmtheorem5.p1.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.1.1.m1.1b"><ci id="S5.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem5.p1.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.1.1.m1.1d">italic_k</annotation></semantics></math>-partite graph</em> <math alttext="G=(\{V_{1},\ldots,V_{k}\},p)" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.2.m1.3"><semantics id="S5.Thmtheorem5.p1.2.m1.3a"><mrow id="S5.Thmtheorem5.p1.2.m1.3.3" xref="S5.Thmtheorem5.p1.2.m1.3.3.cmml"><mi id="S5.Thmtheorem5.p1.2.m1.3.3.3" xref="S5.Thmtheorem5.p1.2.m1.3.3.3.cmml">G</mi><mo id="S5.Thmtheorem5.p1.2.m1.3.3.2" xref="S5.Thmtheorem5.p1.2.m1.3.3.2.cmml">=</mo><mrow id="S5.Thmtheorem5.p1.2.m1.3.3.1.1" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.2.cmml"><mo id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.2" stretchy="false" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.2.cmml">(</mo><mrow id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.3.cmml"><mo id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.3" stretchy="false" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.3.cmml">{</mo><msub id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.1.1" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.1.1.2" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.1.1.2.cmml">V</mi><mn id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.1.1.3" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.4" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.3.cmml">,</mo><mi id="S5.Thmtheorem5.p1.2.m1.1.1" mathvariant="normal" xref="S5.Thmtheorem5.p1.2.m1.1.1.cmml">…</mi><mo id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.5" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.3.cmml">,</mo><msub id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.2" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.2.cmml"><mi id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.2.2" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.2.2.cmml">V</mi><mi id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.2.3" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.2.3.cmml">k</mi></msub><mo id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.6" stretchy="false" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.3.cmml">}</mo></mrow><mo id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.3" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.2.cmml">,</mo><mi id="S5.Thmtheorem5.p1.2.m1.2.2" xref="S5.Thmtheorem5.p1.2.m1.2.2.cmml">p</mi><mo id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.4" stretchy="false" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.2.m1.3b"><apply id="S5.Thmtheorem5.p1.2.m1.3.3.cmml" xref="S5.Thmtheorem5.p1.2.m1.3.3"><eq id="S5.Thmtheorem5.p1.2.m1.3.3.2.cmml" xref="S5.Thmtheorem5.p1.2.m1.3.3.2"></eq><ci id="S5.Thmtheorem5.p1.2.m1.3.3.3.cmml" xref="S5.Thmtheorem5.p1.2.m1.3.3.3">𝐺</ci><interval closure="open" id="S5.Thmtheorem5.p1.2.m1.3.3.1.2.cmml" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1"><set id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.3.cmml" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2"><apply id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.1.1.cmml" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.1.1.2">𝑉</ci><cn id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.1.1.3">1</cn></apply><ci id="S5.Thmtheorem5.p1.2.m1.1.1.cmml" xref="S5.Thmtheorem5.p1.2.m1.1.1">…</ci><apply id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.2.cmml" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.2.1.cmml" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.2">subscript</csymbol><ci id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.2.2.cmml" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.2.2">𝑉</ci><ci id="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.2.3.cmml" xref="S5.Thmtheorem5.p1.2.m1.3.3.1.1.1.2.2.3">𝑘</ci></apply></set><ci id="S5.Thmtheorem5.p1.2.m1.2.2.cmml" xref="S5.Thmtheorem5.p1.2.m1.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.2.m1.3c">G=(\{V_{1},\ldots,V_{k}\},p)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.2.m1.3d">italic_G = ( { italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } , italic_p )</annotation></semantics></math> is a weighted graph where edge weights are sampled independently as follows: each edge between vertices in two different color classes independently takes a weight of <math alttext="-n" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.3.m2.1"><semantics id="S5.Thmtheorem5.p1.3.m2.1a"><mrow id="S5.Thmtheorem5.p1.3.m2.1.1" xref="S5.Thmtheorem5.p1.3.m2.1.1.cmml"><mo id="S5.Thmtheorem5.p1.3.m2.1.1a" xref="S5.Thmtheorem5.p1.3.m2.1.1.cmml">−</mo><mi id="S5.Thmtheorem5.p1.3.m2.1.1.2" xref="S5.Thmtheorem5.p1.3.m2.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.3.m2.1b"><apply id="S5.Thmtheorem5.p1.3.m2.1.1.cmml" xref="S5.Thmtheorem5.p1.3.m2.1.1"><minus id="S5.Thmtheorem5.p1.3.m2.1.1.1.cmml" xref="S5.Thmtheorem5.p1.3.m2.1.1"></minus><ci id="S5.Thmtheorem5.p1.3.m2.1.1.2.cmml" xref="S5.Thmtheorem5.p1.3.m2.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.3.m2.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.3.m2.1d">- italic_n</annotation></semantics></math> with probability <math alttext="p" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.4.m3.1"><semantics id="S5.Thmtheorem5.p1.4.m3.1a"><mi id="S5.Thmtheorem5.p1.4.m3.1.1" xref="S5.Thmtheorem5.p1.4.m3.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.4.m3.1b"><ci id="S5.Thmtheorem5.p1.4.m3.1.1.cmml" xref="S5.Thmtheorem5.p1.4.m3.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.4.m3.1c">p</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.4.m3.1d">italic_p</annotation></semantics></math>, and a weight of <math alttext="1" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.5.m4.1"><semantics id="S5.Thmtheorem5.p1.5.m4.1a"><mn id="S5.Thmtheorem5.p1.5.m4.1.1" xref="S5.Thmtheorem5.p1.5.m4.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.5.m4.1b"><cn id="S5.Thmtheorem5.p1.5.m4.1.1.cmml" type="integer" xref="S5.Thmtheorem5.p1.5.m4.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.5.m4.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.5.m4.1d">1</annotation></semantics></math> with probability <math alttext="1-p" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.6.m5.1"><semantics id="S5.Thmtheorem5.p1.6.m5.1a"><mrow id="S5.Thmtheorem5.p1.6.m5.1.1" xref="S5.Thmtheorem5.p1.6.m5.1.1.cmml"><mn id="S5.Thmtheorem5.p1.6.m5.1.1.2" xref="S5.Thmtheorem5.p1.6.m5.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem5.p1.6.m5.1.1.1" xref="S5.Thmtheorem5.p1.6.m5.1.1.1.cmml">−</mo><mi id="S5.Thmtheorem5.p1.6.m5.1.1.3" xref="S5.Thmtheorem5.p1.6.m5.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.6.m5.1b"><apply id="S5.Thmtheorem5.p1.6.m5.1.1.cmml" xref="S5.Thmtheorem5.p1.6.m5.1.1"><minus id="S5.Thmtheorem5.p1.6.m5.1.1.1.cmml" xref="S5.Thmtheorem5.p1.6.m5.1.1.1"></minus><cn id="S5.Thmtheorem5.p1.6.m5.1.1.2.cmml" type="integer" xref="S5.Thmtheorem5.p1.6.m5.1.1.2">1</cn><ci id="S5.Thmtheorem5.p1.6.m5.1.1.3.cmml" xref="S5.Thmtheorem5.p1.6.m5.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.6.m5.1c">1-p</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.6.m5.1d">1 - italic_p</annotation></semantics></math>; each edge between vertices of the same color class takes a weight of <math alttext="-n" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.7.m6.1"><semantics id="S5.Thmtheorem5.p1.7.m6.1a"><mrow id="S5.Thmtheorem5.p1.7.m6.1.1" xref="S5.Thmtheorem5.p1.7.m6.1.1.cmml"><mo id="S5.Thmtheorem5.p1.7.m6.1.1a" xref="S5.Thmtheorem5.p1.7.m6.1.1.cmml">−</mo><mi id="S5.Thmtheorem5.p1.7.m6.1.1.2" xref="S5.Thmtheorem5.p1.7.m6.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.7.m6.1b"><apply id="S5.Thmtheorem5.p1.7.m6.1.1.cmml" xref="S5.Thmtheorem5.p1.7.m6.1.1"><minus id="S5.Thmtheorem5.p1.7.m6.1.1.1.cmml" xref="S5.Thmtheorem5.p1.7.m6.1.1"></minus><ci id="S5.Thmtheorem5.p1.7.m6.1.1.2.cmml" xref="S5.Thmtheorem5.p1.7.m6.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.7.m6.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.7.m6.1d">- italic_n</annotation></semantics></math> with probability <math alttext="1" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.8.m7.1"><semantics id="S5.Thmtheorem5.p1.8.m7.1a"><mn id="S5.Thmtheorem5.p1.8.m7.1.1" xref="S5.Thmtheorem5.p1.8.m7.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.8.m7.1b"><cn id="S5.Thmtheorem5.p1.8.m7.1.1.cmml" type="integer" xref="S5.Thmtheorem5.p1.8.m7.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.8.m7.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.8.m7.1d">1</annotation></semantics></math>. The input parameter <math alttext="p\in(0,1)" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.9.m8.2"><semantics id="S5.Thmtheorem5.p1.9.m8.2a"><mrow id="S5.Thmtheorem5.p1.9.m8.2.3" xref="S5.Thmtheorem5.p1.9.m8.2.3.cmml"><mi id="S5.Thmtheorem5.p1.9.m8.2.3.2" xref="S5.Thmtheorem5.p1.9.m8.2.3.2.cmml">p</mi><mo id="S5.Thmtheorem5.p1.9.m8.2.3.1" xref="S5.Thmtheorem5.p1.9.m8.2.3.1.cmml">∈</mo><mrow id="S5.Thmtheorem5.p1.9.m8.2.3.3.2" xref="S5.Thmtheorem5.p1.9.m8.2.3.3.1.cmml"><mo id="S5.Thmtheorem5.p1.9.m8.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem5.p1.9.m8.2.3.3.1.cmml">(</mo><mn id="S5.Thmtheorem5.p1.9.m8.1.1" xref="S5.Thmtheorem5.p1.9.m8.1.1.cmml">0</mn><mo id="S5.Thmtheorem5.p1.9.m8.2.3.3.2.2" xref="S5.Thmtheorem5.p1.9.m8.2.3.3.1.cmml">,</mo><mn id="S5.Thmtheorem5.p1.9.m8.2.2" xref="S5.Thmtheorem5.p1.9.m8.2.2.cmml">1</mn><mo id="S5.Thmtheorem5.p1.9.m8.2.3.3.2.3" stretchy="false" xref="S5.Thmtheorem5.p1.9.m8.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.9.m8.2b"><apply id="S5.Thmtheorem5.p1.9.m8.2.3.cmml" xref="S5.Thmtheorem5.p1.9.m8.2.3"><in id="S5.Thmtheorem5.p1.9.m8.2.3.1.cmml" xref="S5.Thmtheorem5.p1.9.m8.2.3.1"></in><ci id="S5.Thmtheorem5.p1.9.m8.2.3.2.cmml" xref="S5.Thmtheorem5.p1.9.m8.2.3.2">𝑝</ci><interval closure="open" id="S5.Thmtheorem5.p1.9.m8.2.3.3.1.cmml" xref="S5.Thmtheorem5.p1.9.m8.2.3.3.2"><cn id="S5.Thmtheorem5.p1.9.m8.1.1.cmml" type="integer" xref="S5.Thmtheorem5.p1.9.m8.1.1">0</cn><cn id="S5.Thmtheorem5.p1.9.m8.2.2.cmml" type="integer" xref="S5.Thmtheorem5.p1.9.m8.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.9.m8.2c">p\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.9.m8.2d">italic_p ∈ ( 0 , 1 )</annotation></semantics></math> is called the <em class="ltx_emph ltx_font_italic" id="S5.Thmtheorem5.p1.10.2">perturbation probability</em>, and it is allowed to depend on <math alttext="n" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.10.m9.1"><semantics id="S5.Thmtheorem5.p1.10.m9.1a"><mi id="S5.Thmtheorem5.p1.10.m9.1.1" xref="S5.Thmtheorem5.p1.10.m9.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.10.m9.1b"><ci id="S5.Thmtheorem5.p1.10.m9.1.1.cmml" xref="S5.Thmtheorem5.p1.10.m9.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.10.m9.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.10.m9.1d">italic_n</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.Thmtheorem5.p2"> <p class="ltx_p" id="S5.Thmtheorem5.p2.4">A <em class="ltx_emph ltx_font_italic" id="S5.Thmtheorem5.p2.4.1">random Turán graph</em> <math alttext="G=(n,k,p)" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p2.1.m1.3"><semantics id="S5.Thmtheorem5.p2.1.m1.3a"><mrow id="S5.Thmtheorem5.p2.1.m1.3.4" xref="S5.Thmtheorem5.p2.1.m1.3.4.cmml"><mi id="S5.Thmtheorem5.p2.1.m1.3.4.2" xref="S5.Thmtheorem5.p2.1.m1.3.4.2.cmml">G</mi><mo id="S5.Thmtheorem5.p2.1.m1.3.4.1" xref="S5.Thmtheorem5.p2.1.m1.3.4.1.cmml">=</mo><mrow id="S5.Thmtheorem5.p2.1.m1.3.4.3.2" xref="S5.Thmtheorem5.p2.1.m1.3.4.3.1.cmml"><mo id="S5.Thmtheorem5.p2.1.m1.3.4.3.2.1" stretchy="false" xref="S5.Thmtheorem5.p2.1.m1.3.4.3.1.cmml">(</mo><mi id="S5.Thmtheorem5.p2.1.m1.1.1" xref="S5.Thmtheorem5.p2.1.m1.1.1.cmml">n</mi><mo id="S5.Thmtheorem5.p2.1.m1.3.4.3.2.2" xref="S5.Thmtheorem5.p2.1.m1.3.4.3.1.cmml">,</mo><mi id="S5.Thmtheorem5.p2.1.m1.2.2" xref="S5.Thmtheorem5.p2.1.m1.2.2.cmml">k</mi><mo id="S5.Thmtheorem5.p2.1.m1.3.4.3.2.3" xref="S5.Thmtheorem5.p2.1.m1.3.4.3.1.cmml">,</mo><mi id="S5.Thmtheorem5.p2.1.m1.3.3" xref="S5.Thmtheorem5.p2.1.m1.3.3.cmml">p</mi><mo id="S5.Thmtheorem5.p2.1.m1.3.4.3.2.4" stretchy="false" xref="S5.Thmtheorem5.p2.1.m1.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p2.1.m1.3b"><apply id="S5.Thmtheorem5.p2.1.m1.3.4.cmml" xref="S5.Thmtheorem5.p2.1.m1.3.4"><eq id="S5.Thmtheorem5.p2.1.m1.3.4.1.cmml" xref="S5.Thmtheorem5.p2.1.m1.3.4.1"></eq><ci id="S5.Thmtheorem5.p2.1.m1.3.4.2.cmml" xref="S5.Thmtheorem5.p2.1.m1.3.4.2">𝐺</ci><vector id="S5.Thmtheorem5.p2.1.m1.3.4.3.1.cmml" xref="S5.Thmtheorem5.p2.1.m1.3.4.3.2"><ci id="S5.Thmtheorem5.p2.1.m1.1.1.cmml" xref="S5.Thmtheorem5.p2.1.m1.1.1">𝑛</ci><ci id="S5.Thmtheorem5.p2.1.m1.2.2.cmml" xref="S5.Thmtheorem5.p2.1.m1.2.2">𝑘</ci><ci id="S5.Thmtheorem5.p2.1.m1.3.3.cmml" xref="S5.Thmtheorem5.p2.1.m1.3.3">𝑝</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p2.1.m1.3c">G=(n,k,p)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p2.1.m1.3d">italic_G = ( italic_n , italic_k , italic_p )</annotation></semantics></math> is a random balanced <math alttext="k" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p2.2.m2.1"><semantics id="S5.Thmtheorem5.p2.2.m2.1a"><mi id="S5.Thmtheorem5.p2.2.m2.1.1" xref="S5.Thmtheorem5.p2.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p2.2.m2.1b"><ci id="S5.Thmtheorem5.p2.2.m2.1.1.cmml" xref="S5.Thmtheorem5.p2.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p2.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p2.2.m2.1d">italic_k</annotation></semantics></math>-partite graph where each color class contains the same number of vertices, i.e., for all <math alttext="i\in[k]" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p2.3.m3.1"><semantics id="S5.Thmtheorem5.p2.3.m3.1a"><mrow id="S5.Thmtheorem5.p2.3.m3.1.2" xref="S5.Thmtheorem5.p2.3.m3.1.2.cmml"><mi id="S5.Thmtheorem5.p2.3.m3.1.2.2" xref="S5.Thmtheorem5.p2.3.m3.1.2.2.cmml">i</mi><mo id="S5.Thmtheorem5.p2.3.m3.1.2.1" xref="S5.Thmtheorem5.p2.3.m3.1.2.1.cmml">∈</mo><mrow id="S5.Thmtheorem5.p2.3.m3.1.2.3.2" xref="S5.Thmtheorem5.p2.3.m3.1.2.3.1.cmml"><mo id="S5.Thmtheorem5.p2.3.m3.1.2.3.2.1" stretchy="false" xref="S5.Thmtheorem5.p2.3.m3.1.2.3.1.1.cmml">[</mo><mi id="S5.Thmtheorem5.p2.3.m3.1.1" xref="S5.Thmtheorem5.p2.3.m3.1.1.cmml">k</mi><mo id="S5.Thmtheorem5.p2.3.m3.1.2.3.2.2" stretchy="false" xref="S5.Thmtheorem5.p2.3.m3.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p2.3.m3.1b"><apply id="S5.Thmtheorem5.p2.3.m3.1.2.cmml" xref="S5.Thmtheorem5.p2.3.m3.1.2"><in id="S5.Thmtheorem5.p2.3.m3.1.2.1.cmml" xref="S5.Thmtheorem5.p2.3.m3.1.2.1"></in><ci id="S5.Thmtheorem5.p2.3.m3.1.2.2.cmml" xref="S5.Thmtheorem5.p2.3.m3.1.2.2">𝑖</ci><apply id="S5.Thmtheorem5.p2.3.m3.1.2.3.1.cmml" xref="S5.Thmtheorem5.p2.3.m3.1.2.3.2"><csymbol cd="latexml" id="S5.Thmtheorem5.p2.3.m3.1.2.3.1.1.cmml" xref="S5.Thmtheorem5.p2.3.m3.1.2.3.2.1">delimited-[]</csymbol><ci id="S5.Thmtheorem5.p2.3.m3.1.1.cmml" xref="S5.Thmtheorem5.p2.3.m3.1.1">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p2.3.m3.1c">i\in[k]</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p2.3.m3.1d">italic_i ∈ [ italic_k ]</annotation></semantics></math> we have <math alttext="|V_{i}|=\frac{n}{k}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p2.4.m4.1"><semantics id="S5.Thmtheorem5.p2.4.m4.1a"><mrow id="S5.Thmtheorem5.p2.4.m4.1.1" xref="S5.Thmtheorem5.p2.4.m4.1.1.cmml"><mrow id="S5.Thmtheorem5.p2.4.m4.1.1.1.1" xref="S5.Thmtheorem5.p2.4.m4.1.1.1.2.cmml"><mo id="S5.Thmtheorem5.p2.4.m4.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem5.p2.4.m4.1.1.1.2.1.cmml">|</mo><msub id="S5.Thmtheorem5.p2.4.m4.1.1.1.1.1" xref="S5.Thmtheorem5.p2.4.m4.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem5.p2.4.m4.1.1.1.1.1.2" xref="S5.Thmtheorem5.p2.4.m4.1.1.1.1.1.2.cmml">V</mi><mi id="S5.Thmtheorem5.p2.4.m4.1.1.1.1.1.3" xref="S5.Thmtheorem5.p2.4.m4.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.Thmtheorem5.p2.4.m4.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem5.p2.4.m4.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.Thmtheorem5.p2.4.m4.1.1.2" xref="S5.Thmtheorem5.p2.4.m4.1.1.2.cmml">=</mo><mfrac id="S5.Thmtheorem5.p2.4.m4.1.1.3" xref="S5.Thmtheorem5.p2.4.m4.1.1.3.cmml"><mi id="S5.Thmtheorem5.p2.4.m4.1.1.3.2" xref="S5.Thmtheorem5.p2.4.m4.1.1.3.2.cmml">n</mi><mi id="S5.Thmtheorem5.p2.4.m4.1.1.3.3" xref="S5.Thmtheorem5.p2.4.m4.1.1.3.3.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p2.4.m4.1b"><apply id="S5.Thmtheorem5.p2.4.m4.1.1.cmml" xref="S5.Thmtheorem5.p2.4.m4.1.1"><eq id="S5.Thmtheorem5.p2.4.m4.1.1.2.cmml" xref="S5.Thmtheorem5.p2.4.m4.1.1.2"></eq><apply id="S5.Thmtheorem5.p2.4.m4.1.1.1.2.cmml" xref="S5.Thmtheorem5.p2.4.m4.1.1.1.1"><abs id="S5.Thmtheorem5.p2.4.m4.1.1.1.2.1.cmml" xref="S5.Thmtheorem5.p2.4.m4.1.1.1.1.2"></abs><apply id="S5.Thmtheorem5.p2.4.m4.1.1.1.1.1.cmml" xref="S5.Thmtheorem5.p2.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p2.4.m4.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem5.p2.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem5.p2.4.m4.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem5.p2.4.m4.1.1.1.1.1.2">𝑉</ci><ci id="S5.Thmtheorem5.p2.4.m4.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem5.p2.4.m4.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S5.Thmtheorem5.p2.4.m4.1.1.3.cmml" xref="S5.Thmtheorem5.p2.4.m4.1.1.3"><divide id="S5.Thmtheorem5.p2.4.m4.1.1.3.1.cmml" xref="S5.Thmtheorem5.p2.4.m4.1.1.3"></divide><ci id="S5.Thmtheorem5.p2.4.m4.1.1.3.2.cmml" xref="S5.Thmtheorem5.p2.4.m4.1.1.3.2">𝑛</ci><ci id="S5.Thmtheorem5.p2.4.m4.1.1.3.3.cmml" xref="S5.Thmtheorem5.p2.4.m4.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p2.4.m4.1c">|V_{i}|=\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p2.4.m4.1d">| italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | = divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S5.SS2.p5"> <p class="ltx_p" id="S5.SS2.p5.16">The goal is to find a partition of maximum welfare for the case when the input is a random balanced <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.p5.1.m1.1"><semantics id="S5.SS2.p5.1.m1.1a"><mi id="S5.SS2.p5.1.m1.1.1" xref="S5.SS2.p5.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.1.m1.1b"><ci id="S5.SS2.p5.1.m1.1.1.cmml" xref="S5.SS2.p5.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.1.m1.1d">italic_k</annotation></semantics></math>-partite graph (or a random Turán graph). Note that when <math alttext="p=0" class="ltx_Math" display="inline" id="S5.SS2.p5.2.m2.1"><semantics id="S5.SS2.p5.2.m2.1a"><mrow id="S5.SS2.p5.2.m2.1.1" xref="S5.SS2.p5.2.m2.1.1.cmml"><mi id="S5.SS2.p5.2.m2.1.1.2" xref="S5.SS2.p5.2.m2.1.1.2.cmml">p</mi><mo id="S5.SS2.p5.2.m2.1.1.1" xref="S5.SS2.p5.2.m2.1.1.1.cmml">=</mo><mn id="S5.SS2.p5.2.m2.1.1.3" xref="S5.SS2.p5.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.2.m2.1b"><apply id="S5.SS2.p5.2.m2.1.1.cmml" xref="S5.SS2.p5.2.m2.1.1"><eq id="S5.SS2.p5.2.m2.1.1.1.cmml" xref="S5.SS2.p5.2.m2.1.1.1"></eq><ci id="S5.SS2.p5.2.m2.1.1.2.cmml" xref="S5.SS2.p5.2.m2.1.1.2">𝑝</ci><cn id="S5.SS2.p5.2.m2.1.1.3.cmml" type="integer" xref="S5.SS2.p5.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.2.m2.1c">p=0</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.2.m2.1d">italic_p = 0</annotation></semantics></math> or <math alttext="p=1" class="ltx_Math" display="inline" id="S5.SS2.p5.3.m3.1"><semantics id="S5.SS2.p5.3.m3.1a"><mrow id="S5.SS2.p5.3.m3.1.1" xref="S5.SS2.p5.3.m3.1.1.cmml"><mi id="S5.SS2.p5.3.m3.1.1.2" xref="S5.SS2.p5.3.m3.1.1.2.cmml">p</mi><mo id="S5.SS2.p5.3.m3.1.1.1" xref="S5.SS2.p5.3.m3.1.1.1.cmml">=</mo><mn id="S5.SS2.p5.3.m3.1.1.3" xref="S5.SS2.p5.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.3.m3.1b"><apply id="S5.SS2.p5.3.m3.1.1.cmml" xref="S5.SS2.p5.3.m3.1.1"><eq id="S5.SS2.p5.3.m3.1.1.1.cmml" xref="S5.SS2.p5.3.m3.1.1.1"></eq><ci id="S5.SS2.p5.3.m3.1.1.2.cmml" xref="S5.SS2.p5.3.m3.1.1.2">𝑝</ci><cn id="S5.SS2.p5.3.m3.1.1.3.cmml" type="integer" xref="S5.SS2.p5.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.3.m3.1c">p=1</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.3.m3.1d">italic_p = 1</annotation></semantics></math>, the problem becomes trivial: When <math alttext="p=0" class="ltx_Math" display="inline" id="S5.SS2.p5.4.m4.1"><semantics id="S5.SS2.p5.4.m4.1a"><mrow id="S5.SS2.p5.4.m4.1.1" xref="S5.SS2.p5.4.m4.1.1.cmml"><mi id="S5.SS2.p5.4.m4.1.1.2" xref="S5.SS2.p5.4.m4.1.1.2.cmml">p</mi><mo id="S5.SS2.p5.4.m4.1.1.1" xref="S5.SS2.p5.4.m4.1.1.1.cmml">=</mo><mn id="S5.SS2.p5.4.m4.1.1.3" xref="S5.SS2.p5.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.4.m4.1b"><apply id="S5.SS2.p5.4.m4.1.1.cmml" xref="S5.SS2.p5.4.m4.1.1"><eq id="S5.SS2.p5.4.m4.1.1.1.cmml" xref="S5.SS2.p5.4.m4.1.1.1"></eq><ci id="S5.SS2.p5.4.m4.1.1.2.cmml" xref="S5.SS2.p5.4.m4.1.1.2">𝑝</ci><cn id="S5.SS2.p5.4.m4.1.1.3.cmml" type="integer" xref="S5.SS2.p5.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.4.m4.1c">p=0</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.4.m4.1d">italic_p = 0</annotation></semantics></math>, all weights between vertices from different color classes are deterministically positive, and the graph <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS2.p5.5.m5.1"><semantics id="S5.SS2.p5.5.m5.1a"><msup id="S5.SS2.p5.5.m5.1.1" xref="S5.SS2.p5.5.m5.1.1.cmml"><mi id="S5.SS2.p5.5.m5.1.1.2" xref="S5.SS2.p5.5.m5.1.1.2.cmml">G</mi><mo id="S5.SS2.p5.5.m5.1.1.3" xref="S5.SS2.p5.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.5.m5.1b"><apply id="S5.SS2.p5.5.m5.1.1.cmml" xref="S5.SS2.p5.5.m5.1.1"><csymbol cd="ambiguous" id="S5.SS2.p5.5.m5.1.1.1.cmml" xref="S5.SS2.p5.5.m5.1.1">superscript</csymbol><ci id="S5.SS2.p5.5.m5.1.1.2.cmml" xref="S5.SS2.p5.5.m5.1.1.2">𝐺</ci><ci id="S5.SS2.p5.5.m5.1.1.3.cmml" xref="S5.SS2.p5.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.5.m5.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.5.m5.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> induced by edges of weight <math alttext="1" class="ltx_Math" display="inline" id="S5.SS2.p5.6.m6.1"><semantics id="S5.SS2.p5.6.m6.1a"><mn id="S5.SS2.p5.6.m6.1.1" xref="S5.SS2.p5.6.m6.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.6.m6.1b"><cn id="S5.SS2.p5.6.m6.1.1.cmml" type="integer" xref="S5.SS2.p5.6.m6.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.6.m6.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.6.m6.1d">1</annotation></semantics></math> is a complete <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.p5.7.m7.1"><semantics id="S5.SS2.p5.7.m7.1a"><mi id="S5.SS2.p5.7.m7.1.1" xref="S5.SS2.p5.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.7.m7.1b"><ci id="S5.SS2.p5.7.m7.1.1.cmml" xref="S5.SS2.p5.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.7.m7.1d">italic_k</annotation></semantics></math>-partite graph. In this case, an optimal partition of a Turán graph consists of <math alttext="\frac{n}{k}" class="ltx_Math" display="inline" id="S5.SS2.p5.8.m8.1"><semantics id="S5.SS2.p5.8.m8.1a"><mfrac id="S5.SS2.p5.8.m8.1.1" xref="S5.SS2.p5.8.m8.1.1.cmml"><mi id="S5.SS2.p5.8.m8.1.1.2" xref="S5.SS2.p5.8.m8.1.1.2.cmml">n</mi><mi id="S5.SS2.p5.8.m8.1.1.3" xref="S5.SS2.p5.8.m8.1.1.3.cmml">k</mi></mfrac><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.8.m8.1b"><apply id="S5.SS2.p5.8.m8.1.1.cmml" xref="S5.SS2.p5.8.m8.1.1"><divide id="S5.SS2.p5.8.m8.1.1.1.cmml" xref="S5.SS2.p5.8.m8.1.1"></divide><ci id="S5.SS2.p5.8.m8.1.1.2.cmml" xref="S5.SS2.p5.8.m8.1.1.2">𝑛</ci><ci id="S5.SS2.p5.8.m8.1.1.3.cmml" xref="S5.SS2.p5.8.m8.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.8.m8.1c">\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.8.m8.1d">divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math> coalitions, each containing a unique member from each of the color classes <math alttext="V_{1},\dots,V_{k}" class="ltx_Math" display="inline" id="S5.SS2.p5.9.m9.3"><semantics id="S5.SS2.p5.9.m9.3a"><mrow id="S5.SS2.p5.9.m9.3.3.2" xref="S5.SS2.p5.9.m9.3.3.3.cmml"><msub id="S5.SS2.p5.9.m9.2.2.1.1" xref="S5.SS2.p5.9.m9.2.2.1.1.cmml"><mi id="S5.SS2.p5.9.m9.2.2.1.1.2" xref="S5.SS2.p5.9.m9.2.2.1.1.2.cmml">V</mi><mn id="S5.SS2.p5.9.m9.2.2.1.1.3" xref="S5.SS2.p5.9.m9.2.2.1.1.3.cmml">1</mn></msub><mo id="S5.SS2.p5.9.m9.3.3.2.3" xref="S5.SS2.p5.9.m9.3.3.3.cmml">,</mo><mi id="S5.SS2.p5.9.m9.1.1" mathvariant="normal" xref="S5.SS2.p5.9.m9.1.1.cmml">…</mi><mo id="S5.SS2.p5.9.m9.3.3.2.4" xref="S5.SS2.p5.9.m9.3.3.3.cmml">,</mo><msub id="S5.SS2.p5.9.m9.3.3.2.2" xref="S5.SS2.p5.9.m9.3.3.2.2.cmml"><mi id="S5.SS2.p5.9.m9.3.3.2.2.2" xref="S5.SS2.p5.9.m9.3.3.2.2.2.cmml">V</mi><mi id="S5.SS2.p5.9.m9.3.3.2.2.3" xref="S5.SS2.p5.9.m9.3.3.2.2.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.9.m9.3b"><list id="S5.SS2.p5.9.m9.3.3.3.cmml" xref="S5.SS2.p5.9.m9.3.3.2"><apply id="S5.SS2.p5.9.m9.2.2.1.1.cmml" xref="S5.SS2.p5.9.m9.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS2.p5.9.m9.2.2.1.1.1.cmml" xref="S5.SS2.p5.9.m9.2.2.1.1">subscript</csymbol><ci id="S5.SS2.p5.9.m9.2.2.1.1.2.cmml" xref="S5.SS2.p5.9.m9.2.2.1.1.2">𝑉</ci><cn id="S5.SS2.p5.9.m9.2.2.1.1.3.cmml" type="integer" xref="S5.SS2.p5.9.m9.2.2.1.1.3">1</cn></apply><ci id="S5.SS2.p5.9.m9.1.1.cmml" xref="S5.SS2.p5.9.m9.1.1">…</ci><apply id="S5.SS2.p5.9.m9.3.3.2.2.cmml" xref="S5.SS2.p5.9.m9.3.3.2.2"><csymbol cd="ambiguous" id="S5.SS2.p5.9.m9.3.3.2.2.1.cmml" xref="S5.SS2.p5.9.m9.3.3.2.2">subscript</csymbol><ci id="S5.SS2.p5.9.m9.3.3.2.2.2.cmml" xref="S5.SS2.p5.9.m9.3.3.2.2.2">𝑉</ci><ci id="S5.SS2.p5.9.m9.3.3.2.2.3.cmml" xref="S5.SS2.p5.9.m9.3.3.2.2.3">𝑘</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.9.m9.3c">V_{1},\dots,V_{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.9.m9.3d">italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. For a general balanced <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.p5.10.m10.1"><semantics id="S5.SS2.p5.10.m10.1a"><mi id="S5.SS2.p5.10.m10.1.1" xref="S5.SS2.p5.10.m10.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.10.m10.1b"><ci id="S5.SS2.p5.10.m10.1.1.cmml" xref="S5.SS2.p5.10.m10.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.10.m10.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.10.m10.1d">italic_k</annotation></semantics></math>-partite graph, the welfare-maximizing partition contains <math alttext="|V_{k}|" class="ltx_Math" display="inline" id="S5.SS2.p5.11.m11.1"><semantics id="S5.SS2.p5.11.m11.1a"><mrow id="S5.SS2.p5.11.m11.1.1.1" xref="S5.SS2.p5.11.m11.1.1.2.cmml"><mo id="S5.SS2.p5.11.m11.1.1.1.2" stretchy="false" xref="S5.SS2.p5.11.m11.1.1.2.1.cmml">|</mo><msub id="S5.SS2.p5.11.m11.1.1.1.1" xref="S5.SS2.p5.11.m11.1.1.1.1.cmml"><mi id="S5.SS2.p5.11.m11.1.1.1.1.2" xref="S5.SS2.p5.11.m11.1.1.1.1.2.cmml">V</mi><mi id="S5.SS2.p5.11.m11.1.1.1.1.3" xref="S5.SS2.p5.11.m11.1.1.1.1.3.cmml">k</mi></msub><mo id="S5.SS2.p5.11.m11.1.1.1.3" stretchy="false" xref="S5.SS2.p5.11.m11.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.11.m11.1b"><apply id="S5.SS2.p5.11.m11.1.1.2.cmml" xref="S5.SS2.p5.11.m11.1.1.1"><abs id="S5.SS2.p5.11.m11.1.1.2.1.cmml" xref="S5.SS2.p5.11.m11.1.1.1.2"></abs><apply id="S5.SS2.p5.11.m11.1.1.1.1.cmml" xref="S5.SS2.p5.11.m11.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p5.11.m11.1.1.1.1.1.cmml" xref="S5.SS2.p5.11.m11.1.1.1.1">subscript</csymbol><ci id="S5.SS2.p5.11.m11.1.1.1.1.2.cmml" xref="S5.SS2.p5.11.m11.1.1.1.1.2">𝑉</ci><ci id="S5.SS2.p5.11.m11.1.1.1.1.3.cmml" xref="S5.SS2.p5.11.m11.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.11.m11.1c">|V_{k}|</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.11.m11.1d">| italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |</annotation></semantics></math> <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.p5.12.m12.1"><semantics id="S5.SS2.p5.12.m12.1a"><mi id="S5.SS2.p5.12.m12.1.1" xref="S5.SS2.p5.12.m12.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.12.m12.1b"><ci id="S5.SS2.p5.12.m12.1.1.cmml" xref="S5.SS2.p5.12.m12.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.12.m12.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.12.m12.1d">italic_k</annotation></semantics></math>-cliques, <math alttext="|V_{k-1}|-|V_{k}|" class="ltx_Math" display="inline" id="S5.SS2.p5.13.m13.2"><semantics id="S5.SS2.p5.13.m13.2a"><mrow id="S5.SS2.p5.13.m13.2.2" xref="S5.SS2.p5.13.m13.2.2.cmml"><mrow id="S5.SS2.p5.13.m13.1.1.1.1" xref="S5.SS2.p5.13.m13.1.1.1.2.cmml"><mo id="S5.SS2.p5.13.m13.1.1.1.1.2" stretchy="false" xref="S5.SS2.p5.13.m13.1.1.1.2.1.cmml">|</mo><msub id="S5.SS2.p5.13.m13.1.1.1.1.1" xref="S5.SS2.p5.13.m13.1.1.1.1.1.cmml"><mi id="S5.SS2.p5.13.m13.1.1.1.1.1.2" xref="S5.SS2.p5.13.m13.1.1.1.1.1.2.cmml">V</mi><mrow id="S5.SS2.p5.13.m13.1.1.1.1.1.3" xref="S5.SS2.p5.13.m13.1.1.1.1.1.3.cmml"><mi id="S5.SS2.p5.13.m13.1.1.1.1.1.3.2" xref="S5.SS2.p5.13.m13.1.1.1.1.1.3.2.cmml">k</mi><mo id="S5.SS2.p5.13.m13.1.1.1.1.1.3.1" xref="S5.SS2.p5.13.m13.1.1.1.1.1.3.1.cmml">−</mo><mn id="S5.SS2.p5.13.m13.1.1.1.1.1.3.3" xref="S5.SS2.p5.13.m13.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S5.SS2.p5.13.m13.1.1.1.1.3" stretchy="false" xref="S5.SS2.p5.13.m13.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.SS2.p5.13.m13.2.2.3" xref="S5.SS2.p5.13.m13.2.2.3.cmml">−</mo><mrow id="S5.SS2.p5.13.m13.2.2.2.1" xref="S5.SS2.p5.13.m13.2.2.2.2.cmml"><mo id="S5.SS2.p5.13.m13.2.2.2.1.2" stretchy="false" xref="S5.SS2.p5.13.m13.2.2.2.2.1.cmml">|</mo><msub id="S5.SS2.p5.13.m13.2.2.2.1.1" xref="S5.SS2.p5.13.m13.2.2.2.1.1.cmml"><mi id="S5.SS2.p5.13.m13.2.2.2.1.1.2" xref="S5.SS2.p5.13.m13.2.2.2.1.1.2.cmml">V</mi><mi id="S5.SS2.p5.13.m13.2.2.2.1.1.3" xref="S5.SS2.p5.13.m13.2.2.2.1.1.3.cmml">k</mi></msub><mo id="S5.SS2.p5.13.m13.2.2.2.1.3" stretchy="false" xref="S5.SS2.p5.13.m13.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.13.m13.2b"><apply id="S5.SS2.p5.13.m13.2.2.cmml" xref="S5.SS2.p5.13.m13.2.2"><minus id="S5.SS2.p5.13.m13.2.2.3.cmml" xref="S5.SS2.p5.13.m13.2.2.3"></minus><apply id="S5.SS2.p5.13.m13.1.1.1.2.cmml" xref="S5.SS2.p5.13.m13.1.1.1.1"><abs id="S5.SS2.p5.13.m13.1.1.1.2.1.cmml" xref="S5.SS2.p5.13.m13.1.1.1.1.2"></abs><apply id="S5.SS2.p5.13.m13.1.1.1.1.1.cmml" xref="S5.SS2.p5.13.m13.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p5.13.m13.1.1.1.1.1.1.cmml" xref="S5.SS2.p5.13.m13.1.1.1.1.1">subscript</csymbol><ci id="S5.SS2.p5.13.m13.1.1.1.1.1.2.cmml" xref="S5.SS2.p5.13.m13.1.1.1.1.1.2">𝑉</ci><apply id="S5.SS2.p5.13.m13.1.1.1.1.1.3.cmml" xref="S5.SS2.p5.13.m13.1.1.1.1.1.3"><minus id="S5.SS2.p5.13.m13.1.1.1.1.1.3.1.cmml" xref="S5.SS2.p5.13.m13.1.1.1.1.1.3.1"></minus><ci id="S5.SS2.p5.13.m13.1.1.1.1.1.3.2.cmml" xref="S5.SS2.p5.13.m13.1.1.1.1.1.3.2">𝑘</ci><cn id="S5.SS2.p5.13.m13.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.SS2.p5.13.m13.1.1.1.1.1.3.3">1</cn></apply></apply></apply><apply id="S5.SS2.p5.13.m13.2.2.2.2.cmml" xref="S5.SS2.p5.13.m13.2.2.2.1"><abs id="S5.SS2.p5.13.m13.2.2.2.2.1.cmml" xref="S5.SS2.p5.13.m13.2.2.2.1.2"></abs><apply id="S5.SS2.p5.13.m13.2.2.2.1.1.cmml" xref="S5.SS2.p5.13.m13.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS2.p5.13.m13.2.2.2.1.1.1.cmml" xref="S5.SS2.p5.13.m13.2.2.2.1.1">subscript</csymbol><ci id="S5.SS2.p5.13.m13.2.2.2.1.1.2.cmml" xref="S5.SS2.p5.13.m13.2.2.2.1.1.2">𝑉</ci><ci id="S5.SS2.p5.13.m13.2.2.2.1.1.3.cmml" xref="S5.SS2.p5.13.m13.2.2.2.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.13.m13.2c">|V_{k-1}|-|V_{k}|</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.13.m13.2d">| italic_V start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT | - | italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |</annotation></semantics></math> <math alttext="(k-1)" class="ltx_Math" display="inline" id="S5.SS2.p5.14.m14.1"><semantics id="S5.SS2.p5.14.m14.1a"><mrow id="S5.SS2.p5.14.m14.1.1.1" xref="S5.SS2.p5.14.m14.1.1.1.1.cmml"><mo id="S5.SS2.p5.14.m14.1.1.1.2" stretchy="false" xref="S5.SS2.p5.14.m14.1.1.1.1.cmml">(</mo><mrow id="S5.SS2.p5.14.m14.1.1.1.1" xref="S5.SS2.p5.14.m14.1.1.1.1.cmml"><mi id="S5.SS2.p5.14.m14.1.1.1.1.2" xref="S5.SS2.p5.14.m14.1.1.1.1.2.cmml">k</mi><mo id="S5.SS2.p5.14.m14.1.1.1.1.1" xref="S5.SS2.p5.14.m14.1.1.1.1.1.cmml">−</mo><mn id="S5.SS2.p5.14.m14.1.1.1.1.3" xref="S5.SS2.p5.14.m14.1.1.1.1.3.cmml">1</mn></mrow><mo id="S5.SS2.p5.14.m14.1.1.1.3" stretchy="false" xref="S5.SS2.p5.14.m14.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.14.m14.1b"><apply id="S5.SS2.p5.14.m14.1.1.1.1.cmml" xref="S5.SS2.p5.14.m14.1.1.1"><minus id="S5.SS2.p5.14.m14.1.1.1.1.1.cmml" xref="S5.SS2.p5.14.m14.1.1.1.1.1"></minus><ci id="S5.SS2.p5.14.m14.1.1.1.1.2.cmml" xref="S5.SS2.p5.14.m14.1.1.1.1.2">𝑘</ci><cn id="S5.SS2.p5.14.m14.1.1.1.1.3.cmml" type="integer" xref="S5.SS2.p5.14.m14.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.14.m14.1c">(k-1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.14.m14.1d">( italic_k - 1 )</annotation></semantics></math>-cliques, etc. Conversely, when <math alttext="p=1" class="ltx_Math" display="inline" id="S5.SS2.p5.15.m15.1"><semantics id="S5.SS2.p5.15.m15.1a"><mrow id="S5.SS2.p5.15.m15.1.1" xref="S5.SS2.p5.15.m15.1.1.cmml"><mi id="S5.SS2.p5.15.m15.1.1.2" xref="S5.SS2.p5.15.m15.1.1.2.cmml">p</mi><mo id="S5.SS2.p5.15.m15.1.1.1" xref="S5.SS2.p5.15.m15.1.1.1.cmml">=</mo><mn id="S5.SS2.p5.15.m15.1.1.3" xref="S5.SS2.p5.15.m15.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.15.m15.1b"><apply id="S5.SS2.p5.15.m15.1.1.cmml" xref="S5.SS2.p5.15.m15.1.1"><eq id="S5.SS2.p5.15.m15.1.1.1.cmml" xref="S5.SS2.p5.15.m15.1.1.1"></eq><ci id="S5.SS2.p5.15.m15.1.1.2.cmml" xref="S5.SS2.p5.15.m15.1.1.2">𝑝</ci><cn id="S5.SS2.p5.15.m15.1.1.3.cmml" type="integer" xref="S5.SS2.p5.15.m15.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.15.m15.1c">p=1</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.15.m15.1d">italic_p = 1</annotation></semantics></math>, then all edges in the graph have weight of <math alttext="-n" class="ltx_Math" display="inline" id="S5.SS2.p5.16.m16.1"><semantics id="S5.SS2.p5.16.m16.1a"><mrow id="S5.SS2.p5.16.m16.1.1" xref="S5.SS2.p5.16.m16.1.1.cmml"><mo id="S5.SS2.p5.16.m16.1.1a" xref="S5.SS2.p5.16.m16.1.1.cmml">−</mo><mi id="S5.SS2.p5.16.m16.1.1.2" xref="S5.SS2.p5.16.m16.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p5.16.m16.1b"><apply id="S5.SS2.p5.16.m16.1.1.cmml" xref="S5.SS2.p5.16.m16.1.1"><minus id="S5.SS2.p5.16.m16.1.1.1.cmml" xref="S5.SS2.p5.16.m16.1.1"></minus><ci id="S5.SS2.p5.16.m16.1.1.2.cmml" xref="S5.SS2.p5.16.m16.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p5.16.m16.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p5.16.m16.1d">- italic_n</annotation></semantics></math>, which implies the maximum welfare is obtained by the singleton partition.</p> </div> <div class="ltx_para" id="S5.SS2.p6"> <p class="ltx_p" id="S5.SS2.p6.6">We now establish a straightforward upper bound on the maximum welfare. Recall that <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS2.p6.1.m1.1"><semantics id="S5.SS2.p6.1.m1.1a"><msup id="S5.SS2.p6.1.m1.1.1" xref="S5.SS2.p6.1.m1.1.1.cmml"><mi id="S5.SS2.p6.1.m1.1.1.2" xref="S5.SS2.p6.1.m1.1.1.2.cmml">G</mi><mo id="S5.SS2.p6.1.m1.1.1.3" xref="S5.SS2.p6.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.p6.1.m1.1b"><apply id="S5.SS2.p6.1.m1.1.1.cmml" xref="S5.SS2.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p6.1.m1.1.1.1.cmml" xref="S5.SS2.p6.1.m1.1.1">superscript</csymbol><ci id="S5.SS2.p6.1.m1.1.1.2.cmml" xref="S5.SS2.p6.1.m1.1.1.2">𝐺</ci><ci id="S5.SS2.p6.1.m1.1.1.3.cmml" xref="S5.SS2.p6.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p6.1.m1.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p6.1.m1.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is the graph obtained by removing all negative edges from <math alttext="G" class="ltx_Math" display="inline" id="S5.SS2.p6.2.m2.1"><semantics id="S5.SS2.p6.2.m2.1a"><mi id="S5.SS2.p6.2.m2.1.1" xref="S5.SS2.p6.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p6.2.m2.1b"><ci id="S5.SS2.p6.2.m2.1.1.cmml" xref="S5.SS2.p6.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p6.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p6.2.m2.1d">italic_G</annotation></semantics></math>. Since <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS2.p6.3.m3.1"><semantics id="S5.SS2.p6.3.m3.1a"><msup id="S5.SS2.p6.3.m3.1.1" xref="S5.SS2.p6.3.m3.1.1.cmml"><mi id="S5.SS2.p6.3.m3.1.1.2" xref="S5.SS2.p6.3.m3.1.1.2.cmml">G</mi><mo id="S5.SS2.p6.3.m3.1.1.3" xref="S5.SS2.p6.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.p6.3.m3.1b"><apply id="S5.SS2.p6.3.m3.1.1.cmml" xref="S5.SS2.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS2.p6.3.m3.1.1.1.cmml" xref="S5.SS2.p6.3.m3.1.1">superscript</csymbol><ci id="S5.SS2.p6.3.m3.1.1.2.cmml" xref="S5.SS2.p6.3.m3.1.1.2">𝐺</ci><ci id="S5.SS2.p6.3.m3.1.1.3.cmml" xref="S5.SS2.p6.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p6.3.m3.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p6.3.m3.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.p6.4.m4.1"><semantics id="S5.SS2.p6.4.m4.1a"><mi id="S5.SS2.p6.4.m4.1.1" xref="S5.SS2.p6.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p6.4.m4.1b"><ci id="S5.SS2.p6.4.m4.1.1.cmml" xref="S5.SS2.p6.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p6.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p6.4.m4.1d">italic_k</annotation></semantics></math>-partite, the maximum clique size in <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS2.p6.5.m5.1"><semantics id="S5.SS2.p6.5.m5.1a"><msup id="S5.SS2.p6.5.m5.1.1" xref="S5.SS2.p6.5.m5.1.1.cmml"><mi id="S5.SS2.p6.5.m5.1.1.2" xref="S5.SS2.p6.5.m5.1.1.2.cmml">G</mi><mo id="S5.SS2.p6.5.m5.1.1.3" xref="S5.SS2.p6.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.p6.5.m5.1b"><apply id="S5.SS2.p6.5.m5.1.1.cmml" xref="S5.SS2.p6.5.m5.1.1"><csymbol cd="ambiguous" id="S5.SS2.p6.5.m5.1.1.1.cmml" xref="S5.SS2.p6.5.m5.1.1">superscript</csymbol><ci id="S5.SS2.p6.5.m5.1.1.2.cmml" xref="S5.SS2.p6.5.m5.1.1.2">𝐺</ci><ci id="S5.SS2.p6.5.m5.1.1.3.cmml" xref="S5.SS2.p6.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p6.5.m5.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p6.5.m5.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is at most <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.p6.6.m6.1"><semantics id="S5.SS2.p6.6.m6.1a"><mi id="S5.SS2.p6.6.m6.1.1" xref="S5.SS2.p6.6.m6.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p6.6.m6.1b"><ci id="S5.SS2.p6.6.m6.1.1.cmml" xref="S5.SS2.p6.6.m6.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p6.6.m6.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p6.6.m6.1d">italic_k</annotation></semantics></math>. Thus, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem1" title="Lemma 5.1. ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5.1</span></a> implies the following proposition.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S5.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem6.1.1.1">Proposition 5.6</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem6.p1"> <p class="ltx_p" id="S5.Thmtheorem6.p1.2"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem6.p1.2.2">In a random balanced <math alttext="k" class="ltx_Math" display="inline" id="S5.Thmtheorem6.p1.1.1.m1.1"><semantics id="S5.Thmtheorem6.p1.1.1.m1.1a"><mi id="S5.Thmtheorem6.p1.1.1.m1.1.1" xref="S5.Thmtheorem6.p1.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem6.p1.1.1.m1.1b"><ci id="S5.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem6.p1.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem6.p1.1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem6.p1.1.1.m1.1d">italic_k</annotation></semantics></math>-partite graph, the maximum welfare is bounded by <math alttext="\mathcal{SW}(\pi^{*})\leq n(k-1)" class="ltx_Math" display="inline" id="S5.Thmtheorem6.p1.2.2.m2.2"><semantics id="S5.Thmtheorem6.p1.2.2.m2.2a"><mrow id="S5.Thmtheorem6.p1.2.2.m2.2.2" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.cmml"><mrow id="S5.Thmtheorem6.p1.2.2.m2.1.1.1" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.3" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.3.cmml">𝒮</mi><mo id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.2" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.4" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.4.cmml">𝒲</mi><mo id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.2a" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1.2" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1.2.cmml">π</mi><mo id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1.3" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem6.p1.2.2.m2.2.2.3" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.3.cmml">≤</mo><mrow id="S5.Thmtheorem6.p1.2.2.m2.2.2.2" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.cmml"><mi id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.3" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.3.cmml">n</mi><mo id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.2" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.2.cmml">⁢</mo><mrow id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.cmml"><mo id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.2" stretchy="false" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.cmml"><mi id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.2" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.2.cmml">k</mi><mo id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.1" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.1.cmml">−</mo><mn id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.3" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.3" stretchy="false" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem6.p1.2.2.m2.2b"><apply id="S5.Thmtheorem6.p1.2.2.m2.2.2.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.2.2"><leq id="S5.Thmtheorem6.p1.2.2.m2.2.2.3.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.3"></leq><apply id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1"><times id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.2.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.2"></times><ci id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.3.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.3">𝒮</ci><ci id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.4.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.4">𝒲</ci><apply id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1.2">𝜋</ci><times id="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.1.1.1.1.3"></times></apply></apply><apply id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2"><times id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.2.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.2"></times><ci id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.3.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.3">𝑛</ci><apply id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1"><minus id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.1"></minus><ci id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.2.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.2">𝑘</ci><cn id="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem6.p1.2.2.m2.2.2.2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem6.p1.2.2.m2.2c">\mathcal{SW}(\pi^{*})\leq n(k-1)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem6.p1.2.2.m2.2d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ≤ italic_n ( italic_k - 1 )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S5.SS2.p7"> <p class="ltx_p" id="S5.SS2.p7.8">In our analysis, both <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.p7.1.m1.1"><semantics id="S5.SS2.p7.1.m1.1a"><mi id="S5.SS2.p7.1.m1.1.1" xref="S5.SS2.p7.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p7.1.m1.1b"><ci id="S5.SS2.p7.1.m1.1.1.cmml" xref="S5.SS2.p7.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p7.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p7.1.m1.1d">italic_k</annotation></semantics></math> and <math alttext="p" class="ltx_Math" display="inline" id="S5.SS2.p7.2.m2.1"><semantics id="S5.SS2.p7.2.m2.1a"><mi id="S5.SS2.p7.2.m2.1.1" xref="S5.SS2.p7.2.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p7.2.m2.1b"><ci id="S5.SS2.p7.2.m2.1.1.cmml" xref="S5.SS2.p7.2.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p7.2.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p7.2.m2.1d">italic_p</annotation></semantics></math> can depend on <math alttext="n" class="ltx_Math" display="inline" id="S5.SS2.p7.3.m3.1"><semantics id="S5.SS2.p7.3.m3.1a"><mi id="S5.SS2.p7.3.m3.1.1" xref="S5.SS2.p7.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p7.3.m3.1b"><ci id="S5.SS2.p7.3.m3.1.1.cmml" xref="S5.SS2.p7.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p7.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p7.3.m3.1d">italic_n</annotation></semantics></math>. We now present polynomial-time algorithms that compute a constant-factor approximation of social welfare when <math alttext="p=\mathcal{O}\left(\frac{1}{k}\right)" class="ltx_Math" display="inline" id="S5.SS2.p7.4.m4.1"><semantics id="S5.SS2.p7.4.m4.1a"><mrow id="S5.SS2.p7.4.m4.1.2" xref="S5.SS2.p7.4.m4.1.2.cmml"><mi id="S5.SS2.p7.4.m4.1.2.2" xref="S5.SS2.p7.4.m4.1.2.2.cmml">p</mi><mo id="S5.SS2.p7.4.m4.1.2.1" xref="S5.SS2.p7.4.m4.1.2.1.cmml">=</mo><mrow id="S5.SS2.p7.4.m4.1.2.3" xref="S5.SS2.p7.4.m4.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.p7.4.m4.1.2.3.2" xref="S5.SS2.p7.4.m4.1.2.3.2.cmml">𝒪</mi><mo id="S5.SS2.p7.4.m4.1.2.3.1" xref="S5.SS2.p7.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS2.p7.4.m4.1.2.3.3.2" xref="S5.SS2.p7.4.m4.1.1.cmml"><mo id="S5.SS2.p7.4.m4.1.2.3.3.2.1" xref="S5.SS2.p7.4.m4.1.1.cmml">(</mo><mfrac id="S5.SS2.p7.4.m4.1.1" xref="S5.SS2.p7.4.m4.1.1.cmml"><mn id="S5.SS2.p7.4.m4.1.1.2" xref="S5.SS2.p7.4.m4.1.1.2.cmml">1</mn><mi id="S5.SS2.p7.4.m4.1.1.3" xref="S5.SS2.p7.4.m4.1.1.3.cmml">k</mi></mfrac><mo id="S5.SS2.p7.4.m4.1.2.3.3.2.2" xref="S5.SS2.p7.4.m4.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p7.4.m4.1b"><apply id="S5.SS2.p7.4.m4.1.2.cmml" xref="S5.SS2.p7.4.m4.1.2"><eq id="S5.SS2.p7.4.m4.1.2.1.cmml" xref="S5.SS2.p7.4.m4.1.2.1"></eq><ci id="S5.SS2.p7.4.m4.1.2.2.cmml" xref="S5.SS2.p7.4.m4.1.2.2">𝑝</ci><apply id="S5.SS2.p7.4.m4.1.2.3.cmml" xref="S5.SS2.p7.4.m4.1.2.3"><times id="S5.SS2.p7.4.m4.1.2.3.1.cmml" xref="S5.SS2.p7.4.m4.1.2.3.1"></times><ci id="S5.SS2.p7.4.m4.1.2.3.2.cmml" xref="S5.SS2.p7.4.m4.1.2.3.2">𝒪</ci><apply id="S5.SS2.p7.4.m4.1.1.cmml" xref="S5.SS2.p7.4.m4.1.2.3.3.2"><divide id="S5.SS2.p7.4.m4.1.1.1.cmml" xref="S5.SS2.p7.4.m4.1.2.3.3.2"></divide><cn id="S5.SS2.p7.4.m4.1.1.2.cmml" type="integer" xref="S5.SS2.p7.4.m4.1.1.2">1</cn><ci id="S5.SS2.p7.4.m4.1.1.3.cmml" xref="S5.SS2.p7.4.m4.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p7.4.m4.1c">p=\mathcal{O}\left(\frac{1}{k}\right)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p7.4.m4.1d">italic_p = caligraphic_O ( divide start_ARG 1 end_ARG start_ARG italic_k end_ARG )</annotation></semantics></math>, and a <math alttext="\log_{e}n" class="ltx_Math" display="inline" id="S5.SS2.p7.5.m5.1"><semantics id="S5.SS2.p7.5.m5.1a"><mrow id="S5.SS2.p7.5.m5.1.1" xref="S5.SS2.p7.5.m5.1.1.cmml"><msub id="S5.SS2.p7.5.m5.1.1.1" xref="S5.SS2.p7.5.m5.1.1.1.cmml"><mi id="S5.SS2.p7.5.m5.1.1.1.2" xref="S5.SS2.p7.5.m5.1.1.1.2.cmml">log</mi><mi id="S5.SS2.p7.5.m5.1.1.1.3" xref="S5.SS2.p7.5.m5.1.1.1.3.cmml">e</mi></msub><mo id="S5.SS2.p7.5.m5.1.1a" lspace="0.167em" xref="S5.SS2.p7.5.m5.1.1.cmml">⁡</mo><mi id="S5.SS2.p7.5.m5.1.1.2" xref="S5.SS2.p7.5.m5.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p7.5.m5.1b"><apply id="S5.SS2.p7.5.m5.1.1.cmml" xref="S5.SS2.p7.5.m5.1.1"><apply id="S5.SS2.p7.5.m5.1.1.1.cmml" xref="S5.SS2.p7.5.m5.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p7.5.m5.1.1.1.1.cmml" xref="S5.SS2.p7.5.m5.1.1.1">subscript</csymbol><log id="S5.SS2.p7.5.m5.1.1.1.2.cmml" xref="S5.SS2.p7.5.m5.1.1.1.2"></log><ci id="S5.SS2.p7.5.m5.1.1.1.3.cmml" xref="S5.SS2.p7.5.m5.1.1.1.3">𝑒</ci></apply><ci id="S5.SS2.p7.5.m5.1.1.2.cmml" xref="S5.SS2.p7.5.m5.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p7.5.m5.1c">\log_{e}n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p7.5.m5.1d">roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n</annotation></semantics></math>-approximation when <math alttext="p" class="ltx_Math" display="inline" id="S5.SS2.p7.6.m6.1"><semantics id="S5.SS2.p7.6.m6.1a"><mi id="S5.SS2.p7.6.m6.1.1" xref="S5.SS2.p7.6.m6.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p7.6.m6.1b"><ci id="S5.SS2.p7.6.m6.1.1.cmml" xref="S5.SS2.p7.6.m6.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p7.6.m6.1c">p</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p7.6.m6.1d">italic_p</annotation></semantics></math> is a constant for random balanced <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.p7.7.m7.1"><semantics id="S5.SS2.p7.7.m7.1a"><mi id="S5.SS2.p7.7.m7.1.1" xref="S5.SS2.p7.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p7.7.m7.1b"><ci id="S5.SS2.p7.7.m7.1.1.cmml" xref="S5.SS2.p7.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p7.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p7.7.m7.1d">italic_k</annotation></semantics></math>-partite graphs. We first illustrate our results by providing proofs for the special case of random Turán graphs. We then extend our results to random balanced <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.p7.8.m8.1"><semantics id="S5.SS2.p7.8.m8.1a"><mi id="S5.SS2.p7.8.m8.1.1" xref="S5.SS2.p7.8.m8.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p7.8.m8.1b"><ci id="S5.SS2.p7.8.m8.1.1.cmml" xref="S5.SS2.p7.8.m8.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p7.8.m8.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p7.8.m8.1d">italic_k</annotation></semantics></math>-partite graphs in <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.SS3" title="5.3 Balanced Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">5.3</span></a>. They are obtained by employing a reduction to Turán graphs.</p> </div> <section class="ltx_subsubsection" id="S5.SS2.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">5.2.1 </span>Low Perturbation Regime for Random Turán Graphs</h4> <div class="ltx_para" id="S5.SS2.SSS1.p1"> <p class="ltx_p" id="S5.SS2.SSS1.p1.8"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg1" title="In 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a> takes as input a random Turán graph, and an accuracy parameter (constant number) <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p1.1.m1.1"><semantics id="S5.SS2.SSS1.p1.1.m1.1a"><mrow id="S5.SS2.SSS1.p1.1.m1.1.1" xref="S5.SS2.SSS1.p1.1.m1.1.1.cmml"><mi id="S5.SS2.SSS1.p1.1.m1.1.1.2" xref="S5.SS2.SSS1.p1.1.m1.1.1.2.cmml">ε</mi><mo id="S5.SS2.SSS1.p1.1.m1.1.1.1" xref="S5.SS2.SSS1.p1.1.m1.1.1.1.cmml">></mo><mn id="S5.SS2.SSS1.p1.1.m1.1.1.3" xref="S5.SS2.SSS1.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p1.1.m1.1b"><apply id="S5.SS2.SSS1.p1.1.m1.1.1.cmml" xref="S5.SS2.SSS1.p1.1.m1.1.1"><gt id="S5.SS2.SSS1.p1.1.m1.1.1.1.cmml" xref="S5.SS2.SSS1.p1.1.m1.1.1.1"></gt><ci id="S5.SS2.SSS1.p1.1.m1.1.1.2.cmml" xref="S5.SS2.SSS1.p1.1.m1.1.1.2">𝜀</ci><cn id="S5.SS2.SSS1.p1.1.m1.1.1.3.cmml" type="integer" xref="S5.SS2.SSS1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p1.1.m1.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p1.1.m1.1d">italic_ε > 0</annotation></semantics></math>. In addition, the algorithm takes as input a subset of color classes denoted by <math alttext="S\subseteq\{V_{1},\dots,V_{k}\}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p1.2.m2.3"><semantics id="S5.SS2.SSS1.p1.2.m2.3a"><mrow id="S5.SS2.SSS1.p1.2.m2.3.3" xref="S5.SS2.SSS1.p1.2.m2.3.3.cmml"><mi id="S5.SS2.SSS1.p1.2.m2.3.3.4" xref="S5.SS2.SSS1.p1.2.m2.3.3.4.cmml">S</mi><mo id="S5.SS2.SSS1.p1.2.m2.3.3.3" xref="S5.SS2.SSS1.p1.2.m2.3.3.3.cmml">⊆</mo><mrow id="S5.SS2.SSS1.p1.2.m2.3.3.2.2" xref="S5.SS2.SSS1.p1.2.m2.3.3.2.3.cmml"><mo id="S5.SS2.SSS1.p1.2.m2.3.3.2.2.3" stretchy="false" xref="S5.SS2.SSS1.p1.2.m2.3.3.2.3.cmml">{</mo><msub id="S5.SS2.SSS1.p1.2.m2.2.2.1.1.1" xref="S5.SS2.SSS1.p1.2.m2.2.2.1.1.1.cmml"><mi id="S5.SS2.SSS1.p1.2.m2.2.2.1.1.1.2" xref="S5.SS2.SSS1.p1.2.m2.2.2.1.1.1.2.cmml">V</mi><mn id="S5.SS2.SSS1.p1.2.m2.2.2.1.1.1.3" xref="S5.SS2.SSS1.p1.2.m2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS2.SSS1.p1.2.m2.3.3.2.2.4" xref="S5.SS2.SSS1.p1.2.m2.3.3.2.3.cmml">,</mo><mi id="S5.SS2.SSS1.p1.2.m2.1.1" mathvariant="normal" xref="S5.SS2.SSS1.p1.2.m2.1.1.cmml">…</mi><mo id="S5.SS2.SSS1.p1.2.m2.3.3.2.2.5" xref="S5.SS2.SSS1.p1.2.m2.3.3.2.3.cmml">,</mo><msub id="S5.SS2.SSS1.p1.2.m2.3.3.2.2.2" xref="S5.SS2.SSS1.p1.2.m2.3.3.2.2.2.cmml"><mi id="S5.SS2.SSS1.p1.2.m2.3.3.2.2.2.2" xref="S5.SS2.SSS1.p1.2.m2.3.3.2.2.2.2.cmml">V</mi><mi id="S5.SS2.SSS1.p1.2.m2.3.3.2.2.2.3" xref="S5.SS2.SSS1.p1.2.m2.3.3.2.2.2.3.cmml">k</mi></msub><mo id="S5.SS2.SSS1.p1.2.m2.3.3.2.2.6" stretchy="false" xref="S5.SS2.SSS1.p1.2.m2.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p1.2.m2.3b"><apply id="S5.SS2.SSS1.p1.2.m2.3.3.cmml" xref="S5.SS2.SSS1.p1.2.m2.3.3"><subset id="S5.SS2.SSS1.p1.2.m2.3.3.3.cmml" xref="S5.SS2.SSS1.p1.2.m2.3.3.3"></subset><ci id="S5.SS2.SSS1.p1.2.m2.3.3.4.cmml" xref="S5.SS2.SSS1.p1.2.m2.3.3.4">𝑆</ci><set id="S5.SS2.SSS1.p1.2.m2.3.3.2.3.cmml" xref="S5.SS2.SSS1.p1.2.m2.3.3.2.2"><apply id="S5.SS2.SSS1.p1.2.m2.2.2.1.1.1.cmml" xref="S5.SS2.SSS1.p1.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p1.2.m2.2.2.1.1.1.1.cmml" xref="S5.SS2.SSS1.p1.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S5.SS2.SSS1.p1.2.m2.2.2.1.1.1.2.cmml" xref="S5.SS2.SSS1.p1.2.m2.2.2.1.1.1.2">𝑉</ci><cn id="S5.SS2.SSS1.p1.2.m2.2.2.1.1.1.3.cmml" type="integer" xref="S5.SS2.SSS1.p1.2.m2.2.2.1.1.1.3">1</cn></apply><ci id="S5.SS2.SSS1.p1.2.m2.1.1.cmml" xref="S5.SS2.SSS1.p1.2.m2.1.1">…</ci><apply id="S5.SS2.SSS1.p1.2.m2.3.3.2.2.2.cmml" xref="S5.SS2.SSS1.p1.2.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p1.2.m2.3.3.2.2.2.1.cmml" xref="S5.SS2.SSS1.p1.2.m2.3.3.2.2.2">subscript</csymbol><ci id="S5.SS2.SSS1.p1.2.m2.3.3.2.2.2.2.cmml" xref="S5.SS2.SSS1.p1.2.m2.3.3.2.2.2.2">𝑉</ci><ci id="S5.SS2.SSS1.p1.2.m2.3.3.2.2.2.3.cmml" xref="S5.SS2.SSS1.p1.2.m2.3.3.2.2.2.3">𝑘</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p1.2.m2.3c">S\subseteq\{V_{1},\dots,V_{k}\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p1.2.m2.3d">italic_S ⊆ { italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }</annotation></semantics></math>. The number of color classes in <math alttext="S" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p1.3.m3.1"><semantics id="S5.SS2.SSS1.p1.3.m3.1a"><mi id="S5.SS2.SSS1.p1.3.m3.1.1" xref="S5.SS2.SSS1.p1.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p1.3.m3.1b"><ci id="S5.SS2.SSS1.p1.3.m3.1.1.cmml" xref="S5.SS2.SSS1.p1.3.m3.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p1.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p1.3.m3.1d">italic_S</annotation></semantics></math> is denoted by <math alttext="k^{\prime}\leq k" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p1.4.m4.1"><semantics id="S5.SS2.SSS1.p1.4.m4.1a"><mrow id="S5.SS2.SSS1.p1.4.m4.1.1" xref="S5.SS2.SSS1.p1.4.m4.1.1.cmml"><msup id="S5.SS2.SSS1.p1.4.m4.1.1.2" xref="S5.SS2.SSS1.p1.4.m4.1.1.2.cmml"><mi id="S5.SS2.SSS1.p1.4.m4.1.1.2.2" xref="S5.SS2.SSS1.p1.4.m4.1.1.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.p1.4.m4.1.1.2.3" xref="S5.SS2.SSS1.p1.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS1.p1.4.m4.1.1.1" xref="S5.SS2.SSS1.p1.4.m4.1.1.1.cmml">≤</mo><mi id="S5.SS2.SSS1.p1.4.m4.1.1.3" xref="S5.SS2.SSS1.p1.4.m4.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p1.4.m4.1b"><apply id="S5.SS2.SSS1.p1.4.m4.1.1.cmml" xref="S5.SS2.SSS1.p1.4.m4.1.1"><leq id="S5.SS2.SSS1.p1.4.m4.1.1.1.cmml" xref="S5.SS2.SSS1.p1.4.m4.1.1.1"></leq><apply id="S5.SS2.SSS1.p1.4.m4.1.1.2.cmml" xref="S5.SS2.SSS1.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p1.4.m4.1.1.2.1.cmml" xref="S5.SS2.SSS1.p1.4.m4.1.1.2">superscript</csymbol><ci id="S5.SS2.SSS1.p1.4.m4.1.1.2.2.cmml" xref="S5.SS2.SSS1.p1.4.m4.1.1.2.2">𝑘</ci><ci id="S5.SS2.SSS1.p1.4.m4.1.1.2.3.cmml" xref="S5.SS2.SSS1.p1.4.m4.1.1.2.3">′</ci></apply><ci id="S5.SS2.SSS1.p1.4.m4.1.1.3.cmml" xref="S5.SS2.SSS1.p1.4.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p1.4.m4.1c">k^{\prime}\leq k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p1.4.m4.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_k</annotation></semantics></math>. Our algorithm outputs a partition of the vertices in <math alttext="S" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p1.5.m5.1"><semantics id="S5.SS2.SSS1.p1.5.m5.1a"><mi id="S5.SS2.SSS1.p1.5.m5.1.1" xref="S5.SS2.SSS1.p1.5.m5.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p1.5.m5.1b"><ci id="S5.SS2.SSS1.p1.5.m5.1.1.cmml" xref="S5.SS2.SSS1.p1.5.m5.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p1.5.m5.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p1.5.m5.1d">italic_S</annotation></semantics></math> into coalitions, meaning that only the vertices in <math alttext="\bigcup_{V_{i}\in S}V_{i}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p1.6.m6.1"><semantics id="S5.SS2.SSS1.p1.6.m6.1a"><mrow id="S5.SS2.SSS1.p1.6.m6.1.1" xref="S5.SS2.SSS1.p1.6.m6.1.1.cmml"><msub id="S5.SS2.SSS1.p1.6.m6.1.1.1" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.cmml"><mo id="S5.SS2.SSS1.p1.6.m6.1.1.1.2" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.2.cmml">⋃</mo><mrow id="S5.SS2.SSS1.p1.6.m6.1.1.1.3" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.3.cmml"><msub id="S5.SS2.SSS1.p1.6.m6.1.1.1.3.2" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.3.2.cmml"><mi id="S5.SS2.SSS1.p1.6.m6.1.1.1.3.2.2" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.3.2.2.cmml">V</mi><mi id="S5.SS2.SSS1.p1.6.m6.1.1.1.3.2.3" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.SS2.SSS1.p1.6.m6.1.1.1.3.1" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.3.1.cmml">∈</mo><mi id="S5.SS2.SSS1.p1.6.m6.1.1.1.3.3" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.3.3.cmml">S</mi></mrow></msub><msub id="S5.SS2.SSS1.p1.6.m6.1.1.2" xref="S5.SS2.SSS1.p1.6.m6.1.1.2.cmml"><mi id="S5.SS2.SSS1.p1.6.m6.1.1.2.2" xref="S5.SS2.SSS1.p1.6.m6.1.1.2.2.cmml">V</mi><mi id="S5.SS2.SSS1.p1.6.m6.1.1.2.3" xref="S5.SS2.SSS1.p1.6.m6.1.1.2.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p1.6.m6.1b"><apply id="S5.SS2.SSS1.p1.6.m6.1.1.cmml" xref="S5.SS2.SSS1.p1.6.m6.1.1"><apply id="S5.SS2.SSS1.p1.6.m6.1.1.1.cmml" xref="S5.SS2.SSS1.p1.6.m6.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p1.6.m6.1.1.1.1.cmml" xref="S5.SS2.SSS1.p1.6.m6.1.1.1">subscript</csymbol><union id="S5.SS2.SSS1.p1.6.m6.1.1.1.2.cmml" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.2"></union><apply id="S5.SS2.SSS1.p1.6.m6.1.1.1.3.cmml" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.3"><in id="S5.SS2.SSS1.p1.6.m6.1.1.1.3.1.cmml" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.3.1"></in><apply id="S5.SS2.SSS1.p1.6.m6.1.1.1.3.2.cmml" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.3.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p1.6.m6.1.1.1.3.2.1.cmml" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.3.2">subscript</csymbol><ci id="S5.SS2.SSS1.p1.6.m6.1.1.1.3.2.2.cmml" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.3.2.2">𝑉</ci><ci id="S5.SS2.SSS1.p1.6.m6.1.1.1.3.2.3.cmml" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.3.2.3">𝑖</ci></apply><ci id="S5.SS2.SSS1.p1.6.m6.1.1.1.3.3.cmml" xref="S5.SS2.SSS1.p1.6.m6.1.1.1.3.3">𝑆</ci></apply></apply><apply id="S5.SS2.SSS1.p1.6.m6.1.1.2.cmml" xref="S5.SS2.SSS1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p1.6.m6.1.1.2.1.cmml" xref="S5.SS2.SSS1.p1.6.m6.1.1.2">subscript</csymbol><ci id="S5.SS2.SSS1.p1.6.m6.1.1.2.2.cmml" xref="S5.SS2.SSS1.p1.6.m6.1.1.2.2">𝑉</ci><ci id="S5.SS2.SSS1.p1.6.m6.1.1.2.3.cmml" xref="S5.SS2.SSS1.p1.6.m6.1.1.2.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p1.6.m6.1c">\bigcup_{V_{i}\in S}V_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p1.6.m6.1d">⋃ start_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_S end_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> are considered, as if the graph consisted of exactly <math alttext="k^{\prime}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p1.7.m7.1"><semantics id="S5.SS2.SSS1.p1.7.m7.1a"><msup id="S5.SS2.SSS1.p1.7.m7.1.1" xref="S5.SS2.SSS1.p1.7.m7.1.1.cmml"><mi id="S5.SS2.SSS1.p1.7.m7.1.1.2" xref="S5.SS2.SSS1.p1.7.m7.1.1.2.cmml">k</mi><mo id="S5.SS2.SSS1.p1.7.m7.1.1.3" xref="S5.SS2.SSS1.p1.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p1.7.m7.1b"><apply id="S5.SS2.SSS1.p1.7.m7.1.1.cmml" xref="S5.SS2.SSS1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p1.7.m7.1.1.1.cmml" xref="S5.SS2.SSS1.p1.7.m7.1.1">superscript</csymbol><ci id="S5.SS2.SSS1.p1.7.m7.1.1.2.cmml" xref="S5.SS2.SSS1.p1.7.m7.1.1.2">𝑘</ci><ci id="S5.SS2.SSS1.p1.7.m7.1.1.3.cmml" xref="S5.SS2.SSS1.p1.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p1.7.m7.1c">k^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p1.7.m7.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> color classes, each containing <math alttext="\frac{n}{k}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p1.8.m8.1"><semantics id="S5.SS2.SSS1.p1.8.m8.1a"><mfrac id="S5.SS2.SSS1.p1.8.m8.1.1" xref="S5.SS2.SSS1.p1.8.m8.1.1.cmml"><mi id="S5.SS2.SSS1.p1.8.m8.1.1.2" xref="S5.SS2.SSS1.p1.8.m8.1.1.2.cmml">n</mi><mi id="S5.SS2.SSS1.p1.8.m8.1.1.3" xref="S5.SS2.SSS1.p1.8.m8.1.1.3.cmml">k</mi></mfrac><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p1.8.m8.1b"><apply id="S5.SS2.SSS1.p1.8.m8.1.1.cmml" xref="S5.SS2.SSS1.p1.8.m8.1.1"><divide id="S5.SS2.SSS1.p1.8.m8.1.1.1.cmml" xref="S5.SS2.SSS1.p1.8.m8.1.1"></divide><ci id="S5.SS2.SSS1.p1.8.m8.1.1.2.cmml" xref="S5.SS2.SSS1.p1.8.m8.1.1.2">𝑛</ci><ci id="S5.SS2.SSS1.p1.8.m8.1.1.3.cmml" xref="S5.SS2.SSS1.p1.8.m8.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p1.8.m8.1c">\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p1.8.m8.1d">divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math> vertices.</p> </div> <div class="ltx_para" id="S5.SS2.SSS1.p2"> <p class="ltx_p" id="S5.SS2.SSS1.p2.7"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg1" title="In 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a> begins by randomly selecting a vertex to initiate the formation of the first coalition. It then iteratively adds a new vertex <math alttext="w" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p2.1.m1.1"><semantics id="S5.SS2.SSS1.p2.1.m1.1a"><mi id="S5.SS2.SSS1.p2.1.m1.1.1" xref="S5.SS2.SSS1.p2.1.m1.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p2.1.m1.1b"><ci id="S5.SS2.SSS1.p2.1.m1.1.1.cmml" xref="S5.SS2.SSS1.p2.1.m1.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p2.1.m1.1c">w</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p2.1.m1.1d">italic_w</annotation></semantics></math> to the coalition if <math alttext="w" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p2.2.m2.1"><semantics id="S5.SS2.SSS1.p2.2.m2.1a"><mi id="S5.SS2.SSS1.p2.2.m2.1.1" xref="S5.SS2.SSS1.p2.2.m2.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p2.2.m2.1b"><ci id="S5.SS2.SSS1.p2.2.m2.1.1.cmml" xref="S5.SS2.SSS1.p2.2.m2.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p2.2.m2.1c">w</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p2.2.m2.1d">italic_w</annotation></semantics></math> has only edges of weight <math alttext="1" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p2.3.m3.1"><semantics id="S5.SS2.SSS1.p2.3.m3.1a"><mn id="S5.SS2.SSS1.p2.3.m3.1.1" xref="S5.SS2.SSS1.p2.3.m3.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p2.3.m3.1b"><cn id="S5.SS2.SSS1.p2.3.m3.1.1.cmml" type="integer" xref="S5.SS2.SSS1.p2.3.m3.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p2.3.m3.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p2.3.m3.1d">1</annotation></semantics></math> towards all current members of the coalition (this ensures there are no <math alttext="-n" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p2.4.m4.1"><semantics id="S5.SS2.SSS1.p2.4.m4.1a"><mrow id="S5.SS2.SSS1.p2.4.m4.1.1" xref="S5.SS2.SSS1.p2.4.m4.1.1.cmml"><mo id="S5.SS2.SSS1.p2.4.m4.1.1a" xref="S5.SS2.SSS1.p2.4.m4.1.1.cmml">−</mo><mi id="S5.SS2.SSS1.p2.4.m4.1.1.2" xref="S5.SS2.SSS1.p2.4.m4.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p2.4.m4.1b"><apply id="S5.SS2.SSS1.p2.4.m4.1.1.cmml" xref="S5.SS2.SSS1.p2.4.m4.1.1"><minus id="S5.SS2.SSS1.p2.4.m4.1.1.1.cmml" xref="S5.SS2.SSS1.p2.4.m4.1.1"></minus><ci id="S5.SS2.SSS1.p2.4.m4.1.1.2.cmml" xref="S5.SS2.SSS1.p2.4.m4.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p2.4.m4.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p2.4.m4.1d">- italic_n</annotation></semantics></math> edges in the created coalitions). This process continues until no additional vertices can be included. Hence, each formed coalition is a maximal clique in the subgraph <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p2.5.m5.1"><semantics id="S5.SS2.SSS1.p2.5.m5.1a"><msup id="S5.SS2.SSS1.p2.5.m5.1.1" xref="S5.SS2.SSS1.p2.5.m5.1.1.cmml"><mi id="S5.SS2.SSS1.p2.5.m5.1.1.2" xref="S5.SS2.SSS1.p2.5.m5.1.1.2.cmml">G</mi><mo id="S5.SS2.SSS1.p2.5.m5.1.1.3" xref="S5.SS2.SSS1.p2.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p2.5.m5.1b"><apply id="S5.SS2.SSS1.p2.5.m5.1.1.cmml" xref="S5.SS2.SSS1.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p2.5.m5.1.1.1.cmml" xref="S5.SS2.SSS1.p2.5.m5.1.1">superscript</csymbol><ci id="S5.SS2.SSS1.p2.5.m5.1.1.2.cmml" xref="S5.SS2.SSS1.p2.5.m5.1.1.2">𝐺</ci><ci id="S5.SS2.SSS1.p2.5.m5.1.1.3.cmml" xref="S5.SS2.SSS1.p2.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p2.5.m5.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p2.5.m5.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> obtained by removing all negative edges, and we refer to these coalitions as maximal cliques. If the size of the resulting maximal clique exceeds <math alttext="k^{\prime}\sqrt{1-\varepsilon}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p2.6.m6.1"><semantics id="S5.SS2.SSS1.p2.6.m6.1a"><mrow id="S5.SS2.SSS1.p2.6.m6.1.1" xref="S5.SS2.SSS1.p2.6.m6.1.1.cmml"><msup id="S5.SS2.SSS1.p2.6.m6.1.1.2" xref="S5.SS2.SSS1.p2.6.m6.1.1.2.cmml"><mi id="S5.SS2.SSS1.p2.6.m6.1.1.2.2" xref="S5.SS2.SSS1.p2.6.m6.1.1.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.p2.6.m6.1.1.2.3" xref="S5.SS2.SSS1.p2.6.m6.1.1.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS1.p2.6.m6.1.1.1" xref="S5.SS2.SSS1.p2.6.m6.1.1.1.cmml">⁢</mo><msqrt id="S5.SS2.SSS1.p2.6.m6.1.1.3" xref="S5.SS2.SSS1.p2.6.m6.1.1.3.cmml"><mrow id="S5.SS2.SSS1.p2.6.m6.1.1.3.2" xref="S5.SS2.SSS1.p2.6.m6.1.1.3.2.cmml"><mn id="S5.SS2.SSS1.p2.6.m6.1.1.3.2.2" xref="S5.SS2.SSS1.p2.6.m6.1.1.3.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.p2.6.m6.1.1.3.2.1" xref="S5.SS2.SSS1.p2.6.m6.1.1.3.2.1.cmml">−</mo><mi id="S5.SS2.SSS1.p2.6.m6.1.1.3.2.3" xref="S5.SS2.SSS1.p2.6.m6.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p2.6.m6.1b"><apply id="S5.SS2.SSS1.p2.6.m6.1.1.cmml" xref="S5.SS2.SSS1.p2.6.m6.1.1"><times id="S5.SS2.SSS1.p2.6.m6.1.1.1.cmml" xref="S5.SS2.SSS1.p2.6.m6.1.1.1"></times><apply id="S5.SS2.SSS1.p2.6.m6.1.1.2.cmml" xref="S5.SS2.SSS1.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p2.6.m6.1.1.2.1.cmml" xref="S5.SS2.SSS1.p2.6.m6.1.1.2">superscript</csymbol><ci id="S5.SS2.SSS1.p2.6.m6.1.1.2.2.cmml" xref="S5.SS2.SSS1.p2.6.m6.1.1.2.2">𝑘</ci><ci id="S5.SS2.SSS1.p2.6.m6.1.1.2.3.cmml" xref="S5.SS2.SSS1.p2.6.m6.1.1.2.3">′</ci></apply><apply id="S5.SS2.SSS1.p2.6.m6.1.1.3.cmml" xref="S5.SS2.SSS1.p2.6.m6.1.1.3"><root id="S5.SS2.SSS1.p2.6.m6.1.1.3a.cmml" xref="S5.SS2.SSS1.p2.6.m6.1.1.3"></root><apply id="S5.SS2.SSS1.p2.6.m6.1.1.3.2.cmml" xref="S5.SS2.SSS1.p2.6.m6.1.1.3.2"><minus id="S5.SS2.SSS1.p2.6.m6.1.1.3.2.1.cmml" xref="S5.SS2.SSS1.p2.6.m6.1.1.3.2.1"></minus><cn id="S5.SS2.SSS1.p2.6.m6.1.1.3.2.2.cmml" type="integer" xref="S5.SS2.SSS1.p2.6.m6.1.1.3.2.2">1</cn><ci id="S5.SS2.SSS1.p2.6.m6.1.1.3.2.3.cmml" xref="S5.SS2.SSS1.p2.6.m6.1.1.3.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p2.6.m6.1c">k^{\prime}\sqrt{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p2.6.m6.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_ε end_ARG</annotation></semantics></math>, the vertices in the clique are removed from the pool of available vertices, and the process is repeated with the remaining vertices. However, if at any point the size of the obtained maximal clique is smaller than <math alttext="k^{\prime}\sqrt{1-\varepsilon}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p2.7.m7.1"><semantics id="S5.SS2.SSS1.p2.7.m7.1a"><mrow id="S5.SS2.SSS1.p2.7.m7.1.1" xref="S5.SS2.SSS1.p2.7.m7.1.1.cmml"><msup id="S5.SS2.SSS1.p2.7.m7.1.1.2" xref="S5.SS2.SSS1.p2.7.m7.1.1.2.cmml"><mi id="S5.SS2.SSS1.p2.7.m7.1.1.2.2" xref="S5.SS2.SSS1.p2.7.m7.1.1.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.p2.7.m7.1.1.2.3" xref="S5.SS2.SSS1.p2.7.m7.1.1.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS1.p2.7.m7.1.1.1" xref="S5.SS2.SSS1.p2.7.m7.1.1.1.cmml">⁢</mo><msqrt id="S5.SS2.SSS1.p2.7.m7.1.1.3" xref="S5.SS2.SSS1.p2.7.m7.1.1.3.cmml"><mrow id="S5.SS2.SSS1.p2.7.m7.1.1.3.2" xref="S5.SS2.SSS1.p2.7.m7.1.1.3.2.cmml"><mn id="S5.SS2.SSS1.p2.7.m7.1.1.3.2.2" xref="S5.SS2.SSS1.p2.7.m7.1.1.3.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.p2.7.m7.1.1.3.2.1" xref="S5.SS2.SSS1.p2.7.m7.1.1.3.2.1.cmml">−</mo><mi id="S5.SS2.SSS1.p2.7.m7.1.1.3.2.3" xref="S5.SS2.SSS1.p2.7.m7.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p2.7.m7.1b"><apply id="S5.SS2.SSS1.p2.7.m7.1.1.cmml" xref="S5.SS2.SSS1.p2.7.m7.1.1"><times id="S5.SS2.SSS1.p2.7.m7.1.1.1.cmml" xref="S5.SS2.SSS1.p2.7.m7.1.1.1"></times><apply id="S5.SS2.SSS1.p2.7.m7.1.1.2.cmml" xref="S5.SS2.SSS1.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p2.7.m7.1.1.2.1.cmml" xref="S5.SS2.SSS1.p2.7.m7.1.1.2">superscript</csymbol><ci id="S5.SS2.SSS1.p2.7.m7.1.1.2.2.cmml" xref="S5.SS2.SSS1.p2.7.m7.1.1.2.2">𝑘</ci><ci id="S5.SS2.SSS1.p2.7.m7.1.1.2.3.cmml" xref="S5.SS2.SSS1.p2.7.m7.1.1.2.3">′</ci></apply><apply id="S5.SS2.SSS1.p2.7.m7.1.1.3.cmml" xref="S5.SS2.SSS1.p2.7.m7.1.1.3"><root id="S5.SS2.SSS1.p2.7.m7.1.1.3a.cmml" xref="S5.SS2.SSS1.p2.7.m7.1.1.3"></root><apply id="S5.SS2.SSS1.p2.7.m7.1.1.3.2.cmml" xref="S5.SS2.SSS1.p2.7.m7.1.1.3.2"><minus id="S5.SS2.SSS1.p2.7.m7.1.1.3.2.1.cmml" xref="S5.SS2.SSS1.p2.7.m7.1.1.3.2.1"></minus><cn id="S5.SS2.SSS1.p2.7.m7.1.1.3.2.2.cmml" type="integer" xref="S5.SS2.SSS1.p2.7.m7.1.1.3.2.2">1</cn><ci id="S5.SS2.SSS1.p2.7.m7.1.1.3.2.3.cmml" xref="S5.SS2.SSS1.p2.7.m7.1.1.3.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p2.7.m7.1c">k^{\prime}\sqrt{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p2.7.m7.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_ε end_ARG</annotation></semantics></math>, the algorithm terminates and returns the current set of coalitions, with any remaining vertices assigned to singletons coalitions.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg1.9.1.1">Algorithm 1</span> </span> Greedy coalition formation</figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_left" id="alg1.5.5"><span class="ltx_text ltx_font_bold" id="alg1.5.5.1">Input:</span> <math alttext="\langle G,S,\varepsilon\rangle" class="ltx_Math" display="inline" id="alg1.1.1.m1.3"><semantics id="alg1.1.1.m1.3a"><mrow id="alg1.1.1.m1.3.4.2" xref="alg1.1.1.m1.3.4.1.cmml"><mo id="alg1.1.1.m1.3.4.2.1" stretchy="false" xref="alg1.1.1.m1.3.4.1.cmml">⟨</mo><mi id="alg1.1.1.m1.1.1" xref="alg1.1.1.m1.1.1.cmml">G</mi><mo id="alg1.1.1.m1.3.4.2.2" xref="alg1.1.1.m1.3.4.1.cmml">,</mo><mi id="alg1.1.1.m1.2.2" xref="alg1.1.1.m1.2.2.cmml">S</mi><mo id="alg1.1.1.m1.3.4.2.3" xref="alg1.1.1.m1.3.4.1.cmml">,</mo><mi id="alg1.1.1.m1.3.3" xref="alg1.1.1.m1.3.3.cmml">ε</mi><mo id="alg1.1.1.m1.3.4.2.4" stretchy="false" xref="alg1.1.1.m1.3.4.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="alg1.1.1.m1.3b"><list id="alg1.1.1.m1.3.4.1.cmml" xref="alg1.1.1.m1.3.4.2"><ci id="alg1.1.1.m1.1.1.cmml" xref="alg1.1.1.m1.1.1">𝐺</ci><ci id="alg1.1.1.m1.2.2.cmml" xref="alg1.1.1.m1.2.2">𝑆</ci><ci id="alg1.1.1.m1.3.3.cmml" xref="alg1.1.1.m1.3.3">𝜀</ci></list></annotation-xml><annotation encoding="application/x-tex" id="alg1.1.1.m1.3c">\langle G,S,\varepsilon\rangle</annotation><annotation encoding="application/x-llamapun" id="alg1.1.1.m1.3d">⟨ italic_G , italic_S , italic_ε ⟩</annotation></semantics></math> where <math alttext="G=(n,k,p)" class="ltx_Math" display="inline" id="alg1.2.2.m2.3"><semantics id="alg1.2.2.m2.3a"><mrow id="alg1.2.2.m2.3.4" xref="alg1.2.2.m2.3.4.cmml"><mi id="alg1.2.2.m2.3.4.2" xref="alg1.2.2.m2.3.4.2.cmml">G</mi><mo id="alg1.2.2.m2.3.4.1" xref="alg1.2.2.m2.3.4.1.cmml">=</mo><mrow id="alg1.2.2.m2.3.4.3.2" xref="alg1.2.2.m2.3.4.3.1.cmml"><mo id="alg1.2.2.m2.3.4.3.2.1" stretchy="false" xref="alg1.2.2.m2.3.4.3.1.cmml">(</mo><mi id="alg1.2.2.m2.1.1" xref="alg1.2.2.m2.1.1.cmml">n</mi><mo id="alg1.2.2.m2.3.4.3.2.2" xref="alg1.2.2.m2.3.4.3.1.cmml">,</mo><mi id="alg1.2.2.m2.2.2" xref="alg1.2.2.m2.2.2.cmml">k</mi><mo id="alg1.2.2.m2.3.4.3.2.3" xref="alg1.2.2.m2.3.4.3.1.cmml">,</mo><mi id="alg1.2.2.m2.3.3" xref="alg1.2.2.m2.3.3.cmml">p</mi><mo id="alg1.2.2.m2.3.4.3.2.4" stretchy="false" xref="alg1.2.2.m2.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.2.2.m2.3b"><apply id="alg1.2.2.m2.3.4.cmml" xref="alg1.2.2.m2.3.4"><eq id="alg1.2.2.m2.3.4.1.cmml" xref="alg1.2.2.m2.3.4.1"></eq><ci id="alg1.2.2.m2.3.4.2.cmml" xref="alg1.2.2.m2.3.4.2">𝐺</ci><vector id="alg1.2.2.m2.3.4.3.1.cmml" xref="alg1.2.2.m2.3.4.3.2"><ci id="alg1.2.2.m2.1.1.cmml" xref="alg1.2.2.m2.1.1">𝑛</ci><ci id="alg1.2.2.m2.2.2.cmml" xref="alg1.2.2.m2.2.2">𝑘</ci><ci id="alg1.2.2.m2.3.3.cmml" xref="alg1.2.2.m2.3.3">𝑝</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.2.2.m2.3c">G=(n,k,p)</annotation><annotation encoding="application/x-llamapun" id="alg1.2.2.m2.3d">italic_G = ( italic_n , italic_k , italic_p )</annotation></semantics></math> is a random Turán graph, <math alttext="S\subseteq\{V_{1},\cdots,V_{k}\}" class="ltx_Math" display="inline" id="alg1.3.3.m3.3"><semantics id="alg1.3.3.m3.3a"><mrow id="alg1.3.3.m3.3.3" xref="alg1.3.3.m3.3.3.cmml"><mi id="alg1.3.3.m3.3.3.4" xref="alg1.3.3.m3.3.3.4.cmml">S</mi><mo id="alg1.3.3.m3.3.3.3" xref="alg1.3.3.m3.3.3.3.cmml">⊆</mo><mrow id="alg1.3.3.m3.3.3.2.2" xref="alg1.3.3.m3.3.3.2.3.cmml"><mo id="alg1.3.3.m3.3.3.2.2.3" stretchy="false" xref="alg1.3.3.m3.3.3.2.3.cmml">{</mo><msub id="alg1.3.3.m3.2.2.1.1.1" xref="alg1.3.3.m3.2.2.1.1.1.cmml"><mi id="alg1.3.3.m3.2.2.1.1.1.2" xref="alg1.3.3.m3.2.2.1.1.1.2.cmml">V</mi><mn id="alg1.3.3.m3.2.2.1.1.1.3" xref="alg1.3.3.m3.2.2.1.1.1.3.cmml">1</mn></msub><mo id="alg1.3.3.m3.3.3.2.2.4" xref="alg1.3.3.m3.3.3.2.3.cmml">,</mo><mi id="alg1.3.3.m3.1.1" mathvariant="normal" xref="alg1.3.3.m3.1.1.cmml">⋯</mi><mo id="alg1.3.3.m3.3.3.2.2.5" xref="alg1.3.3.m3.3.3.2.3.cmml">,</mo><msub id="alg1.3.3.m3.3.3.2.2.2" xref="alg1.3.3.m3.3.3.2.2.2.cmml"><mi id="alg1.3.3.m3.3.3.2.2.2.2" xref="alg1.3.3.m3.3.3.2.2.2.2.cmml">V</mi><mi id="alg1.3.3.m3.3.3.2.2.2.3" xref="alg1.3.3.m3.3.3.2.2.2.3.cmml">k</mi></msub><mo id="alg1.3.3.m3.3.3.2.2.6" stretchy="false" xref="alg1.3.3.m3.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.3.3.m3.3b"><apply id="alg1.3.3.m3.3.3.cmml" xref="alg1.3.3.m3.3.3"><subset id="alg1.3.3.m3.3.3.3.cmml" xref="alg1.3.3.m3.3.3.3"></subset><ci id="alg1.3.3.m3.3.3.4.cmml" xref="alg1.3.3.m3.3.3.4">𝑆</ci><set id="alg1.3.3.m3.3.3.2.3.cmml" xref="alg1.3.3.m3.3.3.2.2"><apply id="alg1.3.3.m3.2.2.1.1.1.cmml" xref="alg1.3.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="alg1.3.3.m3.2.2.1.1.1.1.cmml" xref="alg1.3.3.m3.2.2.1.1.1">subscript</csymbol><ci id="alg1.3.3.m3.2.2.1.1.1.2.cmml" xref="alg1.3.3.m3.2.2.1.1.1.2">𝑉</ci><cn id="alg1.3.3.m3.2.2.1.1.1.3.cmml" type="integer" xref="alg1.3.3.m3.2.2.1.1.1.3">1</cn></apply><ci id="alg1.3.3.m3.1.1.cmml" xref="alg1.3.3.m3.1.1">⋯</ci><apply id="alg1.3.3.m3.3.3.2.2.2.cmml" xref="alg1.3.3.m3.3.3.2.2.2"><csymbol cd="ambiguous" id="alg1.3.3.m3.3.3.2.2.2.1.cmml" xref="alg1.3.3.m3.3.3.2.2.2">subscript</csymbol><ci id="alg1.3.3.m3.3.3.2.2.2.2.cmml" xref="alg1.3.3.m3.3.3.2.2.2.2">𝑉</ci><ci id="alg1.3.3.m3.3.3.2.2.2.3.cmml" xref="alg1.3.3.m3.3.3.2.2.2.3">𝑘</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.3.3.m3.3c">S\subseteq\{V_{1},\cdots,V_{k}\}</annotation><annotation encoding="application/x-llamapun" id="alg1.3.3.m3.3d">italic_S ⊆ { italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }</annotation></semantics></math> is a subset of color classes with <math alttext="|S|=k^{\prime}" class="ltx_Math" display="inline" id="alg1.4.4.m4.1"><semantics id="alg1.4.4.m4.1a"><mrow id="alg1.4.4.m4.1.2" xref="alg1.4.4.m4.1.2.cmml"><mrow id="alg1.4.4.m4.1.2.2.2" xref="alg1.4.4.m4.1.2.2.1.cmml"><mo id="alg1.4.4.m4.1.2.2.2.1" stretchy="false" xref="alg1.4.4.m4.1.2.2.1.1.cmml">|</mo><mi id="alg1.4.4.m4.1.1" xref="alg1.4.4.m4.1.1.cmml">S</mi><mo id="alg1.4.4.m4.1.2.2.2.2" stretchy="false" xref="alg1.4.4.m4.1.2.2.1.1.cmml">|</mo></mrow><mo id="alg1.4.4.m4.1.2.1" xref="alg1.4.4.m4.1.2.1.cmml">=</mo><msup id="alg1.4.4.m4.1.2.3" xref="alg1.4.4.m4.1.2.3.cmml"><mi id="alg1.4.4.m4.1.2.3.2" xref="alg1.4.4.m4.1.2.3.2.cmml">k</mi><mo id="alg1.4.4.m4.1.2.3.3" xref="alg1.4.4.m4.1.2.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="alg1.4.4.m4.1b"><apply id="alg1.4.4.m4.1.2.cmml" xref="alg1.4.4.m4.1.2"><eq id="alg1.4.4.m4.1.2.1.cmml" xref="alg1.4.4.m4.1.2.1"></eq><apply id="alg1.4.4.m4.1.2.2.1.cmml" xref="alg1.4.4.m4.1.2.2.2"><abs id="alg1.4.4.m4.1.2.2.1.1.cmml" xref="alg1.4.4.m4.1.2.2.2.1"></abs><ci id="alg1.4.4.m4.1.1.cmml" xref="alg1.4.4.m4.1.1">𝑆</ci></apply><apply id="alg1.4.4.m4.1.2.3.cmml" xref="alg1.4.4.m4.1.2.3"><csymbol cd="ambiguous" id="alg1.4.4.m4.1.2.3.1.cmml" xref="alg1.4.4.m4.1.2.3">superscript</csymbol><ci id="alg1.4.4.m4.1.2.3.2.cmml" xref="alg1.4.4.m4.1.2.3.2">𝑘</ci><ci id="alg1.4.4.m4.1.2.3.3.cmml" xref="alg1.4.4.m4.1.2.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.4.4.m4.1c">|S|=k^{\prime}</annotation><annotation encoding="application/x-llamapun" id="alg1.4.4.m4.1d">| italic_S | = italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="\varepsilon\in(0,1)" class="ltx_Math" display="inline" id="alg1.5.5.m5.2"><semantics id="alg1.5.5.m5.2a"><mrow id="alg1.5.5.m5.2.3" xref="alg1.5.5.m5.2.3.cmml"><mi id="alg1.5.5.m5.2.3.2" xref="alg1.5.5.m5.2.3.2.cmml">ε</mi><mo id="alg1.5.5.m5.2.3.1" xref="alg1.5.5.m5.2.3.1.cmml">∈</mo><mrow id="alg1.5.5.m5.2.3.3.2" xref="alg1.5.5.m5.2.3.3.1.cmml"><mo id="alg1.5.5.m5.2.3.3.2.1" stretchy="false" xref="alg1.5.5.m5.2.3.3.1.cmml">(</mo><mn id="alg1.5.5.m5.1.1" xref="alg1.5.5.m5.1.1.cmml">0</mn><mo id="alg1.5.5.m5.2.3.3.2.2" xref="alg1.5.5.m5.2.3.3.1.cmml">,</mo><mn id="alg1.5.5.m5.2.2" xref="alg1.5.5.m5.2.2.cmml">1</mn><mo id="alg1.5.5.m5.2.3.3.2.3" stretchy="false" xref="alg1.5.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.5.5.m5.2b"><apply id="alg1.5.5.m5.2.3.cmml" xref="alg1.5.5.m5.2.3"><in id="alg1.5.5.m5.2.3.1.cmml" xref="alg1.5.5.m5.2.3.1"></in><ci id="alg1.5.5.m5.2.3.2.cmml" xref="alg1.5.5.m5.2.3.2">𝜀</ci><interval closure="open" id="alg1.5.5.m5.2.3.3.1.cmml" xref="alg1.5.5.m5.2.3.3.2"><cn id="alg1.5.5.m5.1.1.cmml" type="integer" xref="alg1.5.5.m5.1.1">0</cn><cn id="alg1.5.5.m5.2.2.cmml" type="integer" xref="alg1.5.5.m5.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.5.5.m5.2c">\varepsilon\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="alg1.5.5.m5.2d">italic_ε ∈ ( 0 , 1 )</annotation></semantics></math>.</p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_left" id="alg1.7.7"><span class="ltx_text ltx_font_bold" id="alg1.7.7.1">Output:</span> Partition <math alttext="\pi" class="ltx_Math" display="inline" id="alg1.6.6.m1.1"><semantics id="alg1.6.6.m1.1a"><mi id="alg1.6.6.m1.1.1" xref="alg1.6.6.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="alg1.6.6.m1.1b"><ci id="alg1.6.6.m1.1.1.cmml" xref="alg1.6.6.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.6.6.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="alg1.6.6.m1.1d">italic_π</annotation></semantics></math> on <math alttext="\bigcup_{V_{i}\in S}V_{i}" class="ltx_Math" display="inline" id="alg1.7.7.m2.1"><semantics id="alg1.7.7.m2.1a"><mrow id="alg1.7.7.m2.1.1" xref="alg1.7.7.m2.1.1.cmml"><msub id="alg1.7.7.m2.1.1.1" xref="alg1.7.7.m2.1.1.1.cmml"><mo id="alg1.7.7.m2.1.1.1.2" xref="alg1.7.7.m2.1.1.1.2.cmml">⋃</mo><mrow id="alg1.7.7.m2.1.1.1.3" xref="alg1.7.7.m2.1.1.1.3.cmml"><msub id="alg1.7.7.m2.1.1.1.3.2" xref="alg1.7.7.m2.1.1.1.3.2.cmml"><mi id="alg1.7.7.m2.1.1.1.3.2.2" xref="alg1.7.7.m2.1.1.1.3.2.2.cmml">V</mi><mi id="alg1.7.7.m2.1.1.1.3.2.3" xref="alg1.7.7.m2.1.1.1.3.2.3.cmml">i</mi></msub><mo id="alg1.7.7.m2.1.1.1.3.1" xref="alg1.7.7.m2.1.1.1.3.1.cmml">∈</mo><mi id="alg1.7.7.m2.1.1.1.3.3" xref="alg1.7.7.m2.1.1.1.3.3.cmml">S</mi></mrow></msub><msub id="alg1.7.7.m2.1.1.2" xref="alg1.7.7.m2.1.1.2.cmml"><mi id="alg1.7.7.m2.1.1.2.2" xref="alg1.7.7.m2.1.1.2.2.cmml">V</mi><mi id="alg1.7.7.m2.1.1.2.3" xref="alg1.7.7.m2.1.1.2.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg1.7.7.m2.1b"><apply id="alg1.7.7.m2.1.1.cmml" xref="alg1.7.7.m2.1.1"><apply id="alg1.7.7.m2.1.1.1.cmml" xref="alg1.7.7.m2.1.1.1"><csymbol cd="ambiguous" id="alg1.7.7.m2.1.1.1.1.cmml" xref="alg1.7.7.m2.1.1.1">subscript</csymbol><union id="alg1.7.7.m2.1.1.1.2.cmml" xref="alg1.7.7.m2.1.1.1.2"></union><apply id="alg1.7.7.m2.1.1.1.3.cmml" xref="alg1.7.7.m2.1.1.1.3"><in id="alg1.7.7.m2.1.1.1.3.1.cmml" xref="alg1.7.7.m2.1.1.1.3.1"></in><apply id="alg1.7.7.m2.1.1.1.3.2.cmml" xref="alg1.7.7.m2.1.1.1.3.2"><csymbol cd="ambiguous" id="alg1.7.7.m2.1.1.1.3.2.1.cmml" xref="alg1.7.7.m2.1.1.1.3.2">subscript</csymbol><ci id="alg1.7.7.m2.1.1.1.3.2.2.cmml" xref="alg1.7.7.m2.1.1.1.3.2.2">𝑉</ci><ci id="alg1.7.7.m2.1.1.1.3.2.3.cmml" xref="alg1.7.7.m2.1.1.1.3.2.3">𝑖</ci></apply><ci id="alg1.7.7.m2.1.1.1.3.3.cmml" xref="alg1.7.7.m2.1.1.1.3.3">𝑆</ci></apply></apply><apply id="alg1.7.7.m2.1.1.2.cmml" xref="alg1.7.7.m2.1.1.2"><csymbol cd="ambiguous" id="alg1.7.7.m2.1.1.2.1.cmml" xref="alg1.7.7.m2.1.1.2">subscript</csymbol><ci id="alg1.7.7.m2.1.1.2.2.cmml" xref="alg1.7.7.m2.1.1.2.2">𝑉</ci><ci id="alg1.7.7.m2.1.1.2.3.cmml" xref="alg1.7.7.m2.1.1.2.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.7.7.m2.1c">\bigcup_{V_{i}\in S}V_{i}</annotation><annotation encoding="application/x-llamapun" id="alg1.7.7.m2.1d">⋃ start_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_S end_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <div class="ltx_listing ltx_figure_panel ltx_listing" id="alg1.10"> <div class="ltx_listingline" id="alg1.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l1.1.1.1" style="font-size:80%;">1:</span></span><math alttext="\pi\leftarrow\emptyset" class="ltx_Math" display="inline" id="alg1.l1.m1.1"><semantics id="alg1.l1.m1.1a"><mrow id="alg1.l1.m1.1.1" xref="alg1.l1.m1.1.1.cmml"><mi id="alg1.l1.m1.1.1.2" xref="alg1.l1.m1.1.1.2.cmml">π</mi><mo id="alg1.l1.m1.1.1.1" stretchy="false" xref="alg1.l1.m1.1.1.1.cmml">←</mo><mi id="alg1.l1.m1.1.1.3" mathvariant="normal" xref="alg1.l1.m1.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m1.1b"><apply id="alg1.l1.m1.1.1.cmml" xref="alg1.l1.m1.1.1"><ci id="alg1.l1.m1.1.1.1.cmml" xref="alg1.l1.m1.1.1.1">←</ci><ci id="alg1.l1.m1.1.1.2.cmml" xref="alg1.l1.m1.1.1.2">𝜋</ci><emptyset id="alg1.l1.m1.1.1.3.cmml" xref="alg1.l1.m1.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m1.1c">\pi\leftarrow\emptyset</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m1.1d">italic_π ← ∅</annotation></semantics></math>, <math alttext="R\leftarrow\bigcup_{V_{i}\in S}V_{i}" class="ltx_Math" display="inline" id="alg1.l1.m2.1"><semantics id="alg1.l1.m2.1a"><mrow id="alg1.l1.m2.1.1" xref="alg1.l1.m2.1.1.cmml"><mi id="alg1.l1.m2.1.1.2" xref="alg1.l1.m2.1.1.2.cmml">R</mi><mo id="alg1.l1.m2.1.1.1" rspace="0.111em" stretchy="false" xref="alg1.l1.m2.1.1.1.cmml">←</mo><mrow id="alg1.l1.m2.1.1.3" xref="alg1.l1.m2.1.1.3.cmml"><msub id="alg1.l1.m2.1.1.3.1" xref="alg1.l1.m2.1.1.3.1.cmml"><mo id="alg1.l1.m2.1.1.3.1.2" xref="alg1.l1.m2.1.1.3.1.2.cmml">⋃</mo><mrow id="alg1.l1.m2.1.1.3.1.3" xref="alg1.l1.m2.1.1.3.1.3.cmml"><msub id="alg1.l1.m2.1.1.3.1.3.2" xref="alg1.l1.m2.1.1.3.1.3.2.cmml"><mi id="alg1.l1.m2.1.1.3.1.3.2.2" xref="alg1.l1.m2.1.1.3.1.3.2.2.cmml">V</mi><mi id="alg1.l1.m2.1.1.3.1.3.2.3" xref="alg1.l1.m2.1.1.3.1.3.2.3.cmml">i</mi></msub><mo id="alg1.l1.m2.1.1.3.1.3.1" xref="alg1.l1.m2.1.1.3.1.3.1.cmml">∈</mo><mi id="alg1.l1.m2.1.1.3.1.3.3" xref="alg1.l1.m2.1.1.3.1.3.3.cmml">S</mi></mrow></msub><msub id="alg1.l1.m2.1.1.3.2" xref="alg1.l1.m2.1.1.3.2.cmml"><mi id="alg1.l1.m2.1.1.3.2.2" xref="alg1.l1.m2.1.1.3.2.2.cmml">V</mi><mi id="alg1.l1.m2.1.1.3.2.3" xref="alg1.l1.m2.1.1.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m2.1b"><apply id="alg1.l1.m2.1.1.cmml" xref="alg1.l1.m2.1.1"><ci id="alg1.l1.m2.1.1.1.cmml" xref="alg1.l1.m2.1.1.1">←</ci><ci id="alg1.l1.m2.1.1.2.cmml" xref="alg1.l1.m2.1.1.2">𝑅</ci><apply id="alg1.l1.m2.1.1.3.cmml" xref="alg1.l1.m2.1.1.3"><apply id="alg1.l1.m2.1.1.3.1.cmml" xref="alg1.l1.m2.1.1.3.1"><csymbol cd="ambiguous" id="alg1.l1.m2.1.1.3.1.1.cmml" xref="alg1.l1.m2.1.1.3.1">subscript</csymbol><union id="alg1.l1.m2.1.1.3.1.2.cmml" xref="alg1.l1.m2.1.1.3.1.2"></union><apply id="alg1.l1.m2.1.1.3.1.3.cmml" xref="alg1.l1.m2.1.1.3.1.3"><in id="alg1.l1.m2.1.1.3.1.3.1.cmml" xref="alg1.l1.m2.1.1.3.1.3.1"></in><apply id="alg1.l1.m2.1.1.3.1.3.2.cmml" xref="alg1.l1.m2.1.1.3.1.3.2"><csymbol cd="ambiguous" id="alg1.l1.m2.1.1.3.1.3.2.1.cmml" xref="alg1.l1.m2.1.1.3.1.3.2">subscript</csymbol><ci id="alg1.l1.m2.1.1.3.1.3.2.2.cmml" xref="alg1.l1.m2.1.1.3.1.3.2.2">𝑉</ci><ci id="alg1.l1.m2.1.1.3.1.3.2.3.cmml" xref="alg1.l1.m2.1.1.3.1.3.2.3">𝑖</ci></apply><ci id="alg1.l1.m2.1.1.3.1.3.3.cmml" xref="alg1.l1.m2.1.1.3.1.3.3">𝑆</ci></apply></apply><apply id="alg1.l1.m2.1.1.3.2.cmml" xref="alg1.l1.m2.1.1.3.2"><csymbol cd="ambiguous" id="alg1.l1.m2.1.1.3.2.1.cmml" xref="alg1.l1.m2.1.1.3.2">subscript</csymbol><ci id="alg1.l1.m2.1.1.3.2.2.cmml" xref="alg1.l1.m2.1.1.3.2.2">𝑉</ci><ci id="alg1.l1.m2.1.1.3.2.3.cmml" xref="alg1.l1.m2.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m2.1c">R\leftarrow\bigcup_{V_{i}\in S}V_{i}</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m2.1d">italic_R ← ⋃ start_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_S end_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l2.1.1.1" style="font-size:80%;">2:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l2.2">while</span> <math alttext="R\neq\emptyset" class="ltx_Math" display="inline" id="alg1.l2.m1.1"><semantics id="alg1.l2.m1.1a"><mrow id="alg1.l2.m1.1.1" xref="alg1.l2.m1.1.1.cmml"><mi id="alg1.l2.m1.1.1.2" xref="alg1.l2.m1.1.1.2.cmml">R</mi><mo id="alg1.l2.m1.1.1.1" xref="alg1.l2.m1.1.1.1.cmml">≠</mo><mi id="alg1.l2.m1.1.1.3" mathvariant="normal" xref="alg1.l2.m1.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l2.m1.1b"><apply id="alg1.l2.m1.1.1.cmml" xref="alg1.l2.m1.1.1"><neq id="alg1.l2.m1.1.1.1.cmml" xref="alg1.l2.m1.1.1.1"></neq><ci id="alg1.l2.m1.1.1.2.cmml" xref="alg1.l2.m1.1.1.2">𝑅</ci><emptyset id="alg1.l2.m1.1.1.3.cmml" xref="alg1.l2.m1.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m1.1c">R\neq\emptyset</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m1.1d">italic_R ≠ ∅</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l2.3">do</span> </div> <div class="ltx_listingline" id="alg1.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l3.1.1.1" style="font-size:80%;">3:</span></span> Select <math alttext="v\in R" class="ltx_Math" display="inline" id="alg1.l3.m1.1"><semantics id="alg1.l3.m1.1a"><mrow id="alg1.l3.m1.1.1" xref="alg1.l3.m1.1.1.cmml"><mi id="alg1.l3.m1.1.1.2" xref="alg1.l3.m1.1.1.2.cmml">v</mi><mo id="alg1.l3.m1.1.1.1" xref="alg1.l3.m1.1.1.1.cmml">∈</mo><mi id="alg1.l3.m1.1.1.3" xref="alg1.l3.m1.1.1.3.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l3.m1.1b"><apply id="alg1.l3.m1.1.1.cmml" xref="alg1.l3.m1.1.1"><in id="alg1.l3.m1.1.1.1.cmml" xref="alg1.l3.m1.1.1.1"></in><ci id="alg1.l3.m1.1.1.2.cmml" xref="alg1.l3.m1.1.1.2">𝑣</ci><ci id="alg1.l3.m1.1.1.3.cmml" xref="alg1.l3.m1.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m1.1c">v\in R</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m1.1d">italic_v ∈ italic_R</annotation></semantics></math> to begin coalition <math alttext="C=\{v\}" class="ltx_Math" display="inline" id="alg1.l3.m2.1"><semantics id="alg1.l3.m2.1a"><mrow id="alg1.l3.m2.1.2" xref="alg1.l3.m2.1.2.cmml"><mi id="alg1.l3.m2.1.2.2" xref="alg1.l3.m2.1.2.2.cmml">C</mi><mo id="alg1.l3.m2.1.2.1" xref="alg1.l3.m2.1.2.1.cmml">=</mo><mrow id="alg1.l3.m2.1.2.3.2" xref="alg1.l3.m2.1.2.3.1.cmml"><mo id="alg1.l3.m2.1.2.3.2.1" stretchy="false" xref="alg1.l3.m2.1.2.3.1.cmml">{</mo><mi id="alg1.l3.m2.1.1" xref="alg1.l3.m2.1.1.cmml">v</mi><mo id="alg1.l3.m2.1.2.3.2.2" stretchy="false" xref="alg1.l3.m2.1.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l3.m2.1b"><apply id="alg1.l3.m2.1.2.cmml" xref="alg1.l3.m2.1.2"><eq id="alg1.l3.m2.1.2.1.cmml" xref="alg1.l3.m2.1.2.1"></eq><ci id="alg1.l3.m2.1.2.2.cmml" xref="alg1.l3.m2.1.2.2">𝐶</ci><set id="alg1.l3.m2.1.2.3.1.cmml" xref="alg1.l3.m2.1.2.3.2"><ci id="alg1.l3.m2.1.1.cmml" xref="alg1.l3.m2.1.1">𝑣</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m2.1c">C=\{v\}</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m2.1d">italic_C = { italic_v }</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l4.1.1.1" style="font-size:80%;">4:</span></span> <math alttext="L\leftarrow R" class="ltx_Math" display="inline" id="alg1.l4.m1.1"><semantics id="alg1.l4.m1.1a"><mrow id="alg1.l4.m1.1.1" xref="alg1.l4.m1.1.1.cmml"><mi id="alg1.l4.m1.1.1.2" xref="alg1.l4.m1.1.1.2.cmml">L</mi><mo id="alg1.l4.m1.1.1.1" stretchy="false" xref="alg1.l4.m1.1.1.1.cmml">←</mo><mi id="alg1.l4.m1.1.1.3" xref="alg1.l4.m1.1.1.3.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l4.m1.1b"><apply id="alg1.l4.m1.1.1.cmml" xref="alg1.l4.m1.1.1"><ci id="alg1.l4.m1.1.1.1.cmml" xref="alg1.l4.m1.1.1.1">←</ci><ci id="alg1.l4.m1.1.1.2.cmml" xref="alg1.l4.m1.1.1.2">𝐿</ci><ci id="alg1.l4.m1.1.1.3.cmml" xref="alg1.l4.m1.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m1.1c">L\leftarrow R</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m1.1d">italic_L ← italic_R</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l5.1.1.1" style="font-size:80%;">5:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l5.2">while</span> <math alttext="\exists w\in L" class="ltx_Math" display="inline" id="alg1.l5.m1.1"><semantics id="alg1.l5.m1.1a"><mrow id="alg1.l5.m1.1.1" xref="alg1.l5.m1.1.1.cmml"><mrow id="alg1.l5.m1.1.1.2" xref="alg1.l5.m1.1.1.2.cmml"><mo id="alg1.l5.m1.1.1.2.1" rspace="0.167em" xref="alg1.l5.m1.1.1.2.1.cmml">∃</mo><mi id="alg1.l5.m1.1.1.2.2" xref="alg1.l5.m1.1.1.2.2.cmml">w</mi></mrow><mo id="alg1.l5.m1.1.1.1" xref="alg1.l5.m1.1.1.1.cmml">∈</mo><mi id="alg1.l5.m1.1.1.3" xref="alg1.l5.m1.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l5.m1.1b"><apply id="alg1.l5.m1.1.1.cmml" xref="alg1.l5.m1.1.1"><in id="alg1.l5.m1.1.1.1.cmml" xref="alg1.l5.m1.1.1.1"></in><apply id="alg1.l5.m1.1.1.2.cmml" xref="alg1.l5.m1.1.1.2"><exists id="alg1.l5.m1.1.1.2.1.cmml" xref="alg1.l5.m1.1.1.2.1"></exists><ci id="alg1.l5.m1.1.1.2.2.cmml" xref="alg1.l5.m1.1.1.2.2">𝑤</ci></apply><ci id="alg1.l5.m1.1.1.3.cmml" xref="alg1.l5.m1.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.m1.1c">\exists w\in L</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.m1.1d">∃ italic_w ∈ italic_L</annotation></semantics></math> with all edges towards <math alttext="C" class="ltx_Math" display="inline" id="alg1.l5.m2.1"><semantics id="alg1.l5.m2.1a"><mi id="alg1.l5.m2.1.1" xref="alg1.l5.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="alg1.l5.m2.1b"><ci id="alg1.l5.m2.1.1.cmml" xref="alg1.l5.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.m2.1d">italic_C</annotation></semantics></math> of weight 1 <span class="ltx_text ltx_font_bold" id="alg1.l5.3">do</span> </div> <div class="ltx_listingline" id="alg1.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l6.1.1.1" style="font-size:80%;">6:</span></span> <math alttext="C\leftarrow C\cup\{w\}" class="ltx_Math" display="inline" id="alg1.l6.m1.1"><semantics id="alg1.l6.m1.1a"><mrow id="alg1.l6.m1.1.2" xref="alg1.l6.m1.1.2.cmml"><mi id="alg1.l6.m1.1.2.2" xref="alg1.l6.m1.1.2.2.cmml">C</mi><mo id="alg1.l6.m1.1.2.1" stretchy="false" xref="alg1.l6.m1.1.2.1.cmml">←</mo><mrow id="alg1.l6.m1.1.2.3" xref="alg1.l6.m1.1.2.3.cmml"><mi id="alg1.l6.m1.1.2.3.2" xref="alg1.l6.m1.1.2.3.2.cmml">C</mi><mo id="alg1.l6.m1.1.2.3.1" xref="alg1.l6.m1.1.2.3.1.cmml">∪</mo><mrow id="alg1.l6.m1.1.2.3.3.2" xref="alg1.l6.m1.1.2.3.3.1.cmml"><mo id="alg1.l6.m1.1.2.3.3.2.1" stretchy="false" xref="alg1.l6.m1.1.2.3.3.1.cmml">{</mo><mi id="alg1.l6.m1.1.1" xref="alg1.l6.m1.1.1.cmml">w</mi><mo id="alg1.l6.m1.1.2.3.3.2.2" stretchy="false" xref="alg1.l6.m1.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l6.m1.1b"><apply id="alg1.l6.m1.1.2.cmml" xref="alg1.l6.m1.1.2"><ci id="alg1.l6.m1.1.2.1.cmml" xref="alg1.l6.m1.1.2.1">←</ci><ci id="alg1.l6.m1.1.2.2.cmml" xref="alg1.l6.m1.1.2.2">𝐶</ci><apply id="alg1.l6.m1.1.2.3.cmml" xref="alg1.l6.m1.1.2.3"><union id="alg1.l6.m1.1.2.3.1.cmml" xref="alg1.l6.m1.1.2.3.1"></union><ci id="alg1.l6.m1.1.2.3.2.cmml" xref="alg1.l6.m1.1.2.3.2">𝐶</ci><set id="alg1.l6.m1.1.2.3.3.1.cmml" xref="alg1.l6.m1.1.2.3.3.2"><ci id="alg1.l6.m1.1.1.cmml" xref="alg1.l6.m1.1.1">𝑤</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m1.1c">C\leftarrow C\cup\{w\}</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m1.1d">italic_C ← italic_C ∪ { italic_w }</annotation></semantics></math>, <math alttext="L\leftarrow L\setminus\{w\}" class="ltx_Math" display="inline" id="alg1.l6.m2.1"><semantics id="alg1.l6.m2.1a"><mrow id="alg1.l6.m2.1.2" xref="alg1.l6.m2.1.2.cmml"><mi id="alg1.l6.m2.1.2.2" xref="alg1.l6.m2.1.2.2.cmml">L</mi><mo id="alg1.l6.m2.1.2.1" stretchy="false" xref="alg1.l6.m2.1.2.1.cmml">←</mo><mrow id="alg1.l6.m2.1.2.3" xref="alg1.l6.m2.1.2.3.cmml"><mi id="alg1.l6.m2.1.2.3.2" xref="alg1.l6.m2.1.2.3.2.cmml">L</mi><mo id="alg1.l6.m2.1.2.3.1" xref="alg1.l6.m2.1.2.3.1.cmml">∖</mo><mrow id="alg1.l6.m2.1.2.3.3.2" xref="alg1.l6.m2.1.2.3.3.1.cmml"><mo id="alg1.l6.m2.1.2.3.3.2.1" stretchy="false" xref="alg1.l6.m2.1.2.3.3.1.cmml">{</mo><mi id="alg1.l6.m2.1.1" xref="alg1.l6.m2.1.1.cmml">w</mi><mo id="alg1.l6.m2.1.2.3.3.2.2" stretchy="false" xref="alg1.l6.m2.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l6.m2.1b"><apply id="alg1.l6.m2.1.2.cmml" xref="alg1.l6.m2.1.2"><ci id="alg1.l6.m2.1.2.1.cmml" xref="alg1.l6.m2.1.2.1">←</ci><ci id="alg1.l6.m2.1.2.2.cmml" xref="alg1.l6.m2.1.2.2">𝐿</ci><apply id="alg1.l6.m2.1.2.3.cmml" xref="alg1.l6.m2.1.2.3"><setdiff id="alg1.l6.m2.1.2.3.1.cmml" xref="alg1.l6.m2.1.2.3.1"></setdiff><ci id="alg1.l6.m2.1.2.3.2.cmml" xref="alg1.l6.m2.1.2.3.2">𝐿</ci><set id="alg1.l6.m2.1.2.3.3.1.cmml" xref="alg1.l6.m2.1.2.3.3.2"><ci id="alg1.l6.m2.1.1.cmml" xref="alg1.l6.m2.1.1">𝑤</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m2.1c">L\leftarrow L\setminus\{w\}</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m2.1d">italic_L ← italic_L ∖ { italic_w }</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l7"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l7.1.1.1" style="font-size:80%;">7:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l7.2">if</span> <math alttext="|C|\geq k^{\prime}(\sqrt{1-\varepsilon})" class="ltx_Math" display="inline" id="alg1.l7.m1.2"><semantics id="alg1.l7.m1.2a"><mrow id="alg1.l7.m1.2.3" xref="alg1.l7.m1.2.3.cmml"><mrow id="alg1.l7.m1.2.3.2.2" xref="alg1.l7.m1.2.3.2.1.cmml"><mo id="alg1.l7.m1.2.3.2.2.1" stretchy="false" xref="alg1.l7.m1.2.3.2.1.1.cmml">|</mo><mi id="alg1.l7.m1.1.1" xref="alg1.l7.m1.1.1.cmml">C</mi><mo id="alg1.l7.m1.2.3.2.2.2" stretchy="false" xref="alg1.l7.m1.2.3.2.1.1.cmml">|</mo></mrow><mo id="alg1.l7.m1.2.3.1" xref="alg1.l7.m1.2.3.1.cmml">≥</mo><mrow id="alg1.l7.m1.2.3.3" xref="alg1.l7.m1.2.3.3.cmml"><msup id="alg1.l7.m1.2.3.3.2" xref="alg1.l7.m1.2.3.3.2.cmml"><mi id="alg1.l7.m1.2.3.3.2.2" xref="alg1.l7.m1.2.3.3.2.2.cmml">k</mi><mo id="alg1.l7.m1.2.3.3.2.3" xref="alg1.l7.m1.2.3.3.2.3.cmml">′</mo></msup><mo id="alg1.l7.m1.2.3.3.1" xref="alg1.l7.m1.2.3.3.1.cmml">⁢</mo><mrow id="alg1.l7.m1.2.3.3.3.2" xref="alg1.l7.m1.2.2.cmml"><mo id="alg1.l7.m1.2.3.3.3.2.1" stretchy="false" xref="alg1.l7.m1.2.2.cmml">(</mo><msqrt id="alg1.l7.m1.2.2" xref="alg1.l7.m1.2.2.cmml"><mrow id="alg1.l7.m1.2.2.2" xref="alg1.l7.m1.2.2.2.cmml"><mn id="alg1.l7.m1.2.2.2.2" xref="alg1.l7.m1.2.2.2.2.cmml">1</mn><mo id="alg1.l7.m1.2.2.2.1" xref="alg1.l7.m1.2.2.2.1.cmml">−</mo><mi id="alg1.l7.m1.2.2.2.3" xref="alg1.l7.m1.2.2.2.3.cmml">ε</mi></mrow></msqrt><mo id="alg1.l7.m1.2.3.3.3.2.2" stretchy="false" xref="alg1.l7.m1.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l7.m1.2b"><apply id="alg1.l7.m1.2.3.cmml" xref="alg1.l7.m1.2.3"><geq id="alg1.l7.m1.2.3.1.cmml" xref="alg1.l7.m1.2.3.1"></geq><apply id="alg1.l7.m1.2.3.2.1.cmml" xref="alg1.l7.m1.2.3.2.2"><abs id="alg1.l7.m1.2.3.2.1.1.cmml" xref="alg1.l7.m1.2.3.2.2.1"></abs><ci id="alg1.l7.m1.1.1.cmml" xref="alg1.l7.m1.1.1">𝐶</ci></apply><apply id="alg1.l7.m1.2.3.3.cmml" xref="alg1.l7.m1.2.3.3"><times id="alg1.l7.m1.2.3.3.1.cmml" xref="alg1.l7.m1.2.3.3.1"></times><apply id="alg1.l7.m1.2.3.3.2.cmml" xref="alg1.l7.m1.2.3.3.2"><csymbol cd="ambiguous" id="alg1.l7.m1.2.3.3.2.1.cmml" xref="alg1.l7.m1.2.3.3.2">superscript</csymbol><ci id="alg1.l7.m1.2.3.3.2.2.cmml" xref="alg1.l7.m1.2.3.3.2.2">𝑘</ci><ci id="alg1.l7.m1.2.3.3.2.3.cmml" xref="alg1.l7.m1.2.3.3.2.3">′</ci></apply><apply id="alg1.l7.m1.2.2.cmml" xref="alg1.l7.m1.2.3.3.3.2"><root id="alg1.l7.m1.2.2a.cmml" xref="alg1.l7.m1.2.3.3.3.2"></root><apply id="alg1.l7.m1.2.2.2.cmml" xref="alg1.l7.m1.2.2.2"><minus id="alg1.l7.m1.2.2.2.1.cmml" xref="alg1.l7.m1.2.2.2.1"></minus><cn id="alg1.l7.m1.2.2.2.2.cmml" type="integer" xref="alg1.l7.m1.2.2.2.2">1</cn><ci id="alg1.l7.m1.2.2.2.3.cmml" xref="alg1.l7.m1.2.2.2.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l7.m1.2c">|C|\geq k^{\prime}(\sqrt{1-\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="alg1.l7.m1.2d">| italic_C | ≥ italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( square-root start_ARG 1 - italic_ε end_ARG )</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l7.3">then</span> </div> <div class="ltx_listingline" id="alg1.l8"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l8.1.1.1" style="font-size:80%;">8:</span></span> <math alttext="\pi\leftarrow\pi\cup\{C\}" class="ltx_Math" display="inline" id="alg1.l8.m1.1"><semantics id="alg1.l8.m1.1a"><mrow id="alg1.l8.m1.1.2" xref="alg1.l8.m1.1.2.cmml"><mi id="alg1.l8.m1.1.2.2" xref="alg1.l8.m1.1.2.2.cmml">π</mi><mo id="alg1.l8.m1.1.2.1" stretchy="false" xref="alg1.l8.m1.1.2.1.cmml">←</mo><mrow id="alg1.l8.m1.1.2.3" xref="alg1.l8.m1.1.2.3.cmml"><mi id="alg1.l8.m1.1.2.3.2" xref="alg1.l8.m1.1.2.3.2.cmml">π</mi><mo id="alg1.l8.m1.1.2.3.1" xref="alg1.l8.m1.1.2.3.1.cmml">∪</mo><mrow id="alg1.l8.m1.1.2.3.3.2" xref="alg1.l8.m1.1.2.3.3.1.cmml"><mo id="alg1.l8.m1.1.2.3.3.2.1" stretchy="false" xref="alg1.l8.m1.1.2.3.3.1.cmml">{</mo><mi id="alg1.l8.m1.1.1" xref="alg1.l8.m1.1.1.cmml">C</mi><mo id="alg1.l8.m1.1.2.3.3.2.2" stretchy="false" xref="alg1.l8.m1.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l8.m1.1b"><apply id="alg1.l8.m1.1.2.cmml" xref="alg1.l8.m1.1.2"><ci id="alg1.l8.m1.1.2.1.cmml" xref="alg1.l8.m1.1.2.1">←</ci><ci id="alg1.l8.m1.1.2.2.cmml" xref="alg1.l8.m1.1.2.2">𝜋</ci><apply id="alg1.l8.m1.1.2.3.cmml" xref="alg1.l8.m1.1.2.3"><union id="alg1.l8.m1.1.2.3.1.cmml" xref="alg1.l8.m1.1.2.3.1"></union><ci id="alg1.l8.m1.1.2.3.2.cmml" xref="alg1.l8.m1.1.2.3.2">𝜋</ci><set id="alg1.l8.m1.1.2.3.3.1.cmml" xref="alg1.l8.m1.1.2.3.3.2"><ci id="alg1.l8.m1.1.1.cmml" xref="alg1.l8.m1.1.1">𝐶</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l8.m1.1c">\pi\leftarrow\pi\cup\{C\}</annotation><annotation encoding="application/x-llamapun" id="alg1.l8.m1.1d">italic_π ← italic_π ∪ { italic_C }</annotation></semantics></math>, <math alttext="R\leftarrow R\setminus C" class="ltx_Math" display="inline" id="alg1.l8.m2.1"><semantics id="alg1.l8.m2.1a"><mrow id="alg1.l8.m2.1.1" xref="alg1.l8.m2.1.1.cmml"><mi id="alg1.l8.m2.1.1.2" xref="alg1.l8.m2.1.1.2.cmml">R</mi><mo id="alg1.l8.m2.1.1.1" stretchy="false" xref="alg1.l8.m2.1.1.1.cmml">←</mo><mrow id="alg1.l8.m2.1.1.3" xref="alg1.l8.m2.1.1.3.cmml"><mi id="alg1.l8.m2.1.1.3.2" xref="alg1.l8.m2.1.1.3.2.cmml">R</mi><mo id="alg1.l8.m2.1.1.3.1" xref="alg1.l8.m2.1.1.3.1.cmml">∖</mo><mi id="alg1.l8.m2.1.1.3.3" xref="alg1.l8.m2.1.1.3.3.cmml">C</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l8.m2.1b"><apply id="alg1.l8.m2.1.1.cmml" xref="alg1.l8.m2.1.1"><ci id="alg1.l8.m2.1.1.1.cmml" xref="alg1.l8.m2.1.1.1">←</ci><ci id="alg1.l8.m2.1.1.2.cmml" xref="alg1.l8.m2.1.1.2">𝑅</ci><apply id="alg1.l8.m2.1.1.3.cmml" xref="alg1.l8.m2.1.1.3"><setdiff id="alg1.l8.m2.1.1.3.1.cmml" xref="alg1.l8.m2.1.1.3.1"></setdiff><ci id="alg1.l8.m2.1.1.3.2.cmml" xref="alg1.l8.m2.1.1.3.2">𝑅</ci><ci id="alg1.l8.m2.1.1.3.3.cmml" xref="alg1.l8.m2.1.1.3.3">𝐶</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l8.m2.1c">R\leftarrow R\setminus C</annotation><annotation encoding="application/x-llamapun" id="alg1.l8.m2.1d">italic_R ← italic_R ∖ italic_C</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l9"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l9.1.1.1" style="font-size:80%;">9:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l9.2">else</span> </div> <div class="ltx_listingline" id="alg1.l10"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l10.1.1.1" style="font-size:80%;">10:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l10.2">return</span> <math alttext="\pi\cup\{\{v\}\colon v\in R\}" class="ltx_Math" display="inline" id="alg1.l10.m1.3"><semantics id="alg1.l10.m1.3a"><mrow id="alg1.l10.m1.3.3" xref="alg1.l10.m1.3.3.cmml"><mi id="alg1.l10.m1.3.3.4" xref="alg1.l10.m1.3.3.4.cmml">π</mi><mo id="alg1.l10.m1.3.3.3" xref="alg1.l10.m1.3.3.3.cmml">∪</mo><mrow id="alg1.l10.m1.3.3.2.2" xref="alg1.l10.m1.3.3.2.3.cmml"><mo id="alg1.l10.m1.3.3.2.2.3" stretchy="false" xref="alg1.l10.m1.3.3.2.3.1.cmml">{</mo><mrow id="alg1.l10.m1.2.2.1.1.1.2" xref="alg1.l10.m1.2.2.1.1.1.1.cmml"><mo id="alg1.l10.m1.2.2.1.1.1.2.1" stretchy="false" xref="alg1.l10.m1.2.2.1.1.1.1.cmml">{</mo><mi id="alg1.l10.m1.1.1" xref="alg1.l10.m1.1.1.cmml">v</mi><mo id="alg1.l10.m1.2.2.1.1.1.2.2" rspace="0.278em" stretchy="false" xref="alg1.l10.m1.2.2.1.1.1.1.cmml">}</mo></mrow><mo id="alg1.l10.m1.3.3.2.2.4" rspace="0.278em" xref="alg1.l10.m1.3.3.2.3.1.cmml">:</mo><mrow id="alg1.l10.m1.3.3.2.2.2" xref="alg1.l10.m1.3.3.2.2.2.cmml"><mi id="alg1.l10.m1.3.3.2.2.2.2" xref="alg1.l10.m1.3.3.2.2.2.2.cmml">v</mi><mo id="alg1.l10.m1.3.3.2.2.2.1" xref="alg1.l10.m1.3.3.2.2.2.1.cmml">∈</mo><mi id="alg1.l10.m1.3.3.2.2.2.3" xref="alg1.l10.m1.3.3.2.2.2.3.cmml">R</mi></mrow><mo id="alg1.l10.m1.3.3.2.2.5" stretchy="false" xref="alg1.l10.m1.3.3.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l10.m1.3b"><apply id="alg1.l10.m1.3.3.cmml" xref="alg1.l10.m1.3.3"><union id="alg1.l10.m1.3.3.3.cmml" xref="alg1.l10.m1.3.3.3"></union><ci id="alg1.l10.m1.3.3.4.cmml" xref="alg1.l10.m1.3.3.4">𝜋</ci><apply id="alg1.l10.m1.3.3.2.3.cmml" xref="alg1.l10.m1.3.3.2.2"><csymbol cd="latexml" id="alg1.l10.m1.3.3.2.3.1.cmml" xref="alg1.l10.m1.3.3.2.2.3">conditional-set</csymbol><set id="alg1.l10.m1.2.2.1.1.1.1.cmml" xref="alg1.l10.m1.2.2.1.1.1.2"><ci id="alg1.l10.m1.1.1.cmml" xref="alg1.l10.m1.1.1">𝑣</ci></set><apply id="alg1.l10.m1.3.3.2.2.2.cmml" xref="alg1.l10.m1.3.3.2.2.2"><in id="alg1.l10.m1.3.3.2.2.2.1.cmml" xref="alg1.l10.m1.3.3.2.2.2.1"></in><ci id="alg1.l10.m1.3.3.2.2.2.2.cmml" xref="alg1.l10.m1.3.3.2.2.2.2">𝑣</ci><ci id="alg1.l10.m1.3.3.2.2.2.3.cmml" xref="alg1.l10.m1.3.3.2.2.2.3">𝑅</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l10.m1.3c">\pi\cup\{\{v\}\colon v\in R\}</annotation><annotation encoding="application/x-llamapun" id="alg1.l10.m1.3d">italic_π ∪ { { italic_v } : italic_v ∈ italic_R }</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l11"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l11.1.1.1" style="font-size:80%;">11:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l11.2">return</span> <math alttext="\pi" class="ltx_Math" display="inline" id="alg1.l11.m1.1"><semantics id="alg1.l11.m1.1a"><mi id="alg1.l11.m1.1.1" xref="alg1.l11.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="alg1.l11.m1.1b"><ci id="alg1.l11.m1.1.1.cmml" xref="alg1.l11.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l11.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="alg1.l11.m1.1d">italic_π</annotation></semantics></math> </div> </div> </div> </div> </figure> <div class="ltx_para" id="S5.SS2.SSS1.p3"> <p class="ltx_p" id="S5.SS2.SSS1.p3.13">The following lemma shows that for sufficiently small values of <math alttext="p" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p3.1.m1.1"><semantics id="S5.SS2.SSS1.p3.1.m1.1a"><mi id="S5.SS2.SSS1.p3.1.m1.1.1" xref="S5.SS2.SSS1.p3.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p3.1.m1.1b"><ci id="S5.SS2.SSS1.p3.1.m1.1.1.cmml" xref="S5.SS2.SSS1.p3.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p3.1.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p3.1.m1.1d">italic_p</annotation></semantics></math>, by selecting a subset of color classes <math alttext="S\subseteq\{V_{1},\dots,V_{k}\}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p3.2.m2.3"><semantics id="S5.SS2.SSS1.p3.2.m2.3a"><mrow id="S5.SS2.SSS1.p3.2.m2.3.3" xref="S5.SS2.SSS1.p3.2.m2.3.3.cmml"><mi id="S5.SS2.SSS1.p3.2.m2.3.3.4" xref="S5.SS2.SSS1.p3.2.m2.3.3.4.cmml">S</mi><mo id="S5.SS2.SSS1.p3.2.m2.3.3.3" xref="S5.SS2.SSS1.p3.2.m2.3.3.3.cmml">⊆</mo><mrow id="S5.SS2.SSS1.p3.2.m2.3.3.2.2" xref="S5.SS2.SSS1.p3.2.m2.3.3.2.3.cmml"><mo id="S5.SS2.SSS1.p3.2.m2.3.3.2.2.3" stretchy="false" xref="S5.SS2.SSS1.p3.2.m2.3.3.2.3.cmml">{</mo><msub id="S5.SS2.SSS1.p3.2.m2.2.2.1.1.1" xref="S5.SS2.SSS1.p3.2.m2.2.2.1.1.1.cmml"><mi id="S5.SS2.SSS1.p3.2.m2.2.2.1.1.1.2" xref="S5.SS2.SSS1.p3.2.m2.2.2.1.1.1.2.cmml">V</mi><mn id="S5.SS2.SSS1.p3.2.m2.2.2.1.1.1.3" xref="S5.SS2.SSS1.p3.2.m2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS2.SSS1.p3.2.m2.3.3.2.2.4" xref="S5.SS2.SSS1.p3.2.m2.3.3.2.3.cmml">,</mo><mi id="S5.SS2.SSS1.p3.2.m2.1.1" mathvariant="normal" xref="S5.SS2.SSS1.p3.2.m2.1.1.cmml">…</mi><mo id="S5.SS2.SSS1.p3.2.m2.3.3.2.2.5" xref="S5.SS2.SSS1.p3.2.m2.3.3.2.3.cmml">,</mo><msub id="S5.SS2.SSS1.p3.2.m2.3.3.2.2.2" xref="S5.SS2.SSS1.p3.2.m2.3.3.2.2.2.cmml"><mi id="S5.SS2.SSS1.p3.2.m2.3.3.2.2.2.2" xref="S5.SS2.SSS1.p3.2.m2.3.3.2.2.2.2.cmml">V</mi><mi id="S5.SS2.SSS1.p3.2.m2.3.3.2.2.2.3" xref="S5.SS2.SSS1.p3.2.m2.3.3.2.2.2.3.cmml">k</mi></msub><mo id="S5.SS2.SSS1.p3.2.m2.3.3.2.2.6" stretchy="false" xref="S5.SS2.SSS1.p3.2.m2.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p3.2.m2.3b"><apply id="S5.SS2.SSS1.p3.2.m2.3.3.cmml" xref="S5.SS2.SSS1.p3.2.m2.3.3"><subset id="S5.SS2.SSS1.p3.2.m2.3.3.3.cmml" xref="S5.SS2.SSS1.p3.2.m2.3.3.3"></subset><ci id="S5.SS2.SSS1.p3.2.m2.3.3.4.cmml" xref="S5.SS2.SSS1.p3.2.m2.3.3.4">𝑆</ci><set id="S5.SS2.SSS1.p3.2.m2.3.3.2.3.cmml" xref="S5.SS2.SSS1.p3.2.m2.3.3.2.2"><apply id="S5.SS2.SSS1.p3.2.m2.2.2.1.1.1.cmml" xref="S5.SS2.SSS1.p3.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p3.2.m2.2.2.1.1.1.1.cmml" xref="S5.SS2.SSS1.p3.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S5.SS2.SSS1.p3.2.m2.2.2.1.1.1.2.cmml" xref="S5.SS2.SSS1.p3.2.m2.2.2.1.1.1.2">𝑉</ci><cn id="S5.SS2.SSS1.p3.2.m2.2.2.1.1.1.3.cmml" type="integer" xref="S5.SS2.SSS1.p3.2.m2.2.2.1.1.1.3">1</cn></apply><ci id="S5.SS2.SSS1.p3.2.m2.1.1.cmml" xref="S5.SS2.SSS1.p3.2.m2.1.1">…</ci><apply id="S5.SS2.SSS1.p3.2.m2.3.3.2.2.2.cmml" xref="S5.SS2.SSS1.p3.2.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p3.2.m2.3.3.2.2.2.1.cmml" xref="S5.SS2.SSS1.p3.2.m2.3.3.2.2.2">subscript</csymbol><ci id="S5.SS2.SSS1.p3.2.m2.3.3.2.2.2.2.cmml" xref="S5.SS2.SSS1.p3.2.m2.3.3.2.2.2.2">𝑉</ci><ci id="S5.SS2.SSS1.p3.2.m2.3.3.2.2.2.3.cmml" xref="S5.SS2.SSS1.p3.2.m2.3.3.2.2.2.3">𝑘</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p3.2.m2.3c">S\subseteq\{V_{1},\dots,V_{k}\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p3.2.m2.3d">italic_S ⊆ { italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }</annotation></semantics></math>, where <math alttext="k^{\prime}=|S|" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p3.3.m3.1"><semantics id="S5.SS2.SSS1.p3.3.m3.1a"><mrow id="S5.SS2.SSS1.p3.3.m3.1.2" xref="S5.SS2.SSS1.p3.3.m3.1.2.cmml"><msup id="S5.SS2.SSS1.p3.3.m3.1.2.2" xref="S5.SS2.SSS1.p3.3.m3.1.2.2.cmml"><mi id="S5.SS2.SSS1.p3.3.m3.1.2.2.2" xref="S5.SS2.SSS1.p3.3.m3.1.2.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.p3.3.m3.1.2.2.3" xref="S5.SS2.SSS1.p3.3.m3.1.2.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS1.p3.3.m3.1.2.1" xref="S5.SS2.SSS1.p3.3.m3.1.2.1.cmml">=</mo><mrow id="S5.SS2.SSS1.p3.3.m3.1.2.3.2" xref="S5.SS2.SSS1.p3.3.m3.1.2.3.1.cmml"><mo id="S5.SS2.SSS1.p3.3.m3.1.2.3.2.1" stretchy="false" xref="S5.SS2.SSS1.p3.3.m3.1.2.3.1.1.cmml">|</mo><mi id="S5.SS2.SSS1.p3.3.m3.1.1" xref="S5.SS2.SSS1.p3.3.m3.1.1.cmml">S</mi><mo id="S5.SS2.SSS1.p3.3.m3.1.2.3.2.2" stretchy="false" xref="S5.SS2.SSS1.p3.3.m3.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p3.3.m3.1b"><apply id="S5.SS2.SSS1.p3.3.m3.1.2.cmml" xref="S5.SS2.SSS1.p3.3.m3.1.2"><eq id="S5.SS2.SSS1.p3.3.m3.1.2.1.cmml" xref="S5.SS2.SSS1.p3.3.m3.1.2.1"></eq><apply id="S5.SS2.SSS1.p3.3.m3.1.2.2.cmml" xref="S5.SS2.SSS1.p3.3.m3.1.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p3.3.m3.1.2.2.1.cmml" xref="S5.SS2.SSS1.p3.3.m3.1.2.2">superscript</csymbol><ci id="S5.SS2.SSS1.p3.3.m3.1.2.2.2.cmml" xref="S5.SS2.SSS1.p3.3.m3.1.2.2.2">𝑘</ci><ci id="S5.SS2.SSS1.p3.3.m3.1.2.2.3.cmml" xref="S5.SS2.SSS1.p3.3.m3.1.2.2.3">′</ci></apply><apply id="S5.SS2.SSS1.p3.3.m3.1.2.3.1.cmml" xref="S5.SS2.SSS1.p3.3.m3.1.2.3.2"><abs id="S5.SS2.SSS1.p3.3.m3.1.2.3.1.1.cmml" xref="S5.SS2.SSS1.p3.3.m3.1.2.3.2.1"></abs><ci id="S5.SS2.SSS1.p3.3.m3.1.1.cmml" xref="S5.SS2.SSS1.p3.3.m3.1.1">𝑆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p3.3.m3.1c">k^{\prime}=|S|</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p3.3.m3.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = | italic_S |</annotation></semantics></math>, and running <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg1" title="In 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a>, the algorithm produces nearly <math alttext="\frac{n}{k}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p3.4.m4.1"><semantics id="S5.SS2.SSS1.p3.4.m4.1a"><mfrac id="S5.SS2.SSS1.p3.4.m4.1.1" xref="S5.SS2.SSS1.p3.4.m4.1.1.cmml"><mi id="S5.SS2.SSS1.p3.4.m4.1.1.2" xref="S5.SS2.SSS1.p3.4.m4.1.1.2.cmml">n</mi><mi id="S5.SS2.SSS1.p3.4.m4.1.1.3" xref="S5.SS2.SSS1.p3.4.m4.1.1.3.cmml">k</mi></mfrac><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p3.4.m4.1b"><apply id="S5.SS2.SSS1.p3.4.m4.1.1.cmml" xref="S5.SS2.SSS1.p3.4.m4.1.1"><divide id="S5.SS2.SSS1.p3.4.m4.1.1.1.cmml" xref="S5.SS2.SSS1.p3.4.m4.1.1"></divide><ci id="S5.SS2.SSS1.p3.4.m4.1.1.2.cmml" xref="S5.SS2.SSS1.p3.4.m4.1.1.2">𝑛</ci><ci id="S5.SS2.SSS1.p3.4.m4.1.1.3.cmml" xref="S5.SS2.SSS1.p3.4.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p3.4.m4.1c">\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p3.4.m4.1d">divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math> maximal cliques, each of which exceeds the size <math alttext="k^{\prime}\sqrt{1-\varepsilon}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p3.5.m5.1"><semantics id="S5.SS2.SSS1.p3.5.m5.1a"><mrow id="S5.SS2.SSS1.p3.5.m5.1.1" xref="S5.SS2.SSS1.p3.5.m5.1.1.cmml"><msup id="S5.SS2.SSS1.p3.5.m5.1.1.2" xref="S5.SS2.SSS1.p3.5.m5.1.1.2.cmml"><mi id="S5.SS2.SSS1.p3.5.m5.1.1.2.2" xref="S5.SS2.SSS1.p3.5.m5.1.1.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.p3.5.m5.1.1.2.3" xref="S5.SS2.SSS1.p3.5.m5.1.1.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS1.p3.5.m5.1.1.1" xref="S5.SS2.SSS1.p3.5.m5.1.1.1.cmml">⁢</mo><msqrt id="S5.SS2.SSS1.p3.5.m5.1.1.3" xref="S5.SS2.SSS1.p3.5.m5.1.1.3.cmml"><mrow id="S5.SS2.SSS1.p3.5.m5.1.1.3.2" xref="S5.SS2.SSS1.p3.5.m5.1.1.3.2.cmml"><mn id="S5.SS2.SSS1.p3.5.m5.1.1.3.2.2" xref="S5.SS2.SSS1.p3.5.m5.1.1.3.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.p3.5.m5.1.1.3.2.1" xref="S5.SS2.SSS1.p3.5.m5.1.1.3.2.1.cmml">−</mo><mi id="S5.SS2.SSS1.p3.5.m5.1.1.3.2.3" xref="S5.SS2.SSS1.p3.5.m5.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p3.5.m5.1b"><apply id="S5.SS2.SSS1.p3.5.m5.1.1.cmml" xref="S5.SS2.SSS1.p3.5.m5.1.1"><times id="S5.SS2.SSS1.p3.5.m5.1.1.1.cmml" xref="S5.SS2.SSS1.p3.5.m5.1.1.1"></times><apply id="S5.SS2.SSS1.p3.5.m5.1.1.2.cmml" xref="S5.SS2.SSS1.p3.5.m5.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p3.5.m5.1.1.2.1.cmml" xref="S5.SS2.SSS1.p3.5.m5.1.1.2">superscript</csymbol><ci id="S5.SS2.SSS1.p3.5.m5.1.1.2.2.cmml" xref="S5.SS2.SSS1.p3.5.m5.1.1.2.2">𝑘</ci><ci id="S5.SS2.SSS1.p3.5.m5.1.1.2.3.cmml" xref="S5.SS2.SSS1.p3.5.m5.1.1.2.3">′</ci></apply><apply id="S5.SS2.SSS1.p3.5.m5.1.1.3.cmml" xref="S5.SS2.SSS1.p3.5.m5.1.1.3"><root id="S5.SS2.SSS1.p3.5.m5.1.1.3a.cmml" xref="S5.SS2.SSS1.p3.5.m5.1.1.3"></root><apply id="S5.SS2.SSS1.p3.5.m5.1.1.3.2.cmml" xref="S5.SS2.SSS1.p3.5.m5.1.1.3.2"><minus id="S5.SS2.SSS1.p3.5.m5.1.1.3.2.1.cmml" xref="S5.SS2.SSS1.p3.5.m5.1.1.3.2.1"></minus><cn id="S5.SS2.SSS1.p3.5.m5.1.1.3.2.2.cmml" type="integer" xref="S5.SS2.SSS1.p3.5.m5.1.1.3.2.2">1</cn><ci id="S5.SS2.SSS1.p3.5.m5.1.1.3.2.3.cmml" xref="S5.SS2.SSS1.p3.5.m5.1.1.3.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p3.5.m5.1c">k^{\prime}\sqrt{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p3.5.m5.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_ε end_ARG</annotation></semantics></math> with high probability. When <math alttext="p=\mathcal{O}\left(\frac{1}{k}\right)" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p3.6.m6.1"><semantics id="S5.SS2.SSS1.p3.6.m6.1a"><mrow id="S5.SS2.SSS1.p3.6.m6.1.2" xref="S5.SS2.SSS1.p3.6.m6.1.2.cmml"><mi id="S5.SS2.SSS1.p3.6.m6.1.2.2" xref="S5.SS2.SSS1.p3.6.m6.1.2.2.cmml">p</mi><mo id="S5.SS2.SSS1.p3.6.m6.1.2.1" xref="S5.SS2.SSS1.p3.6.m6.1.2.1.cmml">=</mo><mrow id="S5.SS2.SSS1.p3.6.m6.1.2.3" xref="S5.SS2.SSS1.p3.6.m6.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS1.p3.6.m6.1.2.3.2" xref="S5.SS2.SSS1.p3.6.m6.1.2.3.2.cmml">𝒪</mi><mo id="S5.SS2.SSS1.p3.6.m6.1.2.3.1" xref="S5.SS2.SSS1.p3.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS1.p3.6.m6.1.2.3.3.2" xref="S5.SS2.SSS1.p3.6.m6.1.1.cmml"><mo id="S5.SS2.SSS1.p3.6.m6.1.2.3.3.2.1" xref="S5.SS2.SSS1.p3.6.m6.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS1.p3.6.m6.1.1" xref="S5.SS2.SSS1.p3.6.m6.1.1.cmml"><mn id="S5.SS2.SSS1.p3.6.m6.1.1.2" xref="S5.SS2.SSS1.p3.6.m6.1.1.2.cmml">1</mn><mi id="S5.SS2.SSS1.p3.6.m6.1.1.3" xref="S5.SS2.SSS1.p3.6.m6.1.1.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS1.p3.6.m6.1.2.3.3.2.2" xref="S5.SS2.SSS1.p3.6.m6.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p3.6.m6.1b"><apply id="S5.SS2.SSS1.p3.6.m6.1.2.cmml" xref="S5.SS2.SSS1.p3.6.m6.1.2"><eq id="S5.SS2.SSS1.p3.6.m6.1.2.1.cmml" xref="S5.SS2.SSS1.p3.6.m6.1.2.1"></eq><ci id="S5.SS2.SSS1.p3.6.m6.1.2.2.cmml" xref="S5.SS2.SSS1.p3.6.m6.1.2.2">𝑝</ci><apply id="S5.SS2.SSS1.p3.6.m6.1.2.3.cmml" xref="S5.SS2.SSS1.p3.6.m6.1.2.3"><times id="S5.SS2.SSS1.p3.6.m6.1.2.3.1.cmml" xref="S5.SS2.SSS1.p3.6.m6.1.2.3.1"></times><ci id="S5.SS2.SSS1.p3.6.m6.1.2.3.2.cmml" xref="S5.SS2.SSS1.p3.6.m6.1.2.3.2">𝒪</ci><apply id="S5.SS2.SSS1.p3.6.m6.1.1.cmml" xref="S5.SS2.SSS1.p3.6.m6.1.2.3.3.2"><divide id="S5.SS2.SSS1.p3.6.m6.1.1.1.cmml" xref="S5.SS2.SSS1.p3.6.m6.1.2.3.3.2"></divide><cn id="S5.SS2.SSS1.p3.6.m6.1.1.2.cmml" type="integer" xref="S5.SS2.SSS1.p3.6.m6.1.1.2">1</cn><ci id="S5.SS2.SSS1.p3.6.m6.1.1.3.cmml" xref="S5.SS2.SSS1.p3.6.m6.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p3.6.m6.1c">p=\mathcal{O}\left(\frac{1}{k}\right)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p3.6.m6.1d">italic_p = caligraphic_O ( divide start_ARG 1 end_ARG start_ARG italic_k end_ARG )</annotation></semantics></math>, the input set <math alttext="S" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p3.7.m7.1"><semantics id="S5.SS2.SSS1.p3.7.m7.1a"><mi id="S5.SS2.SSS1.p3.7.m7.1.1" xref="S5.SS2.SSS1.p3.7.m7.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p3.7.m7.1b"><ci id="S5.SS2.SSS1.p3.7.m7.1.1.cmml" xref="S5.SS2.SSS1.p3.7.m7.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p3.7.m7.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p3.7.m7.1d">italic_S</annotation></semantics></math> can include all <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p3.8.m8.1"><semantics id="S5.SS2.SSS1.p3.8.m8.1a"><mi id="S5.SS2.SSS1.p3.8.m8.1.1" xref="S5.SS2.SSS1.p3.8.m8.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p3.8.m8.1b"><ci id="S5.SS2.SSS1.p3.8.m8.1.1.cmml" xref="S5.SS2.SSS1.p3.8.m8.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p3.8.m8.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p3.8.m8.1d">italic_k</annotation></semantics></math> color classes, i.e., <math alttext="S=\{V_{1},\dots,V_{k}\}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p3.9.m9.3"><semantics id="S5.SS2.SSS1.p3.9.m9.3a"><mrow id="S5.SS2.SSS1.p3.9.m9.3.3" xref="S5.SS2.SSS1.p3.9.m9.3.3.cmml"><mi id="S5.SS2.SSS1.p3.9.m9.3.3.4" xref="S5.SS2.SSS1.p3.9.m9.3.3.4.cmml">S</mi><mo id="S5.SS2.SSS1.p3.9.m9.3.3.3" xref="S5.SS2.SSS1.p3.9.m9.3.3.3.cmml">=</mo><mrow id="S5.SS2.SSS1.p3.9.m9.3.3.2.2" xref="S5.SS2.SSS1.p3.9.m9.3.3.2.3.cmml"><mo id="S5.SS2.SSS1.p3.9.m9.3.3.2.2.3" stretchy="false" xref="S5.SS2.SSS1.p3.9.m9.3.3.2.3.cmml">{</mo><msub id="S5.SS2.SSS1.p3.9.m9.2.2.1.1.1" xref="S5.SS2.SSS1.p3.9.m9.2.2.1.1.1.cmml"><mi id="S5.SS2.SSS1.p3.9.m9.2.2.1.1.1.2" xref="S5.SS2.SSS1.p3.9.m9.2.2.1.1.1.2.cmml">V</mi><mn id="S5.SS2.SSS1.p3.9.m9.2.2.1.1.1.3" xref="S5.SS2.SSS1.p3.9.m9.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS2.SSS1.p3.9.m9.3.3.2.2.4" xref="S5.SS2.SSS1.p3.9.m9.3.3.2.3.cmml">,</mo><mi id="S5.SS2.SSS1.p3.9.m9.1.1" mathvariant="normal" xref="S5.SS2.SSS1.p3.9.m9.1.1.cmml">…</mi><mo id="S5.SS2.SSS1.p3.9.m9.3.3.2.2.5" xref="S5.SS2.SSS1.p3.9.m9.3.3.2.3.cmml">,</mo><msub id="S5.SS2.SSS1.p3.9.m9.3.3.2.2.2" xref="S5.SS2.SSS1.p3.9.m9.3.3.2.2.2.cmml"><mi id="S5.SS2.SSS1.p3.9.m9.3.3.2.2.2.2" xref="S5.SS2.SSS1.p3.9.m9.3.3.2.2.2.2.cmml">V</mi><mi id="S5.SS2.SSS1.p3.9.m9.3.3.2.2.2.3" xref="S5.SS2.SSS1.p3.9.m9.3.3.2.2.2.3.cmml">k</mi></msub><mo id="S5.SS2.SSS1.p3.9.m9.3.3.2.2.6" stretchy="false" xref="S5.SS2.SSS1.p3.9.m9.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p3.9.m9.3b"><apply id="S5.SS2.SSS1.p3.9.m9.3.3.cmml" xref="S5.SS2.SSS1.p3.9.m9.3.3"><eq id="S5.SS2.SSS1.p3.9.m9.3.3.3.cmml" xref="S5.SS2.SSS1.p3.9.m9.3.3.3"></eq><ci id="S5.SS2.SSS1.p3.9.m9.3.3.4.cmml" xref="S5.SS2.SSS1.p3.9.m9.3.3.4">𝑆</ci><set id="S5.SS2.SSS1.p3.9.m9.3.3.2.3.cmml" xref="S5.SS2.SSS1.p3.9.m9.3.3.2.2"><apply id="S5.SS2.SSS1.p3.9.m9.2.2.1.1.1.cmml" xref="S5.SS2.SSS1.p3.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p3.9.m9.2.2.1.1.1.1.cmml" xref="S5.SS2.SSS1.p3.9.m9.2.2.1.1.1">subscript</csymbol><ci id="S5.SS2.SSS1.p3.9.m9.2.2.1.1.1.2.cmml" xref="S5.SS2.SSS1.p3.9.m9.2.2.1.1.1.2">𝑉</ci><cn id="S5.SS2.SSS1.p3.9.m9.2.2.1.1.1.3.cmml" type="integer" xref="S5.SS2.SSS1.p3.9.m9.2.2.1.1.1.3">1</cn></apply><ci id="S5.SS2.SSS1.p3.9.m9.1.1.cmml" xref="S5.SS2.SSS1.p3.9.m9.1.1">…</ci><apply id="S5.SS2.SSS1.p3.9.m9.3.3.2.2.2.cmml" xref="S5.SS2.SSS1.p3.9.m9.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.p3.9.m9.3.3.2.2.2.1.cmml" xref="S5.SS2.SSS1.p3.9.m9.3.3.2.2.2">subscript</csymbol><ci id="S5.SS2.SSS1.p3.9.m9.3.3.2.2.2.2.cmml" xref="S5.SS2.SSS1.p3.9.m9.3.3.2.2.2.2">𝑉</ci><ci id="S5.SS2.SSS1.p3.9.m9.3.3.2.2.2.3.cmml" xref="S5.SS2.SSS1.p3.9.m9.3.3.2.2.2.3">𝑘</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p3.9.m9.3c">S=\{V_{1},\dots,V_{k}\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p3.9.m9.3d">italic_S = { italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }</annotation></semantics></math>. In this case, with high probability, the algorithm finds nearly <math alttext="\frac{n}{k}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p3.10.m10.1"><semantics id="S5.SS2.SSS1.p3.10.m10.1a"><mfrac id="S5.SS2.SSS1.p3.10.m10.1.1" xref="S5.SS2.SSS1.p3.10.m10.1.1.cmml"><mi id="S5.SS2.SSS1.p3.10.m10.1.1.2" xref="S5.SS2.SSS1.p3.10.m10.1.1.2.cmml">n</mi><mi id="S5.SS2.SSS1.p3.10.m10.1.1.3" xref="S5.SS2.SSS1.p3.10.m10.1.1.3.cmml">k</mi></mfrac><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p3.10.m10.1b"><apply id="S5.SS2.SSS1.p3.10.m10.1.1.cmml" xref="S5.SS2.SSS1.p3.10.m10.1.1"><divide id="S5.SS2.SSS1.p3.10.m10.1.1.1.cmml" xref="S5.SS2.SSS1.p3.10.m10.1.1"></divide><ci id="S5.SS2.SSS1.p3.10.m10.1.1.2.cmml" xref="S5.SS2.SSS1.p3.10.m10.1.1.2">𝑛</ci><ci id="S5.SS2.SSS1.p3.10.m10.1.1.3.cmml" xref="S5.SS2.SSS1.p3.10.m10.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p3.10.m10.1c">\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p3.10.m10.1d">divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math> maximal cliques, each larger than <math alttext="k\sqrt{1-\varepsilon}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p3.11.m11.1"><semantics id="S5.SS2.SSS1.p3.11.m11.1a"><mrow id="S5.SS2.SSS1.p3.11.m11.1.1" xref="S5.SS2.SSS1.p3.11.m11.1.1.cmml"><mi id="S5.SS2.SSS1.p3.11.m11.1.1.2" xref="S5.SS2.SSS1.p3.11.m11.1.1.2.cmml">k</mi><mo id="S5.SS2.SSS1.p3.11.m11.1.1.1" xref="S5.SS2.SSS1.p3.11.m11.1.1.1.cmml">⁢</mo><msqrt id="S5.SS2.SSS1.p3.11.m11.1.1.3" xref="S5.SS2.SSS1.p3.11.m11.1.1.3.cmml"><mrow id="S5.SS2.SSS1.p3.11.m11.1.1.3.2" xref="S5.SS2.SSS1.p3.11.m11.1.1.3.2.cmml"><mn id="S5.SS2.SSS1.p3.11.m11.1.1.3.2.2" xref="S5.SS2.SSS1.p3.11.m11.1.1.3.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.p3.11.m11.1.1.3.2.1" xref="S5.SS2.SSS1.p3.11.m11.1.1.3.2.1.cmml">−</mo><mi id="S5.SS2.SSS1.p3.11.m11.1.1.3.2.3" xref="S5.SS2.SSS1.p3.11.m11.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p3.11.m11.1b"><apply id="S5.SS2.SSS1.p3.11.m11.1.1.cmml" xref="S5.SS2.SSS1.p3.11.m11.1.1"><times id="S5.SS2.SSS1.p3.11.m11.1.1.1.cmml" xref="S5.SS2.SSS1.p3.11.m11.1.1.1"></times><ci id="S5.SS2.SSS1.p3.11.m11.1.1.2.cmml" xref="S5.SS2.SSS1.p3.11.m11.1.1.2">𝑘</ci><apply id="S5.SS2.SSS1.p3.11.m11.1.1.3.cmml" xref="S5.SS2.SSS1.p3.11.m11.1.1.3"><root id="S5.SS2.SSS1.p3.11.m11.1.1.3a.cmml" xref="S5.SS2.SSS1.p3.11.m11.1.1.3"></root><apply id="S5.SS2.SSS1.p3.11.m11.1.1.3.2.cmml" xref="S5.SS2.SSS1.p3.11.m11.1.1.3.2"><minus id="S5.SS2.SSS1.p3.11.m11.1.1.3.2.1.cmml" xref="S5.SS2.SSS1.p3.11.m11.1.1.3.2.1"></minus><cn id="S5.SS2.SSS1.p3.11.m11.1.1.3.2.2.cmml" type="integer" xref="S5.SS2.SSS1.p3.11.m11.1.1.3.2.2">1</cn><ci id="S5.SS2.SSS1.p3.11.m11.1.1.3.2.3.cmml" xref="S5.SS2.SSS1.p3.11.m11.1.1.3.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p3.11.m11.1c">k\sqrt{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p3.11.m11.1d">italic_k square-root start_ARG 1 - italic_ε end_ARG</annotation></semantics></math>, making the size of these cliques nearly identical to the ideal clique size of <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p3.12.m12.1"><semantics id="S5.SS2.SSS1.p3.12.m12.1a"><mi id="S5.SS2.SSS1.p3.12.m12.1.1" xref="S5.SS2.SSS1.p3.12.m12.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p3.12.m12.1b"><ci id="S5.SS2.SSS1.p3.12.m12.1.1.cmml" xref="S5.SS2.SSS1.p3.12.m12.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p3.12.m12.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p3.12.m12.1d">italic_k</annotation></semantics></math> for small values of <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p3.13.m13.1"><semantics id="S5.SS2.SSS1.p3.13.m13.1a"><mi id="S5.SS2.SSS1.p3.13.m13.1.1" xref="S5.SS2.SSS1.p3.13.m13.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p3.13.m13.1b"><ci id="S5.SS2.SSS1.p3.13.m13.1.1.cmml" xref="S5.SS2.SSS1.p3.13.m13.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p3.13.m13.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p3.13.m13.1d">italic_ε</annotation></semantics></math>. Consequently, this results in a constant approximation of the social welfare.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S5.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem7.1.1.1">Lemma 5.7</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem7.p1"> <p class="ltx_p" id="S5.Thmtheorem7.p1.10"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem7.p1.10.10">Consider a random Turán graph <math alttext="G=(n,k,p)" class="ltx_Math" display="inline" id="S5.Thmtheorem7.p1.1.1.m1.3"><semantics id="S5.Thmtheorem7.p1.1.1.m1.3a"><mrow id="S5.Thmtheorem7.p1.1.1.m1.3.4" xref="S5.Thmtheorem7.p1.1.1.m1.3.4.cmml"><mi id="S5.Thmtheorem7.p1.1.1.m1.3.4.2" xref="S5.Thmtheorem7.p1.1.1.m1.3.4.2.cmml">G</mi><mo id="S5.Thmtheorem7.p1.1.1.m1.3.4.1" xref="S5.Thmtheorem7.p1.1.1.m1.3.4.1.cmml">=</mo><mrow id="S5.Thmtheorem7.p1.1.1.m1.3.4.3.2" xref="S5.Thmtheorem7.p1.1.1.m1.3.4.3.1.cmml"><mo id="S5.Thmtheorem7.p1.1.1.m1.3.4.3.2.1" stretchy="false" xref="S5.Thmtheorem7.p1.1.1.m1.3.4.3.1.cmml">(</mo><mi id="S5.Thmtheorem7.p1.1.1.m1.1.1" xref="S5.Thmtheorem7.p1.1.1.m1.1.1.cmml">n</mi><mo id="S5.Thmtheorem7.p1.1.1.m1.3.4.3.2.2" xref="S5.Thmtheorem7.p1.1.1.m1.3.4.3.1.cmml">,</mo><mi id="S5.Thmtheorem7.p1.1.1.m1.2.2" xref="S5.Thmtheorem7.p1.1.1.m1.2.2.cmml">k</mi><mo id="S5.Thmtheorem7.p1.1.1.m1.3.4.3.2.3" xref="S5.Thmtheorem7.p1.1.1.m1.3.4.3.1.cmml">,</mo><mi id="S5.Thmtheorem7.p1.1.1.m1.3.3" xref="S5.Thmtheorem7.p1.1.1.m1.3.3.cmml">p</mi><mo id="S5.Thmtheorem7.p1.1.1.m1.3.4.3.2.4" stretchy="false" xref="S5.Thmtheorem7.p1.1.1.m1.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem7.p1.1.1.m1.3b"><apply id="S5.Thmtheorem7.p1.1.1.m1.3.4.cmml" xref="S5.Thmtheorem7.p1.1.1.m1.3.4"><eq id="S5.Thmtheorem7.p1.1.1.m1.3.4.1.cmml" xref="S5.Thmtheorem7.p1.1.1.m1.3.4.1"></eq><ci id="S5.Thmtheorem7.p1.1.1.m1.3.4.2.cmml" xref="S5.Thmtheorem7.p1.1.1.m1.3.4.2">𝐺</ci><vector id="S5.Thmtheorem7.p1.1.1.m1.3.4.3.1.cmml" xref="S5.Thmtheorem7.p1.1.1.m1.3.4.3.2"><ci id="S5.Thmtheorem7.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem7.p1.1.1.m1.1.1">𝑛</ci><ci id="S5.Thmtheorem7.p1.1.1.m1.2.2.cmml" xref="S5.Thmtheorem7.p1.1.1.m1.2.2">𝑘</ci><ci id="S5.Thmtheorem7.p1.1.1.m1.3.3.cmml" xref="S5.Thmtheorem7.p1.1.1.m1.3.3">𝑝</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem7.p1.1.1.m1.3c">G=(n,k,p)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem7.p1.1.1.m1.3d">italic_G = ( italic_n , italic_k , italic_p )</annotation></semantics></math>, and a nonempty subset of color classes <math alttext="S\subseteq\{V_{1},\cdots,V_{k}\}" class="ltx_Math" display="inline" id="S5.Thmtheorem7.p1.2.2.m2.3"><semantics id="S5.Thmtheorem7.p1.2.2.m2.3a"><mrow id="S5.Thmtheorem7.p1.2.2.m2.3.3" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.cmml"><mi id="S5.Thmtheorem7.p1.2.2.m2.3.3.4" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.4.cmml">S</mi><mo id="S5.Thmtheorem7.p1.2.2.m2.3.3.3" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.3.cmml">⊆</mo><mrow id="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.2.3.cmml"><mo id="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.3" stretchy="false" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.2.3.cmml">{</mo><msub id="S5.Thmtheorem7.p1.2.2.m2.2.2.1.1.1" xref="S5.Thmtheorem7.p1.2.2.m2.2.2.1.1.1.cmml"><mi id="S5.Thmtheorem7.p1.2.2.m2.2.2.1.1.1.2" xref="S5.Thmtheorem7.p1.2.2.m2.2.2.1.1.1.2.cmml">V</mi><mn id="S5.Thmtheorem7.p1.2.2.m2.2.2.1.1.1.3" xref="S5.Thmtheorem7.p1.2.2.m2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.4" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.2.3.cmml">,</mo><mi id="S5.Thmtheorem7.p1.2.2.m2.1.1" mathvariant="normal" xref="S5.Thmtheorem7.p1.2.2.m2.1.1.cmml">⋯</mi><mo id="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.5" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.2.3.cmml">,</mo><msub id="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.2" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.2.cmml"><mi id="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.2.2" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.2.2.cmml">V</mi><mi id="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.2.3" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.2.3.cmml">k</mi></msub><mo id="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.6" stretchy="false" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem7.p1.2.2.m2.3b"><apply id="S5.Thmtheorem7.p1.2.2.m2.3.3.cmml" xref="S5.Thmtheorem7.p1.2.2.m2.3.3"><subset id="S5.Thmtheorem7.p1.2.2.m2.3.3.3.cmml" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.3"></subset><ci id="S5.Thmtheorem7.p1.2.2.m2.3.3.4.cmml" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.4">𝑆</ci><set id="S5.Thmtheorem7.p1.2.2.m2.3.3.2.3.cmml" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2"><apply id="S5.Thmtheorem7.p1.2.2.m2.2.2.1.1.1.cmml" xref="S5.Thmtheorem7.p1.2.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem7.p1.2.2.m2.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem7.p1.2.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem7.p1.2.2.m2.2.2.1.1.1.2.cmml" xref="S5.Thmtheorem7.p1.2.2.m2.2.2.1.1.1.2">𝑉</ci><cn id="S5.Thmtheorem7.p1.2.2.m2.2.2.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem7.p1.2.2.m2.2.2.1.1.1.3">1</cn></apply><ci id="S5.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem7.p1.2.2.m2.1.1">⋯</ci><apply id="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.2.cmml" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.2.1.cmml" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.2">subscript</csymbol><ci id="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.2.2.cmml" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.2.2">𝑉</ci><ci id="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.2.3.cmml" xref="S5.Thmtheorem7.p1.2.2.m2.3.3.2.2.2.3">𝑘</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem7.p1.2.2.m2.3c">S\subseteq\{V_{1},\cdots,V_{k}\}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem7.p1.2.2.m2.3d">italic_S ⊆ { italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }</annotation></semantics></math>, and <math alttext="p=\mathcal{O}(\frac{1}{k^{\prime}})" class="ltx_Math" display="inline" id="S5.Thmtheorem7.p1.3.3.m3.1"><semantics id="S5.Thmtheorem7.p1.3.3.m3.1a"><mrow id="S5.Thmtheorem7.p1.3.3.m3.1.2" xref="S5.Thmtheorem7.p1.3.3.m3.1.2.cmml"><mi id="S5.Thmtheorem7.p1.3.3.m3.1.2.2" xref="S5.Thmtheorem7.p1.3.3.m3.1.2.2.cmml">p</mi><mo id="S5.Thmtheorem7.p1.3.3.m3.1.2.1" xref="S5.Thmtheorem7.p1.3.3.m3.1.2.1.cmml">=</mo><mrow id="S5.Thmtheorem7.p1.3.3.m3.1.2.3" xref="S5.Thmtheorem7.p1.3.3.m3.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem7.p1.3.3.m3.1.2.3.2" xref="S5.Thmtheorem7.p1.3.3.m3.1.2.3.2.cmml">𝒪</mi><mo id="S5.Thmtheorem7.p1.3.3.m3.1.2.3.1" xref="S5.Thmtheorem7.p1.3.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem7.p1.3.3.m3.1.2.3.3.2" xref="S5.Thmtheorem7.p1.3.3.m3.1.1.cmml"><mo id="S5.Thmtheorem7.p1.3.3.m3.1.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem7.p1.3.3.m3.1.1.cmml">(</mo><mfrac id="S5.Thmtheorem7.p1.3.3.m3.1.1" xref="S5.Thmtheorem7.p1.3.3.m3.1.1.cmml"><mn id="S5.Thmtheorem7.p1.3.3.m3.1.1.2" xref="S5.Thmtheorem7.p1.3.3.m3.1.1.2.cmml">1</mn><msup id="S5.Thmtheorem7.p1.3.3.m3.1.1.3" xref="S5.Thmtheorem7.p1.3.3.m3.1.1.3.cmml"><mi id="S5.Thmtheorem7.p1.3.3.m3.1.1.3.2" xref="S5.Thmtheorem7.p1.3.3.m3.1.1.3.2.cmml">k</mi><mo id="S5.Thmtheorem7.p1.3.3.m3.1.1.3.3" xref="S5.Thmtheorem7.p1.3.3.m3.1.1.3.3.cmml">′</mo></msup></mfrac><mo id="S5.Thmtheorem7.p1.3.3.m3.1.2.3.3.2.2" stretchy="false" xref="S5.Thmtheorem7.p1.3.3.m3.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem7.p1.3.3.m3.1b"><apply id="S5.Thmtheorem7.p1.3.3.m3.1.2.cmml" xref="S5.Thmtheorem7.p1.3.3.m3.1.2"><eq id="S5.Thmtheorem7.p1.3.3.m3.1.2.1.cmml" xref="S5.Thmtheorem7.p1.3.3.m3.1.2.1"></eq><ci id="S5.Thmtheorem7.p1.3.3.m3.1.2.2.cmml" xref="S5.Thmtheorem7.p1.3.3.m3.1.2.2">𝑝</ci><apply id="S5.Thmtheorem7.p1.3.3.m3.1.2.3.cmml" xref="S5.Thmtheorem7.p1.3.3.m3.1.2.3"><times id="S5.Thmtheorem7.p1.3.3.m3.1.2.3.1.cmml" xref="S5.Thmtheorem7.p1.3.3.m3.1.2.3.1"></times><ci id="S5.Thmtheorem7.p1.3.3.m3.1.2.3.2.cmml" xref="S5.Thmtheorem7.p1.3.3.m3.1.2.3.2">𝒪</ci><apply id="S5.Thmtheorem7.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem7.p1.3.3.m3.1.2.3.3.2"><divide id="S5.Thmtheorem7.p1.3.3.m3.1.1.1.cmml" xref="S5.Thmtheorem7.p1.3.3.m3.1.2.3.3.2"></divide><cn id="S5.Thmtheorem7.p1.3.3.m3.1.1.2.cmml" type="integer" xref="S5.Thmtheorem7.p1.3.3.m3.1.1.2">1</cn><apply id="S5.Thmtheorem7.p1.3.3.m3.1.1.3.cmml" xref="S5.Thmtheorem7.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem7.p1.3.3.m3.1.1.3.1.cmml" xref="S5.Thmtheorem7.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="S5.Thmtheorem7.p1.3.3.m3.1.1.3.2.cmml" xref="S5.Thmtheorem7.p1.3.3.m3.1.1.3.2">𝑘</ci><ci id="S5.Thmtheorem7.p1.3.3.m3.1.1.3.3.cmml" xref="S5.Thmtheorem7.p1.3.3.m3.1.1.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem7.p1.3.3.m3.1c">p=\mathcal{O}(\frac{1}{k^{\prime}})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem7.p1.3.3.m3.1d">italic_p = caligraphic_O ( divide start_ARG 1 end_ARG start_ARG italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG )</annotation></semantics></math> for <math alttext="k^{\prime}=|S|" class="ltx_Math" display="inline" id="S5.Thmtheorem7.p1.4.4.m4.1"><semantics id="S5.Thmtheorem7.p1.4.4.m4.1a"><mrow id="S5.Thmtheorem7.p1.4.4.m4.1.2" xref="S5.Thmtheorem7.p1.4.4.m4.1.2.cmml"><msup id="S5.Thmtheorem7.p1.4.4.m4.1.2.2" xref="S5.Thmtheorem7.p1.4.4.m4.1.2.2.cmml"><mi id="S5.Thmtheorem7.p1.4.4.m4.1.2.2.2" xref="S5.Thmtheorem7.p1.4.4.m4.1.2.2.2.cmml">k</mi><mo id="S5.Thmtheorem7.p1.4.4.m4.1.2.2.3" xref="S5.Thmtheorem7.p1.4.4.m4.1.2.2.3.cmml">′</mo></msup><mo id="S5.Thmtheorem7.p1.4.4.m4.1.2.1" xref="S5.Thmtheorem7.p1.4.4.m4.1.2.1.cmml">=</mo><mrow id="S5.Thmtheorem7.p1.4.4.m4.1.2.3.2" xref="S5.Thmtheorem7.p1.4.4.m4.1.2.3.1.cmml"><mo id="S5.Thmtheorem7.p1.4.4.m4.1.2.3.2.1" stretchy="false" xref="S5.Thmtheorem7.p1.4.4.m4.1.2.3.1.1.cmml">|</mo><mi id="S5.Thmtheorem7.p1.4.4.m4.1.1" xref="S5.Thmtheorem7.p1.4.4.m4.1.1.cmml">S</mi><mo id="S5.Thmtheorem7.p1.4.4.m4.1.2.3.2.2" stretchy="false" xref="S5.Thmtheorem7.p1.4.4.m4.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem7.p1.4.4.m4.1b"><apply id="S5.Thmtheorem7.p1.4.4.m4.1.2.cmml" xref="S5.Thmtheorem7.p1.4.4.m4.1.2"><eq id="S5.Thmtheorem7.p1.4.4.m4.1.2.1.cmml" xref="S5.Thmtheorem7.p1.4.4.m4.1.2.1"></eq><apply id="S5.Thmtheorem7.p1.4.4.m4.1.2.2.cmml" xref="S5.Thmtheorem7.p1.4.4.m4.1.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem7.p1.4.4.m4.1.2.2.1.cmml" xref="S5.Thmtheorem7.p1.4.4.m4.1.2.2">superscript</csymbol><ci id="S5.Thmtheorem7.p1.4.4.m4.1.2.2.2.cmml" xref="S5.Thmtheorem7.p1.4.4.m4.1.2.2.2">𝑘</ci><ci id="S5.Thmtheorem7.p1.4.4.m4.1.2.2.3.cmml" xref="S5.Thmtheorem7.p1.4.4.m4.1.2.2.3">′</ci></apply><apply id="S5.Thmtheorem7.p1.4.4.m4.1.2.3.1.cmml" xref="S5.Thmtheorem7.p1.4.4.m4.1.2.3.2"><abs id="S5.Thmtheorem7.p1.4.4.m4.1.2.3.1.1.cmml" xref="S5.Thmtheorem7.p1.4.4.m4.1.2.3.2.1"></abs><ci id="S5.Thmtheorem7.p1.4.4.m4.1.1.cmml" xref="S5.Thmtheorem7.p1.4.4.m4.1.1">𝑆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem7.p1.4.4.m4.1c">k^{\prime}=|S|</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem7.p1.4.4.m4.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = | italic_S |</annotation></semantics></math>. For any fixed <math alttext="\varepsilon\in(0,1)" class="ltx_Math" display="inline" id="S5.Thmtheorem7.p1.5.5.m5.2"><semantics id="S5.Thmtheorem7.p1.5.5.m5.2a"><mrow id="S5.Thmtheorem7.p1.5.5.m5.2.3" xref="S5.Thmtheorem7.p1.5.5.m5.2.3.cmml"><mi id="S5.Thmtheorem7.p1.5.5.m5.2.3.2" xref="S5.Thmtheorem7.p1.5.5.m5.2.3.2.cmml">ε</mi><mo id="S5.Thmtheorem7.p1.5.5.m5.2.3.1" xref="S5.Thmtheorem7.p1.5.5.m5.2.3.1.cmml">∈</mo><mrow id="S5.Thmtheorem7.p1.5.5.m5.2.3.3.2" xref="S5.Thmtheorem7.p1.5.5.m5.2.3.3.1.cmml"><mo id="S5.Thmtheorem7.p1.5.5.m5.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem7.p1.5.5.m5.2.3.3.1.cmml">(</mo><mn id="S5.Thmtheorem7.p1.5.5.m5.1.1" xref="S5.Thmtheorem7.p1.5.5.m5.1.1.cmml">0</mn><mo id="S5.Thmtheorem7.p1.5.5.m5.2.3.3.2.2" xref="S5.Thmtheorem7.p1.5.5.m5.2.3.3.1.cmml">,</mo><mn id="S5.Thmtheorem7.p1.5.5.m5.2.2" xref="S5.Thmtheorem7.p1.5.5.m5.2.2.cmml">1</mn><mo id="S5.Thmtheorem7.p1.5.5.m5.2.3.3.2.3" stretchy="false" xref="S5.Thmtheorem7.p1.5.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem7.p1.5.5.m5.2b"><apply id="S5.Thmtheorem7.p1.5.5.m5.2.3.cmml" xref="S5.Thmtheorem7.p1.5.5.m5.2.3"><in id="S5.Thmtheorem7.p1.5.5.m5.2.3.1.cmml" xref="S5.Thmtheorem7.p1.5.5.m5.2.3.1"></in><ci id="S5.Thmtheorem7.p1.5.5.m5.2.3.2.cmml" xref="S5.Thmtheorem7.p1.5.5.m5.2.3.2">𝜀</ci><interval closure="open" id="S5.Thmtheorem7.p1.5.5.m5.2.3.3.1.cmml" xref="S5.Thmtheorem7.p1.5.5.m5.2.3.3.2"><cn id="S5.Thmtheorem7.p1.5.5.m5.1.1.cmml" type="integer" xref="S5.Thmtheorem7.p1.5.5.m5.1.1">0</cn><cn id="S5.Thmtheorem7.p1.5.5.m5.2.2.cmml" type="integer" xref="S5.Thmtheorem7.p1.5.5.m5.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem7.p1.5.5.m5.2c">\varepsilon\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem7.p1.5.5.m5.2d">italic_ε ∈ ( 0 , 1 )</annotation></semantics></math> and <math alttext="\alpha\in(0,1)" class="ltx_Math" display="inline" id="S5.Thmtheorem7.p1.6.6.m6.2"><semantics id="S5.Thmtheorem7.p1.6.6.m6.2a"><mrow id="S5.Thmtheorem7.p1.6.6.m6.2.3" xref="S5.Thmtheorem7.p1.6.6.m6.2.3.cmml"><mi id="S5.Thmtheorem7.p1.6.6.m6.2.3.2" xref="S5.Thmtheorem7.p1.6.6.m6.2.3.2.cmml">α</mi><mo id="S5.Thmtheorem7.p1.6.6.m6.2.3.1" xref="S5.Thmtheorem7.p1.6.6.m6.2.3.1.cmml">∈</mo><mrow id="S5.Thmtheorem7.p1.6.6.m6.2.3.3.2" xref="S5.Thmtheorem7.p1.6.6.m6.2.3.3.1.cmml"><mo id="S5.Thmtheorem7.p1.6.6.m6.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem7.p1.6.6.m6.2.3.3.1.cmml">(</mo><mn id="S5.Thmtheorem7.p1.6.6.m6.1.1" xref="S5.Thmtheorem7.p1.6.6.m6.1.1.cmml">0</mn><mo id="S5.Thmtheorem7.p1.6.6.m6.2.3.3.2.2" xref="S5.Thmtheorem7.p1.6.6.m6.2.3.3.1.cmml">,</mo><mn id="S5.Thmtheorem7.p1.6.6.m6.2.2" xref="S5.Thmtheorem7.p1.6.6.m6.2.2.cmml">1</mn><mo id="S5.Thmtheorem7.p1.6.6.m6.2.3.3.2.3" stretchy="false" xref="S5.Thmtheorem7.p1.6.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem7.p1.6.6.m6.2b"><apply id="S5.Thmtheorem7.p1.6.6.m6.2.3.cmml" xref="S5.Thmtheorem7.p1.6.6.m6.2.3"><in id="S5.Thmtheorem7.p1.6.6.m6.2.3.1.cmml" xref="S5.Thmtheorem7.p1.6.6.m6.2.3.1"></in><ci id="S5.Thmtheorem7.p1.6.6.m6.2.3.2.cmml" xref="S5.Thmtheorem7.p1.6.6.m6.2.3.2">𝛼</ci><interval closure="open" id="S5.Thmtheorem7.p1.6.6.m6.2.3.3.1.cmml" xref="S5.Thmtheorem7.p1.6.6.m6.2.3.3.2"><cn id="S5.Thmtheorem7.p1.6.6.m6.1.1.cmml" type="integer" xref="S5.Thmtheorem7.p1.6.6.m6.1.1">0</cn><cn id="S5.Thmtheorem7.p1.6.6.m6.2.2.cmml" type="integer" xref="S5.Thmtheorem7.p1.6.6.m6.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem7.p1.6.6.m6.2c">\alpha\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem7.p1.6.6.m6.2d">italic_α ∈ ( 0 , 1 )</annotation></semantics></math>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg1" title="In 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a> returns a partition <math alttext="\pi" class="ltx_Math" display="inline" id="S5.Thmtheorem7.p1.7.7.m7.1"><semantics id="S5.Thmtheorem7.p1.7.7.m7.1a"><mi id="S5.Thmtheorem7.p1.7.7.m7.1.1" xref="S5.Thmtheorem7.p1.7.7.m7.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem7.p1.7.7.m7.1b"><ci id="S5.Thmtheorem7.p1.7.7.m7.1.1.cmml" xref="S5.Thmtheorem7.p1.7.7.m7.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem7.p1.7.7.m7.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem7.p1.7.7.m7.1d">italic_π</annotation></semantics></math> with <math alttext="\alpha\frac{n}{k}" class="ltx_Math" display="inline" id="S5.Thmtheorem7.p1.8.8.m8.1"><semantics id="S5.Thmtheorem7.p1.8.8.m8.1a"><mrow id="S5.Thmtheorem7.p1.8.8.m8.1.1" xref="S5.Thmtheorem7.p1.8.8.m8.1.1.cmml"><mi id="S5.Thmtheorem7.p1.8.8.m8.1.1.2" xref="S5.Thmtheorem7.p1.8.8.m8.1.1.2.cmml">α</mi><mo id="S5.Thmtheorem7.p1.8.8.m8.1.1.1" xref="S5.Thmtheorem7.p1.8.8.m8.1.1.1.cmml">⁢</mo><mfrac id="S5.Thmtheorem7.p1.8.8.m8.1.1.3" xref="S5.Thmtheorem7.p1.8.8.m8.1.1.3.cmml"><mi id="S5.Thmtheorem7.p1.8.8.m8.1.1.3.2" xref="S5.Thmtheorem7.p1.8.8.m8.1.1.3.2.cmml">n</mi><mi id="S5.Thmtheorem7.p1.8.8.m8.1.1.3.3" xref="S5.Thmtheorem7.p1.8.8.m8.1.1.3.3.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem7.p1.8.8.m8.1b"><apply id="S5.Thmtheorem7.p1.8.8.m8.1.1.cmml" xref="S5.Thmtheorem7.p1.8.8.m8.1.1"><times id="S5.Thmtheorem7.p1.8.8.m8.1.1.1.cmml" xref="S5.Thmtheorem7.p1.8.8.m8.1.1.1"></times><ci id="S5.Thmtheorem7.p1.8.8.m8.1.1.2.cmml" xref="S5.Thmtheorem7.p1.8.8.m8.1.1.2">𝛼</ci><apply id="S5.Thmtheorem7.p1.8.8.m8.1.1.3.cmml" xref="S5.Thmtheorem7.p1.8.8.m8.1.1.3"><divide id="S5.Thmtheorem7.p1.8.8.m8.1.1.3.1.cmml" xref="S5.Thmtheorem7.p1.8.8.m8.1.1.3"></divide><ci id="S5.Thmtheorem7.p1.8.8.m8.1.1.3.2.cmml" xref="S5.Thmtheorem7.p1.8.8.m8.1.1.3.2">𝑛</ci><ci id="S5.Thmtheorem7.p1.8.8.m8.1.1.3.3.cmml" xref="S5.Thmtheorem7.p1.8.8.m8.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem7.p1.8.8.m8.1c">\alpha\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem7.p1.8.8.m8.1d">italic_α divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math> cliques of size at least <math alttext="k^{\prime}\sqrt{1-\varepsilon}" class="ltx_Math" display="inline" id="S5.Thmtheorem7.p1.9.9.m9.1"><semantics id="S5.Thmtheorem7.p1.9.9.m9.1a"><mrow id="S5.Thmtheorem7.p1.9.9.m9.1.1" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.cmml"><msup id="S5.Thmtheorem7.p1.9.9.m9.1.1.2" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.2.cmml"><mi id="S5.Thmtheorem7.p1.9.9.m9.1.1.2.2" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.2.2.cmml">k</mi><mo id="S5.Thmtheorem7.p1.9.9.m9.1.1.2.3" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.2.3.cmml">′</mo></msup><mo id="S5.Thmtheorem7.p1.9.9.m9.1.1.1" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.1.cmml">⁢</mo><msqrt id="S5.Thmtheorem7.p1.9.9.m9.1.1.3" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.3.cmml"><mrow id="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2.cmml"><mn id="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2.2" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2.2.cmml">1</mn><mo id="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2.1" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2.1.cmml">−</mo><mi id="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2.3" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem7.p1.9.9.m9.1b"><apply id="S5.Thmtheorem7.p1.9.9.m9.1.1.cmml" xref="S5.Thmtheorem7.p1.9.9.m9.1.1"><times id="S5.Thmtheorem7.p1.9.9.m9.1.1.1.cmml" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.1"></times><apply id="S5.Thmtheorem7.p1.9.9.m9.1.1.2.cmml" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.2"><csymbol cd="ambiguous" id="S5.Thmtheorem7.p1.9.9.m9.1.1.2.1.cmml" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.2">superscript</csymbol><ci id="S5.Thmtheorem7.p1.9.9.m9.1.1.2.2.cmml" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.2.2">𝑘</ci><ci id="S5.Thmtheorem7.p1.9.9.m9.1.1.2.3.cmml" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.2.3">′</ci></apply><apply id="S5.Thmtheorem7.p1.9.9.m9.1.1.3.cmml" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.3"><root id="S5.Thmtheorem7.p1.9.9.m9.1.1.3a.cmml" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.3"></root><apply id="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2.cmml" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2"><minus id="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2.1.cmml" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2.1"></minus><cn id="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2.2.cmml" type="integer" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2.2">1</cn><ci id="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2.3.cmml" xref="S5.Thmtheorem7.p1.9.9.m9.1.1.3.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem7.p1.9.9.m9.1c">k^{\prime}\sqrt{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem7.p1.9.9.m9.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_ε end_ARG</annotation></semantics></math> with probability <math alttext="1-e^{-\Theta\left(\frac{nk^{\prime}}{k}\right)}" class="ltx_Math" display="inline" id="S5.Thmtheorem7.p1.10.10.m10.1"><semantics id="S5.Thmtheorem7.p1.10.10.m10.1a"><mrow id="S5.Thmtheorem7.p1.10.10.m10.1.2" xref="S5.Thmtheorem7.p1.10.10.m10.1.2.cmml"><mn id="S5.Thmtheorem7.p1.10.10.m10.1.2.2" xref="S5.Thmtheorem7.p1.10.10.m10.1.2.2.cmml">1</mn><mo id="S5.Thmtheorem7.p1.10.10.m10.1.2.1" xref="S5.Thmtheorem7.p1.10.10.m10.1.2.1.cmml">−</mo><msup id="S5.Thmtheorem7.p1.10.10.m10.1.2.3" xref="S5.Thmtheorem7.p1.10.10.m10.1.2.3.cmml"><mi id="S5.Thmtheorem7.p1.10.10.m10.1.2.3.2" xref="S5.Thmtheorem7.p1.10.10.m10.1.2.3.2.cmml">e</mi><mrow id="S5.Thmtheorem7.p1.10.10.m10.1.1.1" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.cmml"><mo id="S5.Thmtheorem7.p1.10.10.m10.1.1.1a" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.cmml">−</mo><mrow id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3.cmml"><mi id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3.2" mathvariant="normal" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3.2.cmml">Θ</mi><mo id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3.1" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3.3.2" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.cmml"><mo id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3.3.2.1" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.cmml">(</mo><mfrac id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.cmml"><mrow id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.cmml"><mi id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.2" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.2.cmml">n</mi><mo id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.1" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.1.cmml">⁢</mo><msup id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.3" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.3.cmml"><mi id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.3.2" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.3.2.cmml">k</mi><mo id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.3.3" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.3.3.cmml">′</mo></msup></mrow><mi id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.3" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.3.cmml">k</mi></mfrac><mo id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3.3.2.2" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem7.p1.10.10.m10.1b"><apply id="S5.Thmtheorem7.p1.10.10.m10.1.2.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.2"><minus id="S5.Thmtheorem7.p1.10.10.m10.1.2.1.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.2.1"></minus><cn id="S5.Thmtheorem7.p1.10.10.m10.1.2.2.cmml" type="integer" xref="S5.Thmtheorem7.p1.10.10.m10.1.2.2">1</cn><apply id="S5.Thmtheorem7.p1.10.10.m10.1.2.3.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem7.p1.10.10.m10.1.2.3.1.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.2.3">superscript</csymbol><ci id="S5.Thmtheorem7.p1.10.10.m10.1.2.3.2.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.2.3.2">𝑒</ci><apply id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1"><minus id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.2.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1"></minus><apply id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3"><times id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3.1.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3.1"></times><ci id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3.2.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3.2">Θ</ci><apply id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3.3.2"><divide id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.1.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.3.3.2"></divide><apply id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2"><times id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.1.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.1"></times><ci id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.2.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.2">𝑛</ci><apply id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.3.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.3.1.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.3">superscript</csymbol><ci id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.3.2.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.3.2">𝑘</ci><ci id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.3.3.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.2.3.3">′</ci></apply></apply><ci id="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.3.cmml" xref="S5.Thmtheorem7.p1.10.10.m10.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem7.p1.10.10.m10.1c">1-e^{-\Theta\left(\frac{nk^{\prime}}{k}\right)}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem7.p1.10.10.m10.1d">1 - italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S5.SS2.SSS1.8"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.SS2.SSS1.1.p1"> <p class="ltx_p" id="S5.SS2.SSS1.1.p1.13">We prove that the size of the first <math alttext="\frac{n}{k}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.1.p1.1.m1.1"><semantics id="S5.SS2.SSS1.1.p1.1.m1.1a"><mfrac id="S5.SS2.SSS1.1.p1.1.m1.1.1" xref="S5.SS2.SSS1.1.p1.1.m1.1.1.cmml"><mi id="S5.SS2.SSS1.1.p1.1.m1.1.1.2" xref="S5.SS2.SSS1.1.p1.1.m1.1.1.2.cmml">n</mi><mi id="S5.SS2.SSS1.1.p1.1.m1.1.1.3" xref="S5.SS2.SSS1.1.p1.1.m1.1.1.3.cmml">k</mi></mfrac><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.1.p1.1.m1.1b"><apply id="S5.SS2.SSS1.1.p1.1.m1.1.1.cmml" xref="S5.SS2.SSS1.1.p1.1.m1.1.1"><divide id="S5.SS2.SSS1.1.p1.1.m1.1.1.1.cmml" xref="S5.SS2.SSS1.1.p1.1.m1.1.1"></divide><ci id="S5.SS2.SSS1.1.p1.1.m1.1.1.2.cmml" xref="S5.SS2.SSS1.1.p1.1.m1.1.1.2">𝑛</ci><ci id="S5.SS2.SSS1.1.p1.1.m1.1.1.3.cmml" xref="S5.SS2.SSS1.1.p1.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.1.p1.1.m1.1c">\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.1.p1.1.m1.1d">divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math> maximal cliques exceeds <math alttext="k^{\prime}(\sqrt{1-\varepsilon})" class="ltx_Math" display="inline" id="S5.SS2.SSS1.1.p1.2.m2.1"><semantics id="S5.SS2.SSS1.1.p1.2.m2.1a"><mrow id="S5.SS2.SSS1.1.p1.2.m2.1.2" xref="S5.SS2.SSS1.1.p1.2.m2.1.2.cmml"><msup id="S5.SS2.SSS1.1.p1.2.m2.1.2.2" xref="S5.SS2.SSS1.1.p1.2.m2.1.2.2.cmml"><mi id="S5.SS2.SSS1.1.p1.2.m2.1.2.2.2" xref="S5.SS2.SSS1.1.p1.2.m2.1.2.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.1.p1.2.m2.1.2.2.3" xref="S5.SS2.SSS1.1.p1.2.m2.1.2.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS1.1.p1.2.m2.1.2.1" xref="S5.SS2.SSS1.1.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S5.SS2.SSS1.1.p1.2.m2.1.2.3.2" xref="S5.SS2.SSS1.1.p1.2.m2.1.1.cmml"><mo id="S5.SS2.SSS1.1.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S5.SS2.SSS1.1.p1.2.m2.1.1.cmml">(</mo><msqrt id="S5.SS2.SSS1.1.p1.2.m2.1.1" xref="S5.SS2.SSS1.1.p1.2.m2.1.1.cmml"><mrow id="S5.SS2.SSS1.1.p1.2.m2.1.1.2" xref="S5.SS2.SSS1.1.p1.2.m2.1.1.2.cmml"><mn id="S5.SS2.SSS1.1.p1.2.m2.1.1.2.2" xref="S5.SS2.SSS1.1.p1.2.m2.1.1.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.1.p1.2.m2.1.1.2.1" xref="S5.SS2.SSS1.1.p1.2.m2.1.1.2.1.cmml">−</mo><mi id="S5.SS2.SSS1.1.p1.2.m2.1.1.2.3" xref="S5.SS2.SSS1.1.p1.2.m2.1.1.2.3.cmml">ε</mi></mrow></msqrt><mo id="S5.SS2.SSS1.1.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S5.SS2.SSS1.1.p1.2.m2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.1.p1.2.m2.1b"><apply id="S5.SS2.SSS1.1.p1.2.m2.1.2.cmml" xref="S5.SS2.SSS1.1.p1.2.m2.1.2"><times id="S5.SS2.SSS1.1.p1.2.m2.1.2.1.cmml" xref="S5.SS2.SSS1.1.p1.2.m2.1.2.1"></times><apply id="S5.SS2.SSS1.1.p1.2.m2.1.2.2.cmml" xref="S5.SS2.SSS1.1.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.1.p1.2.m2.1.2.2.1.cmml" xref="S5.SS2.SSS1.1.p1.2.m2.1.2.2">superscript</csymbol><ci id="S5.SS2.SSS1.1.p1.2.m2.1.2.2.2.cmml" xref="S5.SS2.SSS1.1.p1.2.m2.1.2.2.2">𝑘</ci><ci id="S5.SS2.SSS1.1.p1.2.m2.1.2.2.3.cmml" xref="S5.SS2.SSS1.1.p1.2.m2.1.2.2.3">′</ci></apply><apply id="S5.SS2.SSS1.1.p1.2.m2.1.1.cmml" xref="S5.SS2.SSS1.1.p1.2.m2.1.2.3.2"><root id="S5.SS2.SSS1.1.p1.2.m2.1.1a.cmml" xref="S5.SS2.SSS1.1.p1.2.m2.1.2.3.2"></root><apply id="S5.SS2.SSS1.1.p1.2.m2.1.1.2.cmml" xref="S5.SS2.SSS1.1.p1.2.m2.1.1.2"><minus id="S5.SS2.SSS1.1.p1.2.m2.1.1.2.1.cmml" xref="S5.SS2.SSS1.1.p1.2.m2.1.1.2.1"></minus><cn id="S5.SS2.SSS1.1.p1.2.m2.1.1.2.2.cmml" type="integer" xref="S5.SS2.SSS1.1.p1.2.m2.1.1.2.2">1</cn><ci id="S5.SS2.SSS1.1.p1.2.m2.1.1.2.3.cmml" xref="S5.SS2.SSS1.1.p1.2.m2.1.1.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.1.p1.2.m2.1c">k^{\prime}(\sqrt{1-\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.1.p1.2.m2.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( square-root start_ARG 1 - italic_ε end_ARG )</annotation></semantics></math> with high probability. Let <math alttext="C" class="ltx_Math" display="inline" id="S5.SS2.SSS1.1.p1.3.m3.1"><semantics id="S5.SS2.SSS1.1.p1.3.m3.1a"><mi id="S5.SS2.SSS1.1.p1.3.m3.1.1" xref="S5.SS2.SSS1.1.p1.3.m3.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.1.p1.3.m3.1b"><ci id="S5.SS2.SSS1.1.p1.3.m3.1.1.cmml" xref="S5.SS2.SSS1.1.p1.3.m3.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.1.p1.3.m3.1c">C</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.1.p1.3.m3.1d">italic_C</annotation></semantics></math> denote a clique that is formed during the <math alttext="i" class="ltx_Math" display="inline" id="S5.SS2.SSS1.1.p1.4.m4.1"><semantics id="S5.SS2.SSS1.1.p1.4.m4.1a"><mi id="S5.SS2.SSS1.1.p1.4.m4.1.1" xref="S5.SS2.SSS1.1.p1.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.1.p1.4.m4.1b"><ci id="S5.SS2.SSS1.1.p1.4.m4.1.1.cmml" xref="S5.SS2.SSS1.1.p1.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.1.p1.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.1.p1.4.m4.1d">italic_i</annotation></semantics></math>th iteration of the while loop, with a current size of <math alttext="t" class="ltx_Math" display="inline" id="S5.SS2.SSS1.1.p1.5.m5.1"><semantics id="S5.SS2.SSS1.1.p1.5.m5.1a"><mi id="S5.SS2.SSS1.1.p1.5.m5.1.1" xref="S5.SS2.SSS1.1.p1.5.m5.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.1.p1.5.m5.1b"><ci id="S5.SS2.SSS1.1.p1.5.m5.1.1.cmml" xref="S5.SS2.SSS1.1.p1.5.m5.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.1.p1.5.m5.1c">t</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.1.p1.5.m5.1d">italic_t</annotation></semantics></math>. A color class is said to be <em class="ltx_emph ltx_font_italic" id="S5.SS2.SSS1.1.p1.13.1">available</em> if no vertex from that class has been added to <math alttext="C" class="ltx_Math" display="inline" id="S5.SS2.SSS1.1.p1.6.m6.1"><semantics id="S5.SS2.SSS1.1.p1.6.m6.1a"><mi id="S5.SS2.SSS1.1.p1.6.m6.1.1" xref="S5.SS2.SSS1.1.p1.6.m6.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.1.p1.6.m6.1b"><ci id="S5.SS2.SSS1.1.p1.6.m6.1.1.cmml" xref="S5.SS2.SSS1.1.p1.6.m6.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.1.p1.6.m6.1c">C</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.1.p1.6.m6.1d">italic_C</annotation></semantics></math>, yet. The probability that <math alttext="C" class="ltx_Math" display="inline" id="S5.SS2.SSS1.1.p1.7.m7.1"><semantics id="S5.SS2.SSS1.1.p1.7.m7.1a"><mi id="S5.SS2.SSS1.1.p1.7.m7.1.1" xref="S5.SS2.SSS1.1.p1.7.m7.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.1.p1.7.m7.1b"><ci id="S5.SS2.SSS1.1.p1.7.m7.1.1.cmml" xref="S5.SS2.SSS1.1.p1.7.m7.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.1.p1.7.m7.1c">C</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.1.p1.7.m7.1d">italic_C</annotation></semantics></math> is maximal is <math alttext="\left(1-(1-p)^{t}\right)^{(k^{\prime}-t)\left(\frac{n}{k}-i\right)}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.1.p1.8.m8.3"><semantics id="S5.SS2.SSS1.1.p1.8.m8.3a"><msup id="S5.SS2.SSS1.1.p1.8.m8.3.3" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.cmml"><mrow id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.cmml"><mo id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.2" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.cmml">(</mo><mrow id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.cmml"><mn id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.3" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.3.cmml">1</mn><mo id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.2" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.2.cmml">−</mo><msup id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.cmml"><mrow id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.cmml"><mo id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.cmml"><mn id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.2" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.1" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.3" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.3" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.3.cmml">t</mi></msup></mrow><mo id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.3" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.cmml">)</mo></mrow><mrow id="S5.SS2.SSS1.1.p1.8.m8.2.2.2" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.cmml"><mrow id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.cmml"><mo id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.cmml"><msup id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.2" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.2.cmml"><mi id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.2.2" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.2.3" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.1" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.1.cmml">−</mo><mi id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.3" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.3.cmml">t</mi></mrow><mo id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.3" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.3.cmml">⁢</mo><mrow id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.cmml"><mo id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.2" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.cmml">(</mo><mrow id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.cmml"><mfrac id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.2" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.2.cmml"><mi id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.2.2" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.2.2.cmml">n</mi><mi id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.2.3" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.2.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.1" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.1.cmml">−</mo><mi id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.3" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.3.cmml">i</mi></mrow><mo id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.3" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.1.p1.8.m8.3b"><apply id="S5.SS2.SSS1.1.p1.8.m8.3.3.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.3.3"><csymbol cd="ambiguous" id="S5.SS2.SSS1.1.p1.8.m8.3.3.2.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.3.3">superscript</csymbol><apply id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1"><minus id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.2.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.2"></minus><cn id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.3.cmml" type="integer" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.3">1</cn><apply id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.2.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1">superscript</csymbol><apply id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1"><minus id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.1"></minus><cn id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.2">1</cn><ci id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.3.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.1.1.1.3">𝑝</ci></apply><ci id="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.3.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.3.3.1.1.1.1.3">𝑡</ci></apply></apply><apply id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2"><times id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.3.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.3"></times><apply id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1"><minus id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.1"></minus><apply id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.2.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.2.1.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.2">superscript</csymbol><ci id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.2.2.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.2.2">𝑘</ci><ci id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.2.3.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.2.3">′</ci></apply><ci id="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.3.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.1.1.1.1.1.1.3">𝑡</ci></apply><apply id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1"><minus id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.1.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.1"></minus><apply id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.2.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.2"><divide id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.2.1.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.2"></divide><ci id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.2.2.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.2.2">𝑛</ci><ci id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.2.3.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.2.3">𝑘</ci></apply><ci id="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.3.cmml" xref="S5.SS2.SSS1.1.p1.8.m8.2.2.2.2.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.1.p1.8.m8.3c">\left(1-(1-p)^{t}\right)^{(k^{\prime}-t)\left(\frac{n}{k}-i\right)}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.1.p1.8.m8.3d">( 1 - ( 1 - italic_p ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_t ) ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG - italic_i ) end_POSTSUPERSCRIPT</annotation></semantics></math>, as this represents the probability that none of the <math alttext="\frac{n}{k}-i" class="ltx_Math" display="inline" id="S5.SS2.SSS1.1.p1.9.m9.1"><semantics id="S5.SS2.SSS1.1.p1.9.m9.1a"><mrow id="S5.SS2.SSS1.1.p1.9.m9.1.1" xref="S5.SS2.SSS1.1.p1.9.m9.1.1.cmml"><mfrac id="S5.SS2.SSS1.1.p1.9.m9.1.1.2" xref="S5.SS2.SSS1.1.p1.9.m9.1.1.2.cmml"><mi id="S5.SS2.SSS1.1.p1.9.m9.1.1.2.2" xref="S5.SS2.SSS1.1.p1.9.m9.1.1.2.2.cmml">n</mi><mi id="S5.SS2.SSS1.1.p1.9.m9.1.1.2.3" xref="S5.SS2.SSS1.1.p1.9.m9.1.1.2.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS1.1.p1.9.m9.1.1.1" xref="S5.SS2.SSS1.1.p1.9.m9.1.1.1.cmml">−</mo><mi id="S5.SS2.SSS1.1.p1.9.m9.1.1.3" xref="S5.SS2.SSS1.1.p1.9.m9.1.1.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.1.p1.9.m9.1b"><apply id="S5.SS2.SSS1.1.p1.9.m9.1.1.cmml" xref="S5.SS2.SSS1.1.p1.9.m9.1.1"><minus id="S5.SS2.SSS1.1.p1.9.m9.1.1.1.cmml" xref="S5.SS2.SSS1.1.p1.9.m9.1.1.1"></minus><apply id="S5.SS2.SSS1.1.p1.9.m9.1.1.2.cmml" xref="S5.SS2.SSS1.1.p1.9.m9.1.1.2"><divide id="S5.SS2.SSS1.1.p1.9.m9.1.1.2.1.cmml" xref="S5.SS2.SSS1.1.p1.9.m9.1.1.2"></divide><ci id="S5.SS2.SSS1.1.p1.9.m9.1.1.2.2.cmml" xref="S5.SS2.SSS1.1.p1.9.m9.1.1.2.2">𝑛</ci><ci id="S5.SS2.SSS1.1.p1.9.m9.1.1.2.3.cmml" xref="S5.SS2.SSS1.1.p1.9.m9.1.1.2.3">𝑘</ci></apply><ci id="S5.SS2.SSS1.1.p1.9.m9.1.1.3.cmml" xref="S5.SS2.SSS1.1.p1.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.1.p1.9.m9.1c">\frac{n}{k}-i</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.1.p1.9.m9.1d">divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG - italic_i</annotation></semantics></math> remaining vertices in each of the <math alttext="k^{\prime}-t" class="ltx_Math" display="inline" id="S5.SS2.SSS1.1.p1.10.m10.1"><semantics id="S5.SS2.SSS1.1.p1.10.m10.1a"><mrow id="S5.SS2.SSS1.1.p1.10.m10.1.1" xref="S5.SS2.SSS1.1.p1.10.m10.1.1.cmml"><msup id="S5.SS2.SSS1.1.p1.10.m10.1.1.2" xref="S5.SS2.SSS1.1.p1.10.m10.1.1.2.cmml"><mi id="S5.SS2.SSS1.1.p1.10.m10.1.1.2.2" xref="S5.SS2.SSS1.1.p1.10.m10.1.1.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.1.p1.10.m10.1.1.2.3" xref="S5.SS2.SSS1.1.p1.10.m10.1.1.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS1.1.p1.10.m10.1.1.1" xref="S5.SS2.SSS1.1.p1.10.m10.1.1.1.cmml">−</mo><mi id="S5.SS2.SSS1.1.p1.10.m10.1.1.3" xref="S5.SS2.SSS1.1.p1.10.m10.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.1.p1.10.m10.1b"><apply id="S5.SS2.SSS1.1.p1.10.m10.1.1.cmml" xref="S5.SS2.SSS1.1.p1.10.m10.1.1"><minus id="S5.SS2.SSS1.1.p1.10.m10.1.1.1.cmml" xref="S5.SS2.SSS1.1.p1.10.m10.1.1.1"></minus><apply id="S5.SS2.SSS1.1.p1.10.m10.1.1.2.cmml" xref="S5.SS2.SSS1.1.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.1.p1.10.m10.1.1.2.1.cmml" xref="S5.SS2.SSS1.1.p1.10.m10.1.1.2">superscript</csymbol><ci id="S5.SS2.SSS1.1.p1.10.m10.1.1.2.2.cmml" xref="S5.SS2.SSS1.1.p1.10.m10.1.1.2.2">𝑘</ci><ci id="S5.SS2.SSS1.1.p1.10.m10.1.1.2.3.cmml" xref="S5.SS2.SSS1.1.p1.10.m10.1.1.2.3">′</ci></apply><ci id="S5.SS2.SSS1.1.p1.10.m10.1.1.3.cmml" xref="S5.SS2.SSS1.1.p1.10.m10.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.1.p1.10.m10.1c">k^{\prime}-t</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.1.p1.10.m10.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_t</annotation></semantics></math> available color classes can be added to <math alttext="C" class="ltx_Math" display="inline" id="S5.SS2.SSS1.1.p1.11.m11.1"><semantics id="S5.SS2.SSS1.1.p1.11.m11.1a"><mi id="S5.SS2.SSS1.1.p1.11.m11.1.1" xref="S5.SS2.SSS1.1.p1.11.m11.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.1.p1.11.m11.1b"><ci id="S5.SS2.SSS1.1.p1.11.m11.1.1.cmml" xref="S5.SS2.SSS1.1.p1.11.m11.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.1.p1.11.m11.1c">C</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.1.p1.11.m11.1d">italic_C</annotation></semantics></math>, due to having at least one edge of weight <math alttext="-n" class="ltx_Math" display="inline" id="S5.SS2.SSS1.1.p1.12.m12.1"><semantics id="S5.SS2.SSS1.1.p1.12.m12.1a"><mrow id="S5.SS2.SSS1.1.p1.12.m12.1.1" xref="S5.SS2.SSS1.1.p1.12.m12.1.1.cmml"><mo id="S5.SS2.SSS1.1.p1.12.m12.1.1a" xref="S5.SS2.SSS1.1.p1.12.m12.1.1.cmml">−</mo><mi id="S5.SS2.SSS1.1.p1.12.m12.1.1.2" xref="S5.SS2.SSS1.1.p1.12.m12.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.1.p1.12.m12.1b"><apply id="S5.SS2.SSS1.1.p1.12.m12.1.1.cmml" xref="S5.SS2.SSS1.1.p1.12.m12.1.1"><minus id="S5.SS2.SSS1.1.p1.12.m12.1.1.1.cmml" xref="S5.SS2.SSS1.1.p1.12.m12.1.1"></minus><ci id="S5.SS2.SSS1.1.p1.12.m12.1.1.2.cmml" xref="S5.SS2.SSS1.1.p1.12.m12.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.1.p1.12.m12.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.1.p1.12.m12.1d">- italic_n</annotation></semantics></math> with a vertex in <math alttext="C" class="ltx_Math" display="inline" id="S5.SS2.SSS1.1.p1.13.m13.1"><semantics id="S5.SS2.SSS1.1.p1.13.m13.1a"><mi id="S5.SS2.SSS1.1.p1.13.m13.1.1" xref="S5.SS2.SSS1.1.p1.13.m13.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.1.p1.13.m13.1b"><ci id="S5.SS2.SSS1.1.p1.13.m13.1.1.cmml" xref="S5.SS2.SSS1.1.p1.13.m13.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.1.p1.13.m13.1c">C</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.1.p1.13.m13.1d">italic_C</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS2.SSS1.2.p2"> <p class="ltx_p" id="S5.SS2.SSS1.2.p2.6">Let <math alttext="X" class="ltx_Math" display="inline" id="S5.SS2.SSS1.2.p2.1.m1.1"><semantics id="S5.SS2.SSS1.2.p2.1.m1.1a"><mi id="S5.SS2.SSS1.2.p2.1.m1.1.1" xref="S5.SS2.SSS1.2.p2.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.2.p2.1.m1.1b"><ci id="S5.SS2.SSS1.2.p2.1.m1.1.1.cmml" xref="S5.SS2.SSS1.2.p2.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.2.p2.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.2.p2.1.m1.1d">italic_X</annotation></semantics></math> denote the number of maximal cliques in the subgraph induced by the color classes in <math alttext="S" class="ltx_Math" display="inline" id="S5.SS2.SSS1.2.p2.2.m2.1"><semantics id="S5.SS2.SSS1.2.p2.2.m2.1a"><mi id="S5.SS2.SSS1.2.p2.2.m2.1.1" xref="S5.SS2.SSS1.2.p2.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.2.p2.2.m2.1b"><ci id="S5.SS2.SSS1.2.p2.2.m2.1.1.cmml" xref="S5.SS2.SSS1.2.p2.2.m2.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.2.p2.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.2.p2.2.m2.1d">italic_S</annotation></semantics></math>, where each clique has size at most <math alttext="t_{0}=k^{\prime}\sqrt{1-\varepsilon}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.2.p2.3.m3.1"><semantics id="S5.SS2.SSS1.2.p2.3.m3.1a"><mrow id="S5.SS2.SSS1.2.p2.3.m3.1.1" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.cmml"><msub id="S5.SS2.SSS1.2.p2.3.m3.1.1.2" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.2.cmml"><mi id="S5.SS2.SSS1.2.p2.3.m3.1.1.2.2" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.2.2.cmml">t</mi><mn id="S5.SS2.SSS1.2.p2.3.m3.1.1.2.3" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.2.3.cmml">0</mn></msub><mo id="S5.SS2.SSS1.2.p2.3.m3.1.1.1" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.1.cmml">=</mo><mrow id="S5.SS2.SSS1.2.p2.3.m3.1.1.3" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.cmml"><msup id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.2" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.2.cmml"><mi id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.2.2" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.2.3" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.1" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.1.cmml">⁢</mo><msqrt id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.cmml"><mrow id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2.cmml"><mn id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2.2" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2.1" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2.1.cmml">−</mo><mi id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2.3" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2.3.cmml">ε</mi></mrow></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.2.p2.3.m3.1b"><apply id="S5.SS2.SSS1.2.p2.3.m3.1.1.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1"><eq id="S5.SS2.SSS1.2.p2.3.m3.1.1.1.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.1"></eq><apply id="S5.SS2.SSS1.2.p2.3.m3.1.1.2.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.2.p2.3.m3.1.1.2.1.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.2">subscript</csymbol><ci id="S5.SS2.SSS1.2.p2.3.m3.1.1.2.2.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.2.2">𝑡</ci><cn id="S5.SS2.SSS1.2.p2.3.m3.1.1.2.3.cmml" type="integer" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.2.3">0</cn></apply><apply id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3"><times id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.1.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.1"></times><apply id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.2.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.2.1.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.2">superscript</csymbol><ci id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.2.2.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.2.2">𝑘</ci><ci id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.2.3.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.2.3">′</ci></apply><apply id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3"><root id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3a.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3"></root><apply id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2"><minus id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2.1.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2.1"></minus><cn id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2.2.cmml" type="integer" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2.2">1</cn><ci id="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2.3.cmml" xref="S5.SS2.SSS1.2.p2.3.m3.1.1.3.3.2.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.2.p2.3.m3.1c">t_{0}=k^{\prime}\sqrt{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.2.p2.3.m3.1d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_ε end_ARG</annotation></semantics></math>. As there are <math alttext="\binom{k^{\prime}}{t}(\frac{n}{k})^{t}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.2.p2.4.m4.3"><semantics id="S5.SS2.SSS1.2.p2.4.m4.3a"><mrow id="S5.SS2.SSS1.2.p2.4.m4.3.4" xref="S5.SS2.SSS1.2.p2.4.m4.3.4.cmml"><mrow id="S5.SS2.SSS1.2.p2.4.m4.2.2.4" xref="S5.SS2.SSS1.2.p2.4.m4.2.2.3.cmml"><mo id="S5.SS2.SSS1.2.p2.4.m4.2.2.4.1" xref="S5.SS2.SSS1.2.p2.4.m4.2.2.3.1.cmml">(</mo><mfrac id="S5.SS2.SSS1.2.p2.4.m4.2.2.2.2" linethickness="0pt" xref="S5.SS2.SSS1.2.p2.4.m4.2.2.3.cmml"><msup id="S5.SS2.SSS1.2.p2.4.m4.1.1.1.1.1.1" xref="S5.SS2.SSS1.2.p2.4.m4.1.1.1.1.1.1.cmml"><mi id="S5.SS2.SSS1.2.p2.4.m4.1.1.1.1.1.1.3" xref="S5.SS2.SSS1.2.p2.4.m4.1.1.1.1.1.1.3.cmml">k</mi><mo id="S5.SS2.SSS1.2.p2.4.m4.1.1.1.1.1.1.4" xref="S5.SS2.SSS1.2.p2.4.m4.1.1.1.1.1.1.4.cmml">′</mo></msup><mi id="S5.SS2.SSS1.2.p2.4.m4.2.2.2.2.2.1" xref="S5.SS2.SSS1.2.p2.4.m4.2.2.2.2.2.1.cmml">t</mi></mfrac><mo id="S5.SS2.SSS1.2.p2.4.m4.2.2.4.2" xref="S5.SS2.SSS1.2.p2.4.m4.2.2.3.1.cmml">)</mo></mrow><mo id="S5.SS2.SSS1.2.p2.4.m4.3.4.1" xref="S5.SS2.SSS1.2.p2.4.m4.3.4.1.cmml">⁢</mo><msup id="S5.SS2.SSS1.2.p2.4.m4.3.4.2" xref="S5.SS2.SSS1.2.p2.4.m4.3.4.2.cmml"><mrow id="S5.SS2.SSS1.2.p2.4.m4.3.4.2.2.2" xref="S5.SS2.SSS1.2.p2.4.m4.3.3.cmml"><mo id="S5.SS2.SSS1.2.p2.4.m4.3.4.2.2.2.1" stretchy="false" xref="S5.SS2.SSS1.2.p2.4.m4.3.3.cmml">(</mo><mfrac id="S5.SS2.SSS1.2.p2.4.m4.3.3" xref="S5.SS2.SSS1.2.p2.4.m4.3.3.cmml"><mi id="S5.SS2.SSS1.2.p2.4.m4.3.3.2" xref="S5.SS2.SSS1.2.p2.4.m4.3.3.2.cmml">n</mi><mi id="S5.SS2.SSS1.2.p2.4.m4.3.3.3" xref="S5.SS2.SSS1.2.p2.4.m4.3.3.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS1.2.p2.4.m4.3.4.2.2.2.2" stretchy="false" xref="S5.SS2.SSS1.2.p2.4.m4.3.3.cmml">)</mo></mrow><mi id="S5.SS2.SSS1.2.p2.4.m4.3.4.2.3" xref="S5.SS2.SSS1.2.p2.4.m4.3.4.2.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.2.p2.4.m4.3b"><apply id="S5.SS2.SSS1.2.p2.4.m4.3.4.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.3.4"><times id="S5.SS2.SSS1.2.p2.4.m4.3.4.1.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.3.4.1"></times><apply id="S5.SS2.SSS1.2.p2.4.m4.2.2.3.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.2.2.4"><csymbol cd="latexml" id="S5.SS2.SSS1.2.p2.4.m4.2.2.3.1.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.2.2.4.1">binomial</csymbol><apply id="S5.SS2.SSS1.2.p2.4.m4.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS1.2.p2.4.m4.1.1.1.1.1.1.2.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.1.1.1.1.1.1">superscript</csymbol><ci id="S5.SS2.SSS1.2.p2.4.m4.1.1.1.1.1.1.3.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.1.1.1.1.1.1.3">𝑘</ci><ci id="S5.SS2.SSS1.2.p2.4.m4.1.1.1.1.1.1.4.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.1.1.1.1.1.1.4">′</ci></apply><ci id="S5.SS2.SSS1.2.p2.4.m4.2.2.2.2.2.1.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.2.2.2.2.2.1">𝑡</ci></apply><apply id="S5.SS2.SSS1.2.p2.4.m4.3.4.2.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.3.4.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.2.p2.4.m4.3.4.2.1.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.3.4.2">superscript</csymbol><apply id="S5.SS2.SSS1.2.p2.4.m4.3.3.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.3.4.2.2.2"><divide id="S5.SS2.SSS1.2.p2.4.m4.3.3.1.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.3.4.2.2.2"></divide><ci id="S5.SS2.SSS1.2.p2.4.m4.3.3.2.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.3.3.2">𝑛</ci><ci id="S5.SS2.SSS1.2.p2.4.m4.3.3.3.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.3.3.3">𝑘</ci></apply><ci id="S5.SS2.SSS1.2.p2.4.m4.3.4.2.3.cmml" xref="S5.SS2.SSS1.2.p2.4.m4.3.4.2.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.2.p2.4.m4.3c">\binom{k^{\prime}}{t}(\frac{n}{k})^{t}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.2.p2.4.m4.3d">( FRACOP start_ARG italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_t end_ARG ) ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> cliques of size <math alttext="t" class="ltx_Math" display="inline" id="S5.SS2.SSS1.2.p2.5.m5.1"><semantics id="S5.SS2.SSS1.2.p2.5.m5.1a"><mi id="S5.SS2.SSS1.2.p2.5.m5.1.1" xref="S5.SS2.SSS1.2.p2.5.m5.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.2.p2.5.m5.1b"><ci id="S5.SS2.SSS1.2.p2.5.m5.1.1.cmml" xref="S5.SS2.SSS1.2.p2.5.m5.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.2.p2.5.m5.1c">t</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.2.p2.5.m5.1d">italic_t</annotation></semantics></math>, and since <math alttext="\binom{k^{\prime}}{t}(\frac{n}{k})^{t}\leq\binom{k}{t}(\frac{n}{k})^{t}\leq k^% {t}(\frac{n}{k})^{t}\leq n^{t}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.2.p2.6.m6.7"><semantics id="S5.SS2.SSS1.2.p2.6.m6.7a"><mrow id="S5.SS2.SSS1.2.p2.6.m6.7.8" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.cmml"><mrow id="S5.SS2.SSS1.2.p2.6.m6.7.8.2" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.2.cmml"><mrow id="S5.SS2.SSS1.2.p2.6.m6.2.2.4" xref="S5.SS2.SSS1.2.p2.6.m6.2.2.3.cmml"><mo id="S5.SS2.SSS1.2.p2.6.m6.2.2.4.1" xref="S5.SS2.SSS1.2.p2.6.m6.2.2.3.1.cmml">(</mo><mfrac id="S5.SS2.SSS1.2.p2.6.m6.2.2.2.2" linethickness="0pt" xref="S5.SS2.SSS1.2.p2.6.m6.2.2.3.cmml"><msup id="S5.SS2.SSS1.2.p2.6.m6.1.1.1.1.1.1" xref="S5.SS2.SSS1.2.p2.6.m6.1.1.1.1.1.1.cmml"><mi id="S5.SS2.SSS1.2.p2.6.m6.1.1.1.1.1.1.3" xref="S5.SS2.SSS1.2.p2.6.m6.1.1.1.1.1.1.3.cmml">k</mi><mo id="S5.SS2.SSS1.2.p2.6.m6.1.1.1.1.1.1.4" xref="S5.SS2.SSS1.2.p2.6.m6.1.1.1.1.1.1.4.cmml">′</mo></msup><mi id="S5.SS2.SSS1.2.p2.6.m6.2.2.2.2.2.1" xref="S5.SS2.SSS1.2.p2.6.m6.2.2.2.2.2.1.cmml">t</mi></mfrac><mo id="S5.SS2.SSS1.2.p2.6.m6.2.2.4.2" xref="S5.SS2.SSS1.2.p2.6.m6.2.2.3.1.cmml">)</mo></mrow><mo id="S5.SS2.SSS1.2.p2.6.m6.7.8.2.1" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.2.1.cmml">⁢</mo><msup id="S5.SS2.SSS1.2.p2.6.m6.7.8.2.2" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.2.2.cmml"><mrow id="S5.SS2.SSS1.2.p2.6.m6.7.8.2.2.2.2" xref="S5.SS2.SSS1.2.p2.6.m6.5.5.cmml"><mo id="S5.SS2.SSS1.2.p2.6.m6.7.8.2.2.2.2.1" stretchy="false" xref="S5.SS2.SSS1.2.p2.6.m6.5.5.cmml">(</mo><mfrac id="S5.SS2.SSS1.2.p2.6.m6.5.5" xref="S5.SS2.SSS1.2.p2.6.m6.5.5.cmml"><mi id="S5.SS2.SSS1.2.p2.6.m6.5.5.2" xref="S5.SS2.SSS1.2.p2.6.m6.5.5.2.cmml">n</mi><mi id="S5.SS2.SSS1.2.p2.6.m6.5.5.3" xref="S5.SS2.SSS1.2.p2.6.m6.5.5.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS1.2.p2.6.m6.7.8.2.2.2.2.2" stretchy="false" xref="S5.SS2.SSS1.2.p2.6.m6.5.5.cmml">)</mo></mrow><mi id="S5.SS2.SSS1.2.p2.6.m6.7.8.2.2.3" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.2.2.3.cmml">t</mi></msup></mrow><mo id="S5.SS2.SSS1.2.p2.6.m6.7.8.3" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.3.cmml">≤</mo><mrow id="S5.SS2.SSS1.2.p2.6.m6.7.8.4" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.4.cmml"><mrow id="S5.SS2.SSS1.2.p2.6.m6.4.4.4" xref="S5.SS2.SSS1.2.p2.6.m6.4.4.3.cmml"><mo id="S5.SS2.SSS1.2.p2.6.m6.4.4.4.1" xref="S5.SS2.SSS1.2.p2.6.m6.4.4.3.1.cmml">(</mo><mfrac id="S5.SS2.SSS1.2.p2.6.m6.4.4.2.2" linethickness="0pt" xref="S5.SS2.SSS1.2.p2.6.m6.4.4.3.cmml"><mi id="S5.SS2.SSS1.2.p2.6.m6.3.3.1.1.1.1" xref="S5.SS2.SSS1.2.p2.6.m6.3.3.1.1.1.1.cmml">k</mi><mi id="S5.SS2.SSS1.2.p2.6.m6.4.4.2.2.2.1" xref="S5.SS2.SSS1.2.p2.6.m6.4.4.2.2.2.1.cmml">t</mi></mfrac><mo id="S5.SS2.SSS1.2.p2.6.m6.4.4.4.2" xref="S5.SS2.SSS1.2.p2.6.m6.4.4.3.1.cmml">)</mo></mrow><mo id="S5.SS2.SSS1.2.p2.6.m6.7.8.4.1" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.4.1.cmml">⁢</mo><msup id="S5.SS2.SSS1.2.p2.6.m6.7.8.4.2" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.4.2.cmml"><mrow id="S5.SS2.SSS1.2.p2.6.m6.7.8.4.2.2.2" xref="S5.SS2.SSS1.2.p2.6.m6.6.6.cmml"><mo id="S5.SS2.SSS1.2.p2.6.m6.7.8.4.2.2.2.1" stretchy="false" xref="S5.SS2.SSS1.2.p2.6.m6.6.6.cmml">(</mo><mfrac id="S5.SS2.SSS1.2.p2.6.m6.6.6" xref="S5.SS2.SSS1.2.p2.6.m6.6.6.cmml"><mi id="S5.SS2.SSS1.2.p2.6.m6.6.6.2" xref="S5.SS2.SSS1.2.p2.6.m6.6.6.2.cmml">n</mi><mi id="S5.SS2.SSS1.2.p2.6.m6.6.6.3" xref="S5.SS2.SSS1.2.p2.6.m6.6.6.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS1.2.p2.6.m6.7.8.4.2.2.2.2" stretchy="false" xref="S5.SS2.SSS1.2.p2.6.m6.6.6.cmml">)</mo></mrow><mi id="S5.SS2.SSS1.2.p2.6.m6.7.8.4.2.3" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.4.2.3.cmml">t</mi></msup></mrow><mo id="S5.SS2.SSS1.2.p2.6.m6.7.8.5" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.5.cmml">≤</mo><mrow id="S5.SS2.SSS1.2.p2.6.m6.7.8.6" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.cmml"><msup id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.2" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.2.cmml"><mi id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.2.2" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.2.2.cmml">k</mi><mi id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.2.3" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.2.3.cmml">t</mi></msup><mo id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.1" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.1.cmml">⁢</mo><msup id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.3" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.3.cmml"><mrow id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.3.2.2" xref="S5.SS2.SSS1.2.p2.6.m6.7.7.cmml"><mo id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.3.2.2.1" stretchy="false" xref="S5.SS2.SSS1.2.p2.6.m6.7.7.cmml">(</mo><mfrac id="S5.SS2.SSS1.2.p2.6.m6.7.7" xref="S5.SS2.SSS1.2.p2.6.m6.7.7.cmml"><mi id="S5.SS2.SSS1.2.p2.6.m6.7.7.2" xref="S5.SS2.SSS1.2.p2.6.m6.7.7.2.cmml">n</mi><mi id="S5.SS2.SSS1.2.p2.6.m6.7.7.3" xref="S5.SS2.SSS1.2.p2.6.m6.7.7.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.3.2.2.2" stretchy="false" xref="S5.SS2.SSS1.2.p2.6.m6.7.7.cmml">)</mo></mrow><mi id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.3.3" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.3.3.cmml">t</mi></msup></mrow><mo id="S5.SS2.SSS1.2.p2.6.m6.7.8.7" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.7.cmml">≤</mo><msup id="S5.SS2.SSS1.2.p2.6.m6.7.8.8" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.8.cmml"><mi id="S5.SS2.SSS1.2.p2.6.m6.7.8.8.2" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.8.2.cmml">n</mi><mi id="S5.SS2.SSS1.2.p2.6.m6.7.8.8.3" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.8.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.2.p2.6.m6.7b"><apply id="S5.SS2.SSS1.2.p2.6.m6.7.8.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8"><and id="S5.SS2.SSS1.2.p2.6.m6.7.8a.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8"></and><apply id="S5.SS2.SSS1.2.p2.6.m6.7.8b.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8"><leq id="S5.SS2.SSS1.2.p2.6.m6.7.8.3.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.3"></leq><apply id="S5.SS2.SSS1.2.p2.6.m6.7.8.2.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.2"><times id="S5.SS2.SSS1.2.p2.6.m6.7.8.2.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.2.1"></times><apply id="S5.SS2.SSS1.2.p2.6.m6.2.2.3.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.2.2.4"><csymbol cd="latexml" id="S5.SS2.SSS1.2.p2.6.m6.2.2.3.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.2.2.4.1">binomial</csymbol><apply id="S5.SS2.SSS1.2.p2.6.m6.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS1.2.p2.6.m6.1.1.1.1.1.1.2.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.1.1.1.1.1.1">superscript</csymbol><ci id="S5.SS2.SSS1.2.p2.6.m6.1.1.1.1.1.1.3.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.1.1.1.1.1.1.3">𝑘</ci><ci id="S5.SS2.SSS1.2.p2.6.m6.1.1.1.1.1.1.4.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.1.1.1.1.1.1.4">′</ci></apply><ci id="S5.SS2.SSS1.2.p2.6.m6.2.2.2.2.2.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.2.2.2.2.2.1">𝑡</ci></apply><apply id="S5.SS2.SSS1.2.p2.6.m6.7.8.2.2.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.2.p2.6.m6.7.8.2.2.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.2.2">superscript</csymbol><apply id="S5.SS2.SSS1.2.p2.6.m6.5.5.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.2.2.2.2"><divide id="S5.SS2.SSS1.2.p2.6.m6.5.5.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.2.2.2.2"></divide><ci id="S5.SS2.SSS1.2.p2.6.m6.5.5.2.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.5.5.2">𝑛</ci><ci id="S5.SS2.SSS1.2.p2.6.m6.5.5.3.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.5.5.3">𝑘</ci></apply><ci id="S5.SS2.SSS1.2.p2.6.m6.7.8.2.2.3.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.2.2.3">𝑡</ci></apply></apply><apply id="S5.SS2.SSS1.2.p2.6.m6.7.8.4.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.4"><times id="S5.SS2.SSS1.2.p2.6.m6.7.8.4.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.4.1"></times><apply id="S5.SS2.SSS1.2.p2.6.m6.4.4.3.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.4.4.4"><csymbol cd="latexml" id="S5.SS2.SSS1.2.p2.6.m6.4.4.3.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.4.4.4.1">binomial</csymbol><ci id="S5.SS2.SSS1.2.p2.6.m6.3.3.1.1.1.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.3.3.1.1.1.1">𝑘</ci><ci id="S5.SS2.SSS1.2.p2.6.m6.4.4.2.2.2.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.4.4.2.2.2.1">𝑡</ci></apply><apply id="S5.SS2.SSS1.2.p2.6.m6.7.8.4.2.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.4.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.2.p2.6.m6.7.8.4.2.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.4.2">superscript</csymbol><apply id="S5.SS2.SSS1.2.p2.6.m6.6.6.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.4.2.2.2"><divide id="S5.SS2.SSS1.2.p2.6.m6.6.6.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.4.2.2.2"></divide><ci id="S5.SS2.SSS1.2.p2.6.m6.6.6.2.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.6.6.2">𝑛</ci><ci id="S5.SS2.SSS1.2.p2.6.m6.6.6.3.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.6.6.3">𝑘</ci></apply><ci id="S5.SS2.SSS1.2.p2.6.m6.7.8.4.2.3.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.4.2.3">𝑡</ci></apply></apply></apply><apply id="S5.SS2.SSS1.2.p2.6.m6.7.8c.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8"><leq id="S5.SS2.SSS1.2.p2.6.m6.7.8.5.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.5"></leq><share href="https://arxiv.org/html/2503.06017v1#S5.SS2.SSS1.2.p2.6.m6.7.8.4.cmml" id="S5.SS2.SSS1.2.p2.6.m6.7.8d.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8"></share><apply id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6"><times id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.1"></times><apply id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.2.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.2.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.2">superscript</csymbol><ci id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.2.2.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.2.2">𝑘</ci><ci id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.2.3.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.2.3">𝑡</ci></apply><apply id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.3.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.3"><csymbol cd="ambiguous" id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.3.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.3">superscript</csymbol><apply id="S5.SS2.SSS1.2.p2.6.m6.7.7.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.3.2.2"><divide id="S5.SS2.SSS1.2.p2.6.m6.7.7.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.3.2.2"></divide><ci id="S5.SS2.SSS1.2.p2.6.m6.7.7.2.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.7.2">𝑛</ci><ci id="S5.SS2.SSS1.2.p2.6.m6.7.7.3.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.7.3">𝑘</ci></apply><ci id="S5.SS2.SSS1.2.p2.6.m6.7.8.6.3.3.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.6.3.3">𝑡</ci></apply></apply></apply><apply id="S5.SS2.SSS1.2.p2.6.m6.7.8e.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8"><leq id="S5.SS2.SSS1.2.p2.6.m6.7.8.7.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.7"></leq><share href="https://arxiv.org/html/2503.06017v1#S5.SS2.SSS1.2.p2.6.m6.7.8.6.cmml" id="S5.SS2.SSS1.2.p2.6.m6.7.8f.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8"></share><apply id="S5.SS2.SSS1.2.p2.6.m6.7.8.8.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.8"><csymbol cd="ambiguous" id="S5.SS2.SSS1.2.p2.6.m6.7.8.8.1.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.8">superscript</csymbol><ci id="S5.SS2.SSS1.2.p2.6.m6.7.8.8.2.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.8.2">𝑛</ci><ci id="S5.SS2.SSS1.2.p2.6.m6.7.8.8.3.cmml" xref="S5.SS2.SSS1.2.p2.6.m6.7.8.8.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.2.p2.6.m6.7c">\binom{k^{\prime}}{t}(\frac{n}{k})^{t}\leq\binom{k}{t}(\frac{n}{k})^{t}\leq k^% {t}(\frac{n}{k})^{t}\leq n^{t}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.2.p2.6.m6.7d">( FRACOP start_ARG italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_t end_ARG ) ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ≤ ( FRACOP start_ARG italic_k end_ARG start_ARG italic_t end_ARG ) ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ≤ italic_k start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ≤ italic_n start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>, the following holds:</p> </div> <div class="ltx_para" id="S5.SS2.SSS1.3.p3"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx5"> <tbody id="S5.Ex22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mathbb{E}[X]\leq\sum_{t=1}^{t_{0}}n^{t}\left(1-(1-p)^{t}\right)^% {(k^{\prime}-t)\left(\frac{n}{k}-i\right)}" class="ltx_Math" display="inline" id="S5.Ex22.m1.4"><semantics id="S5.Ex22.m1.4a"><mrow id="S5.Ex22.m1.4.4" xref="S5.Ex22.m1.4.4.cmml"><mrow id="S5.Ex22.m1.4.4.3" xref="S5.Ex22.m1.4.4.3.cmml"><mi id="S5.Ex22.m1.4.4.3.2" xref="S5.Ex22.m1.4.4.3.2.cmml">𝔼</mi><mo id="S5.Ex22.m1.4.4.3.1" xref="S5.Ex22.m1.4.4.3.1.cmml">⁢</mo><mrow id="S5.Ex22.m1.4.4.3.3.2" xref="S5.Ex22.m1.4.4.3.3.1.cmml"><mo id="S5.Ex22.m1.4.4.3.3.2.1" stretchy="false" xref="S5.Ex22.m1.4.4.3.3.1.1.cmml">[</mo><mi id="S5.Ex22.m1.3.3" xref="S5.Ex22.m1.3.3.cmml">X</mi><mo id="S5.Ex22.m1.4.4.3.3.2.2" stretchy="false" xref="S5.Ex22.m1.4.4.3.3.1.1.cmml">]</mo></mrow></mrow><mo id="S5.Ex22.m1.4.4.2" xref="S5.Ex22.m1.4.4.2.cmml">≤</mo><mrow id="S5.Ex22.m1.4.4.1" xref="S5.Ex22.m1.4.4.1.cmml"><mstyle displaystyle="true" id="S5.Ex22.m1.4.4.1.2" xref="S5.Ex22.m1.4.4.1.2.cmml"><munderover id="S5.Ex22.m1.4.4.1.2a" xref="S5.Ex22.m1.4.4.1.2.cmml"><mo id="S5.Ex22.m1.4.4.1.2.2.2" movablelimits="false" xref="S5.Ex22.m1.4.4.1.2.2.2.cmml">∑</mo><mrow id="S5.Ex22.m1.4.4.1.2.2.3" xref="S5.Ex22.m1.4.4.1.2.2.3.cmml"><mi id="S5.Ex22.m1.4.4.1.2.2.3.2" xref="S5.Ex22.m1.4.4.1.2.2.3.2.cmml">t</mi><mo id="S5.Ex22.m1.4.4.1.2.2.3.1" xref="S5.Ex22.m1.4.4.1.2.2.3.1.cmml">=</mo><mn id="S5.Ex22.m1.4.4.1.2.2.3.3" xref="S5.Ex22.m1.4.4.1.2.2.3.3.cmml">1</mn></mrow><msub id="S5.Ex22.m1.4.4.1.2.3" xref="S5.Ex22.m1.4.4.1.2.3.cmml"><mi id="S5.Ex22.m1.4.4.1.2.3.2" xref="S5.Ex22.m1.4.4.1.2.3.2.cmml">t</mi><mn id="S5.Ex22.m1.4.4.1.2.3.3" xref="S5.Ex22.m1.4.4.1.2.3.3.cmml">0</mn></msub></munderover></mstyle><mrow id="S5.Ex22.m1.4.4.1.1" xref="S5.Ex22.m1.4.4.1.1.cmml"><msup id="S5.Ex22.m1.4.4.1.1.3" xref="S5.Ex22.m1.4.4.1.1.3.cmml"><mi id="S5.Ex22.m1.4.4.1.1.3.2" xref="S5.Ex22.m1.4.4.1.1.3.2.cmml">n</mi><mi id="S5.Ex22.m1.4.4.1.1.3.3" xref="S5.Ex22.m1.4.4.1.1.3.3.cmml">t</mi></msup><mo id="S5.Ex22.m1.4.4.1.1.2" xref="S5.Ex22.m1.4.4.1.1.2.cmml">⁢</mo><msup id="S5.Ex22.m1.4.4.1.1.1" xref="S5.Ex22.m1.4.4.1.1.1.cmml"><mrow id="S5.Ex22.m1.4.4.1.1.1.1.1" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.cmml"><mo id="S5.Ex22.m1.4.4.1.1.1.1.1.2" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex22.m1.4.4.1.1.1.1.1.1" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.cmml"><mn id="S5.Ex22.m1.4.4.1.1.1.1.1.1.3" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.3.cmml">1</mn><mo id="S5.Ex22.m1.4.4.1.1.1.1.1.1.2" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.2.cmml">−</mo><msup id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.cmml"><mrow id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.cmml"><mn id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.2" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.1" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.3" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.3" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.3.cmml">t</mi></msup></mrow><mo id="S5.Ex22.m1.4.4.1.1.1.1.1.3" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S5.Ex22.m1.2.2.2" xref="S5.Ex22.m1.2.2.2.cmml"><mrow id="S5.Ex22.m1.1.1.1.1.1" xref="S5.Ex22.m1.1.1.1.1.1.1.cmml"><mo id="S5.Ex22.m1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex22.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex22.m1.1.1.1.1.1.1" xref="S5.Ex22.m1.1.1.1.1.1.1.cmml"><msup id="S5.Ex22.m1.1.1.1.1.1.1.2" xref="S5.Ex22.m1.1.1.1.1.1.1.2.cmml"><mi id="S5.Ex22.m1.1.1.1.1.1.1.2.2" xref="S5.Ex22.m1.1.1.1.1.1.1.2.2.cmml">k</mi><mo id="S5.Ex22.m1.1.1.1.1.1.1.2.3" xref="S5.Ex22.m1.1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S5.Ex22.m1.1.1.1.1.1.1.1" xref="S5.Ex22.m1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S5.Ex22.m1.1.1.1.1.1.1.3" xref="S5.Ex22.m1.1.1.1.1.1.1.3.cmml">t</mi></mrow><mo id="S5.Ex22.m1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex22.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Ex22.m1.2.2.2.3" xref="S5.Ex22.m1.2.2.2.3.cmml">⁢</mo><mrow id="S5.Ex22.m1.2.2.2.2.1" xref="S5.Ex22.m1.2.2.2.2.1.1.cmml"><mo id="S5.Ex22.m1.2.2.2.2.1.2" xref="S5.Ex22.m1.2.2.2.2.1.1.cmml">(</mo><mrow id="S5.Ex22.m1.2.2.2.2.1.1" xref="S5.Ex22.m1.2.2.2.2.1.1.cmml"><mfrac id="S5.Ex22.m1.2.2.2.2.1.1.2" xref="S5.Ex22.m1.2.2.2.2.1.1.2.cmml"><mi id="S5.Ex22.m1.2.2.2.2.1.1.2.2" xref="S5.Ex22.m1.2.2.2.2.1.1.2.2.cmml">n</mi><mi id="S5.Ex22.m1.2.2.2.2.1.1.2.3" xref="S5.Ex22.m1.2.2.2.2.1.1.2.3.cmml">k</mi></mfrac><mo id="S5.Ex22.m1.2.2.2.2.1.1.1" xref="S5.Ex22.m1.2.2.2.2.1.1.1.cmml">−</mo><mi id="S5.Ex22.m1.2.2.2.2.1.1.3" xref="S5.Ex22.m1.2.2.2.2.1.1.3.cmml">i</mi></mrow><mo id="S5.Ex22.m1.2.2.2.2.1.3" xref="S5.Ex22.m1.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></msup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex22.m1.4b"><apply id="S5.Ex22.m1.4.4.cmml" xref="S5.Ex22.m1.4.4"><leq id="S5.Ex22.m1.4.4.2.cmml" xref="S5.Ex22.m1.4.4.2"></leq><apply id="S5.Ex22.m1.4.4.3.cmml" xref="S5.Ex22.m1.4.4.3"><times id="S5.Ex22.m1.4.4.3.1.cmml" xref="S5.Ex22.m1.4.4.3.1"></times><ci id="S5.Ex22.m1.4.4.3.2.cmml" xref="S5.Ex22.m1.4.4.3.2">𝔼</ci><apply id="S5.Ex22.m1.4.4.3.3.1.cmml" xref="S5.Ex22.m1.4.4.3.3.2"><csymbol cd="latexml" id="S5.Ex22.m1.4.4.3.3.1.1.cmml" xref="S5.Ex22.m1.4.4.3.3.2.1">delimited-[]</csymbol><ci id="S5.Ex22.m1.3.3.cmml" xref="S5.Ex22.m1.3.3">𝑋</ci></apply></apply><apply id="S5.Ex22.m1.4.4.1.cmml" xref="S5.Ex22.m1.4.4.1"><apply id="S5.Ex22.m1.4.4.1.2.cmml" xref="S5.Ex22.m1.4.4.1.2"><csymbol cd="ambiguous" id="S5.Ex22.m1.4.4.1.2.1.cmml" xref="S5.Ex22.m1.4.4.1.2">superscript</csymbol><apply id="S5.Ex22.m1.4.4.1.2.2.cmml" xref="S5.Ex22.m1.4.4.1.2"><csymbol cd="ambiguous" id="S5.Ex22.m1.4.4.1.2.2.1.cmml" xref="S5.Ex22.m1.4.4.1.2">subscript</csymbol><sum id="S5.Ex22.m1.4.4.1.2.2.2.cmml" xref="S5.Ex22.m1.4.4.1.2.2.2"></sum><apply id="S5.Ex22.m1.4.4.1.2.2.3.cmml" xref="S5.Ex22.m1.4.4.1.2.2.3"><eq id="S5.Ex22.m1.4.4.1.2.2.3.1.cmml" xref="S5.Ex22.m1.4.4.1.2.2.3.1"></eq><ci id="S5.Ex22.m1.4.4.1.2.2.3.2.cmml" xref="S5.Ex22.m1.4.4.1.2.2.3.2">𝑡</ci><cn id="S5.Ex22.m1.4.4.1.2.2.3.3.cmml" type="integer" xref="S5.Ex22.m1.4.4.1.2.2.3.3">1</cn></apply></apply><apply id="S5.Ex22.m1.4.4.1.2.3.cmml" xref="S5.Ex22.m1.4.4.1.2.3"><csymbol cd="ambiguous" id="S5.Ex22.m1.4.4.1.2.3.1.cmml" xref="S5.Ex22.m1.4.4.1.2.3">subscript</csymbol><ci id="S5.Ex22.m1.4.4.1.2.3.2.cmml" xref="S5.Ex22.m1.4.4.1.2.3.2">𝑡</ci><cn id="S5.Ex22.m1.4.4.1.2.3.3.cmml" type="integer" xref="S5.Ex22.m1.4.4.1.2.3.3">0</cn></apply></apply><apply id="S5.Ex22.m1.4.4.1.1.cmml" xref="S5.Ex22.m1.4.4.1.1"><times id="S5.Ex22.m1.4.4.1.1.2.cmml" xref="S5.Ex22.m1.4.4.1.1.2"></times><apply id="S5.Ex22.m1.4.4.1.1.3.cmml" xref="S5.Ex22.m1.4.4.1.1.3"><csymbol cd="ambiguous" id="S5.Ex22.m1.4.4.1.1.3.1.cmml" xref="S5.Ex22.m1.4.4.1.1.3">superscript</csymbol><ci id="S5.Ex22.m1.4.4.1.1.3.2.cmml" xref="S5.Ex22.m1.4.4.1.1.3.2">𝑛</ci><ci id="S5.Ex22.m1.4.4.1.1.3.3.cmml" xref="S5.Ex22.m1.4.4.1.1.3.3">𝑡</ci></apply><apply id="S5.Ex22.m1.4.4.1.1.1.cmml" xref="S5.Ex22.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S5.Ex22.m1.4.4.1.1.1.2.cmml" xref="S5.Ex22.m1.4.4.1.1.1">superscript</csymbol><apply id="S5.Ex22.m1.4.4.1.1.1.1.1.1.cmml" xref="S5.Ex22.m1.4.4.1.1.1.1.1"><minus id="S5.Ex22.m1.4.4.1.1.1.1.1.1.2.cmml" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.2"></minus><cn id="S5.Ex22.m1.4.4.1.1.1.1.1.1.3.cmml" type="integer" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.3">1</cn><apply id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.2.cmml" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1">superscript</csymbol><apply id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1"><minus id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.1"></minus><cn id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.2">1</cn><ci id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.1.1.1.3">𝑝</ci></apply><ci id="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex22.m1.4.4.1.1.1.1.1.1.1.3">𝑡</ci></apply></apply><apply id="S5.Ex22.m1.2.2.2.cmml" xref="S5.Ex22.m1.2.2.2"><times id="S5.Ex22.m1.2.2.2.3.cmml" xref="S5.Ex22.m1.2.2.2.3"></times><apply id="S5.Ex22.m1.1.1.1.1.1.1.cmml" xref="S5.Ex22.m1.1.1.1.1.1"><minus id="S5.Ex22.m1.1.1.1.1.1.1.1.cmml" xref="S5.Ex22.m1.1.1.1.1.1.1.1"></minus><apply id="S5.Ex22.m1.1.1.1.1.1.1.2.cmml" xref="S5.Ex22.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.Ex22.m1.1.1.1.1.1.1.2.1.cmml" xref="S5.Ex22.m1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S5.Ex22.m1.1.1.1.1.1.1.2.2.cmml" xref="S5.Ex22.m1.1.1.1.1.1.1.2.2">𝑘</ci><ci id="S5.Ex22.m1.1.1.1.1.1.1.2.3.cmml" xref="S5.Ex22.m1.1.1.1.1.1.1.2.3">′</ci></apply><ci id="S5.Ex22.m1.1.1.1.1.1.1.3.cmml" xref="S5.Ex22.m1.1.1.1.1.1.1.3">𝑡</ci></apply><apply id="S5.Ex22.m1.2.2.2.2.1.1.cmml" xref="S5.Ex22.m1.2.2.2.2.1"><minus id="S5.Ex22.m1.2.2.2.2.1.1.1.cmml" xref="S5.Ex22.m1.2.2.2.2.1.1.1"></minus><apply id="S5.Ex22.m1.2.2.2.2.1.1.2.cmml" xref="S5.Ex22.m1.2.2.2.2.1.1.2"><divide id="S5.Ex22.m1.2.2.2.2.1.1.2.1.cmml" xref="S5.Ex22.m1.2.2.2.2.1.1.2"></divide><ci id="S5.Ex22.m1.2.2.2.2.1.1.2.2.cmml" xref="S5.Ex22.m1.2.2.2.2.1.1.2.2">𝑛</ci><ci id="S5.Ex22.m1.2.2.2.2.1.1.2.3.cmml" xref="S5.Ex22.m1.2.2.2.2.1.1.2.3">𝑘</ci></apply><ci id="S5.Ex22.m1.2.2.2.2.1.1.3.cmml" xref="S5.Ex22.m1.2.2.2.2.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex22.m1.4c">\displaystyle\mathbb{E}[X]\leq\sum_{t=1}^{t_{0}}n^{t}\left(1-(1-p)^{t}\right)^% {(k^{\prime}-t)\left(\frac{n}{k}-i\right)}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex22.m1.4d">blackboard_E [ italic_X ] ≤ ∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( 1 - ( 1 - italic_p ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_t ) ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG - italic_i ) end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S5.Ex23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq t_{0}n^{t_{0}}\left(1-(1-p)^{t_{0}}\right)^{(k^{\prime}-t_{0% })\left(\frac{n}{k}-i\right)}" class="ltx_Math" display="inline" id="S5.Ex23.m1.3"><semantics id="S5.Ex23.m1.3a"><mrow id="S5.Ex23.m1.3.3" xref="S5.Ex23.m1.3.3.cmml"><mi id="S5.Ex23.m1.3.3.3" xref="S5.Ex23.m1.3.3.3.cmml"></mi><mo id="S5.Ex23.m1.3.3.2" xref="S5.Ex23.m1.3.3.2.cmml">≤</mo><mrow id="S5.Ex23.m1.3.3.1" xref="S5.Ex23.m1.3.3.1.cmml"><msub id="S5.Ex23.m1.3.3.1.3" xref="S5.Ex23.m1.3.3.1.3.cmml"><mi id="S5.Ex23.m1.3.3.1.3.2" xref="S5.Ex23.m1.3.3.1.3.2.cmml">t</mi><mn id="S5.Ex23.m1.3.3.1.3.3" xref="S5.Ex23.m1.3.3.1.3.3.cmml">0</mn></msub><mo id="S5.Ex23.m1.3.3.1.2" xref="S5.Ex23.m1.3.3.1.2.cmml">⁢</mo><msup id="S5.Ex23.m1.3.3.1.4" xref="S5.Ex23.m1.3.3.1.4.cmml"><mi id="S5.Ex23.m1.3.3.1.4.2" xref="S5.Ex23.m1.3.3.1.4.2.cmml">n</mi><msub id="S5.Ex23.m1.3.3.1.4.3" xref="S5.Ex23.m1.3.3.1.4.3.cmml"><mi id="S5.Ex23.m1.3.3.1.4.3.2" xref="S5.Ex23.m1.3.3.1.4.3.2.cmml">t</mi><mn id="S5.Ex23.m1.3.3.1.4.3.3" xref="S5.Ex23.m1.3.3.1.4.3.3.cmml">0</mn></msub></msup><mo id="S5.Ex23.m1.3.3.1.2a" xref="S5.Ex23.m1.3.3.1.2.cmml">⁢</mo><msup id="S5.Ex23.m1.3.3.1.1" xref="S5.Ex23.m1.3.3.1.1.cmml"><mrow id="S5.Ex23.m1.3.3.1.1.1.1" xref="S5.Ex23.m1.3.3.1.1.1.1.1.cmml"><mo id="S5.Ex23.m1.3.3.1.1.1.1.2" xref="S5.Ex23.m1.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex23.m1.3.3.1.1.1.1.1" xref="S5.Ex23.m1.3.3.1.1.1.1.1.cmml"><mn id="S5.Ex23.m1.3.3.1.1.1.1.1.3" xref="S5.Ex23.m1.3.3.1.1.1.1.1.3.cmml">1</mn><mo id="S5.Ex23.m1.3.3.1.1.1.1.1.2" xref="S5.Ex23.m1.3.3.1.1.1.1.1.2.cmml">−</mo><msup id="S5.Ex23.m1.3.3.1.1.1.1.1.1" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.cmml"><mrow id="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.cmml"><mn id="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.2" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.1" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.3" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><msub id="S5.Ex23.m1.3.3.1.1.1.1.1.1.3" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.3.cmml"><mi id="S5.Ex23.m1.3.3.1.1.1.1.1.1.3.2" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.3.2.cmml">t</mi><mn id="S5.Ex23.m1.3.3.1.1.1.1.1.1.3.3" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.3.3.cmml">0</mn></msub></msup></mrow><mo id="S5.Ex23.m1.3.3.1.1.1.1.3" xref="S5.Ex23.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S5.Ex23.m1.2.2.2" xref="S5.Ex23.m1.2.2.2.cmml"><mrow id="S5.Ex23.m1.1.1.1.1.1" xref="S5.Ex23.m1.1.1.1.1.1.1.cmml"><mo id="S5.Ex23.m1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex23.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex23.m1.1.1.1.1.1.1" xref="S5.Ex23.m1.1.1.1.1.1.1.cmml"><msup id="S5.Ex23.m1.1.1.1.1.1.1.2" xref="S5.Ex23.m1.1.1.1.1.1.1.2.cmml"><mi id="S5.Ex23.m1.1.1.1.1.1.1.2.2" xref="S5.Ex23.m1.1.1.1.1.1.1.2.2.cmml">k</mi><mo id="S5.Ex23.m1.1.1.1.1.1.1.2.3" xref="S5.Ex23.m1.1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S5.Ex23.m1.1.1.1.1.1.1.1" xref="S5.Ex23.m1.1.1.1.1.1.1.1.cmml">−</mo><msub id="S5.Ex23.m1.1.1.1.1.1.1.3" xref="S5.Ex23.m1.1.1.1.1.1.1.3.cmml"><mi id="S5.Ex23.m1.1.1.1.1.1.1.3.2" xref="S5.Ex23.m1.1.1.1.1.1.1.3.2.cmml">t</mi><mn id="S5.Ex23.m1.1.1.1.1.1.1.3.3" xref="S5.Ex23.m1.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S5.Ex23.m1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex23.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Ex23.m1.2.2.2.3" xref="S5.Ex23.m1.2.2.2.3.cmml">⁢</mo><mrow id="S5.Ex23.m1.2.2.2.2.1" xref="S5.Ex23.m1.2.2.2.2.1.1.cmml"><mo id="S5.Ex23.m1.2.2.2.2.1.2" xref="S5.Ex23.m1.2.2.2.2.1.1.cmml">(</mo><mrow id="S5.Ex23.m1.2.2.2.2.1.1" xref="S5.Ex23.m1.2.2.2.2.1.1.cmml"><mfrac id="S5.Ex23.m1.2.2.2.2.1.1.2" xref="S5.Ex23.m1.2.2.2.2.1.1.2.cmml"><mi id="S5.Ex23.m1.2.2.2.2.1.1.2.2" xref="S5.Ex23.m1.2.2.2.2.1.1.2.2.cmml">n</mi><mi id="S5.Ex23.m1.2.2.2.2.1.1.2.3" xref="S5.Ex23.m1.2.2.2.2.1.1.2.3.cmml">k</mi></mfrac><mo id="S5.Ex23.m1.2.2.2.2.1.1.1" xref="S5.Ex23.m1.2.2.2.2.1.1.1.cmml">−</mo><mi id="S5.Ex23.m1.2.2.2.2.1.1.3" xref="S5.Ex23.m1.2.2.2.2.1.1.3.cmml">i</mi></mrow><mo id="S5.Ex23.m1.2.2.2.2.1.3" xref="S5.Ex23.m1.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex23.m1.3b"><apply id="S5.Ex23.m1.3.3.cmml" xref="S5.Ex23.m1.3.3"><leq id="S5.Ex23.m1.3.3.2.cmml" xref="S5.Ex23.m1.3.3.2"></leq><csymbol cd="latexml" id="S5.Ex23.m1.3.3.3.cmml" xref="S5.Ex23.m1.3.3.3">absent</csymbol><apply id="S5.Ex23.m1.3.3.1.cmml" xref="S5.Ex23.m1.3.3.1"><times id="S5.Ex23.m1.3.3.1.2.cmml" xref="S5.Ex23.m1.3.3.1.2"></times><apply id="S5.Ex23.m1.3.3.1.3.cmml" xref="S5.Ex23.m1.3.3.1.3"><csymbol cd="ambiguous" id="S5.Ex23.m1.3.3.1.3.1.cmml" xref="S5.Ex23.m1.3.3.1.3">subscript</csymbol><ci id="S5.Ex23.m1.3.3.1.3.2.cmml" xref="S5.Ex23.m1.3.3.1.3.2">𝑡</ci><cn id="S5.Ex23.m1.3.3.1.3.3.cmml" type="integer" xref="S5.Ex23.m1.3.3.1.3.3">0</cn></apply><apply id="S5.Ex23.m1.3.3.1.4.cmml" xref="S5.Ex23.m1.3.3.1.4"><csymbol cd="ambiguous" id="S5.Ex23.m1.3.3.1.4.1.cmml" xref="S5.Ex23.m1.3.3.1.4">superscript</csymbol><ci id="S5.Ex23.m1.3.3.1.4.2.cmml" xref="S5.Ex23.m1.3.3.1.4.2">𝑛</ci><apply id="S5.Ex23.m1.3.3.1.4.3.cmml" xref="S5.Ex23.m1.3.3.1.4.3"><csymbol cd="ambiguous" id="S5.Ex23.m1.3.3.1.4.3.1.cmml" xref="S5.Ex23.m1.3.3.1.4.3">subscript</csymbol><ci id="S5.Ex23.m1.3.3.1.4.3.2.cmml" xref="S5.Ex23.m1.3.3.1.4.3.2">𝑡</ci><cn id="S5.Ex23.m1.3.3.1.4.3.3.cmml" type="integer" xref="S5.Ex23.m1.3.3.1.4.3.3">0</cn></apply></apply><apply id="S5.Ex23.m1.3.3.1.1.cmml" xref="S5.Ex23.m1.3.3.1.1"><csymbol cd="ambiguous" id="S5.Ex23.m1.3.3.1.1.2.cmml" xref="S5.Ex23.m1.3.3.1.1">superscript</csymbol><apply id="S5.Ex23.m1.3.3.1.1.1.1.1.cmml" xref="S5.Ex23.m1.3.3.1.1.1.1"><minus id="S5.Ex23.m1.3.3.1.1.1.1.1.2.cmml" xref="S5.Ex23.m1.3.3.1.1.1.1.1.2"></minus><cn id="S5.Ex23.m1.3.3.1.1.1.1.1.3.cmml" type="integer" xref="S5.Ex23.m1.3.3.1.1.1.1.1.3">1</cn><apply id="S5.Ex23.m1.3.3.1.1.1.1.1.1.cmml" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Ex23.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1">superscript</csymbol><apply id="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1"><minus id="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.1"></minus><cn id="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.2">1</cn><ci id="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="S5.Ex23.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Ex23.m1.3.3.1.1.1.1.1.1.3.1.cmml" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.3">subscript</csymbol><ci id="S5.Ex23.m1.3.3.1.1.1.1.1.1.3.2.cmml" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.3.2">𝑡</ci><cn id="S5.Ex23.m1.3.3.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.Ex23.m1.3.3.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S5.Ex23.m1.2.2.2.cmml" xref="S5.Ex23.m1.2.2.2"><times id="S5.Ex23.m1.2.2.2.3.cmml" xref="S5.Ex23.m1.2.2.2.3"></times><apply id="S5.Ex23.m1.1.1.1.1.1.1.cmml" xref="S5.Ex23.m1.1.1.1.1.1"><minus id="S5.Ex23.m1.1.1.1.1.1.1.1.cmml" xref="S5.Ex23.m1.1.1.1.1.1.1.1"></minus><apply id="S5.Ex23.m1.1.1.1.1.1.1.2.cmml" xref="S5.Ex23.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.Ex23.m1.1.1.1.1.1.1.2.1.cmml" xref="S5.Ex23.m1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S5.Ex23.m1.1.1.1.1.1.1.2.2.cmml" xref="S5.Ex23.m1.1.1.1.1.1.1.2.2">𝑘</ci><ci id="S5.Ex23.m1.1.1.1.1.1.1.2.3.cmml" xref="S5.Ex23.m1.1.1.1.1.1.1.2.3">′</ci></apply><apply id="S5.Ex23.m1.1.1.1.1.1.1.3.cmml" xref="S5.Ex23.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Ex23.m1.1.1.1.1.1.1.3.1.cmml" xref="S5.Ex23.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S5.Ex23.m1.1.1.1.1.1.1.3.2.cmml" xref="S5.Ex23.m1.1.1.1.1.1.1.3.2">𝑡</ci><cn id="S5.Ex23.m1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.Ex23.m1.1.1.1.1.1.1.3.3">0</cn></apply></apply><apply id="S5.Ex23.m1.2.2.2.2.1.1.cmml" xref="S5.Ex23.m1.2.2.2.2.1"><minus id="S5.Ex23.m1.2.2.2.2.1.1.1.cmml" xref="S5.Ex23.m1.2.2.2.2.1.1.1"></minus><apply id="S5.Ex23.m1.2.2.2.2.1.1.2.cmml" xref="S5.Ex23.m1.2.2.2.2.1.1.2"><divide id="S5.Ex23.m1.2.2.2.2.1.1.2.1.cmml" xref="S5.Ex23.m1.2.2.2.2.1.1.2"></divide><ci id="S5.Ex23.m1.2.2.2.2.1.1.2.2.cmml" xref="S5.Ex23.m1.2.2.2.2.1.1.2.2">𝑛</ci><ci id="S5.Ex23.m1.2.2.2.2.1.1.2.3.cmml" xref="S5.Ex23.m1.2.2.2.2.1.1.2.3">𝑘</ci></apply><ci id="S5.Ex23.m1.2.2.2.2.1.1.3.cmml" xref="S5.Ex23.m1.2.2.2.2.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex23.m1.3c">\displaystyle\leq t_{0}n^{t_{0}}\left(1-(1-p)^{t_{0}}\right)^{(k^{\prime}-t_{0% })\left(\frac{n}{k}-i\right)}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex23.m1.3d">≤ italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( 1 - ( 1 - italic_p ) start_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG - italic_i ) end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S5.Ex24"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=t_{0}n^{t_{0}}(1-(1-p)^{t_{0}})^{n(\frac{k^{\prime}-t_{0}}{k}-% \frac{(k^{\prime}-t_{0})i}{n})}" class="ltx_Math" display="inline" id="S5.Ex24.m1.3"><semantics id="S5.Ex24.m1.3a"><mrow id="S5.Ex24.m1.3.3" xref="S5.Ex24.m1.3.3.cmml"><mi id="S5.Ex24.m1.3.3.3" xref="S5.Ex24.m1.3.3.3.cmml"></mi><mo id="S5.Ex24.m1.3.3.2" xref="S5.Ex24.m1.3.3.2.cmml">=</mo><mrow id="S5.Ex24.m1.3.3.1" xref="S5.Ex24.m1.3.3.1.cmml"><msub id="S5.Ex24.m1.3.3.1.3" xref="S5.Ex24.m1.3.3.1.3.cmml"><mi id="S5.Ex24.m1.3.3.1.3.2" xref="S5.Ex24.m1.3.3.1.3.2.cmml">t</mi><mn id="S5.Ex24.m1.3.3.1.3.3" xref="S5.Ex24.m1.3.3.1.3.3.cmml">0</mn></msub><mo id="S5.Ex24.m1.3.3.1.2" xref="S5.Ex24.m1.3.3.1.2.cmml">⁢</mo><msup id="S5.Ex24.m1.3.3.1.4" xref="S5.Ex24.m1.3.3.1.4.cmml"><mi id="S5.Ex24.m1.3.3.1.4.2" xref="S5.Ex24.m1.3.3.1.4.2.cmml">n</mi><msub id="S5.Ex24.m1.3.3.1.4.3" xref="S5.Ex24.m1.3.3.1.4.3.cmml"><mi id="S5.Ex24.m1.3.3.1.4.3.2" xref="S5.Ex24.m1.3.3.1.4.3.2.cmml">t</mi><mn id="S5.Ex24.m1.3.3.1.4.3.3" xref="S5.Ex24.m1.3.3.1.4.3.3.cmml">0</mn></msub></msup><mo id="S5.Ex24.m1.3.3.1.2a" xref="S5.Ex24.m1.3.3.1.2.cmml">⁢</mo><msup id="S5.Ex24.m1.3.3.1.1" xref="S5.Ex24.m1.3.3.1.1.cmml"><mrow id="S5.Ex24.m1.3.3.1.1.1.1" xref="S5.Ex24.m1.3.3.1.1.1.1.1.cmml"><mo id="S5.Ex24.m1.3.3.1.1.1.1.2" stretchy="false" xref="S5.Ex24.m1.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex24.m1.3.3.1.1.1.1.1" xref="S5.Ex24.m1.3.3.1.1.1.1.1.cmml"><mn id="S5.Ex24.m1.3.3.1.1.1.1.1.3" xref="S5.Ex24.m1.3.3.1.1.1.1.1.3.cmml">1</mn><mo id="S5.Ex24.m1.3.3.1.1.1.1.1.2" xref="S5.Ex24.m1.3.3.1.1.1.1.1.2.cmml">−</mo><msup id="S5.Ex24.m1.3.3.1.1.1.1.1.1" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.cmml"><mrow id="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.cmml"><mn id="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.2" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.1" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.3" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><msub id="S5.Ex24.m1.3.3.1.1.1.1.1.1.3" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.3.cmml"><mi id="S5.Ex24.m1.3.3.1.1.1.1.1.1.3.2" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.3.2.cmml">t</mi><mn id="S5.Ex24.m1.3.3.1.1.1.1.1.1.3.3" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.3.3.cmml">0</mn></msub></msup></mrow><mo id="S5.Ex24.m1.3.3.1.1.1.1.3" stretchy="false" xref="S5.Ex24.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S5.Ex24.m1.2.2.2" xref="S5.Ex24.m1.2.2.2.cmml"><mi id="S5.Ex24.m1.2.2.2.4" xref="S5.Ex24.m1.2.2.2.4.cmml">n</mi><mo id="S5.Ex24.m1.2.2.2.3" xref="S5.Ex24.m1.2.2.2.3.cmml">⁢</mo><mrow id="S5.Ex24.m1.2.2.2.2.1" xref="S5.Ex24.m1.2.2.2.2.1.1.cmml"><mo id="S5.Ex24.m1.2.2.2.2.1.2" stretchy="false" xref="S5.Ex24.m1.2.2.2.2.1.1.cmml">(</mo><mrow id="S5.Ex24.m1.2.2.2.2.1.1" xref="S5.Ex24.m1.2.2.2.2.1.1.cmml"><mfrac id="S5.Ex24.m1.2.2.2.2.1.1.2" xref="S5.Ex24.m1.2.2.2.2.1.1.2.cmml"><mrow id="S5.Ex24.m1.2.2.2.2.1.1.2.2" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.cmml"><msup id="S5.Ex24.m1.2.2.2.2.1.1.2.2.2" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.2.cmml"><mi id="S5.Ex24.m1.2.2.2.2.1.1.2.2.2.2" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.2.2.cmml">k</mi><mo id="S5.Ex24.m1.2.2.2.2.1.1.2.2.2.3" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.2.3.cmml">′</mo></msup><mo id="S5.Ex24.m1.2.2.2.2.1.1.2.2.1" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.1.cmml">−</mo><msub id="S5.Ex24.m1.2.2.2.2.1.1.2.2.3" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.3.cmml"><mi id="S5.Ex24.m1.2.2.2.2.1.1.2.2.3.2" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.3.2.cmml">t</mi><mn id="S5.Ex24.m1.2.2.2.2.1.1.2.2.3.3" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.3.3.cmml">0</mn></msub></mrow><mi id="S5.Ex24.m1.2.2.2.2.1.1.2.3" xref="S5.Ex24.m1.2.2.2.2.1.1.2.3.cmml">k</mi></mfrac><mo id="S5.Ex24.m1.2.2.2.2.1.1.1" xref="S5.Ex24.m1.2.2.2.2.1.1.1.cmml">−</mo><mfrac id="S5.Ex24.m1.1.1.1.1" xref="S5.Ex24.m1.1.1.1.1.cmml"><mrow id="S5.Ex24.m1.1.1.1.1.1" xref="S5.Ex24.m1.1.1.1.1.1.cmml"><mrow id="S5.Ex24.m1.1.1.1.1.1.1.1" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex24.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex24.m1.1.1.1.1.1.1.1.1" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.cmml"><msup id="S5.Ex24.m1.1.1.1.1.1.1.1.1.2" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S5.Ex24.m1.1.1.1.1.1.1.1.1.2.2" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.2.2.cmml">k</mi><mo id="S5.Ex24.m1.1.1.1.1.1.1.1.1.2.3" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S5.Ex24.m1.1.1.1.1.1.1.1.1.1" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.1.cmml">−</mo><msub id="S5.Ex24.m1.1.1.1.1.1.1.1.1.3" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S5.Ex24.m1.1.1.1.1.1.1.1.1.3.2" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.3.2.cmml">t</mi><mn id="S5.Ex24.m1.1.1.1.1.1.1.1.1.3.3" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S5.Ex24.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Ex24.m1.1.1.1.1.1.2" xref="S5.Ex24.m1.1.1.1.1.1.2.cmml">⁢</mo><mi id="S5.Ex24.m1.1.1.1.1.1.3" xref="S5.Ex24.m1.1.1.1.1.1.3.cmml">i</mi></mrow><mi id="S5.Ex24.m1.1.1.1.1.3" xref="S5.Ex24.m1.1.1.1.1.3.cmml">n</mi></mfrac></mrow><mo id="S5.Ex24.m1.2.2.2.2.1.3" stretchy="false" xref="S5.Ex24.m1.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex24.m1.3b"><apply id="S5.Ex24.m1.3.3.cmml" xref="S5.Ex24.m1.3.3"><eq id="S5.Ex24.m1.3.3.2.cmml" xref="S5.Ex24.m1.3.3.2"></eq><csymbol cd="latexml" id="S5.Ex24.m1.3.3.3.cmml" xref="S5.Ex24.m1.3.3.3">absent</csymbol><apply id="S5.Ex24.m1.3.3.1.cmml" xref="S5.Ex24.m1.3.3.1"><times id="S5.Ex24.m1.3.3.1.2.cmml" xref="S5.Ex24.m1.3.3.1.2"></times><apply id="S5.Ex24.m1.3.3.1.3.cmml" xref="S5.Ex24.m1.3.3.1.3"><csymbol cd="ambiguous" id="S5.Ex24.m1.3.3.1.3.1.cmml" xref="S5.Ex24.m1.3.3.1.3">subscript</csymbol><ci id="S5.Ex24.m1.3.3.1.3.2.cmml" xref="S5.Ex24.m1.3.3.1.3.2">𝑡</ci><cn id="S5.Ex24.m1.3.3.1.3.3.cmml" type="integer" xref="S5.Ex24.m1.3.3.1.3.3">0</cn></apply><apply id="S5.Ex24.m1.3.3.1.4.cmml" xref="S5.Ex24.m1.3.3.1.4"><csymbol cd="ambiguous" id="S5.Ex24.m1.3.3.1.4.1.cmml" xref="S5.Ex24.m1.3.3.1.4">superscript</csymbol><ci id="S5.Ex24.m1.3.3.1.4.2.cmml" xref="S5.Ex24.m1.3.3.1.4.2">𝑛</ci><apply id="S5.Ex24.m1.3.3.1.4.3.cmml" xref="S5.Ex24.m1.3.3.1.4.3"><csymbol cd="ambiguous" id="S5.Ex24.m1.3.3.1.4.3.1.cmml" xref="S5.Ex24.m1.3.3.1.4.3">subscript</csymbol><ci id="S5.Ex24.m1.3.3.1.4.3.2.cmml" xref="S5.Ex24.m1.3.3.1.4.3.2">𝑡</ci><cn id="S5.Ex24.m1.3.3.1.4.3.3.cmml" type="integer" xref="S5.Ex24.m1.3.3.1.4.3.3">0</cn></apply></apply><apply id="S5.Ex24.m1.3.3.1.1.cmml" xref="S5.Ex24.m1.3.3.1.1"><csymbol cd="ambiguous" id="S5.Ex24.m1.3.3.1.1.2.cmml" xref="S5.Ex24.m1.3.3.1.1">superscript</csymbol><apply id="S5.Ex24.m1.3.3.1.1.1.1.1.cmml" xref="S5.Ex24.m1.3.3.1.1.1.1"><minus id="S5.Ex24.m1.3.3.1.1.1.1.1.2.cmml" xref="S5.Ex24.m1.3.3.1.1.1.1.1.2"></minus><cn id="S5.Ex24.m1.3.3.1.1.1.1.1.3.cmml" type="integer" xref="S5.Ex24.m1.3.3.1.1.1.1.1.3">1</cn><apply id="S5.Ex24.m1.3.3.1.1.1.1.1.1.cmml" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Ex24.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1">superscript</csymbol><apply id="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1"><minus id="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.1"></minus><cn id="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.2">1</cn><ci id="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="S5.Ex24.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Ex24.m1.3.3.1.1.1.1.1.1.3.1.cmml" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.3">subscript</csymbol><ci id="S5.Ex24.m1.3.3.1.1.1.1.1.1.3.2.cmml" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.3.2">𝑡</ci><cn id="S5.Ex24.m1.3.3.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.Ex24.m1.3.3.1.1.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S5.Ex24.m1.2.2.2.cmml" xref="S5.Ex24.m1.2.2.2"><times id="S5.Ex24.m1.2.2.2.3.cmml" xref="S5.Ex24.m1.2.2.2.3"></times><ci id="S5.Ex24.m1.2.2.2.4.cmml" xref="S5.Ex24.m1.2.2.2.4">𝑛</ci><apply id="S5.Ex24.m1.2.2.2.2.1.1.cmml" xref="S5.Ex24.m1.2.2.2.2.1"><minus id="S5.Ex24.m1.2.2.2.2.1.1.1.cmml" xref="S5.Ex24.m1.2.2.2.2.1.1.1"></minus><apply id="S5.Ex24.m1.2.2.2.2.1.1.2.cmml" xref="S5.Ex24.m1.2.2.2.2.1.1.2"><divide id="S5.Ex24.m1.2.2.2.2.1.1.2.1.cmml" xref="S5.Ex24.m1.2.2.2.2.1.1.2"></divide><apply id="S5.Ex24.m1.2.2.2.2.1.1.2.2.cmml" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2"><minus id="S5.Ex24.m1.2.2.2.2.1.1.2.2.1.cmml" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.1"></minus><apply id="S5.Ex24.m1.2.2.2.2.1.1.2.2.2.cmml" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.2"><csymbol cd="ambiguous" id="S5.Ex24.m1.2.2.2.2.1.1.2.2.2.1.cmml" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.2">superscript</csymbol><ci id="S5.Ex24.m1.2.2.2.2.1.1.2.2.2.2.cmml" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.2.2">𝑘</ci><ci id="S5.Ex24.m1.2.2.2.2.1.1.2.2.2.3.cmml" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.2.3">′</ci></apply><apply id="S5.Ex24.m1.2.2.2.2.1.1.2.2.3.cmml" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.3"><csymbol cd="ambiguous" id="S5.Ex24.m1.2.2.2.2.1.1.2.2.3.1.cmml" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.3">subscript</csymbol><ci id="S5.Ex24.m1.2.2.2.2.1.1.2.2.3.2.cmml" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.3.2">𝑡</ci><cn id="S5.Ex24.m1.2.2.2.2.1.1.2.2.3.3.cmml" type="integer" xref="S5.Ex24.m1.2.2.2.2.1.1.2.2.3.3">0</cn></apply></apply><ci id="S5.Ex24.m1.2.2.2.2.1.1.2.3.cmml" xref="S5.Ex24.m1.2.2.2.2.1.1.2.3">𝑘</ci></apply><apply id="S5.Ex24.m1.1.1.1.1.cmml" xref="S5.Ex24.m1.1.1.1.1"><divide id="S5.Ex24.m1.1.1.1.1.2.cmml" xref="S5.Ex24.m1.1.1.1.1"></divide><apply id="S5.Ex24.m1.1.1.1.1.1.cmml" xref="S5.Ex24.m1.1.1.1.1.1"><times id="S5.Ex24.m1.1.1.1.1.1.2.cmml" xref="S5.Ex24.m1.1.1.1.1.1.2"></times><apply id="S5.Ex24.m1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex24.m1.1.1.1.1.1.1.1"><minus id="S5.Ex24.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.1"></minus><apply id="S5.Ex24.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.Ex24.m1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S5.Ex24.m1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.2.2">𝑘</ci><ci id="S5.Ex24.m1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.2.3">′</ci></apply><apply id="S5.Ex24.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Ex24.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S5.Ex24.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.3.2">𝑡</ci><cn id="S5.Ex24.m1.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.Ex24.m1.1.1.1.1.1.1.1.1.3.3">0</cn></apply></apply><ci id="S5.Ex24.m1.1.1.1.1.1.3.cmml" xref="S5.Ex24.m1.1.1.1.1.1.3">𝑖</ci></apply><ci id="S5.Ex24.m1.1.1.1.1.3.cmml" xref="S5.Ex24.m1.1.1.1.1.3">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex24.m1.3c">\displaystyle=t_{0}n^{t_{0}}(1-(1-p)^{t_{0}})^{n(\frac{k^{\prime}-t_{0}}{k}-% \frac{(k^{\prime}-t_{0})i}{n})}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex24.m1.3d">= italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( 1 - ( 1 - italic_p ) start_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n ( divide start_ARG italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_k end_ARG - divide start_ARG ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_i end_ARG start_ARG italic_n end_ARG ) end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S5.Ex25"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq t_{0}n^{t_{0}}e^{-(1-p)^{t_{0}}\left[n(\frac{k^{\prime}-t_{0% }}{k}-\frac{(k^{\prime}-t_{0})i}{n})\right]}" class="ltx_Math" display="inline" id="S5.Ex25.m1.3"><semantics id="S5.Ex25.m1.3a"><mrow id="S5.Ex25.m1.3.4" xref="S5.Ex25.m1.3.4.cmml"><mi id="S5.Ex25.m1.3.4.2" xref="S5.Ex25.m1.3.4.2.cmml"></mi><mo id="S5.Ex25.m1.3.4.1" xref="S5.Ex25.m1.3.4.1.cmml">≤</mo><mrow id="S5.Ex25.m1.3.4.3" xref="S5.Ex25.m1.3.4.3.cmml"><msub id="S5.Ex25.m1.3.4.3.2" xref="S5.Ex25.m1.3.4.3.2.cmml"><mi id="S5.Ex25.m1.3.4.3.2.2" xref="S5.Ex25.m1.3.4.3.2.2.cmml">t</mi><mn id="S5.Ex25.m1.3.4.3.2.3" xref="S5.Ex25.m1.3.4.3.2.3.cmml">0</mn></msub><mo id="S5.Ex25.m1.3.4.3.1" xref="S5.Ex25.m1.3.4.3.1.cmml">⁢</mo><msup id="S5.Ex25.m1.3.4.3.3" xref="S5.Ex25.m1.3.4.3.3.cmml"><mi id="S5.Ex25.m1.3.4.3.3.2" xref="S5.Ex25.m1.3.4.3.3.2.cmml">n</mi><msub id="S5.Ex25.m1.3.4.3.3.3" xref="S5.Ex25.m1.3.4.3.3.3.cmml"><mi id="S5.Ex25.m1.3.4.3.3.3.2" xref="S5.Ex25.m1.3.4.3.3.3.2.cmml">t</mi><mn id="S5.Ex25.m1.3.4.3.3.3.3" xref="S5.Ex25.m1.3.4.3.3.3.3.cmml">0</mn></msub></msup><mo id="S5.Ex25.m1.3.4.3.1a" xref="S5.Ex25.m1.3.4.3.1.cmml">⁢</mo><msup id="S5.Ex25.m1.3.4.3.4" xref="S5.Ex25.m1.3.4.3.4.cmml"><mi id="S5.Ex25.m1.3.4.3.4.2" xref="S5.Ex25.m1.3.4.3.4.2.cmml">e</mi><mrow id="S5.Ex25.m1.3.3.3" xref="S5.Ex25.m1.3.3.3.cmml"><mo id="S5.Ex25.m1.3.3.3a" xref="S5.Ex25.m1.3.3.3.cmml">−</mo><mrow id="S5.Ex25.m1.3.3.3.3" xref="S5.Ex25.m1.3.3.3.3.cmml"><msup id="S5.Ex25.m1.2.2.2.2.1" xref="S5.Ex25.m1.2.2.2.2.1.cmml"><mrow id="S5.Ex25.m1.2.2.2.2.1.1.1" xref="S5.Ex25.m1.2.2.2.2.1.1.1.1.cmml"><mo id="S5.Ex25.m1.2.2.2.2.1.1.1.2" stretchy="false" xref="S5.Ex25.m1.2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.Ex25.m1.2.2.2.2.1.1.1.1" xref="S5.Ex25.m1.2.2.2.2.1.1.1.1.cmml"><mn id="S5.Ex25.m1.2.2.2.2.1.1.1.1.2" xref="S5.Ex25.m1.2.2.2.2.1.1.1.1.2.cmml">1</mn><mo id="S5.Ex25.m1.2.2.2.2.1.1.1.1.1" xref="S5.Ex25.m1.2.2.2.2.1.1.1.1.1.cmml">−</mo><mi id="S5.Ex25.m1.2.2.2.2.1.1.1.1.3" xref="S5.Ex25.m1.2.2.2.2.1.1.1.1.3.cmml">p</mi></mrow><mo id="S5.Ex25.m1.2.2.2.2.1.1.1.3" stretchy="false" xref="S5.Ex25.m1.2.2.2.2.1.1.1.1.cmml">)</mo></mrow><msub id="S5.Ex25.m1.2.2.2.2.1.3" xref="S5.Ex25.m1.2.2.2.2.1.3.cmml"><mi id="S5.Ex25.m1.2.2.2.2.1.3.2" xref="S5.Ex25.m1.2.2.2.2.1.3.2.cmml">t</mi><mn id="S5.Ex25.m1.2.2.2.2.1.3.3" xref="S5.Ex25.m1.2.2.2.2.1.3.3.cmml">0</mn></msub></msup><mo id="S5.Ex25.m1.3.3.3.3.3" xref="S5.Ex25.m1.3.3.3.3.3.cmml">⁢</mo><mrow id="S5.Ex25.m1.3.3.3.3.2.1" xref="S5.Ex25.m1.3.3.3.3.2.2.cmml"><mo id="S5.Ex25.m1.3.3.3.3.2.1.2" xref="S5.Ex25.m1.3.3.3.3.2.2.1.cmml">[</mo><mrow id="S5.Ex25.m1.3.3.3.3.2.1.1" xref="S5.Ex25.m1.3.3.3.3.2.1.1.cmml"><mi id="S5.Ex25.m1.3.3.3.3.2.1.1.3" xref="S5.Ex25.m1.3.3.3.3.2.1.1.3.cmml">n</mi><mo id="S5.Ex25.m1.3.3.3.3.2.1.1.2" xref="S5.Ex25.m1.3.3.3.3.2.1.1.2.cmml">⁢</mo><mrow id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.cmml"><mo id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.2" stretchy="false" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.cmml"><mfrac id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.cmml"><mrow id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.cmml"><msup id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.2" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.cmml"><mi id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.2" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.2.cmml">k</mi><mo id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.3" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.3.cmml">′</mo></msup><mo id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.1" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.1.cmml">−</mo><msub id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.3" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.cmml"><mi id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.2" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.2.cmml">t</mi><mn id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.3" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.3.cmml">0</mn></msub></mrow><mi id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.3" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.3.cmml">k</mi></mfrac><mo id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.1" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.1.cmml">−</mo><mfrac id="S5.Ex25.m1.1.1.1.1" xref="S5.Ex25.m1.1.1.1.1.cmml"><mrow id="S5.Ex25.m1.1.1.1.1.1" xref="S5.Ex25.m1.1.1.1.1.1.cmml"><mrow id="S5.Ex25.m1.1.1.1.1.1.1.1" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex25.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex25.m1.1.1.1.1.1.1.1.1" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.cmml"><msup id="S5.Ex25.m1.1.1.1.1.1.1.1.1.2" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S5.Ex25.m1.1.1.1.1.1.1.1.1.2.2" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.2.2.cmml">k</mi><mo id="S5.Ex25.m1.1.1.1.1.1.1.1.1.2.3" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S5.Ex25.m1.1.1.1.1.1.1.1.1.1" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.1.cmml">−</mo><msub id="S5.Ex25.m1.1.1.1.1.1.1.1.1.3" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S5.Ex25.m1.1.1.1.1.1.1.1.1.3.2" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.3.2.cmml">t</mi><mn id="S5.Ex25.m1.1.1.1.1.1.1.1.1.3.3" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S5.Ex25.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Ex25.m1.1.1.1.1.1.2" xref="S5.Ex25.m1.1.1.1.1.1.2.cmml">⁢</mo><mi id="S5.Ex25.m1.1.1.1.1.1.3" xref="S5.Ex25.m1.1.1.1.1.1.3.cmml">i</mi></mrow><mi id="S5.Ex25.m1.1.1.1.1.3" xref="S5.Ex25.m1.1.1.1.1.3.cmml">n</mi></mfrac></mrow><mo id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.3" stretchy="false" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex25.m1.3.3.3.3.2.1.3" xref="S5.Ex25.m1.3.3.3.3.2.2.1.cmml">]</mo></mrow></mrow></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex25.m1.3b"><apply id="S5.Ex25.m1.3.4.cmml" xref="S5.Ex25.m1.3.4"><leq id="S5.Ex25.m1.3.4.1.cmml" xref="S5.Ex25.m1.3.4.1"></leq><csymbol cd="latexml" id="S5.Ex25.m1.3.4.2.cmml" xref="S5.Ex25.m1.3.4.2">absent</csymbol><apply id="S5.Ex25.m1.3.4.3.cmml" xref="S5.Ex25.m1.3.4.3"><times id="S5.Ex25.m1.3.4.3.1.cmml" xref="S5.Ex25.m1.3.4.3.1"></times><apply id="S5.Ex25.m1.3.4.3.2.cmml" xref="S5.Ex25.m1.3.4.3.2"><csymbol cd="ambiguous" id="S5.Ex25.m1.3.4.3.2.1.cmml" xref="S5.Ex25.m1.3.4.3.2">subscript</csymbol><ci id="S5.Ex25.m1.3.4.3.2.2.cmml" xref="S5.Ex25.m1.3.4.3.2.2">𝑡</ci><cn id="S5.Ex25.m1.3.4.3.2.3.cmml" type="integer" xref="S5.Ex25.m1.3.4.3.2.3">0</cn></apply><apply id="S5.Ex25.m1.3.4.3.3.cmml" xref="S5.Ex25.m1.3.4.3.3"><csymbol cd="ambiguous" id="S5.Ex25.m1.3.4.3.3.1.cmml" xref="S5.Ex25.m1.3.4.3.3">superscript</csymbol><ci id="S5.Ex25.m1.3.4.3.3.2.cmml" xref="S5.Ex25.m1.3.4.3.3.2">𝑛</ci><apply id="S5.Ex25.m1.3.4.3.3.3.cmml" xref="S5.Ex25.m1.3.4.3.3.3"><csymbol cd="ambiguous" id="S5.Ex25.m1.3.4.3.3.3.1.cmml" xref="S5.Ex25.m1.3.4.3.3.3">subscript</csymbol><ci id="S5.Ex25.m1.3.4.3.3.3.2.cmml" xref="S5.Ex25.m1.3.4.3.3.3.2">𝑡</ci><cn id="S5.Ex25.m1.3.4.3.3.3.3.cmml" type="integer" xref="S5.Ex25.m1.3.4.3.3.3.3">0</cn></apply></apply><apply id="S5.Ex25.m1.3.4.3.4.cmml" xref="S5.Ex25.m1.3.4.3.4"><csymbol cd="ambiguous" id="S5.Ex25.m1.3.4.3.4.1.cmml" xref="S5.Ex25.m1.3.4.3.4">superscript</csymbol><ci id="S5.Ex25.m1.3.4.3.4.2.cmml" xref="S5.Ex25.m1.3.4.3.4.2">𝑒</ci><apply id="S5.Ex25.m1.3.3.3.cmml" xref="S5.Ex25.m1.3.3.3"><minus id="S5.Ex25.m1.3.3.3.4.cmml" xref="S5.Ex25.m1.3.3.3"></minus><apply id="S5.Ex25.m1.3.3.3.3.cmml" xref="S5.Ex25.m1.3.3.3.3"><times id="S5.Ex25.m1.3.3.3.3.3.cmml" xref="S5.Ex25.m1.3.3.3.3.3"></times><apply id="S5.Ex25.m1.2.2.2.2.1.cmml" xref="S5.Ex25.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="S5.Ex25.m1.2.2.2.2.1.2.cmml" xref="S5.Ex25.m1.2.2.2.2.1">superscript</csymbol><apply id="S5.Ex25.m1.2.2.2.2.1.1.1.1.cmml" xref="S5.Ex25.m1.2.2.2.2.1.1.1"><minus id="S5.Ex25.m1.2.2.2.2.1.1.1.1.1.cmml" xref="S5.Ex25.m1.2.2.2.2.1.1.1.1.1"></minus><cn id="S5.Ex25.m1.2.2.2.2.1.1.1.1.2.cmml" type="integer" xref="S5.Ex25.m1.2.2.2.2.1.1.1.1.2">1</cn><ci id="S5.Ex25.m1.2.2.2.2.1.1.1.1.3.cmml" xref="S5.Ex25.m1.2.2.2.2.1.1.1.1.3">𝑝</ci></apply><apply id="S5.Ex25.m1.2.2.2.2.1.3.cmml" xref="S5.Ex25.m1.2.2.2.2.1.3"><csymbol cd="ambiguous" id="S5.Ex25.m1.2.2.2.2.1.3.1.cmml" xref="S5.Ex25.m1.2.2.2.2.1.3">subscript</csymbol><ci id="S5.Ex25.m1.2.2.2.2.1.3.2.cmml" xref="S5.Ex25.m1.2.2.2.2.1.3.2">𝑡</ci><cn id="S5.Ex25.m1.2.2.2.2.1.3.3.cmml" type="integer" xref="S5.Ex25.m1.2.2.2.2.1.3.3">0</cn></apply></apply><apply id="S5.Ex25.m1.3.3.3.3.2.2.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1"><csymbol cd="latexml" id="S5.Ex25.m1.3.3.3.3.2.2.1.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.2">delimited-[]</csymbol><apply id="S5.Ex25.m1.3.3.3.3.2.1.1.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1"><times id="S5.Ex25.m1.3.3.3.3.2.1.1.2.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.2"></times><ci id="S5.Ex25.m1.3.3.3.3.2.1.1.3.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.3">𝑛</ci><apply id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1"><minus id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.1.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.1"></minus><apply id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2"><divide id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.1.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2"></divide><apply id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2"><minus id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.1.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.1"></minus><apply id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.1.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.2">superscript</csymbol><ci id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.2.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.2">𝑘</ci><ci id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.3.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.3">′</ci></apply><apply id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.3"><csymbol cd="ambiguous" id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.1.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.3">subscript</csymbol><ci id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.2.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.2">𝑡</ci><cn id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.3.cmml" type="integer" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.3">0</cn></apply></apply><ci id="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.3.cmml" xref="S5.Ex25.m1.3.3.3.3.2.1.1.1.1.1.2.3">𝑘</ci></apply><apply id="S5.Ex25.m1.1.1.1.1.cmml" xref="S5.Ex25.m1.1.1.1.1"><divide id="S5.Ex25.m1.1.1.1.1.2.cmml" xref="S5.Ex25.m1.1.1.1.1"></divide><apply id="S5.Ex25.m1.1.1.1.1.1.cmml" xref="S5.Ex25.m1.1.1.1.1.1"><times id="S5.Ex25.m1.1.1.1.1.1.2.cmml" xref="S5.Ex25.m1.1.1.1.1.1.2"></times><apply id="S5.Ex25.m1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex25.m1.1.1.1.1.1.1.1"><minus id="S5.Ex25.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.1"></minus><apply id="S5.Ex25.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.Ex25.m1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S5.Ex25.m1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.2.2">𝑘</ci><ci id="S5.Ex25.m1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.2.3">′</ci></apply><apply id="S5.Ex25.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Ex25.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S5.Ex25.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.3.2">𝑡</ci><cn id="S5.Ex25.m1.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.Ex25.m1.1.1.1.1.1.1.1.1.3.3">0</cn></apply></apply><ci id="S5.Ex25.m1.1.1.1.1.1.3.cmml" xref="S5.Ex25.m1.1.1.1.1.1.3">𝑖</ci></apply><ci id="S5.Ex25.m1.1.1.1.1.3.cmml" xref="S5.Ex25.m1.1.1.1.1.3">𝑛</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex25.m1.3c">\displaystyle\leq t_{0}n^{t_{0}}e^{-(1-p)^{t_{0}}\left[n(\frac{k^{\prime}-t_{0% }}{k}-\frac{(k^{\prime}-t_{0})i}{n})\right]}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex25.m1.3d">≤ italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - ( 1 - italic_p ) start_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT [ italic_n ( divide start_ARG italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_k end_ARG - divide start_ARG ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_i end_ARG start_ARG italic_n end_ARG ) ] end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S5.Ex26"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=t_{0}e^{t_{0}\log_{e}{n}}\cdot e^{-(1-p)^{t_{0}}\left[n(\frac{k^% {\prime}-t_{0}}{k}-\frac{(k^{\prime}-t_{0})i}{n})\right]}" class="ltx_Math" display="inline" id="S5.Ex26.m1.3"><semantics id="S5.Ex26.m1.3a"><mrow id="S5.Ex26.m1.3.4" xref="S5.Ex26.m1.3.4.cmml"><mi id="S5.Ex26.m1.3.4.2" xref="S5.Ex26.m1.3.4.2.cmml"></mi><mo id="S5.Ex26.m1.3.4.1" xref="S5.Ex26.m1.3.4.1.cmml">=</mo><mrow id="S5.Ex26.m1.3.4.3" xref="S5.Ex26.m1.3.4.3.cmml"><mrow id="S5.Ex26.m1.3.4.3.2" xref="S5.Ex26.m1.3.4.3.2.cmml"><msub id="S5.Ex26.m1.3.4.3.2.2" xref="S5.Ex26.m1.3.4.3.2.2.cmml"><mi id="S5.Ex26.m1.3.4.3.2.2.2" xref="S5.Ex26.m1.3.4.3.2.2.2.cmml">t</mi><mn id="S5.Ex26.m1.3.4.3.2.2.3" xref="S5.Ex26.m1.3.4.3.2.2.3.cmml">0</mn></msub><mo id="S5.Ex26.m1.3.4.3.2.1" xref="S5.Ex26.m1.3.4.3.2.1.cmml">⁢</mo><msup id="S5.Ex26.m1.3.4.3.2.3" xref="S5.Ex26.m1.3.4.3.2.3.cmml"><mi id="S5.Ex26.m1.3.4.3.2.3.2" xref="S5.Ex26.m1.3.4.3.2.3.2.cmml">e</mi><mrow id="S5.Ex26.m1.3.4.3.2.3.3" xref="S5.Ex26.m1.3.4.3.2.3.3.cmml"><msub id="S5.Ex26.m1.3.4.3.2.3.3.2" xref="S5.Ex26.m1.3.4.3.2.3.3.2.cmml"><mi id="S5.Ex26.m1.3.4.3.2.3.3.2.2" xref="S5.Ex26.m1.3.4.3.2.3.3.2.2.cmml">t</mi><mn id="S5.Ex26.m1.3.4.3.2.3.3.2.3" xref="S5.Ex26.m1.3.4.3.2.3.3.2.3.cmml">0</mn></msub><mo id="S5.Ex26.m1.3.4.3.2.3.3.1" lspace="0.167em" xref="S5.Ex26.m1.3.4.3.2.3.3.1.cmml">⁢</mo><mrow id="S5.Ex26.m1.3.4.3.2.3.3.3" xref="S5.Ex26.m1.3.4.3.2.3.3.3.cmml"><msub id="S5.Ex26.m1.3.4.3.2.3.3.3.1" xref="S5.Ex26.m1.3.4.3.2.3.3.3.1.cmml"><mi id="S5.Ex26.m1.3.4.3.2.3.3.3.1.2" xref="S5.Ex26.m1.3.4.3.2.3.3.3.1.2.cmml">log</mi><mi id="S5.Ex26.m1.3.4.3.2.3.3.3.1.3" xref="S5.Ex26.m1.3.4.3.2.3.3.3.1.3.cmml">e</mi></msub><mo id="S5.Ex26.m1.3.4.3.2.3.3.3a" lspace="0.167em" xref="S5.Ex26.m1.3.4.3.2.3.3.3.cmml">⁡</mo><mi id="S5.Ex26.m1.3.4.3.2.3.3.3.2" xref="S5.Ex26.m1.3.4.3.2.3.3.3.2.cmml">n</mi></mrow></mrow></msup></mrow><mo id="S5.Ex26.m1.3.4.3.1" lspace="0.222em" rspace="0.222em" xref="S5.Ex26.m1.3.4.3.1.cmml">⋅</mo><msup id="S5.Ex26.m1.3.4.3.3" xref="S5.Ex26.m1.3.4.3.3.cmml"><mi id="S5.Ex26.m1.3.4.3.3.2" xref="S5.Ex26.m1.3.4.3.3.2.cmml">e</mi><mrow id="S5.Ex26.m1.3.3.3" xref="S5.Ex26.m1.3.3.3.cmml"><mo id="S5.Ex26.m1.3.3.3a" xref="S5.Ex26.m1.3.3.3.cmml">−</mo><mrow id="S5.Ex26.m1.3.3.3.3" xref="S5.Ex26.m1.3.3.3.3.cmml"><msup id="S5.Ex26.m1.2.2.2.2.1" xref="S5.Ex26.m1.2.2.2.2.1.cmml"><mrow id="S5.Ex26.m1.2.2.2.2.1.1.1" xref="S5.Ex26.m1.2.2.2.2.1.1.1.1.cmml"><mo id="S5.Ex26.m1.2.2.2.2.1.1.1.2" stretchy="false" xref="S5.Ex26.m1.2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.Ex26.m1.2.2.2.2.1.1.1.1" xref="S5.Ex26.m1.2.2.2.2.1.1.1.1.cmml"><mn id="S5.Ex26.m1.2.2.2.2.1.1.1.1.2" xref="S5.Ex26.m1.2.2.2.2.1.1.1.1.2.cmml">1</mn><mo id="S5.Ex26.m1.2.2.2.2.1.1.1.1.1" xref="S5.Ex26.m1.2.2.2.2.1.1.1.1.1.cmml">−</mo><mi id="S5.Ex26.m1.2.2.2.2.1.1.1.1.3" xref="S5.Ex26.m1.2.2.2.2.1.1.1.1.3.cmml">p</mi></mrow><mo id="S5.Ex26.m1.2.2.2.2.1.1.1.3" stretchy="false" xref="S5.Ex26.m1.2.2.2.2.1.1.1.1.cmml">)</mo></mrow><msub id="S5.Ex26.m1.2.2.2.2.1.3" xref="S5.Ex26.m1.2.2.2.2.1.3.cmml"><mi id="S5.Ex26.m1.2.2.2.2.1.3.2" xref="S5.Ex26.m1.2.2.2.2.1.3.2.cmml">t</mi><mn id="S5.Ex26.m1.2.2.2.2.1.3.3" xref="S5.Ex26.m1.2.2.2.2.1.3.3.cmml">0</mn></msub></msup><mo id="S5.Ex26.m1.3.3.3.3.3" xref="S5.Ex26.m1.3.3.3.3.3.cmml">⁢</mo><mrow id="S5.Ex26.m1.3.3.3.3.2.1" xref="S5.Ex26.m1.3.3.3.3.2.2.cmml"><mo id="S5.Ex26.m1.3.3.3.3.2.1.2" xref="S5.Ex26.m1.3.3.3.3.2.2.1.cmml">[</mo><mrow id="S5.Ex26.m1.3.3.3.3.2.1.1" xref="S5.Ex26.m1.3.3.3.3.2.1.1.cmml"><mi id="S5.Ex26.m1.3.3.3.3.2.1.1.3" xref="S5.Ex26.m1.3.3.3.3.2.1.1.3.cmml">n</mi><mo id="S5.Ex26.m1.3.3.3.3.2.1.1.2" xref="S5.Ex26.m1.3.3.3.3.2.1.1.2.cmml">⁢</mo><mrow id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.cmml"><mo id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.2" stretchy="false" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.cmml"><mfrac id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.cmml"><mrow id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.cmml"><msup id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.2" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.cmml"><mi id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.2" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.2.cmml">k</mi><mo id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.3" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.3.cmml">′</mo></msup><mo id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.1" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.1.cmml">−</mo><msub id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.3" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.cmml"><mi id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.2" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.2.cmml">t</mi><mn id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.3" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.3.cmml">0</mn></msub></mrow><mi id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.3" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.3.cmml">k</mi></mfrac><mo id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.1" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.1.cmml">−</mo><mfrac id="S5.Ex26.m1.1.1.1.1" xref="S5.Ex26.m1.1.1.1.1.cmml"><mrow id="S5.Ex26.m1.1.1.1.1.1" xref="S5.Ex26.m1.1.1.1.1.1.cmml"><mrow id="S5.Ex26.m1.1.1.1.1.1.1.1" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex26.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex26.m1.1.1.1.1.1.1.1.1" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.cmml"><msup id="S5.Ex26.m1.1.1.1.1.1.1.1.1.2" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S5.Ex26.m1.1.1.1.1.1.1.1.1.2.2" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.2.2.cmml">k</mi><mo id="S5.Ex26.m1.1.1.1.1.1.1.1.1.2.3" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S5.Ex26.m1.1.1.1.1.1.1.1.1.1" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.1.cmml">−</mo><msub id="S5.Ex26.m1.1.1.1.1.1.1.1.1.3" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S5.Ex26.m1.1.1.1.1.1.1.1.1.3.2" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.3.2.cmml">t</mi><mn id="S5.Ex26.m1.1.1.1.1.1.1.1.1.3.3" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S5.Ex26.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Ex26.m1.1.1.1.1.1.2" xref="S5.Ex26.m1.1.1.1.1.1.2.cmml">⁢</mo><mi id="S5.Ex26.m1.1.1.1.1.1.3" xref="S5.Ex26.m1.1.1.1.1.1.3.cmml">i</mi></mrow><mi id="S5.Ex26.m1.1.1.1.1.3" xref="S5.Ex26.m1.1.1.1.1.3.cmml">n</mi></mfrac></mrow><mo id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.3" stretchy="false" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex26.m1.3.3.3.3.2.1.3" xref="S5.Ex26.m1.3.3.3.3.2.2.1.cmml">]</mo></mrow></mrow></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex26.m1.3b"><apply id="S5.Ex26.m1.3.4.cmml" xref="S5.Ex26.m1.3.4"><eq id="S5.Ex26.m1.3.4.1.cmml" xref="S5.Ex26.m1.3.4.1"></eq><csymbol cd="latexml" id="S5.Ex26.m1.3.4.2.cmml" xref="S5.Ex26.m1.3.4.2">absent</csymbol><apply id="S5.Ex26.m1.3.4.3.cmml" xref="S5.Ex26.m1.3.4.3"><ci id="S5.Ex26.m1.3.4.3.1.cmml" xref="S5.Ex26.m1.3.4.3.1">⋅</ci><apply id="S5.Ex26.m1.3.4.3.2.cmml" xref="S5.Ex26.m1.3.4.3.2"><times id="S5.Ex26.m1.3.4.3.2.1.cmml" xref="S5.Ex26.m1.3.4.3.2.1"></times><apply id="S5.Ex26.m1.3.4.3.2.2.cmml" xref="S5.Ex26.m1.3.4.3.2.2"><csymbol cd="ambiguous" id="S5.Ex26.m1.3.4.3.2.2.1.cmml" xref="S5.Ex26.m1.3.4.3.2.2">subscript</csymbol><ci id="S5.Ex26.m1.3.4.3.2.2.2.cmml" xref="S5.Ex26.m1.3.4.3.2.2.2">𝑡</ci><cn id="S5.Ex26.m1.3.4.3.2.2.3.cmml" type="integer" xref="S5.Ex26.m1.3.4.3.2.2.3">0</cn></apply><apply id="S5.Ex26.m1.3.4.3.2.3.cmml" xref="S5.Ex26.m1.3.4.3.2.3"><csymbol cd="ambiguous" id="S5.Ex26.m1.3.4.3.2.3.1.cmml" xref="S5.Ex26.m1.3.4.3.2.3">superscript</csymbol><ci id="S5.Ex26.m1.3.4.3.2.3.2.cmml" xref="S5.Ex26.m1.3.4.3.2.3.2">𝑒</ci><apply id="S5.Ex26.m1.3.4.3.2.3.3.cmml" xref="S5.Ex26.m1.3.4.3.2.3.3"><times id="S5.Ex26.m1.3.4.3.2.3.3.1.cmml" xref="S5.Ex26.m1.3.4.3.2.3.3.1"></times><apply id="S5.Ex26.m1.3.4.3.2.3.3.2.cmml" xref="S5.Ex26.m1.3.4.3.2.3.3.2"><csymbol cd="ambiguous" id="S5.Ex26.m1.3.4.3.2.3.3.2.1.cmml" xref="S5.Ex26.m1.3.4.3.2.3.3.2">subscript</csymbol><ci id="S5.Ex26.m1.3.4.3.2.3.3.2.2.cmml" xref="S5.Ex26.m1.3.4.3.2.3.3.2.2">𝑡</ci><cn id="S5.Ex26.m1.3.4.3.2.3.3.2.3.cmml" type="integer" xref="S5.Ex26.m1.3.4.3.2.3.3.2.3">0</cn></apply><apply id="S5.Ex26.m1.3.4.3.2.3.3.3.cmml" xref="S5.Ex26.m1.3.4.3.2.3.3.3"><apply id="S5.Ex26.m1.3.4.3.2.3.3.3.1.cmml" xref="S5.Ex26.m1.3.4.3.2.3.3.3.1"><csymbol cd="ambiguous" id="S5.Ex26.m1.3.4.3.2.3.3.3.1.1.cmml" xref="S5.Ex26.m1.3.4.3.2.3.3.3.1">subscript</csymbol><log id="S5.Ex26.m1.3.4.3.2.3.3.3.1.2.cmml" xref="S5.Ex26.m1.3.4.3.2.3.3.3.1.2"></log><ci id="S5.Ex26.m1.3.4.3.2.3.3.3.1.3.cmml" xref="S5.Ex26.m1.3.4.3.2.3.3.3.1.3">𝑒</ci></apply><ci id="S5.Ex26.m1.3.4.3.2.3.3.3.2.cmml" xref="S5.Ex26.m1.3.4.3.2.3.3.3.2">𝑛</ci></apply></apply></apply></apply><apply id="S5.Ex26.m1.3.4.3.3.cmml" xref="S5.Ex26.m1.3.4.3.3"><csymbol cd="ambiguous" id="S5.Ex26.m1.3.4.3.3.1.cmml" xref="S5.Ex26.m1.3.4.3.3">superscript</csymbol><ci id="S5.Ex26.m1.3.4.3.3.2.cmml" xref="S5.Ex26.m1.3.4.3.3.2">𝑒</ci><apply id="S5.Ex26.m1.3.3.3.cmml" xref="S5.Ex26.m1.3.3.3"><minus id="S5.Ex26.m1.3.3.3.4.cmml" xref="S5.Ex26.m1.3.3.3"></minus><apply id="S5.Ex26.m1.3.3.3.3.cmml" xref="S5.Ex26.m1.3.3.3.3"><times id="S5.Ex26.m1.3.3.3.3.3.cmml" xref="S5.Ex26.m1.3.3.3.3.3"></times><apply id="S5.Ex26.m1.2.2.2.2.1.cmml" xref="S5.Ex26.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="S5.Ex26.m1.2.2.2.2.1.2.cmml" xref="S5.Ex26.m1.2.2.2.2.1">superscript</csymbol><apply id="S5.Ex26.m1.2.2.2.2.1.1.1.1.cmml" xref="S5.Ex26.m1.2.2.2.2.1.1.1"><minus id="S5.Ex26.m1.2.2.2.2.1.1.1.1.1.cmml" xref="S5.Ex26.m1.2.2.2.2.1.1.1.1.1"></minus><cn id="S5.Ex26.m1.2.2.2.2.1.1.1.1.2.cmml" type="integer" xref="S5.Ex26.m1.2.2.2.2.1.1.1.1.2">1</cn><ci id="S5.Ex26.m1.2.2.2.2.1.1.1.1.3.cmml" xref="S5.Ex26.m1.2.2.2.2.1.1.1.1.3">𝑝</ci></apply><apply id="S5.Ex26.m1.2.2.2.2.1.3.cmml" xref="S5.Ex26.m1.2.2.2.2.1.3"><csymbol cd="ambiguous" id="S5.Ex26.m1.2.2.2.2.1.3.1.cmml" xref="S5.Ex26.m1.2.2.2.2.1.3">subscript</csymbol><ci id="S5.Ex26.m1.2.2.2.2.1.3.2.cmml" xref="S5.Ex26.m1.2.2.2.2.1.3.2">𝑡</ci><cn id="S5.Ex26.m1.2.2.2.2.1.3.3.cmml" type="integer" xref="S5.Ex26.m1.2.2.2.2.1.3.3">0</cn></apply></apply><apply id="S5.Ex26.m1.3.3.3.3.2.2.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1"><csymbol cd="latexml" id="S5.Ex26.m1.3.3.3.3.2.2.1.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.2">delimited-[]</csymbol><apply id="S5.Ex26.m1.3.3.3.3.2.1.1.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1"><times id="S5.Ex26.m1.3.3.3.3.2.1.1.2.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.2"></times><ci id="S5.Ex26.m1.3.3.3.3.2.1.1.3.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.3">𝑛</ci><apply id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1"><minus id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.1.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.1"></minus><apply id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2"><divide id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.1.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2"></divide><apply id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2"><minus id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.1.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.1"></minus><apply id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.1.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.2">superscript</csymbol><ci id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.2.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.2">𝑘</ci><ci id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.3.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.2.3">′</ci></apply><apply id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.3"><csymbol cd="ambiguous" id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.1.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.3">subscript</csymbol><ci id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.2.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.2">𝑡</ci><cn id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.3.cmml" type="integer" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.2.3.3">0</cn></apply></apply><ci id="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.3.cmml" xref="S5.Ex26.m1.3.3.3.3.2.1.1.1.1.1.2.3">𝑘</ci></apply><apply id="S5.Ex26.m1.1.1.1.1.cmml" xref="S5.Ex26.m1.1.1.1.1"><divide id="S5.Ex26.m1.1.1.1.1.2.cmml" xref="S5.Ex26.m1.1.1.1.1"></divide><apply id="S5.Ex26.m1.1.1.1.1.1.cmml" xref="S5.Ex26.m1.1.1.1.1.1"><times id="S5.Ex26.m1.1.1.1.1.1.2.cmml" xref="S5.Ex26.m1.1.1.1.1.1.2"></times><apply id="S5.Ex26.m1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex26.m1.1.1.1.1.1.1.1"><minus id="S5.Ex26.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.1"></minus><apply id="S5.Ex26.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.Ex26.m1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S5.Ex26.m1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.2.2">𝑘</ci><ci id="S5.Ex26.m1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.2.3">′</ci></apply><apply id="S5.Ex26.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Ex26.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S5.Ex26.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.3.2">𝑡</ci><cn id="S5.Ex26.m1.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.3.3">0</cn></apply></apply><ci id="S5.Ex26.m1.1.1.1.1.1.3.cmml" xref="S5.Ex26.m1.1.1.1.1.1.3">𝑖</ci></apply><ci id="S5.Ex26.m1.1.1.1.1.3.cmml" xref="S5.Ex26.m1.1.1.1.1.3">𝑛</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex26.m1.3c">\displaystyle=t_{0}e^{t_{0}\log_{e}{n}}\cdot e^{-(1-p)^{t_{0}}\left[n(\frac{k^% {\prime}-t_{0}}{k}-\frac{(k^{\prime}-t_{0})i}{n})\right]}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex26.m1.3d">= italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_POSTSUPERSCRIPT ⋅ italic_e start_POSTSUPERSCRIPT - ( 1 - italic_p ) start_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT [ italic_n ( divide start_ARG italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_k end_ARG - divide start_ARG ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_i end_ARG start_ARG italic_n end_ARG ) ] end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS2.SSS1.3.p3.5">where we used <math alttext="1-x\leq e^{-x}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.3.p3.1.m1.1"><semantics id="S5.SS2.SSS1.3.p3.1.m1.1a"><mrow id="S5.SS2.SSS1.3.p3.1.m1.1.1" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.cmml"><mrow id="S5.SS2.SSS1.3.p3.1.m1.1.1.2" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.2.cmml"><mn id="S5.SS2.SSS1.3.p3.1.m1.1.1.2.2" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.3.p3.1.m1.1.1.2.1" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.2.1.cmml">−</mo><mi id="S5.SS2.SSS1.3.p3.1.m1.1.1.2.3" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.2.3.cmml">x</mi></mrow><mo id="S5.SS2.SSS1.3.p3.1.m1.1.1.1" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.1.cmml">≤</mo><msup id="S5.SS2.SSS1.3.p3.1.m1.1.1.3" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.3.cmml"><mi id="S5.SS2.SSS1.3.p3.1.m1.1.1.3.2" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.3.2.cmml">e</mi><mrow id="S5.SS2.SSS1.3.p3.1.m1.1.1.3.3" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.3.3.cmml"><mo id="S5.SS2.SSS1.3.p3.1.m1.1.1.3.3a" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.3.3.cmml">−</mo><mi id="S5.SS2.SSS1.3.p3.1.m1.1.1.3.3.2" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.3.3.2.cmml">x</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.3.p3.1.m1.1b"><apply id="S5.SS2.SSS1.3.p3.1.m1.1.1.cmml" xref="S5.SS2.SSS1.3.p3.1.m1.1.1"><leq id="S5.SS2.SSS1.3.p3.1.m1.1.1.1.cmml" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.1"></leq><apply id="S5.SS2.SSS1.3.p3.1.m1.1.1.2.cmml" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.2"><minus id="S5.SS2.SSS1.3.p3.1.m1.1.1.2.1.cmml" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.2.1"></minus><cn id="S5.SS2.SSS1.3.p3.1.m1.1.1.2.2.cmml" type="integer" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.2.2">1</cn><ci id="S5.SS2.SSS1.3.p3.1.m1.1.1.2.3.cmml" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.2.3">𝑥</ci></apply><apply id="S5.SS2.SSS1.3.p3.1.m1.1.1.3.cmml" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.SSS1.3.p3.1.m1.1.1.3.1.cmml" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.3">superscript</csymbol><ci id="S5.SS2.SSS1.3.p3.1.m1.1.1.3.2.cmml" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.3.2">𝑒</ci><apply id="S5.SS2.SSS1.3.p3.1.m1.1.1.3.3.cmml" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.3.3"><minus id="S5.SS2.SSS1.3.p3.1.m1.1.1.3.3.1.cmml" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.3.3"></minus><ci id="S5.SS2.SSS1.3.p3.1.m1.1.1.3.3.2.cmml" xref="S5.SS2.SSS1.3.p3.1.m1.1.1.3.3.2">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.3.p3.1.m1.1c">1-x\leq e^{-x}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.3.p3.1.m1.1d">1 - italic_x ≤ italic_e start_POSTSUPERSCRIPT - italic_x end_POSTSUPERSCRIPT</annotation></semantics></math>. Since <math alttext="t_{0}=k^{\prime}(\sqrt{1-\varepsilon})" class="ltx_Math" display="inline" id="S5.SS2.SSS1.3.p3.2.m2.1"><semantics id="S5.SS2.SSS1.3.p3.2.m2.1a"><mrow id="S5.SS2.SSS1.3.p3.2.m2.1.2" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.cmml"><msub id="S5.SS2.SSS1.3.p3.2.m2.1.2.2" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.2.cmml"><mi id="S5.SS2.SSS1.3.p3.2.m2.1.2.2.2" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.2.2.cmml">t</mi><mn id="S5.SS2.SSS1.3.p3.2.m2.1.2.2.3" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.2.3.cmml">0</mn></msub><mo id="S5.SS2.SSS1.3.p3.2.m2.1.2.1" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.1.cmml">=</mo><mrow id="S5.SS2.SSS1.3.p3.2.m2.1.2.3" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.3.cmml"><msup id="S5.SS2.SSS1.3.p3.2.m2.1.2.3.2" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.3.2.cmml"><mi id="S5.SS2.SSS1.3.p3.2.m2.1.2.3.2.2" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.3.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.3.p3.2.m2.1.2.3.2.3" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.3.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS1.3.p3.2.m2.1.2.3.1" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS1.3.p3.2.m2.1.2.3.3.2" xref="S5.SS2.SSS1.3.p3.2.m2.1.1.cmml"><mo id="S5.SS2.SSS1.3.p3.2.m2.1.2.3.3.2.1" stretchy="false" xref="S5.SS2.SSS1.3.p3.2.m2.1.1.cmml">(</mo><msqrt id="S5.SS2.SSS1.3.p3.2.m2.1.1" xref="S5.SS2.SSS1.3.p3.2.m2.1.1.cmml"><mrow id="S5.SS2.SSS1.3.p3.2.m2.1.1.2" xref="S5.SS2.SSS1.3.p3.2.m2.1.1.2.cmml"><mn id="S5.SS2.SSS1.3.p3.2.m2.1.1.2.2" xref="S5.SS2.SSS1.3.p3.2.m2.1.1.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.3.p3.2.m2.1.1.2.1" xref="S5.SS2.SSS1.3.p3.2.m2.1.1.2.1.cmml">−</mo><mi id="S5.SS2.SSS1.3.p3.2.m2.1.1.2.3" xref="S5.SS2.SSS1.3.p3.2.m2.1.1.2.3.cmml">ε</mi></mrow></msqrt><mo id="S5.SS2.SSS1.3.p3.2.m2.1.2.3.3.2.2" stretchy="false" xref="S5.SS2.SSS1.3.p3.2.m2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.3.p3.2.m2.1b"><apply id="S5.SS2.SSS1.3.p3.2.m2.1.2.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.2"><eq id="S5.SS2.SSS1.3.p3.2.m2.1.2.1.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.1"></eq><apply id="S5.SS2.SSS1.3.p3.2.m2.1.2.2.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.3.p3.2.m2.1.2.2.1.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.2">subscript</csymbol><ci id="S5.SS2.SSS1.3.p3.2.m2.1.2.2.2.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.2.2">𝑡</ci><cn id="S5.SS2.SSS1.3.p3.2.m2.1.2.2.3.cmml" type="integer" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.2.3">0</cn></apply><apply id="S5.SS2.SSS1.3.p3.2.m2.1.2.3.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.3"><times id="S5.SS2.SSS1.3.p3.2.m2.1.2.3.1.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.3.1"></times><apply id="S5.SS2.SSS1.3.p3.2.m2.1.2.3.2.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.3.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.3.p3.2.m2.1.2.3.2.1.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.3.2">superscript</csymbol><ci id="S5.SS2.SSS1.3.p3.2.m2.1.2.3.2.2.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.3.2.2">𝑘</ci><ci id="S5.SS2.SSS1.3.p3.2.m2.1.2.3.2.3.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.3.2.3">′</ci></apply><apply id="S5.SS2.SSS1.3.p3.2.m2.1.1.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.3.3.2"><root id="S5.SS2.SSS1.3.p3.2.m2.1.1a.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.2.3.3.2"></root><apply id="S5.SS2.SSS1.3.p3.2.m2.1.1.2.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.1.2"><minus id="S5.SS2.SSS1.3.p3.2.m2.1.1.2.1.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.1.2.1"></minus><cn id="S5.SS2.SSS1.3.p3.2.m2.1.1.2.2.cmml" type="integer" xref="S5.SS2.SSS1.3.p3.2.m2.1.1.2.2">1</cn><ci id="S5.SS2.SSS1.3.p3.2.m2.1.1.2.3.cmml" xref="S5.SS2.SSS1.3.p3.2.m2.1.1.2.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.3.p3.2.m2.1c">t_{0}=k^{\prime}(\sqrt{1-\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.3.p3.2.m2.1d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( square-root start_ARG 1 - italic_ε end_ARG )</annotation></semantics></math> and <math alttext="p=\mathcal{O}(\frac{1}{k^{\prime}})" class="ltx_Math" display="inline" id="S5.SS2.SSS1.3.p3.3.m3.1"><semantics id="S5.SS2.SSS1.3.p3.3.m3.1a"><mrow id="S5.SS2.SSS1.3.p3.3.m3.1.2" xref="S5.SS2.SSS1.3.p3.3.m3.1.2.cmml"><mi id="S5.SS2.SSS1.3.p3.3.m3.1.2.2" xref="S5.SS2.SSS1.3.p3.3.m3.1.2.2.cmml">p</mi><mo id="S5.SS2.SSS1.3.p3.3.m3.1.2.1" xref="S5.SS2.SSS1.3.p3.3.m3.1.2.1.cmml">=</mo><mrow id="S5.SS2.SSS1.3.p3.3.m3.1.2.3" xref="S5.SS2.SSS1.3.p3.3.m3.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS1.3.p3.3.m3.1.2.3.2" xref="S5.SS2.SSS1.3.p3.3.m3.1.2.3.2.cmml">𝒪</mi><mo id="S5.SS2.SSS1.3.p3.3.m3.1.2.3.1" xref="S5.SS2.SSS1.3.p3.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS1.3.p3.3.m3.1.2.3.3.2" xref="S5.SS2.SSS1.3.p3.3.m3.1.1.cmml"><mo id="S5.SS2.SSS1.3.p3.3.m3.1.2.3.3.2.1" stretchy="false" xref="S5.SS2.SSS1.3.p3.3.m3.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS1.3.p3.3.m3.1.1" xref="S5.SS2.SSS1.3.p3.3.m3.1.1.cmml"><mn id="S5.SS2.SSS1.3.p3.3.m3.1.1.2" xref="S5.SS2.SSS1.3.p3.3.m3.1.1.2.cmml">1</mn><msup id="S5.SS2.SSS1.3.p3.3.m3.1.1.3" xref="S5.SS2.SSS1.3.p3.3.m3.1.1.3.cmml"><mi id="S5.SS2.SSS1.3.p3.3.m3.1.1.3.2" xref="S5.SS2.SSS1.3.p3.3.m3.1.1.3.2.cmml">k</mi><mo id="S5.SS2.SSS1.3.p3.3.m3.1.1.3.3" xref="S5.SS2.SSS1.3.p3.3.m3.1.1.3.3.cmml">′</mo></msup></mfrac><mo id="S5.SS2.SSS1.3.p3.3.m3.1.2.3.3.2.2" stretchy="false" xref="S5.SS2.SSS1.3.p3.3.m3.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.3.p3.3.m3.1b"><apply id="S5.SS2.SSS1.3.p3.3.m3.1.2.cmml" xref="S5.SS2.SSS1.3.p3.3.m3.1.2"><eq id="S5.SS2.SSS1.3.p3.3.m3.1.2.1.cmml" xref="S5.SS2.SSS1.3.p3.3.m3.1.2.1"></eq><ci id="S5.SS2.SSS1.3.p3.3.m3.1.2.2.cmml" xref="S5.SS2.SSS1.3.p3.3.m3.1.2.2">𝑝</ci><apply id="S5.SS2.SSS1.3.p3.3.m3.1.2.3.cmml" xref="S5.SS2.SSS1.3.p3.3.m3.1.2.3"><times id="S5.SS2.SSS1.3.p3.3.m3.1.2.3.1.cmml" xref="S5.SS2.SSS1.3.p3.3.m3.1.2.3.1"></times><ci id="S5.SS2.SSS1.3.p3.3.m3.1.2.3.2.cmml" xref="S5.SS2.SSS1.3.p3.3.m3.1.2.3.2">𝒪</ci><apply id="S5.SS2.SSS1.3.p3.3.m3.1.1.cmml" xref="S5.SS2.SSS1.3.p3.3.m3.1.2.3.3.2"><divide id="S5.SS2.SSS1.3.p3.3.m3.1.1.1.cmml" xref="S5.SS2.SSS1.3.p3.3.m3.1.2.3.3.2"></divide><cn id="S5.SS2.SSS1.3.p3.3.m3.1.1.2.cmml" type="integer" xref="S5.SS2.SSS1.3.p3.3.m3.1.1.2">1</cn><apply id="S5.SS2.SSS1.3.p3.3.m3.1.1.3.cmml" xref="S5.SS2.SSS1.3.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.SSS1.3.p3.3.m3.1.1.3.1.cmml" xref="S5.SS2.SSS1.3.p3.3.m3.1.1.3">superscript</csymbol><ci id="S5.SS2.SSS1.3.p3.3.m3.1.1.3.2.cmml" xref="S5.SS2.SSS1.3.p3.3.m3.1.1.3.2">𝑘</ci><ci id="S5.SS2.SSS1.3.p3.3.m3.1.1.3.3.cmml" xref="S5.SS2.SSS1.3.p3.3.m3.1.1.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.3.p3.3.m3.1c">p=\mathcal{O}(\frac{1}{k^{\prime}})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.3.p3.3.m3.1d">italic_p = caligraphic_O ( divide start_ARG 1 end_ARG start_ARG italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG )</annotation></semantics></math>, <math alttext="(1-p)^{t_{0}}\geq e^{t_{0}(-p-p^{2})}\to c_{0}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.3.p3.4.m4.2"><semantics id="S5.SS2.SSS1.3.p3.4.m4.2a"><mrow id="S5.SS2.SSS1.3.p3.4.m4.2.2" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.cmml"><msup id="S5.SS2.SSS1.3.p3.4.m4.2.2.1" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.cmml"><mrow id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.cmml"><mo id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.2" stretchy="false" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.cmml"><mn id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.2" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.2.cmml">1</mn><mo id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.1" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.1.cmml">−</mo><mi id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.3" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.3.cmml">p</mi></mrow><mo id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.3" stretchy="false" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.cmml">)</mo></mrow><msub id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.3" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.3.cmml"><mi id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.3.2" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.3.2.cmml">t</mi><mn id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.3.3" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.3.3.cmml">0</mn></msub></msup><mo id="S5.SS2.SSS1.3.p3.4.m4.2.2.3" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.3.cmml">≥</mo><msup id="S5.SS2.SSS1.3.p3.4.m4.2.2.4" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.4.cmml"><mi id="S5.SS2.SSS1.3.p3.4.m4.2.2.4.2" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.4.2.cmml">e</mi><mrow id="S5.SS2.SSS1.3.p3.4.m4.1.1.1" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.cmml"><msub id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.3" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.3.cmml"><mi id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.3.2" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.3.2.cmml">t</mi><mn id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.3.3" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.3.3.cmml">0</mn></msub><mo id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.2" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.cmml"><mo id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.cmml"><mrow id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.2" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.2.cmml"><mo id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.2a" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.2.cmml">−</mo><mi id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.2.2" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.2.2.cmml">p</mi></mrow><mo id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.1" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.1.cmml">−</mo><msup id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.3" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.3.cmml"><mi id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.3.2" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.3.2.cmml">p</mi><mn id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.3.3" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.3.3.cmml">2</mn></msup></mrow><mo id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msup><mo id="S5.SS2.SSS1.3.p3.4.m4.2.2.5" stretchy="false" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.5.cmml">→</mo><msub id="S5.SS2.SSS1.3.p3.4.m4.2.2.6" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.6.cmml"><mi id="S5.SS2.SSS1.3.p3.4.m4.2.2.6.2" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.6.2.cmml">c</mi><mn id="S5.SS2.SSS1.3.p3.4.m4.2.2.6.3" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.6.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.3.p3.4.m4.2b"><apply id="S5.SS2.SSS1.3.p3.4.m4.2.2.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2"><and id="S5.SS2.SSS1.3.p3.4.m4.2.2a.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2"></and><apply id="S5.SS2.SSS1.3.p3.4.m4.2.2b.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2"><geq id="S5.SS2.SSS1.3.p3.4.m4.2.2.3.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.3"></geq><apply id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1"><csymbol cd="ambiguous" id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.2.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1">superscript</csymbol><apply id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1"><minus id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.1"></minus><cn id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.2.cmml" type="integer" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.2">1</cn><ci id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.3.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.1.1.1.3">𝑝</ci></apply><apply id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.3.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.3"><csymbol cd="ambiguous" id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.3.1.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.3">subscript</csymbol><ci id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.3.2.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.3.2">𝑡</ci><cn id="S5.SS2.SSS1.3.p3.4.m4.2.2.1.3.3.cmml" type="integer" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.1.3.3">0</cn></apply></apply><apply id="S5.SS2.SSS1.3.p3.4.m4.2.2.4.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.4"><csymbol cd="ambiguous" id="S5.SS2.SSS1.3.p3.4.m4.2.2.4.1.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.4">superscript</csymbol><ci id="S5.SS2.SSS1.3.p3.4.m4.2.2.4.2.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.4.2">𝑒</ci><apply id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1"><times id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.2.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.2"></times><apply id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.3.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.3.1.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.3">subscript</csymbol><ci id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.3.2.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.3.2">𝑡</ci><cn id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.3.3.cmml" type="integer" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.3.3">0</cn></apply><apply id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1"><minus id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.1"></minus><apply id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.2.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.2"><minus id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.2.1.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.2"></minus><ci id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.2.2.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.2.2">𝑝</ci></apply><apply id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.3.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.3.1.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.3">superscript</csymbol><ci id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.3.2.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.3.2">𝑝</ci><cn id="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.SS2.SSS1.3.p3.4.m4.1.1.1.1.1.1.3.3">2</cn></apply></apply></apply></apply></apply><apply id="S5.SS2.SSS1.3.p3.4.m4.2.2c.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2"><ci id="S5.SS2.SSS1.3.p3.4.m4.2.2.5.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.5">→</ci><share href="https://arxiv.org/html/2503.06017v1#S5.SS2.SSS1.3.p3.4.m4.2.2.4.cmml" id="S5.SS2.SSS1.3.p3.4.m4.2.2d.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2"></share><apply id="S5.SS2.SSS1.3.p3.4.m4.2.2.6.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.6"><csymbol cd="ambiguous" id="S5.SS2.SSS1.3.p3.4.m4.2.2.6.1.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.6">subscript</csymbol><ci id="S5.SS2.SSS1.3.p3.4.m4.2.2.6.2.cmml" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.6.2">𝑐</ci><cn id="S5.SS2.SSS1.3.p3.4.m4.2.2.6.3.cmml" type="integer" xref="S5.SS2.SSS1.3.p3.4.m4.2.2.6.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.3.p3.4.m4.2c">(1-p)^{t_{0}}\geq e^{t_{0}(-p-p^{2})}\to c_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.3.p3.4.m4.2d">( 1 - italic_p ) start_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ≥ italic_e start_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( - italic_p - italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT → italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> for some positive constant <math alttext="c_{0}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.3.p3.5.m5.1"><semantics id="S5.SS2.SSS1.3.p3.5.m5.1a"><msub id="S5.SS2.SSS1.3.p3.5.m5.1.1" xref="S5.SS2.SSS1.3.p3.5.m5.1.1.cmml"><mi id="S5.SS2.SSS1.3.p3.5.m5.1.1.2" xref="S5.SS2.SSS1.3.p3.5.m5.1.1.2.cmml">c</mi><mn id="S5.SS2.SSS1.3.p3.5.m5.1.1.3" xref="S5.SS2.SSS1.3.p3.5.m5.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.3.p3.5.m5.1b"><apply id="S5.SS2.SSS1.3.p3.5.m5.1.1.cmml" xref="S5.SS2.SSS1.3.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS1.3.p3.5.m5.1.1.1.cmml" xref="S5.SS2.SSS1.3.p3.5.m5.1.1">subscript</csymbol><ci id="S5.SS2.SSS1.3.p3.5.m5.1.1.2.cmml" xref="S5.SS2.SSS1.3.p3.5.m5.1.1.2">𝑐</ci><cn id="S5.SS2.SSS1.3.p3.5.m5.1.1.3.cmml" type="integer" xref="S5.SS2.SSS1.3.p3.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.3.p3.5.m5.1c">c_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.3.p3.5.m5.1d">italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, the expression can be rewritten as:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx6"> <tbody id="S5.Ex27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mathbb{E}[X]\leq t_{0}e^{t_{0}\log_{e}{n}}\cdot e^{c_{0}\left[-n% (\frac{k^{\prime}-t_{0}}{k}-\frac{(k^{\prime}-t_{0})i}{n})\right]}" class="ltx_Math" display="inline" id="S5.Ex27.m1.3"><semantics id="S5.Ex27.m1.3a"><mrow id="S5.Ex27.m1.3.4" xref="S5.Ex27.m1.3.4.cmml"><mrow id="S5.Ex27.m1.3.4.2" xref="S5.Ex27.m1.3.4.2.cmml"><mi id="S5.Ex27.m1.3.4.2.2" xref="S5.Ex27.m1.3.4.2.2.cmml">𝔼</mi><mo id="S5.Ex27.m1.3.4.2.1" xref="S5.Ex27.m1.3.4.2.1.cmml">⁢</mo><mrow id="S5.Ex27.m1.3.4.2.3.2" xref="S5.Ex27.m1.3.4.2.3.1.cmml"><mo id="S5.Ex27.m1.3.4.2.3.2.1" stretchy="false" xref="S5.Ex27.m1.3.4.2.3.1.1.cmml">[</mo><mi id="S5.Ex27.m1.3.3" xref="S5.Ex27.m1.3.3.cmml">X</mi><mo id="S5.Ex27.m1.3.4.2.3.2.2" stretchy="false" xref="S5.Ex27.m1.3.4.2.3.1.1.cmml">]</mo></mrow></mrow><mo id="S5.Ex27.m1.3.4.1" xref="S5.Ex27.m1.3.4.1.cmml">≤</mo><mrow id="S5.Ex27.m1.3.4.3" xref="S5.Ex27.m1.3.4.3.cmml"><mrow id="S5.Ex27.m1.3.4.3.2" xref="S5.Ex27.m1.3.4.3.2.cmml"><msub id="S5.Ex27.m1.3.4.3.2.2" xref="S5.Ex27.m1.3.4.3.2.2.cmml"><mi id="S5.Ex27.m1.3.4.3.2.2.2" xref="S5.Ex27.m1.3.4.3.2.2.2.cmml">t</mi><mn id="S5.Ex27.m1.3.4.3.2.2.3" xref="S5.Ex27.m1.3.4.3.2.2.3.cmml">0</mn></msub><mo id="S5.Ex27.m1.3.4.3.2.1" xref="S5.Ex27.m1.3.4.3.2.1.cmml">⁢</mo><msup id="S5.Ex27.m1.3.4.3.2.3" xref="S5.Ex27.m1.3.4.3.2.3.cmml"><mi id="S5.Ex27.m1.3.4.3.2.3.2" xref="S5.Ex27.m1.3.4.3.2.3.2.cmml">e</mi><mrow id="S5.Ex27.m1.3.4.3.2.3.3" xref="S5.Ex27.m1.3.4.3.2.3.3.cmml"><msub id="S5.Ex27.m1.3.4.3.2.3.3.2" xref="S5.Ex27.m1.3.4.3.2.3.3.2.cmml"><mi id="S5.Ex27.m1.3.4.3.2.3.3.2.2" xref="S5.Ex27.m1.3.4.3.2.3.3.2.2.cmml">t</mi><mn id="S5.Ex27.m1.3.4.3.2.3.3.2.3" xref="S5.Ex27.m1.3.4.3.2.3.3.2.3.cmml">0</mn></msub><mo id="S5.Ex27.m1.3.4.3.2.3.3.1" lspace="0.167em" xref="S5.Ex27.m1.3.4.3.2.3.3.1.cmml">⁢</mo><mrow id="S5.Ex27.m1.3.4.3.2.3.3.3" xref="S5.Ex27.m1.3.4.3.2.3.3.3.cmml"><msub id="S5.Ex27.m1.3.4.3.2.3.3.3.1" xref="S5.Ex27.m1.3.4.3.2.3.3.3.1.cmml"><mi id="S5.Ex27.m1.3.4.3.2.3.3.3.1.2" xref="S5.Ex27.m1.3.4.3.2.3.3.3.1.2.cmml">log</mi><mi id="S5.Ex27.m1.3.4.3.2.3.3.3.1.3" xref="S5.Ex27.m1.3.4.3.2.3.3.3.1.3.cmml">e</mi></msub><mo id="S5.Ex27.m1.3.4.3.2.3.3.3a" lspace="0.167em" xref="S5.Ex27.m1.3.4.3.2.3.3.3.cmml">⁡</mo><mi id="S5.Ex27.m1.3.4.3.2.3.3.3.2" xref="S5.Ex27.m1.3.4.3.2.3.3.3.2.cmml">n</mi></mrow></mrow></msup></mrow><mo id="S5.Ex27.m1.3.4.3.1" lspace="0.222em" rspace="0.222em" xref="S5.Ex27.m1.3.4.3.1.cmml">⋅</mo><msup id="S5.Ex27.m1.3.4.3.3" xref="S5.Ex27.m1.3.4.3.3.cmml"><mi id="S5.Ex27.m1.3.4.3.3.2" xref="S5.Ex27.m1.3.4.3.3.2.cmml">e</mi><mrow id="S5.Ex27.m1.2.2.2" xref="S5.Ex27.m1.2.2.2.cmml"><msub id="S5.Ex27.m1.2.2.2.4" xref="S5.Ex27.m1.2.2.2.4.cmml"><mi id="S5.Ex27.m1.2.2.2.4.2" xref="S5.Ex27.m1.2.2.2.4.2.cmml">c</mi><mn id="S5.Ex27.m1.2.2.2.4.3" xref="S5.Ex27.m1.2.2.2.4.3.cmml">0</mn></msub><mo id="S5.Ex27.m1.2.2.2.3" xref="S5.Ex27.m1.2.2.2.3.cmml">⁢</mo><mrow id="S5.Ex27.m1.2.2.2.2.1" xref="S5.Ex27.m1.2.2.2.2.2.cmml"><mo id="S5.Ex27.m1.2.2.2.2.1.2" xref="S5.Ex27.m1.2.2.2.2.2.1.cmml">[</mo><mrow id="S5.Ex27.m1.2.2.2.2.1.1" xref="S5.Ex27.m1.2.2.2.2.1.1.cmml"><mo id="S5.Ex27.m1.2.2.2.2.1.1a" xref="S5.Ex27.m1.2.2.2.2.1.1.cmml">−</mo><mrow id="S5.Ex27.m1.2.2.2.2.1.1.1" xref="S5.Ex27.m1.2.2.2.2.1.1.1.cmml"><mi id="S5.Ex27.m1.2.2.2.2.1.1.1.3" xref="S5.Ex27.m1.2.2.2.2.1.1.1.3.cmml">n</mi><mo id="S5.Ex27.m1.2.2.2.2.1.1.1.2" xref="S5.Ex27.m1.2.2.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.cmml"><mo id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.2" stretchy="false" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.cmml"><mfrac id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.cmml"><mrow id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.cmml"><msup id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.2" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.2.cmml"><mi id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.2.2" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.2.2.cmml">k</mi><mo id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.2.3" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.2.3.cmml">′</mo></msup><mo id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.1" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.1.cmml">−</mo><msub id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.3" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.3.cmml"><mi id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.3.2" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.3.2.cmml">t</mi><mn id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.3.3" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.3.3.cmml">0</mn></msub></mrow><mi id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.3" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.3.cmml">k</mi></mfrac><mo id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.1" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.1.cmml">−</mo><mfrac id="S5.Ex27.m1.1.1.1.1" xref="S5.Ex27.m1.1.1.1.1.cmml"><mrow id="S5.Ex27.m1.1.1.1.1.1" xref="S5.Ex27.m1.1.1.1.1.1.cmml"><mrow id="S5.Ex27.m1.1.1.1.1.1.1.1" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex27.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex27.m1.1.1.1.1.1.1.1.1" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.cmml"><msup id="S5.Ex27.m1.1.1.1.1.1.1.1.1.2" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S5.Ex27.m1.1.1.1.1.1.1.1.1.2.2" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.2.2.cmml">k</mi><mo id="S5.Ex27.m1.1.1.1.1.1.1.1.1.2.3" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S5.Ex27.m1.1.1.1.1.1.1.1.1.1" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.1.cmml">−</mo><msub id="S5.Ex27.m1.1.1.1.1.1.1.1.1.3" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S5.Ex27.m1.1.1.1.1.1.1.1.1.3.2" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.3.2.cmml">t</mi><mn id="S5.Ex27.m1.1.1.1.1.1.1.1.1.3.3" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S5.Ex27.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Ex27.m1.1.1.1.1.1.2" xref="S5.Ex27.m1.1.1.1.1.1.2.cmml">⁢</mo><mi id="S5.Ex27.m1.1.1.1.1.1.3" xref="S5.Ex27.m1.1.1.1.1.1.3.cmml">i</mi></mrow><mi id="S5.Ex27.m1.1.1.1.1.3" xref="S5.Ex27.m1.1.1.1.1.3.cmml">n</mi></mfrac></mrow><mo id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.3" stretchy="false" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Ex27.m1.2.2.2.2.1.3" xref="S5.Ex27.m1.2.2.2.2.2.1.cmml">]</mo></mrow></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex27.m1.3b"><apply id="S5.Ex27.m1.3.4.cmml" xref="S5.Ex27.m1.3.4"><leq id="S5.Ex27.m1.3.4.1.cmml" xref="S5.Ex27.m1.3.4.1"></leq><apply id="S5.Ex27.m1.3.4.2.cmml" xref="S5.Ex27.m1.3.4.2"><times id="S5.Ex27.m1.3.4.2.1.cmml" xref="S5.Ex27.m1.3.4.2.1"></times><ci id="S5.Ex27.m1.3.4.2.2.cmml" xref="S5.Ex27.m1.3.4.2.2">𝔼</ci><apply id="S5.Ex27.m1.3.4.2.3.1.cmml" xref="S5.Ex27.m1.3.4.2.3.2"><csymbol cd="latexml" id="S5.Ex27.m1.3.4.2.3.1.1.cmml" xref="S5.Ex27.m1.3.4.2.3.2.1">delimited-[]</csymbol><ci id="S5.Ex27.m1.3.3.cmml" xref="S5.Ex27.m1.3.3">𝑋</ci></apply></apply><apply id="S5.Ex27.m1.3.4.3.cmml" xref="S5.Ex27.m1.3.4.3"><ci id="S5.Ex27.m1.3.4.3.1.cmml" xref="S5.Ex27.m1.3.4.3.1">⋅</ci><apply id="S5.Ex27.m1.3.4.3.2.cmml" xref="S5.Ex27.m1.3.4.3.2"><times id="S5.Ex27.m1.3.4.3.2.1.cmml" xref="S5.Ex27.m1.3.4.3.2.1"></times><apply id="S5.Ex27.m1.3.4.3.2.2.cmml" xref="S5.Ex27.m1.3.4.3.2.2"><csymbol cd="ambiguous" id="S5.Ex27.m1.3.4.3.2.2.1.cmml" xref="S5.Ex27.m1.3.4.3.2.2">subscript</csymbol><ci id="S5.Ex27.m1.3.4.3.2.2.2.cmml" xref="S5.Ex27.m1.3.4.3.2.2.2">𝑡</ci><cn id="S5.Ex27.m1.3.4.3.2.2.3.cmml" type="integer" xref="S5.Ex27.m1.3.4.3.2.2.3">0</cn></apply><apply id="S5.Ex27.m1.3.4.3.2.3.cmml" xref="S5.Ex27.m1.3.4.3.2.3"><csymbol cd="ambiguous" id="S5.Ex27.m1.3.4.3.2.3.1.cmml" xref="S5.Ex27.m1.3.4.3.2.3">superscript</csymbol><ci id="S5.Ex27.m1.3.4.3.2.3.2.cmml" xref="S5.Ex27.m1.3.4.3.2.3.2">𝑒</ci><apply id="S5.Ex27.m1.3.4.3.2.3.3.cmml" xref="S5.Ex27.m1.3.4.3.2.3.3"><times id="S5.Ex27.m1.3.4.3.2.3.3.1.cmml" xref="S5.Ex27.m1.3.4.3.2.3.3.1"></times><apply id="S5.Ex27.m1.3.4.3.2.3.3.2.cmml" xref="S5.Ex27.m1.3.4.3.2.3.3.2"><csymbol cd="ambiguous" id="S5.Ex27.m1.3.4.3.2.3.3.2.1.cmml" xref="S5.Ex27.m1.3.4.3.2.3.3.2">subscript</csymbol><ci id="S5.Ex27.m1.3.4.3.2.3.3.2.2.cmml" xref="S5.Ex27.m1.3.4.3.2.3.3.2.2">𝑡</ci><cn id="S5.Ex27.m1.3.4.3.2.3.3.2.3.cmml" type="integer" xref="S5.Ex27.m1.3.4.3.2.3.3.2.3">0</cn></apply><apply id="S5.Ex27.m1.3.4.3.2.3.3.3.cmml" xref="S5.Ex27.m1.3.4.3.2.3.3.3"><apply id="S5.Ex27.m1.3.4.3.2.3.3.3.1.cmml" xref="S5.Ex27.m1.3.4.3.2.3.3.3.1"><csymbol cd="ambiguous" id="S5.Ex27.m1.3.4.3.2.3.3.3.1.1.cmml" xref="S5.Ex27.m1.3.4.3.2.3.3.3.1">subscript</csymbol><log id="S5.Ex27.m1.3.4.3.2.3.3.3.1.2.cmml" xref="S5.Ex27.m1.3.4.3.2.3.3.3.1.2"></log><ci id="S5.Ex27.m1.3.4.3.2.3.3.3.1.3.cmml" xref="S5.Ex27.m1.3.4.3.2.3.3.3.1.3">𝑒</ci></apply><ci id="S5.Ex27.m1.3.4.3.2.3.3.3.2.cmml" xref="S5.Ex27.m1.3.4.3.2.3.3.3.2">𝑛</ci></apply></apply></apply></apply><apply id="S5.Ex27.m1.3.4.3.3.cmml" xref="S5.Ex27.m1.3.4.3.3"><csymbol cd="ambiguous" id="S5.Ex27.m1.3.4.3.3.1.cmml" xref="S5.Ex27.m1.3.4.3.3">superscript</csymbol><ci id="S5.Ex27.m1.3.4.3.3.2.cmml" xref="S5.Ex27.m1.3.4.3.3.2">𝑒</ci><apply id="S5.Ex27.m1.2.2.2.cmml" xref="S5.Ex27.m1.2.2.2"><times id="S5.Ex27.m1.2.2.2.3.cmml" xref="S5.Ex27.m1.2.2.2.3"></times><apply id="S5.Ex27.m1.2.2.2.4.cmml" xref="S5.Ex27.m1.2.2.2.4"><csymbol cd="ambiguous" id="S5.Ex27.m1.2.2.2.4.1.cmml" xref="S5.Ex27.m1.2.2.2.4">subscript</csymbol><ci id="S5.Ex27.m1.2.2.2.4.2.cmml" xref="S5.Ex27.m1.2.2.2.4.2">𝑐</ci><cn id="S5.Ex27.m1.2.2.2.4.3.cmml" type="integer" xref="S5.Ex27.m1.2.2.2.4.3">0</cn></apply><apply id="S5.Ex27.m1.2.2.2.2.2.cmml" xref="S5.Ex27.m1.2.2.2.2.1"><csymbol cd="latexml" id="S5.Ex27.m1.2.2.2.2.2.1.cmml" xref="S5.Ex27.m1.2.2.2.2.1.2">delimited-[]</csymbol><apply id="S5.Ex27.m1.2.2.2.2.1.1.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1"><minus id="S5.Ex27.m1.2.2.2.2.1.1.2.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1"></minus><apply id="S5.Ex27.m1.2.2.2.2.1.1.1.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1"><times id="S5.Ex27.m1.2.2.2.2.1.1.1.2.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.2"></times><ci id="S5.Ex27.m1.2.2.2.2.1.1.1.3.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.3">𝑛</ci><apply id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1"><minus id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.1.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.1"></minus><apply id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2"><divide id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.1.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2"></divide><apply id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2"><minus id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.1.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.1"></minus><apply id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.2.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.2.1.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.2">superscript</csymbol><ci id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.2.2.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.2.2">𝑘</ci><ci id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.2.3.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.2.3">′</ci></apply><apply id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.3.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.3"><csymbol cd="ambiguous" id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.3.1.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.3">subscript</csymbol><ci id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.3.2.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.3.2">𝑡</ci><cn id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.3.3.cmml" type="integer" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.2.3.3">0</cn></apply></apply><ci id="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.3.cmml" xref="S5.Ex27.m1.2.2.2.2.1.1.1.1.1.1.2.3">𝑘</ci></apply><apply id="S5.Ex27.m1.1.1.1.1.cmml" xref="S5.Ex27.m1.1.1.1.1"><divide id="S5.Ex27.m1.1.1.1.1.2.cmml" xref="S5.Ex27.m1.1.1.1.1"></divide><apply id="S5.Ex27.m1.1.1.1.1.1.cmml" xref="S5.Ex27.m1.1.1.1.1.1"><times id="S5.Ex27.m1.1.1.1.1.1.2.cmml" xref="S5.Ex27.m1.1.1.1.1.1.2"></times><apply id="S5.Ex27.m1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex27.m1.1.1.1.1.1.1.1"><minus id="S5.Ex27.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.1"></minus><apply id="S5.Ex27.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.Ex27.m1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S5.Ex27.m1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.2.2">𝑘</ci><ci id="S5.Ex27.m1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.2.3">′</ci></apply><apply id="S5.Ex27.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Ex27.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S5.Ex27.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.3.2">𝑡</ci><cn id="S5.Ex27.m1.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.Ex27.m1.1.1.1.1.1.1.1.1.3.3">0</cn></apply></apply><ci id="S5.Ex27.m1.1.1.1.1.1.3.cmml" xref="S5.Ex27.m1.1.1.1.1.1.3">𝑖</ci></apply><ci id="S5.Ex27.m1.1.1.1.1.3.cmml" xref="S5.Ex27.m1.1.1.1.1.3">𝑛</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex27.m1.3c">\displaystyle\mathbb{E}[X]\leq t_{0}e^{t_{0}\log_{e}{n}}\cdot e^{c_{0}\left[-n% (\frac{k^{\prime}-t_{0}}{k}-\frac{(k^{\prime}-t_{0})i}{n})\right]}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex27.m1.3d">blackboard_E [ italic_X ] ≤ italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_POSTSUPERSCRIPT ⋅ italic_e start_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT [ - italic_n ( divide start_ARG italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_k end_ARG - divide start_ARG ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) italic_i end_ARG start_ARG italic_n end_ARG ) ] end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S5.Ex28"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=k^{\prime}(\sqrt{1-\varepsilon})e^{k^{\prime}\sqrt{1-\varepsilon% }\log_{e}{n}}\cdot e^{c_{0}\left[-n\left(\frac{k^{\prime}}{k}(1-\sqrt{1-% \varepsilon})-\frac{k^{\prime}(1-\sqrt{1-\varepsilon})i}{n}\right)\right]}." class="ltx_Math" display="inline" id="S5.Ex28.m1.4"><semantics id="S5.Ex28.m1.4a"><mrow id="S5.Ex28.m1.4.4.1" xref="S5.Ex28.m1.4.4.1.1.cmml"><mrow id="S5.Ex28.m1.4.4.1.1" xref="S5.Ex28.m1.4.4.1.1.cmml"><mi id="S5.Ex28.m1.4.4.1.1.2" xref="S5.Ex28.m1.4.4.1.1.2.cmml"></mi><mo id="S5.Ex28.m1.4.4.1.1.1" xref="S5.Ex28.m1.4.4.1.1.1.cmml">=</mo><mrow id="S5.Ex28.m1.4.4.1.1.3" xref="S5.Ex28.m1.4.4.1.1.3.cmml"><mrow id="S5.Ex28.m1.4.4.1.1.3.2" xref="S5.Ex28.m1.4.4.1.1.3.2.cmml"><msup id="S5.Ex28.m1.4.4.1.1.3.2.2" xref="S5.Ex28.m1.4.4.1.1.3.2.2.cmml"><mi id="S5.Ex28.m1.4.4.1.1.3.2.2.2" xref="S5.Ex28.m1.4.4.1.1.3.2.2.2.cmml">k</mi><mo id="S5.Ex28.m1.4.4.1.1.3.2.2.3" xref="S5.Ex28.m1.4.4.1.1.3.2.2.3.cmml">′</mo></msup><mo id="S5.Ex28.m1.4.4.1.1.3.2.1" xref="S5.Ex28.m1.4.4.1.1.3.2.1.cmml">⁢</mo><mrow id="S5.Ex28.m1.4.4.1.1.3.2.3.2" xref="S5.Ex28.m1.3.3.cmml"><mo id="S5.Ex28.m1.4.4.1.1.3.2.3.2.1" stretchy="false" xref="S5.Ex28.m1.3.3.cmml">(</mo><msqrt id="S5.Ex28.m1.3.3" xref="S5.Ex28.m1.3.3.cmml"><mrow id="S5.Ex28.m1.3.3.2" xref="S5.Ex28.m1.3.3.2.cmml"><mn id="S5.Ex28.m1.3.3.2.2" xref="S5.Ex28.m1.3.3.2.2.cmml">1</mn><mo id="S5.Ex28.m1.3.3.2.1" xref="S5.Ex28.m1.3.3.2.1.cmml">−</mo><mi id="S5.Ex28.m1.3.3.2.3" xref="S5.Ex28.m1.3.3.2.3.cmml">ε</mi></mrow></msqrt><mo id="S5.Ex28.m1.4.4.1.1.3.2.3.2.2" stretchy="false" xref="S5.Ex28.m1.3.3.cmml">)</mo></mrow><mo id="S5.Ex28.m1.4.4.1.1.3.2.1a" xref="S5.Ex28.m1.4.4.1.1.3.2.1.cmml">⁢</mo><msup id="S5.Ex28.m1.4.4.1.1.3.2.4" xref="S5.Ex28.m1.4.4.1.1.3.2.4.cmml"><mi id="S5.Ex28.m1.4.4.1.1.3.2.4.2" xref="S5.Ex28.m1.4.4.1.1.3.2.4.2.cmml">e</mi><mrow id="S5.Ex28.m1.4.4.1.1.3.2.4.3" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.cmml"><msup id="S5.Ex28.m1.4.4.1.1.3.2.4.3.2" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.2.cmml"><mi id="S5.Ex28.m1.4.4.1.1.3.2.4.3.2.2" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.2.2.cmml">k</mi><mo id="S5.Ex28.m1.4.4.1.1.3.2.4.3.2.3" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.2.3.cmml">′</mo></msup><mo id="S5.Ex28.m1.4.4.1.1.3.2.4.3.1" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.1.cmml">⁢</mo><msqrt id="S5.Ex28.m1.4.4.1.1.3.2.4.3.3" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.cmml"><mrow id="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2.cmml"><mn id="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2.2" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2.2.cmml">1</mn><mo id="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2.1" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2.1.cmml">−</mo><mi id="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2.3" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2.3.cmml">ε</mi></mrow></msqrt><mo id="S5.Ex28.m1.4.4.1.1.3.2.4.3.1a" lspace="0.167em" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.1.cmml">⁢</mo><mrow id="S5.Ex28.m1.4.4.1.1.3.2.4.3.4" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.cmml"><msub id="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.1" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.1.cmml"><mi id="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.1.2" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.1.2.cmml">log</mi><mi id="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.1.3" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.1.3.cmml">e</mi></msub><mo id="S5.Ex28.m1.4.4.1.1.3.2.4.3.4a" lspace="0.167em" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.cmml">⁡</mo><mi id="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.2" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.2.cmml">n</mi></mrow></mrow></msup></mrow><mo id="S5.Ex28.m1.4.4.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S5.Ex28.m1.4.4.1.1.3.1.cmml">⋅</mo><msup id="S5.Ex28.m1.4.4.1.1.3.3" xref="S5.Ex28.m1.4.4.1.1.3.3.cmml"><mi id="S5.Ex28.m1.4.4.1.1.3.3.2" xref="S5.Ex28.m1.4.4.1.1.3.3.2.cmml">e</mi><mrow id="S5.Ex28.m1.2.2.2" xref="S5.Ex28.m1.2.2.2.cmml"><msub id="S5.Ex28.m1.2.2.2.4" xref="S5.Ex28.m1.2.2.2.4.cmml"><mi id="S5.Ex28.m1.2.2.2.4.2" xref="S5.Ex28.m1.2.2.2.4.2.cmml">c</mi><mn id="S5.Ex28.m1.2.2.2.4.3" xref="S5.Ex28.m1.2.2.2.4.3.cmml">0</mn></msub><mo id="S5.Ex28.m1.2.2.2.3" xref="S5.Ex28.m1.2.2.2.3.cmml">⁢</mo><mrow id="S5.Ex28.m1.2.2.2.2.1" xref="S5.Ex28.m1.2.2.2.2.2.cmml"><mo id="S5.Ex28.m1.2.2.2.2.1.2" xref="S5.Ex28.m1.2.2.2.2.2.1.cmml">[</mo><mrow id="S5.Ex28.m1.2.2.2.2.1.1" xref="S5.Ex28.m1.2.2.2.2.1.1.cmml"><mo id="S5.Ex28.m1.2.2.2.2.1.1a" xref="S5.Ex28.m1.2.2.2.2.1.1.cmml">−</mo><mrow id="S5.Ex28.m1.2.2.2.2.1.1.1" xref="S5.Ex28.m1.2.2.2.2.1.1.1.cmml"><mi id="S5.Ex28.m1.2.2.2.2.1.1.1.3" xref="S5.Ex28.m1.2.2.2.2.1.1.1.3.cmml">n</mi><mo id="S5.Ex28.m1.2.2.2.2.1.1.1.2" xref="S5.Ex28.m1.2.2.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.cmml"><mo id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.2" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.cmml"><mrow id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.cmml"><mfrac id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.cmml"><msup id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.2" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.2.cmml"><mi id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.2.2" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.2.2.cmml">k</mi><mo id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.2.3" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.2.3.cmml">′</mo></msup><mi id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.3" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.3.cmml">k</mi></mfrac><mo id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.2" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mn id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.2" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><msqrt id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml"><mrow id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.cmml"><mn id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.2" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.2.cmml">1</mn><mo id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.1" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.1.cmml">−</mo><mi id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.3" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><mo id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.2" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.2.cmml">−</mo><mfrac id="S5.Ex28.m1.1.1.1.1" xref="S5.Ex28.m1.1.1.1.1.cmml"><mrow id="S5.Ex28.m1.1.1.1.1.1" xref="S5.Ex28.m1.1.1.1.1.1.cmml"><msup id="S5.Ex28.m1.1.1.1.1.1.3" xref="S5.Ex28.m1.1.1.1.1.1.3.cmml"><mi id="S5.Ex28.m1.1.1.1.1.1.3.2" xref="S5.Ex28.m1.1.1.1.1.1.3.2.cmml">k</mi><mo id="S5.Ex28.m1.1.1.1.1.1.3.3" xref="S5.Ex28.m1.1.1.1.1.1.3.3.cmml">′</mo></msup><mo id="S5.Ex28.m1.1.1.1.1.1.2" xref="S5.Ex28.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex28.m1.1.1.1.1.1.1.1" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex28.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex28.m1.1.1.1.1.1.1.1.1" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.cmml"><mn id="S5.Ex28.m1.1.1.1.1.1.1.1.1.2" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Ex28.m1.1.1.1.1.1.1.1.1.1" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.1.cmml">−</mo><msqrt id="S5.Ex28.m1.1.1.1.1.1.1.1.1.3" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.cmml"><mrow id="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2.cmml"><mn id="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2.2" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2.2.cmml">1</mn><mo id="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2.1" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2.1.cmml">−</mo><mi id="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2.3" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><mo id="S5.Ex28.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Ex28.m1.1.1.1.1.1.2a" xref="S5.Ex28.m1.1.1.1.1.1.2.cmml">⁢</mo><mi id="S5.Ex28.m1.1.1.1.1.1.4" xref="S5.Ex28.m1.1.1.1.1.1.4.cmml">i</mi></mrow><mi id="S5.Ex28.m1.1.1.1.1.3" xref="S5.Ex28.m1.1.1.1.1.3.cmml">n</mi></mfrac></mrow><mo id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.3" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Ex28.m1.2.2.2.2.1.3" xref="S5.Ex28.m1.2.2.2.2.2.1.cmml">]</mo></mrow></mrow></msup></mrow></mrow><mo id="S5.Ex28.m1.4.4.1.2" lspace="0em" xref="S5.Ex28.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex28.m1.4b"><apply id="S5.Ex28.m1.4.4.1.1.cmml" xref="S5.Ex28.m1.4.4.1"><eq id="S5.Ex28.m1.4.4.1.1.1.cmml" xref="S5.Ex28.m1.4.4.1.1.1"></eq><csymbol cd="latexml" id="S5.Ex28.m1.4.4.1.1.2.cmml" xref="S5.Ex28.m1.4.4.1.1.2">absent</csymbol><apply id="S5.Ex28.m1.4.4.1.1.3.cmml" xref="S5.Ex28.m1.4.4.1.1.3"><ci id="S5.Ex28.m1.4.4.1.1.3.1.cmml" xref="S5.Ex28.m1.4.4.1.1.3.1">⋅</ci><apply id="S5.Ex28.m1.4.4.1.1.3.2.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2"><times id="S5.Ex28.m1.4.4.1.1.3.2.1.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.1"></times><apply id="S5.Ex28.m1.4.4.1.1.3.2.2.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.2"><csymbol cd="ambiguous" id="S5.Ex28.m1.4.4.1.1.3.2.2.1.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.2">superscript</csymbol><ci id="S5.Ex28.m1.4.4.1.1.3.2.2.2.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.2.2">𝑘</ci><ci id="S5.Ex28.m1.4.4.1.1.3.2.2.3.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.2.3">′</ci></apply><apply id="S5.Ex28.m1.3.3.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.3.2"><root id="S5.Ex28.m1.3.3a.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.3.2"></root><apply id="S5.Ex28.m1.3.3.2.cmml" xref="S5.Ex28.m1.3.3.2"><minus id="S5.Ex28.m1.3.3.2.1.cmml" xref="S5.Ex28.m1.3.3.2.1"></minus><cn id="S5.Ex28.m1.3.3.2.2.cmml" type="integer" xref="S5.Ex28.m1.3.3.2.2">1</cn><ci id="S5.Ex28.m1.3.3.2.3.cmml" xref="S5.Ex28.m1.3.3.2.3">𝜀</ci></apply></apply><apply id="S5.Ex28.m1.4.4.1.1.3.2.4.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4"><csymbol cd="ambiguous" id="S5.Ex28.m1.4.4.1.1.3.2.4.1.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4">superscript</csymbol><ci id="S5.Ex28.m1.4.4.1.1.3.2.4.2.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.2">𝑒</ci><apply id="S5.Ex28.m1.4.4.1.1.3.2.4.3.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3"><times id="S5.Ex28.m1.4.4.1.1.3.2.4.3.1.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.1"></times><apply id="S5.Ex28.m1.4.4.1.1.3.2.4.3.2.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.2"><csymbol cd="ambiguous" id="S5.Ex28.m1.4.4.1.1.3.2.4.3.2.1.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.2">superscript</csymbol><ci id="S5.Ex28.m1.4.4.1.1.3.2.4.3.2.2.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.2.2">𝑘</ci><ci id="S5.Ex28.m1.4.4.1.1.3.2.4.3.2.3.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.2.3">′</ci></apply><apply id="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.3"><root id="S5.Ex28.m1.4.4.1.1.3.2.4.3.3a.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.3"></root><apply id="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2"><minus id="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2.1.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2.1"></minus><cn id="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2.2.cmml" type="integer" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2.2">1</cn><ci id="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2.3.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.3.2.3">𝜀</ci></apply></apply><apply id="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.4"><apply id="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.1.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.1"><csymbol cd="ambiguous" id="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.1.1.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.1">subscript</csymbol><log id="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.1.2.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.1.2"></log><ci id="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.1.3.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.1.3">𝑒</ci></apply><ci id="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.2.cmml" xref="S5.Ex28.m1.4.4.1.1.3.2.4.3.4.2">𝑛</ci></apply></apply></apply></apply><apply id="S5.Ex28.m1.4.4.1.1.3.3.cmml" xref="S5.Ex28.m1.4.4.1.1.3.3"><csymbol cd="ambiguous" id="S5.Ex28.m1.4.4.1.1.3.3.1.cmml" xref="S5.Ex28.m1.4.4.1.1.3.3">superscript</csymbol><ci id="S5.Ex28.m1.4.4.1.1.3.3.2.cmml" xref="S5.Ex28.m1.4.4.1.1.3.3.2">𝑒</ci><apply id="S5.Ex28.m1.2.2.2.cmml" xref="S5.Ex28.m1.2.2.2"><times id="S5.Ex28.m1.2.2.2.3.cmml" xref="S5.Ex28.m1.2.2.2.3"></times><apply id="S5.Ex28.m1.2.2.2.4.cmml" xref="S5.Ex28.m1.2.2.2.4"><csymbol cd="ambiguous" id="S5.Ex28.m1.2.2.2.4.1.cmml" xref="S5.Ex28.m1.2.2.2.4">subscript</csymbol><ci id="S5.Ex28.m1.2.2.2.4.2.cmml" xref="S5.Ex28.m1.2.2.2.4.2">𝑐</ci><cn id="S5.Ex28.m1.2.2.2.4.3.cmml" type="integer" xref="S5.Ex28.m1.2.2.2.4.3">0</cn></apply><apply id="S5.Ex28.m1.2.2.2.2.2.cmml" xref="S5.Ex28.m1.2.2.2.2.1"><csymbol cd="latexml" id="S5.Ex28.m1.2.2.2.2.2.1.cmml" xref="S5.Ex28.m1.2.2.2.2.1.2">delimited-[]</csymbol><apply id="S5.Ex28.m1.2.2.2.2.1.1.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1"><minus id="S5.Ex28.m1.2.2.2.2.1.1.2.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1"></minus><apply id="S5.Ex28.m1.2.2.2.2.1.1.1.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1"><times id="S5.Ex28.m1.2.2.2.2.1.1.1.2.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.2"></times><ci id="S5.Ex28.m1.2.2.2.2.1.1.1.3.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.3">𝑛</ci><apply id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1"><minus id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.2.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.2"></minus><apply id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1"><times id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.2.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.2"></times><apply id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3"><divide id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.1.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3"></divide><apply id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.2.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.2.1.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.2">superscript</csymbol><ci id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.2.2.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.2.2">𝑘</ci><ci id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.2.3.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.2.3">′</ci></apply><ci id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.3.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.3.3">𝑘</ci></apply><apply id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1"><minus id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1"></minus><cn id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.2">1</cn><apply id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3"><root id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3a.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3"></root><apply id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2"><minus id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.1"></minus><cn id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.2.cmml" type="integer" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.2">1</cn><ci id="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="S5.Ex28.m1.2.2.2.2.1.1.1.1.1.1.1.1.1.1.3.2.3">𝜀</ci></apply></apply></apply></apply><apply id="S5.Ex28.m1.1.1.1.1.cmml" xref="S5.Ex28.m1.1.1.1.1"><divide id="S5.Ex28.m1.1.1.1.1.2.cmml" xref="S5.Ex28.m1.1.1.1.1"></divide><apply id="S5.Ex28.m1.1.1.1.1.1.cmml" xref="S5.Ex28.m1.1.1.1.1.1"><times id="S5.Ex28.m1.1.1.1.1.1.2.cmml" xref="S5.Ex28.m1.1.1.1.1.1.2"></times><apply id="S5.Ex28.m1.1.1.1.1.1.3.cmml" xref="S5.Ex28.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Ex28.m1.1.1.1.1.1.3.1.cmml" xref="S5.Ex28.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S5.Ex28.m1.1.1.1.1.1.3.2.cmml" xref="S5.Ex28.m1.1.1.1.1.1.3.2">𝑘</ci><ci id="S5.Ex28.m1.1.1.1.1.1.3.3.cmml" xref="S5.Ex28.m1.1.1.1.1.1.3.3">′</ci></apply><apply id="S5.Ex28.m1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex28.m1.1.1.1.1.1.1.1"><minus id="S5.Ex28.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.1"></minus><cn id="S5.Ex28.m1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.2">1</cn><apply id="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.3"><root id="S5.Ex28.m1.1.1.1.1.1.1.1.1.3a.cmml" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.3"></root><apply id="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2"><minus id="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2.1"></minus><cn id="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2.2.cmml" type="integer" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2.2">1</cn><ci id="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="S5.Ex28.m1.1.1.1.1.1.1.1.1.3.2.3">𝜀</ci></apply></apply></apply><ci id="S5.Ex28.m1.1.1.1.1.1.4.cmml" xref="S5.Ex28.m1.1.1.1.1.1.4">𝑖</ci></apply><ci id="S5.Ex28.m1.1.1.1.1.3.cmml" xref="S5.Ex28.m1.1.1.1.1.3">𝑛</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex28.m1.4c">\displaystyle=k^{\prime}(\sqrt{1-\varepsilon})e^{k^{\prime}\sqrt{1-\varepsilon% }\log_{e}{n}}\cdot e^{c_{0}\left[-n\left(\frac{k^{\prime}}{k}(1-\sqrt{1-% \varepsilon})-\frac{k^{\prime}(1-\sqrt{1-\varepsilon})i}{n}\right)\right]}.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex28.m1.4d">= italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( square-root start_ARG 1 - italic_ε end_ARG ) italic_e start_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_ε end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_POSTSUPERSCRIPT ⋅ italic_e start_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT [ - italic_n ( divide start_ARG italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_k end_ARG ( 1 - square-root start_ARG 1 - italic_ε end_ARG ) - divide start_ARG italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( 1 - square-root start_ARG 1 - italic_ε end_ARG ) italic_i end_ARG start_ARG italic_n end_ARG ) ] end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S5.SS2.SSS1.4.p4"> <p class="ltx_p" id="S5.SS2.SSS1.4.p4.2">For any constant <math alttext="\alpha\in(0,1)" class="ltx_Math" display="inline" id="S5.SS2.SSS1.4.p4.1.m1.2"><semantics id="S5.SS2.SSS1.4.p4.1.m1.2a"><mrow id="S5.SS2.SSS1.4.p4.1.m1.2.3" xref="S5.SS2.SSS1.4.p4.1.m1.2.3.cmml"><mi id="S5.SS2.SSS1.4.p4.1.m1.2.3.2" xref="S5.SS2.SSS1.4.p4.1.m1.2.3.2.cmml">α</mi><mo id="S5.SS2.SSS1.4.p4.1.m1.2.3.1" xref="S5.SS2.SSS1.4.p4.1.m1.2.3.1.cmml">∈</mo><mrow id="S5.SS2.SSS1.4.p4.1.m1.2.3.3.2" xref="S5.SS2.SSS1.4.p4.1.m1.2.3.3.1.cmml"><mo id="S5.SS2.SSS1.4.p4.1.m1.2.3.3.2.1" stretchy="false" xref="S5.SS2.SSS1.4.p4.1.m1.2.3.3.1.cmml">(</mo><mn id="S5.SS2.SSS1.4.p4.1.m1.1.1" xref="S5.SS2.SSS1.4.p4.1.m1.1.1.cmml">0</mn><mo id="S5.SS2.SSS1.4.p4.1.m1.2.3.3.2.2" xref="S5.SS2.SSS1.4.p4.1.m1.2.3.3.1.cmml">,</mo><mn id="S5.SS2.SSS1.4.p4.1.m1.2.2" xref="S5.SS2.SSS1.4.p4.1.m1.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.4.p4.1.m1.2.3.3.2.3" stretchy="false" xref="S5.SS2.SSS1.4.p4.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.4.p4.1.m1.2b"><apply id="S5.SS2.SSS1.4.p4.1.m1.2.3.cmml" xref="S5.SS2.SSS1.4.p4.1.m1.2.3"><in id="S5.SS2.SSS1.4.p4.1.m1.2.3.1.cmml" xref="S5.SS2.SSS1.4.p4.1.m1.2.3.1"></in><ci id="S5.SS2.SSS1.4.p4.1.m1.2.3.2.cmml" xref="S5.SS2.SSS1.4.p4.1.m1.2.3.2">𝛼</ci><interval closure="open" id="S5.SS2.SSS1.4.p4.1.m1.2.3.3.1.cmml" xref="S5.SS2.SSS1.4.p4.1.m1.2.3.3.2"><cn id="S5.SS2.SSS1.4.p4.1.m1.1.1.cmml" type="integer" xref="S5.SS2.SSS1.4.p4.1.m1.1.1">0</cn><cn id="S5.SS2.SSS1.4.p4.1.m1.2.2.cmml" type="integer" xref="S5.SS2.SSS1.4.p4.1.m1.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.4.p4.1.m1.2c">\alpha\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.4.p4.1.m1.2d">italic_α ∈ ( 0 , 1 )</annotation></semantics></math>, while the current iteration satisfies <math alttext="i\leq\alpha\frac{n}{k}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.4.p4.2.m2.1"><semantics id="S5.SS2.SSS1.4.p4.2.m2.1a"><mrow id="S5.SS2.SSS1.4.p4.2.m2.1.1" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.cmml"><mi id="S5.SS2.SSS1.4.p4.2.m2.1.1.2" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.2.cmml">i</mi><mo id="S5.SS2.SSS1.4.p4.2.m2.1.1.1" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.1.cmml">≤</mo><mrow id="S5.SS2.SSS1.4.p4.2.m2.1.1.3" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.3.cmml"><mi id="S5.SS2.SSS1.4.p4.2.m2.1.1.3.2" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.3.2.cmml">α</mi><mo id="S5.SS2.SSS1.4.p4.2.m2.1.1.3.1" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.3.1.cmml">⁢</mo><mfrac id="S5.SS2.SSS1.4.p4.2.m2.1.1.3.3" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.3.3.cmml"><mi id="S5.SS2.SSS1.4.p4.2.m2.1.1.3.3.2" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.3.3.2.cmml">n</mi><mi id="S5.SS2.SSS1.4.p4.2.m2.1.1.3.3.3" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.3.3.3.cmml">k</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.4.p4.2.m2.1b"><apply id="S5.SS2.SSS1.4.p4.2.m2.1.1.cmml" xref="S5.SS2.SSS1.4.p4.2.m2.1.1"><leq id="S5.SS2.SSS1.4.p4.2.m2.1.1.1.cmml" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.1"></leq><ci id="S5.SS2.SSS1.4.p4.2.m2.1.1.2.cmml" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.2">𝑖</ci><apply id="S5.SS2.SSS1.4.p4.2.m2.1.1.3.cmml" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.3"><times id="S5.SS2.SSS1.4.p4.2.m2.1.1.3.1.cmml" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.3.1"></times><ci id="S5.SS2.SSS1.4.p4.2.m2.1.1.3.2.cmml" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.3.2">𝛼</ci><apply id="S5.SS2.SSS1.4.p4.2.m2.1.1.3.3.cmml" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.3.3"><divide id="S5.SS2.SSS1.4.p4.2.m2.1.1.3.3.1.cmml" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.3.3"></divide><ci id="S5.SS2.SSS1.4.p4.2.m2.1.1.3.3.2.cmml" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.3.3.2">𝑛</ci><ci id="S5.SS2.SSS1.4.p4.2.m2.1.1.3.3.3.cmml" xref="S5.SS2.SSS1.4.p4.2.m2.1.1.3.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.4.p4.2.m2.1c">i\leq\alpha\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.4.p4.2.m2.1d">italic_i ≤ italic_α divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math>, it holds that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx7"> <tbody id="S5.Ex29"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mathbb{E}[X]\leq k^{\prime}(\sqrt{1-\varepsilon})e^{k^{\prime}(% \sqrt{1-\varepsilon})\log_{e}{n}}e^{c_{0}\left[-n\frac{k^{\prime}}{k}(1-\alpha% )(1-\sqrt{1-\varepsilon})\right]}." class="ltx_Math" display="inline" id="S5.Ex29.m1.5"><semantics id="S5.Ex29.m1.5a"><mrow id="S5.Ex29.m1.5.5.1" xref="S5.Ex29.m1.5.5.1.1.cmml"><mrow id="S5.Ex29.m1.5.5.1.1" xref="S5.Ex29.m1.5.5.1.1.cmml"><mrow id="S5.Ex29.m1.5.5.1.1.2" xref="S5.Ex29.m1.5.5.1.1.2.cmml"><mi id="S5.Ex29.m1.5.5.1.1.2.2" xref="S5.Ex29.m1.5.5.1.1.2.2.cmml">𝔼</mi><mo id="S5.Ex29.m1.5.5.1.1.2.1" xref="S5.Ex29.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="S5.Ex29.m1.5.5.1.1.2.3.2" xref="S5.Ex29.m1.5.5.1.1.2.3.1.cmml"><mo id="S5.Ex29.m1.5.5.1.1.2.3.2.1" stretchy="false" xref="S5.Ex29.m1.5.5.1.1.2.3.1.1.cmml">[</mo><mi id="S5.Ex29.m1.3.3" xref="S5.Ex29.m1.3.3.cmml">X</mi><mo id="S5.Ex29.m1.5.5.1.1.2.3.2.2" stretchy="false" xref="S5.Ex29.m1.5.5.1.1.2.3.1.1.cmml">]</mo></mrow></mrow><mo id="S5.Ex29.m1.5.5.1.1.1" xref="S5.Ex29.m1.5.5.1.1.1.cmml">≤</mo><mrow id="S5.Ex29.m1.5.5.1.1.3" xref="S5.Ex29.m1.5.5.1.1.3.cmml"><msup id="S5.Ex29.m1.5.5.1.1.3.2" xref="S5.Ex29.m1.5.5.1.1.3.2.cmml"><mi id="S5.Ex29.m1.5.5.1.1.3.2.2" xref="S5.Ex29.m1.5.5.1.1.3.2.2.cmml">k</mi><mo id="S5.Ex29.m1.5.5.1.1.3.2.3" xref="S5.Ex29.m1.5.5.1.1.3.2.3.cmml">′</mo></msup><mo id="S5.Ex29.m1.5.5.1.1.3.1" xref="S5.Ex29.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S5.Ex29.m1.5.5.1.1.3.3.2" xref="S5.Ex29.m1.4.4.cmml"><mo id="S5.Ex29.m1.5.5.1.1.3.3.2.1" stretchy="false" xref="S5.Ex29.m1.4.4.cmml">(</mo><msqrt id="S5.Ex29.m1.4.4" xref="S5.Ex29.m1.4.4.cmml"><mrow id="S5.Ex29.m1.4.4.2" xref="S5.Ex29.m1.4.4.2.cmml"><mn id="S5.Ex29.m1.4.4.2.2" xref="S5.Ex29.m1.4.4.2.2.cmml">1</mn><mo id="S5.Ex29.m1.4.4.2.1" xref="S5.Ex29.m1.4.4.2.1.cmml">−</mo><mi id="S5.Ex29.m1.4.4.2.3" xref="S5.Ex29.m1.4.4.2.3.cmml">ε</mi></mrow></msqrt><mo id="S5.Ex29.m1.5.5.1.1.3.3.2.2" stretchy="false" xref="S5.Ex29.m1.4.4.cmml">)</mo></mrow><mo id="S5.Ex29.m1.5.5.1.1.3.1a" xref="S5.Ex29.m1.5.5.1.1.3.1.cmml">⁢</mo><msup id="S5.Ex29.m1.5.5.1.1.3.4" xref="S5.Ex29.m1.5.5.1.1.3.4.cmml"><mi id="S5.Ex29.m1.5.5.1.1.3.4.2" xref="S5.Ex29.m1.5.5.1.1.3.4.2.cmml">e</mi><mrow id="S5.Ex29.m1.1.1.1" xref="S5.Ex29.m1.1.1.1.cmml"><msup id="S5.Ex29.m1.1.1.1.3" xref="S5.Ex29.m1.1.1.1.3.cmml"><mi id="S5.Ex29.m1.1.1.1.3.2" xref="S5.Ex29.m1.1.1.1.3.2.cmml">k</mi><mo id="S5.Ex29.m1.1.1.1.3.3" xref="S5.Ex29.m1.1.1.1.3.3.cmml">′</mo></msup><mo id="S5.Ex29.m1.1.1.1.2" xref="S5.Ex29.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex29.m1.1.1.1.4.2" xref="S5.Ex29.m1.1.1.1.1.cmml"><mo id="S5.Ex29.m1.1.1.1.4.2.1" stretchy="false" xref="S5.Ex29.m1.1.1.1.1.cmml">(</mo><msqrt id="S5.Ex29.m1.1.1.1.1" xref="S5.Ex29.m1.1.1.1.1.cmml"><mrow id="S5.Ex29.m1.1.1.1.1.2" xref="S5.Ex29.m1.1.1.1.1.2.cmml"><mn id="S5.Ex29.m1.1.1.1.1.2.2" xref="S5.Ex29.m1.1.1.1.1.2.2.cmml">1</mn><mo id="S5.Ex29.m1.1.1.1.1.2.1" xref="S5.Ex29.m1.1.1.1.1.2.1.cmml">−</mo><mi id="S5.Ex29.m1.1.1.1.1.2.3" xref="S5.Ex29.m1.1.1.1.1.2.3.cmml">ε</mi></mrow></msqrt><mo id="S5.Ex29.m1.1.1.1.4.2.2" stretchy="false" xref="S5.Ex29.m1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Ex29.m1.1.1.1.2a" lspace="0.167em" xref="S5.Ex29.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex29.m1.1.1.1.5" xref="S5.Ex29.m1.1.1.1.5.cmml"><msub id="S5.Ex29.m1.1.1.1.5.1" xref="S5.Ex29.m1.1.1.1.5.1.cmml"><mi id="S5.Ex29.m1.1.1.1.5.1.2" xref="S5.Ex29.m1.1.1.1.5.1.2.cmml">log</mi><mi id="S5.Ex29.m1.1.1.1.5.1.3" xref="S5.Ex29.m1.1.1.1.5.1.3.cmml">e</mi></msub><mo id="S5.Ex29.m1.1.1.1.5a" lspace="0.167em" xref="S5.Ex29.m1.1.1.1.5.cmml">⁡</mo><mi id="S5.Ex29.m1.1.1.1.5.2" xref="S5.Ex29.m1.1.1.1.5.2.cmml">n</mi></mrow></mrow></msup><mo id="S5.Ex29.m1.5.5.1.1.3.1b" xref="S5.Ex29.m1.5.5.1.1.3.1.cmml">⁢</mo><msup id="S5.Ex29.m1.5.5.1.1.3.5" xref="S5.Ex29.m1.5.5.1.1.3.5.cmml"><mi id="S5.Ex29.m1.5.5.1.1.3.5.2" xref="S5.Ex29.m1.5.5.1.1.3.5.2.cmml">e</mi><mrow id="S5.Ex29.m1.2.2.1" xref="S5.Ex29.m1.2.2.1.cmml"><msub id="S5.Ex29.m1.2.2.1.3" xref="S5.Ex29.m1.2.2.1.3.cmml"><mi id="S5.Ex29.m1.2.2.1.3.2" xref="S5.Ex29.m1.2.2.1.3.2.cmml">c</mi><mn id="S5.Ex29.m1.2.2.1.3.3" xref="S5.Ex29.m1.2.2.1.3.3.cmml">0</mn></msub><mo id="S5.Ex29.m1.2.2.1.2" xref="S5.Ex29.m1.2.2.1.2.cmml">⁢</mo><mrow id="S5.Ex29.m1.2.2.1.1.1" xref="S5.Ex29.m1.2.2.1.1.2.cmml"><mo id="S5.Ex29.m1.2.2.1.1.1.2" xref="S5.Ex29.m1.2.2.1.1.2.1.cmml">[</mo><mrow id="S5.Ex29.m1.2.2.1.1.1.1" xref="S5.Ex29.m1.2.2.1.1.1.1.cmml"><mo id="S5.Ex29.m1.2.2.1.1.1.1a" xref="S5.Ex29.m1.2.2.1.1.1.1.cmml">−</mo><mrow id="S5.Ex29.m1.2.2.1.1.1.1.2" xref="S5.Ex29.m1.2.2.1.1.1.1.2.cmml"><mi id="S5.Ex29.m1.2.2.1.1.1.1.2.4" xref="S5.Ex29.m1.2.2.1.1.1.1.2.4.cmml">n</mi><mo id="S5.Ex29.m1.2.2.1.1.1.1.2.3" xref="S5.Ex29.m1.2.2.1.1.1.1.2.3.cmml">⁢</mo><mfrac id="S5.Ex29.m1.2.2.1.1.1.1.2.5" xref="S5.Ex29.m1.2.2.1.1.1.1.2.5.cmml"><msup id="S5.Ex29.m1.2.2.1.1.1.1.2.5.2" xref="S5.Ex29.m1.2.2.1.1.1.1.2.5.2.cmml"><mi id="S5.Ex29.m1.2.2.1.1.1.1.2.5.2.2" xref="S5.Ex29.m1.2.2.1.1.1.1.2.5.2.2.cmml">k</mi><mo id="S5.Ex29.m1.2.2.1.1.1.1.2.5.2.3" xref="S5.Ex29.m1.2.2.1.1.1.1.2.5.2.3.cmml">′</mo></msup><mi id="S5.Ex29.m1.2.2.1.1.1.1.2.5.3" xref="S5.Ex29.m1.2.2.1.1.1.1.2.5.3.cmml">k</mi></mfrac><mo id="S5.Ex29.m1.2.2.1.1.1.1.2.3a" xref="S5.Ex29.m1.2.2.1.1.1.1.2.3.cmml">⁢</mo><mrow id="S5.Ex29.m1.2.2.1.1.1.1.1.1.1" xref="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1" xref="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.cmml"><mn id="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.2" xref="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.1" xref="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.3" xref="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.3.cmml">α</mi></mrow><mo id="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Ex29.m1.2.2.1.1.1.1.2.3b" xref="S5.Ex29.m1.2.2.1.1.1.1.2.3.cmml">⁢</mo><mrow id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.cmml"><mo id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.2" stretchy="false" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.cmml">(</mo><mrow id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.cmml"><mn id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.2" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.2.cmml">1</mn><mo id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.1" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.1.cmml">−</mo><msqrt id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.cmml"><mrow id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2.cmml"><mn id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2.2" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2.2.cmml">1</mn><mo id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2.1" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2.1.cmml">−</mo><mi id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2.3" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><mo id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.3" stretchy="false" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Ex29.m1.2.2.1.1.1.3" xref="S5.Ex29.m1.2.2.1.1.2.1.cmml">]</mo></mrow></mrow></msup></mrow></mrow><mo id="S5.Ex29.m1.5.5.1.2" lspace="0em" xref="S5.Ex29.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex29.m1.5b"><apply id="S5.Ex29.m1.5.5.1.1.cmml" xref="S5.Ex29.m1.5.5.1"><leq id="S5.Ex29.m1.5.5.1.1.1.cmml" xref="S5.Ex29.m1.5.5.1.1.1"></leq><apply id="S5.Ex29.m1.5.5.1.1.2.cmml" xref="S5.Ex29.m1.5.5.1.1.2"><times id="S5.Ex29.m1.5.5.1.1.2.1.cmml" xref="S5.Ex29.m1.5.5.1.1.2.1"></times><ci id="S5.Ex29.m1.5.5.1.1.2.2.cmml" xref="S5.Ex29.m1.5.5.1.1.2.2">𝔼</ci><apply id="S5.Ex29.m1.5.5.1.1.2.3.1.cmml" xref="S5.Ex29.m1.5.5.1.1.2.3.2"><csymbol cd="latexml" id="S5.Ex29.m1.5.5.1.1.2.3.1.1.cmml" xref="S5.Ex29.m1.5.5.1.1.2.3.2.1">delimited-[]</csymbol><ci id="S5.Ex29.m1.3.3.cmml" xref="S5.Ex29.m1.3.3">𝑋</ci></apply></apply><apply id="S5.Ex29.m1.5.5.1.1.3.cmml" xref="S5.Ex29.m1.5.5.1.1.3"><times id="S5.Ex29.m1.5.5.1.1.3.1.cmml" xref="S5.Ex29.m1.5.5.1.1.3.1"></times><apply id="S5.Ex29.m1.5.5.1.1.3.2.cmml" xref="S5.Ex29.m1.5.5.1.1.3.2"><csymbol cd="ambiguous" id="S5.Ex29.m1.5.5.1.1.3.2.1.cmml" xref="S5.Ex29.m1.5.5.1.1.3.2">superscript</csymbol><ci id="S5.Ex29.m1.5.5.1.1.3.2.2.cmml" xref="S5.Ex29.m1.5.5.1.1.3.2.2">𝑘</ci><ci id="S5.Ex29.m1.5.5.1.1.3.2.3.cmml" xref="S5.Ex29.m1.5.5.1.1.3.2.3">′</ci></apply><apply id="S5.Ex29.m1.4.4.cmml" xref="S5.Ex29.m1.5.5.1.1.3.3.2"><root id="S5.Ex29.m1.4.4a.cmml" xref="S5.Ex29.m1.5.5.1.1.3.3.2"></root><apply id="S5.Ex29.m1.4.4.2.cmml" xref="S5.Ex29.m1.4.4.2"><minus id="S5.Ex29.m1.4.4.2.1.cmml" xref="S5.Ex29.m1.4.4.2.1"></minus><cn id="S5.Ex29.m1.4.4.2.2.cmml" type="integer" xref="S5.Ex29.m1.4.4.2.2">1</cn><ci id="S5.Ex29.m1.4.4.2.3.cmml" xref="S5.Ex29.m1.4.4.2.3">𝜀</ci></apply></apply><apply id="S5.Ex29.m1.5.5.1.1.3.4.cmml" xref="S5.Ex29.m1.5.5.1.1.3.4"><csymbol cd="ambiguous" id="S5.Ex29.m1.5.5.1.1.3.4.1.cmml" xref="S5.Ex29.m1.5.5.1.1.3.4">superscript</csymbol><ci id="S5.Ex29.m1.5.5.1.1.3.4.2.cmml" xref="S5.Ex29.m1.5.5.1.1.3.4.2">𝑒</ci><apply id="S5.Ex29.m1.1.1.1.cmml" xref="S5.Ex29.m1.1.1.1"><times id="S5.Ex29.m1.1.1.1.2.cmml" xref="S5.Ex29.m1.1.1.1.2"></times><apply id="S5.Ex29.m1.1.1.1.3.cmml" xref="S5.Ex29.m1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Ex29.m1.1.1.1.3.1.cmml" xref="S5.Ex29.m1.1.1.1.3">superscript</csymbol><ci id="S5.Ex29.m1.1.1.1.3.2.cmml" xref="S5.Ex29.m1.1.1.1.3.2">𝑘</ci><ci id="S5.Ex29.m1.1.1.1.3.3.cmml" xref="S5.Ex29.m1.1.1.1.3.3">′</ci></apply><apply id="S5.Ex29.m1.1.1.1.1.cmml" xref="S5.Ex29.m1.1.1.1.4.2"><root id="S5.Ex29.m1.1.1.1.1a.cmml" xref="S5.Ex29.m1.1.1.1.4.2"></root><apply id="S5.Ex29.m1.1.1.1.1.2.cmml" xref="S5.Ex29.m1.1.1.1.1.2"><minus id="S5.Ex29.m1.1.1.1.1.2.1.cmml" xref="S5.Ex29.m1.1.1.1.1.2.1"></minus><cn id="S5.Ex29.m1.1.1.1.1.2.2.cmml" type="integer" xref="S5.Ex29.m1.1.1.1.1.2.2">1</cn><ci id="S5.Ex29.m1.1.1.1.1.2.3.cmml" xref="S5.Ex29.m1.1.1.1.1.2.3">𝜀</ci></apply></apply><apply id="S5.Ex29.m1.1.1.1.5.cmml" xref="S5.Ex29.m1.1.1.1.5"><apply id="S5.Ex29.m1.1.1.1.5.1.cmml" xref="S5.Ex29.m1.1.1.1.5.1"><csymbol cd="ambiguous" id="S5.Ex29.m1.1.1.1.5.1.1.cmml" xref="S5.Ex29.m1.1.1.1.5.1">subscript</csymbol><log id="S5.Ex29.m1.1.1.1.5.1.2.cmml" xref="S5.Ex29.m1.1.1.1.5.1.2"></log><ci id="S5.Ex29.m1.1.1.1.5.1.3.cmml" xref="S5.Ex29.m1.1.1.1.5.1.3">𝑒</ci></apply><ci id="S5.Ex29.m1.1.1.1.5.2.cmml" xref="S5.Ex29.m1.1.1.1.5.2">𝑛</ci></apply></apply></apply><apply id="S5.Ex29.m1.5.5.1.1.3.5.cmml" xref="S5.Ex29.m1.5.5.1.1.3.5"><csymbol cd="ambiguous" id="S5.Ex29.m1.5.5.1.1.3.5.1.cmml" xref="S5.Ex29.m1.5.5.1.1.3.5">superscript</csymbol><ci id="S5.Ex29.m1.5.5.1.1.3.5.2.cmml" xref="S5.Ex29.m1.5.5.1.1.3.5.2">𝑒</ci><apply id="S5.Ex29.m1.2.2.1.cmml" xref="S5.Ex29.m1.2.2.1"><times id="S5.Ex29.m1.2.2.1.2.cmml" xref="S5.Ex29.m1.2.2.1.2"></times><apply id="S5.Ex29.m1.2.2.1.3.cmml" xref="S5.Ex29.m1.2.2.1.3"><csymbol cd="ambiguous" id="S5.Ex29.m1.2.2.1.3.1.cmml" xref="S5.Ex29.m1.2.2.1.3">subscript</csymbol><ci id="S5.Ex29.m1.2.2.1.3.2.cmml" xref="S5.Ex29.m1.2.2.1.3.2">𝑐</ci><cn id="S5.Ex29.m1.2.2.1.3.3.cmml" type="integer" xref="S5.Ex29.m1.2.2.1.3.3">0</cn></apply><apply id="S5.Ex29.m1.2.2.1.1.2.cmml" xref="S5.Ex29.m1.2.2.1.1.1"><csymbol cd="latexml" id="S5.Ex29.m1.2.2.1.1.2.1.cmml" xref="S5.Ex29.m1.2.2.1.1.1.2">delimited-[]</csymbol><apply id="S5.Ex29.m1.2.2.1.1.1.1.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1"><minus id="S5.Ex29.m1.2.2.1.1.1.1.3.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1"></minus><apply id="S5.Ex29.m1.2.2.1.1.1.1.2.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2"><times id="S5.Ex29.m1.2.2.1.1.1.1.2.3.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.3"></times><ci id="S5.Ex29.m1.2.2.1.1.1.1.2.4.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.4">𝑛</ci><apply id="S5.Ex29.m1.2.2.1.1.1.1.2.5.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.5"><divide id="S5.Ex29.m1.2.2.1.1.1.1.2.5.1.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.5"></divide><apply id="S5.Ex29.m1.2.2.1.1.1.1.2.5.2.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.5.2"><csymbol cd="ambiguous" id="S5.Ex29.m1.2.2.1.1.1.1.2.5.2.1.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.5.2">superscript</csymbol><ci id="S5.Ex29.m1.2.2.1.1.1.1.2.5.2.2.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.5.2.2">𝑘</ci><ci id="S5.Ex29.m1.2.2.1.1.1.1.2.5.2.3.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.5.2.3">′</ci></apply><ci id="S5.Ex29.m1.2.2.1.1.1.1.2.5.3.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.5.3">𝑘</ci></apply><apply id="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.1.1.1"><minus id="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.1"></minus><cn id="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.2">1</cn><ci id="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.1.1.1.1.3">𝛼</ci></apply><apply id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1"><minus id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.1.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.1"></minus><cn id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.2.cmml" type="integer" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.2">1</cn><apply id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3"><root id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3a.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3"></root><apply id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2"><minus id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2.1.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2.1"></minus><cn id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2.2.cmml" type="integer" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2.2">1</cn><ci id="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2.3.cmml" xref="S5.Ex29.m1.2.2.1.1.1.1.2.2.1.1.3.2.3">𝜀</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex29.m1.5c">\displaystyle\mathbb{E}[X]\leq k^{\prime}(\sqrt{1-\varepsilon})e^{k^{\prime}(% \sqrt{1-\varepsilon})\log_{e}{n}}e^{c_{0}\left[-n\frac{k^{\prime}}{k}(1-\alpha% )(1-\sqrt{1-\varepsilon})\right]}.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex29.m1.5d">blackboard_E [ italic_X ] ≤ italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( square-root start_ARG 1 - italic_ε end_ARG ) italic_e start_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( square-root start_ARG 1 - italic_ε end_ARG ) roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT [ - italic_n divide start_ARG italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_k end_ARG ( 1 - italic_α ) ( 1 - square-root start_ARG 1 - italic_ε end_ARG ) ] end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S5.SS2.SSS1.5.p5"> <p class="ltx_p" id="S5.SS2.SSS1.5.p5.2">Let <math alttext="a_{0}=\sqrt{1-\varepsilon}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.5.p5.1.m1.1"><semantics id="S5.SS2.SSS1.5.p5.1.m1.1a"><mrow id="S5.SS2.SSS1.5.p5.1.m1.1.1" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.cmml"><msub id="S5.SS2.SSS1.5.p5.1.m1.1.1.2" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.2.cmml"><mi id="S5.SS2.SSS1.5.p5.1.m1.1.1.2.2" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.2.2.cmml">a</mi><mn id="S5.SS2.SSS1.5.p5.1.m1.1.1.2.3" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.2.3.cmml">0</mn></msub><mo id="S5.SS2.SSS1.5.p5.1.m1.1.1.1" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.1.cmml">=</mo><msqrt id="S5.SS2.SSS1.5.p5.1.m1.1.1.3" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.3.cmml"><mrow id="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2.cmml"><mn id="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2.2" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2.1" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2.1.cmml">−</mo><mi id="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2.3" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.5.p5.1.m1.1b"><apply id="S5.SS2.SSS1.5.p5.1.m1.1.1.cmml" xref="S5.SS2.SSS1.5.p5.1.m1.1.1"><eq id="S5.SS2.SSS1.5.p5.1.m1.1.1.1.cmml" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.1"></eq><apply id="S5.SS2.SSS1.5.p5.1.m1.1.1.2.cmml" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.5.p5.1.m1.1.1.2.1.cmml" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.2">subscript</csymbol><ci id="S5.SS2.SSS1.5.p5.1.m1.1.1.2.2.cmml" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.2.2">𝑎</ci><cn id="S5.SS2.SSS1.5.p5.1.m1.1.1.2.3.cmml" type="integer" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.2.3">0</cn></apply><apply id="S5.SS2.SSS1.5.p5.1.m1.1.1.3.cmml" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.3"><root id="S5.SS2.SSS1.5.p5.1.m1.1.1.3a.cmml" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.3"></root><apply id="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2.cmml" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2"><minus id="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2.1.cmml" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2.1"></minus><cn id="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2.2.cmml" type="integer" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2.2">1</cn><ci id="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2.3.cmml" xref="S5.SS2.SSS1.5.p5.1.m1.1.1.3.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.5.p5.1.m1.1c">a_{0}=\sqrt{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.5.p5.1.m1.1d">italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = square-root start_ARG 1 - italic_ε end_ARG</annotation></semantics></math> and <math alttext="b_{0}=c_{0}(1-\alpha)(1-\sqrt{1-\varepsilon})" class="ltx_Math" display="inline" id="S5.SS2.SSS1.5.p5.2.m2.2"><semantics id="S5.SS2.SSS1.5.p5.2.m2.2a"><mrow id="S5.SS2.SSS1.5.p5.2.m2.2.2" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.cmml"><msub id="S5.SS2.SSS1.5.p5.2.m2.2.2.4" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.4.cmml"><mi id="S5.SS2.SSS1.5.p5.2.m2.2.2.4.2" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.4.2.cmml">b</mi><mn id="S5.SS2.SSS1.5.p5.2.m2.2.2.4.3" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.4.3.cmml">0</mn></msub><mo id="S5.SS2.SSS1.5.p5.2.m2.2.2.3" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.3.cmml">=</mo><mrow id="S5.SS2.SSS1.5.p5.2.m2.2.2.2" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.cmml"><msub id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.4" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.4.cmml"><mi id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.4.2" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.4.2.cmml">c</mi><mn id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.4.3" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.4.3.cmml">0</mn></msub><mo id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.3" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.3.cmml">⁢</mo><mrow id="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1" xref="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.cmml"><mo id="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1" xref="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.cmml"><mn id="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.2" xref="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.1" xref="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.1.cmml">−</mo><mi id="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.3" xref="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.3.cmml">α</mi></mrow><mo id="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.3a" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.3.cmml">⁢</mo><mrow id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.cmml"><mo id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.2" stretchy="false" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.cmml">(</mo><mrow id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.cmml"><mn id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.2" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.2.cmml">1</mn><mo id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.1" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.1.cmml">−</mo><msqrt id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.cmml"><mrow id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2.cmml"><mn id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2.2" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2.1" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2.1.cmml">−</mo><mi id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2.3" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><mo id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.3" stretchy="false" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.5.p5.2.m2.2b"><apply id="S5.SS2.SSS1.5.p5.2.m2.2.2.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2"><eq id="S5.SS2.SSS1.5.p5.2.m2.2.2.3.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.3"></eq><apply id="S5.SS2.SSS1.5.p5.2.m2.2.2.4.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.4"><csymbol cd="ambiguous" id="S5.SS2.SSS1.5.p5.2.m2.2.2.4.1.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.4">subscript</csymbol><ci id="S5.SS2.SSS1.5.p5.2.m2.2.2.4.2.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.4.2">𝑏</ci><cn id="S5.SS2.SSS1.5.p5.2.m2.2.2.4.3.cmml" type="integer" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.4.3">0</cn></apply><apply id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2"><times id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.3.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.3"></times><apply id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.4.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.4"><csymbol cd="ambiguous" id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.4.1.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.4">subscript</csymbol><ci id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.4.2.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.4.2">𝑐</ci><cn id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.4.3.cmml" type="integer" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.4.3">0</cn></apply><apply id="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1"><minus id="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.1"></minus><cn id="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.2">1</cn><ci id="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.3.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.1.1.1.1.1.1.3">𝛼</ci></apply><apply id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1"><minus id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.1.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.1"></minus><cn id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.2.cmml" type="integer" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.2">1</cn><apply id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3"><root id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3a.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3"></root><apply id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2"><minus id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2.1.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2.1"></minus><cn id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2.2.cmml" type="integer" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2.2">1</cn><ci id="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2.3.cmml" xref="S5.SS2.SSS1.5.p5.2.m2.2.2.2.2.1.1.3.2.3">𝜀</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.5.p5.2.m2.2c">b_{0}=c_{0}(1-\alpha)(1-\sqrt{1-\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.5.p5.2.m2.2d">italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( 1 - italic_α ) ( 1 - square-root start_ARG 1 - italic_ε end_ARG )</annotation></semantics></math> be two constant numbers,</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex30"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathbb{E}[X]\leq k^{\prime}e^{k^{\prime}[a_{0}\log_{e}n-b_{0}\frac{n}{k}]}% \text{.}" class="ltx_Math" display="block" id="S5.Ex30.m1.2"><semantics id="S5.Ex30.m1.2a"><mrow id="S5.Ex30.m1.2.3" xref="S5.Ex30.m1.2.3.cmml"><mrow id="S5.Ex30.m1.2.3.2" xref="S5.Ex30.m1.2.3.2.cmml"><mi id="S5.Ex30.m1.2.3.2.2" xref="S5.Ex30.m1.2.3.2.2.cmml">𝔼</mi><mo id="S5.Ex30.m1.2.3.2.1" xref="S5.Ex30.m1.2.3.2.1.cmml">⁢</mo><mrow id="S5.Ex30.m1.2.3.2.3.2" xref="S5.Ex30.m1.2.3.2.3.1.cmml"><mo id="S5.Ex30.m1.2.3.2.3.2.1" stretchy="false" xref="S5.Ex30.m1.2.3.2.3.1.1.cmml">[</mo><mi id="S5.Ex30.m1.2.2" xref="S5.Ex30.m1.2.2.cmml">X</mi><mo id="S5.Ex30.m1.2.3.2.3.2.2" stretchy="false" xref="S5.Ex30.m1.2.3.2.3.1.1.cmml">]</mo></mrow></mrow><mo id="S5.Ex30.m1.2.3.1" xref="S5.Ex30.m1.2.3.1.cmml">≤</mo><mrow id="S5.Ex30.m1.2.3.3" xref="S5.Ex30.m1.2.3.3.cmml"><msup id="S5.Ex30.m1.2.3.3.2" xref="S5.Ex30.m1.2.3.3.2.cmml"><mi id="S5.Ex30.m1.2.3.3.2.2" xref="S5.Ex30.m1.2.3.3.2.2.cmml">k</mi><mo id="S5.Ex30.m1.2.3.3.2.3" xref="S5.Ex30.m1.2.3.3.2.3.cmml">′</mo></msup><mo id="S5.Ex30.m1.2.3.3.1" xref="S5.Ex30.m1.2.3.3.1.cmml">⁢</mo><msup id="S5.Ex30.m1.2.3.3.3" xref="S5.Ex30.m1.2.3.3.3.cmml"><mi id="S5.Ex30.m1.2.3.3.3.2" xref="S5.Ex30.m1.2.3.3.3.2.cmml">e</mi><mrow id="S5.Ex30.m1.1.1.1" xref="S5.Ex30.m1.1.1.1.cmml"><msup id="S5.Ex30.m1.1.1.1.3" xref="S5.Ex30.m1.1.1.1.3.cmml"><mi id="S5.Ex30.m1.1.1.1.3.2" xref="S5.Ex30.m1.1.1.1.3.2.cmml">k</mi><mo id="S5.Ex30.m1.1.1.1.3.3" xref="S5.Ex30.m1.1.1.1.3.3.cmml">′</mo></msup><mo id="S5.Ex30.m1.1.1.1.2" xref="S5.Ex30.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex30.m1.1.1.1.1.1" xref="S5.Ex30.m1.1.1.1.1.2.cmml"><mo id="S5.Ex30.m1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex30.m1.1.1.1.1.2.1.cmml">[</mo><mrow id="S5.Ex30.m1.1.1.1.1.1.1" xref="S5.Ex30.m1.1.1.1.1.1.1.cmml"><mrow id="S5.Ex30.m1.1.1.1.1.1.1.2" xref="S5.Ex30.m1.1.1.1.1.1.1.2.cmml"><msub id="S5.Ex30.m1.1.1.1.1.1.1.2.2" xref="S5.Ex30.m1.1.1.1.1.1.1.2.2.cmml"><mi id="S5.Ex30.m1.1.1.1.1.1.1.2.2.2" xref="S5.Ex30.m1.1.1.1.1.1.1.2.2.2.cmml">a</mi><mn id="S5.Ex30.m1.1.1.1.1.1.1.2.2.3" xref="S5.Ex30.m1.1.1.1.1.1.1.2.2.3.cmml">0</mn></msub><mo id="S5.Ex30.m1.1.1.1.1.1.1.2.1" lspace="0.167em" xref="S5.Ex30.m1.1.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S5.Ex30.m1.1.1.1.1.1.1.2.3" xref="S5.Ex30.m1.1.1.1.1.1.1.2.3.cmml"><msub id="S5.Ex30.m1.1.1.1.1.1.1.2.3.1" xref="S5.Ex30.m1.1.1.1.1.1.1.2.3.1.cmml"><mi id="S5.Ex30.m1.1.1.1.1.1.1.2.3.1.2" xref="S5.Ex30.m1.1.1.1.1.1.1.2.3.1.2.cmml">log</mi><mi id="S5.Ex30.m1.1.1.1.1.1.1.2.3.1.3" xref="S5.Ex30.m1.1.1.1.1.1.1.2.3.1.3.cmml">e</mi></msub><mo id="S5.Ex30.m1.1.1.1.1.1.1.2.3a" lspace="0.167em" xref="S5.Ex30.m1.1.1.1.1.1.1.2.3.cmml">⁡</mo><mi id="S5.Ex30.m1.1.1.1.1.1.1.2.3.2" xref="S5.Ex30.m1.1.1.1.1.1.1.2.3.2.cmml">n</mi></mrow></mrow><mo id="S5.Ex30.m1.1.1.1.1.1.1.1" xref="S5.Ex30.m1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S5.Ex30.m1.1.1.1.1.1.1.3" xref="S5.Ex30.m1.1.1.1.1.1.1.3.cmml"><msub id="S5.Ex30.m1.1.1.1.1.1.1.3.2" xref="S5.Ex30.m1.1.1.1.1.1.1.3.2.cmml"><mi id="S5.Ex30.m1.1.1.1.1.1.1.3.2.2" xref="S5.Ex30.m1.1.1.1.1.1.1.3.2.2.cmml">b</mi><mn id="S5.Ex30.m1.1.1.1.1.1.1.3.2.3" xref="S5.Ex30.m1.1.1.1.1.1.1.3.2.3.cmml">0</mn></msub><mo id="S5.Ex30.m1.1.1.1.1.1.1.3.1" xref="S5.Ex30.m1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mfrac id="S5.Ex30.m1.1.1.1.1.1.1.3.3" xref="S5.Ex30.m1.1.1.1.1.1.1.3.3.cmml"><mi id="S5.Ex30.m1.1.1.1.1.1.1.3.3.2" xref="S5.Ex30.m1.1.1.1.1.1.1.3.3.2.cmml">n</mi><mi id="S5.Ex30.m1.1.1.1.1.1.1.3.3.3" xref="S5.Ex30.m1.1.1.1.1.1.1.3.3.3.cmml">k</mi></mfrac></mrow></mrow><mo id="S5.Ex30.m1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex30.m1.1.1.1.1.2.1.cmml">]</mo></mrow></mrow></msup><mo id="S5.Ex30.m1.2.3.3.1a" xref="S5.Ex30.m1.2.3.3.1.cmml">⁢</mo><mtext id="S5.Ex30.m1.2.3.3.4" xref="S5.Ex30.m1.2.3.3.4a.cmml">.</mtext></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex30.m1.2b"><apply id="S5.Ex30.m1.2.3.cmml" xref="S5.Ex30.m1.2.3"><leq id="S5.Ex30.m1.2.3.1.cmml" xref="S5.Ex30.m1.2.3.1"></leq><apply id="S5.Ex30.m1.2.3.2.cmml" xref="S5.Ex30.m1.2.3.2"><times id="S5.Ex30.m1.2.3.2.1.cmml" xref="S5.Ex30.m1.2.3.2.1"></times><ci id="S5.Ex30.m1.2.3.2.2.cmml" xref="S5.Ex30.m1.2.3.2.2">𝔼</ci><apply id="S5.Ex30.m1.2.3.2.3.1.cmml" xref="S5.Ex30.m1.2.3.2.3.2"><csymbol cd="latexml" id="S5.Ex30.m1.2.3.2.3.1.1.cmml" xref="S5.Ex30.m1.2.3.2.3.2.1">delimited-[]</csymbol><ci id="S5.Ex30.m1.2.2.cmml" xref="S5.Ex30.m1.2.2">𝑋</ci></apply></apply><apply id="S5.Ex30.m1.2.3.3.cmml" xref="S5.Ex30.m1.2.3.3"><times id="S5.Ex30.m1.2.3.3.1.cmml" xref="S5.Ex30.m1.2.3.3.1"></times><apply id="S5.Ex30.m1.2.3.3.2.cmml" xref="S5.Ex30.m1.2.3.3.2"><csymbol cd="ambiguous" id="S5.Ex30.m1.2.3.3.2.1.cmml" xref="S5.Ex30.m1.2.3.3.2">superscript</csymbol><ci id="S5.Ex30.m1.2.3.3.2.2.cmml" xref="S5.Ex30.m1.2.3.3.2.2">𝑘</ci><ci id="S5.Ex30.m1.2.3.3.2.3.cmml" xref="S5.Ex30.m1.2.3.3.2.3">′</ci></apply><apply id="S5.Ex30.m1.2.3.3.3.cmml" xref="S5.Ex30.m1.2.3.3.3"><csymbol cd="ambiguous" id="S5.Ex30.m1.2.3.3.3.1.cmml" xref="S5.Ex30.m1.2.3.3.3">superscript</csymbol><ci id="S5.Ex30.m1.2.3.3.3.2.cmml" xref="S5.Ex30.m1.2.3.3.3.2">𝑒</ci><apply id="S5.Ex30.m1.1.1.1.cmml" xref="S5.Ex30.m1.1.1.1"><times id="S5.Ex30.m1.1.1.1.2.cmml" xref="S5.Ex30.m1.1.1.1.2"></times><apply id="S5.Ex30.m1.1.1.1.3.cmml" xref="S5.Ex30.m1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Ex30.m1.1.1.1.3.1.cmml" xref="S5.Ex30.m1.1.1.1.3">superscript</csymbol><ci id="S5.Ex30.m1.1.1.1.3.2.cmml" xref="S5.Ex30.m1.1.1.1.3.2">𝑘</ci><ci id="S5.Ex30.m1.1.1.1.3.3.cmml" xref="S5.Ex30.m1.1.1.1.3.3">′</ci></apply><apply id="S5.Ex30.m1.1.1.1.1.2.cmml" xref="S5.Ex30.m1.1.1.1.1.1"><csymbol cd="latexml" id="S5.Ex30.m1.1.1.1.1.2.1.cmml" xref="S5.Ex30.m1.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S5.Ex30.m1.1.1.1.1.1.1.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1"><minus id="S5.Ex30.m1.1.1.1.1.1.1.1.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.1"></minus><apply id="S5.Ex30.m1.1.1.1.1.1.1.2.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.2"><times id="S5.Ex30.m1.1.1.1.1.1.1.2.1.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.2.1"></times><apply id="S5.Ex30.m1.1.1.1.1.1.1.2.2.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S5.Ex30.m1.1.1.1.1.1.1.2.2.1.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.2.2">subscript</csymbol><ci id="S5.Ex30.m1.1.1.1.1.1.1.2.2.2.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.2.2.2">𝑎</ci><cn id="S5.Ex30.m1.1.1.1.1.1.1.2.2.3.cmml" type="integer" xref="S5.Ex30.m1.1.1.1.1.1.1.2.2.3">0</cn></apply><apply id="S5.Ex30.m1.1.1.1.1.1.1.2.3.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.2.3"><apply id="S5.Ex30.m1.1.1.1.1.1.1.2.3.1.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.2.3.1"><csymbol cd="ambiguous" id="S5.Ex30.m1.1.1.1.1.1.1.2.3.1.1.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.2.3.1">subscript</csymbol><log id="S5.Ex30.m1.1.1.1.1.1.1.2.3.1.2.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.2.3.1.2"></log><ci id="S5.Ex30.m1.1.1.1.1.1.1.2.3.1.3.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.2.3.1.3">𝑒</ci></apply><ci id="S5.Ex30.m1.1.1.1.1.1.1.2.3.2.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.2.3.2">𝑛</ci></apply></apply><apply id="S5.Ex30.m1.1.1.1.1.1.1.3.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.3"><times id="S5.Ex30.m1.1.1.1.1.1.1.3.1.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.3.1"></times><apply id="S5.Ex30.m1.1.1.1.1.1.1.3.2.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S5.Ex30.m1.1.1.1.1.1.1.3.2.1.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.3.2">subscript</csymbol><ci id="S5.Ex30.m1.1.1.1.1.1.1.3.2.2.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.3.2.2">𝑏</ci><cn id="S5.Ex30.m1.1.1.1.1.1.1.3.2.3.cmml" type="integer" xref="S5.Ex30.m1.1.1.1.1.1.1.3.2.3">0</cn></apply><apply id="S5.Ex30.m1.1.1.1.1.1.1.3.3.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.3.3"><divide id="S5.Ex30.m1.1.1.1.1.1.1.3.3.1.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.3.3"></divide><ci id="S5.Ex30.m1.1.1.1.1.1.1.3.3.2.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.3.3.2">𝑛</ci><ci id="S5.Ex30.m1.1.1.1.1.1.1.3.3.3.cmml" xref="S5.Ex30.m1.1.1.1.1.1.1.3.3.3">𝑘</ci></apply></apply></apply></apply></apply></apply><ci id="S5.Ex30.m1.2.3.3.4a.cmml" xref="S5.Ex30.m1.2.3.3.4"><mtext id="S5.Ex30.m1.2.3.3.4.cmml" xref="S5.Ex30.m1.2.3.3.4">.</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex30.m1.2c">\mathbb{E}[X]\leq k^{\prime}e^{k^{\prime}[a_{0}\log_{e}n-b_{0}\frac{n}{k}]}% \text{.}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex30.m1.2d">blackboard_E [ italic_X ] ≤ italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT [ italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n - italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ] end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S5.SS2.SSS1.6.p6"> <p class="ltx_p" id="S5.SS2.SSS1.6.p6.5">Since <math alttext="k=o\left(\tfrac{n}{\log n}\right)" class="ltx_Math" display="inline" id="S5.SS2.SSS1.6.p6.1.m1.1"><semantics id="S5.SS2.SSS1.6.p6.1.m1.1a"><mrow id="S5.SS2.SSS1.6.p6.1.m1.1.2" xref="S5.SS2.SSS1.6.p6.1.m1.1.2.cmml"><mi id="S5.SS2.SSS1.6.p6.1.m1.1.2.2" xref="S5.SS2.SSS1.6.p6.1.m1.1.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.6.p6.1.m1.1.2.1" xref="S5.SS2.SSS1.6.p6.1.m1.1.2.1.cmml">=</mo><mrow id="S5.SS2.SSS1.6.p6.1.m1.1.2.3" xref="S5.SS2.SSS1.6.p6.1.m1.1.2.3.cmml"><mi id="S5.SS2.SSS1.6.p6.1.m1.1.2.3.2" xref="S5.SS2.SSS1.6.p6.1.m1.1.2.3.2.cmml">o</mi><mo id="S5.SS2.SSS1.6.p6.1.m1.1.2.3.1" xref="S5.SS2.SSS1.6.p6.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS1.6.p6.1.m1.1.2.3.3.2" xref="S5.SS2.SSS1.6.p6.1.m1.1.1.cmml"><mo id="S5.SS2.SSS1.6.p6.1.m1.1.2.3.3.2.1" xref="S5.SS2.SSS1.6.p6.1.m1.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS1.6.p6.1.m1.1.1" xref="S5.SS2.SSS1.6.p6.1.m1.1.1.cmml"><mi id="S5.SS2.SSS1.6.p6.1.m1.1.1.2" xref="S5.SS2.SSS1.6.p6.1.m1.1.1.2.cmml">n</mi><mrow id="S5.SS2.SSS1.6.p6.1.m1.1.1.3" xref="S5.SS2.SSS1.6.p6.1.m1.1.1.3.cmml"><mi id="S5.SS2.SSS1.6.p6.1.m1.1.1.3.1" xref="S5.SS2.SSS1.6.p6.1.m1.1.1.3.1.cmml">log</mi><mo id="S5.SS2.SSS1.6.p6.1.m1.1.1.3a" lspace="0.167em" xref="S5.SS2.SSS1.6.p6.1.m1.1.1.3.cmml">⁡</mo><mi id="S5.SS2.SSS1.6.p6.1.m1.1.1.3.2" xref="S5.SS2.SSS1.6.p6.1.m1.1.1.3.2.cmml">n</mi></mrow></mfrac><mo id="S5.SS2.SSS1.6.p6.1.m1.1.2.3.3.2.2" xref="S5.SS2.SSS1.6.p6.1.m1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.6.p6.1.m1.1b"><apply id="S5.SS2.SSS1.6.p6.1.m1.1.2.cmml" xref="S5.SS2.SSS1.6.p6.1.m1.1.2"><eq id="S5.SS2.SSS1.6.p6.1.m1.1.2.1.cmml" xref="S5.SS2.SSS1.6.p6.1.m1.1.2.1"></eq><ci id="S5.SS2.SSS1.6.p6.1.m1.1.2.2.cmml" xref="S5.SS2.SSS1.6.p6.1.m1.1.2.2">𝑘</ci><apply id="S5.SS2.SSS1.6.p6.1.m1.1.2.3.cmml" xref="S5.SS2.SSS1.6.p6.1.m1.1.2.3"><times id="S5.SS2.SSS1.6.p6.1.m1.1.2.3.1.cmml" xref="S5.SS2.SSS1.6.p6.1.m1.1.2.3.1"></times><ci id="S5.SS2.SSS1.6.p6.1.m1.1.2.3.2.cmml" xref="S5.SS2.SSS1.6.p6.1.m1.1.2.3.2">𝑜</ci><apply id="S5.SS2.SSS1.6.p6.1.m1.1.1.cmml" xref="S5.SS2.SSS1.6.p6.1.m1.1.2.3.3.2"><divide id="S5.SS2.SSS1.6.p6.1.m1.1.1.1.cmml" xref="S5.SS2.SSS1.6.p6.1.m1.1.2.3.3.2"></divide><ci id="S5.SS2.SSS1.6.p6.1.m1.1.1.2.cmml" xref="S5.SS2.SSS1.6.p6.1.m1.1.1.2">𝑛</ci><apply id="S5.SS2.SSS1.6.p6.1.m1.1.1.3.cmml" xref="S5.SS2.SSS1.6.p6.1.m1.1.1.3"><log id="S5.SS2.SSS1.6.p6.1.m1.1.1.3.1.cmml" xref="S5.SS2.SSS1.6.p6.1.m1.1.1.3.1"></log><ci id="S5.SS2.SSS1.6.p6.1.m1.1.1.3.2.cmml" xref="S5.SS2.SSS1.6.p6.1.m1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.6.p6.1.m1.1c">k=o\left(\tfrac{n}{\log n}\right)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.6.p6.1.m1.1d">italic_k = italic_o ( divide start_ARG italic_n end_ARG start_ARG roman_log italic_n end_ARG )</annotation></semantics></math>, we have that <math alttext="\mathbb{E}[X]" class="ltx_Math" display="inline" id="S5.SS2.SSS1.6.p6.2.m2.1"><semantics id="S5.SS2.SSS1.6.p6.2.m2.1a"><mrow id="S5.SS2.SSS1.6.p6.2.m2.1.2" xref="S5.SS2.SSS1.6.p6.2.m2.1.2.cmml"><mi id="S5.SS2.SSS1.6.p6.2.m2.1.2.2" xref="S5.SS2.SSS1.6.p6.2.m2.1.2.2.cmml">𝔼</mi><mo id="S5.SS2.SSS1.6.p6.2.m2.1.2.1" xref="S5.SS2.SSS1.6.p6.2.m2.1.2.1.cmml">⁢</mo><mrow id="S5.SS2.SSS1.6.p6.2.m2.1.2.3.2" xref="S5.SS2.SSS1.6.p6.2.m2.1.2.3.1.cmml"><mo id="S5.SS2.SSS1.6.p6.2.m2.1.2.3.2.1" stretchy="false" xref="S5.SS2.SSS1.6.p6.2.m2.1.2.3.1.1.cmml">[</mo><mi id="S5.SS2.SSS1.6.p6.2.m2.1.1" xref="S5.SS2.SSS1.6.p6.2.m2.1.1.cmml">X</mi><mo id="S5.SS2.SSS1.6.p6.2.m2.1.2.3.2.2" stretchy="false" xref="S5.SS2.SSS1.6.p6.2.m2.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.6.p6.2.m2.1b"><apply id="S5.SS2.SSS1.6.p6.2.m2.1.2.cmml" xref="S5.SS2.SSS1.6.p6.2.m2.1.2"><times id="S5.SS2.SSS1.6.p6.2.m2.1.2.1.cmml" xref="S5.SS2.SSS1.6.p6.2.m2.1.2.1"></times><ci id="S5.SS2.SSS1.6.p6.2.m2.1.2.2.cmml" xref="S5.SS2.SSS1.6.p6.2.m2.1.2.2">𝔼</ci><apply id="S5.SS2.SSS1.6.p6.2.m2.1.2.3.1.cmml" xref="S5.SS2.SSS1.6.p6.2.m2.1.2.3.2"><csymbol cd="latexml" id="S5.SS2.SSS1.6.p6.2.m2.1.2.3.1.1.cmml" xref="S5.SS2.SSS1.6.p6.2.m2.1.2.3.2.1">delimited-[]</csymbol><ci id="S5.SS2.SSS1.6.p6.2.m2.1.1.cmml" xref="S5.SS2.SSS1.6.p6.2.m2.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.6.p6.2.m2.1c">\mathbb{E}[X]</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.6.p6.2.m2.1d">blackboard_E [ italic_X ]</annotation></semantics></math> tends to zero as <math alttext="n" class="ltx_Math" display="inline" id="S5.SS2.SSS1.6.p6.3.m3.1"><semantics id="S5.SS2.SSS1.6.p6.3.m3.1a"><mi id="S5.SS2.SSS1.6.p6.3.m3.1.1" xref="S5.SS2.SSS1.6.p6.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.6.p6.3.m3.1b"><ci id="S5.SS2.SSS1.6.p6.3.m3.1.1.cmml" xref="S5.SS2.SSS1.6.p6.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.6.p6.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.6.p6.3.m3.1d">italic_n</annotation></semantics></math> tends to infinity. By Markov’s inequality, the probability of having at least one maximal clique of size at most <math alttext="t_{0}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.6.p6.4.m4.1"><semantics id="S5.SS2.SSS1.6.p6.4.m4.1a"><msub id="S5.SS2.SSS1.6.p6.4.m4.1.1" xref="S5.SS2.SSS1.6.p6.4.m4.1.1.cmml"><mi id="S5.SS2.SSS1.6.p6.4.m4.1.1.2" xref="S5.SS2.SSS1.6.p6.4.m4.1.1.2.cmml">t</mi><mn id="S5.SS2.SSS1.6.p6.4.m4.1.1.3" xref="S5.SS2.SSS1.6.p6.4.m4.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.6.p6.4.m4.1b"><apply id="S5.SS2.SSS1.6.p6.4.m4.1.1.cmml" xref="S5.SS2.SSS1.6.p6.4.m4.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS1.6.p6.4.m4.1.1.1.cmml" xref="S5.SS2.SSS1.6.p6.4.m4.1.1">subscript</csymbol><ci id="S5.SS2.SSS1.6.p6.4.m4.1.1.2.cmml" xref="S5.SS2.SSS1.6.p6.4.m4.1.1.2">𝑡</ci><cn id="S5.SS2.SSS1.6.p6.4.m4.1.1.3.cmml" type="integer" xref="S5.SS2.SSS1.6.p6.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.6.p6.4.m4.1c">t_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.6.p6.4.m4.1d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is at most <math alttext="k^{\prime}e^{-\Theta\left(\frac{nk^{\prime}}{k}\right)}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.6.p6.5.m5.1"><semantics id="S5.SS2.SSS1.6.p6.5.m5.1a"><mrow id="S5.SS2.SSS1.6.p6.5.m5.1.2" xref="S5.SS2.SSS1.6.p6.5.m5.1.2.cmml"><msup id="S5.SS2.SSS1.6.p6.5.m5.1.2.2" xref="S5.SS2.SSS1.6.p6.5.m5.1.2.2.cmml"><mi id="S5.SS2.SSS1.6.p6.5.m5.1.2.2.2" xref="S5.SS2.SSS1.6.p6.5.m5.1.2.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.6.p6.5.m5.1.2.2.3" xref="S5.SS2.SSS1.6.p6.5.m5.1.2.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS1.6.p6.5.m5.1.2.1" xref="S5.SS2.SSS1.6.p6.5.m5.1.2.1.cmml">⁢</mo><msup id="S5.SS2.SSS1.6.p6.5.m5.1.2.3" xref="S5.SS2.SSS1.6.p6.5.m5.1.2.3.cmml"><mi id="S5.SS2.SSS1.6.p6.5.m5.1.2.3.2" xref="S5.SS2.SSS1.6.p6.5.m5.1.2.3.2.cmml">e</mi><mrow id="S5.SS2.SSS1.6.p6.5.m5.1.1.1" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.cmml"><mo id="S5.SS2.SSS1.6.p6.5.m5.1.1.1a" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.cmml">−</mo><mrow id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3.cmml"><mi id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3.2" mathvariant="normal" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3.2.cmml">Θ</mi><mo id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3.1" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3.3.2" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.cmml"><mo id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3.3.2.1" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.cmml"><mrow id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.cmml"><mi id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.2" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.2.cmml">n</mi><mo id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.1" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.1.cmml">⁢</mo><msup id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.3" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.3.cmml"><mi id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.3.2" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.3.2.cmml">k</mi><mo id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.3.3" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.3.3.cmml">′</mo></msup></mrow><mi id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.3" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3.3.2.2" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.6.p6.5.m5.1b"><apply id="S5.SS2.SSS1.6.p6.5.m5.1.2.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.2"><times id="S5.SS2.SSS1.6.p6.5.m5.1.2.1.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.2.1"></times><apply id="S5.SS2.SSS1.6.p6.5.m5.1.2.2.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.6.p6.5.m5.1.2.2.1.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.2.2">superscript</csymbol><ci id="S5.SS2.SSS1.6.p6.5.m5.1.2.2.2.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.2.2.2">𝑘</ci><ci id="S5.SS2.SSS1.6.p6.5.m5.1.2.2.3.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.2.2.3">′</ci></apply><apply id="S5.SS2.SSS1.6.p6.5.m5.1.2.3.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.2.3"><csymbol cd="ambiguous" id="S5.SS2.SSS1.6.p6.5.m5.1.2.3.1.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.2.3">superscript</csymbol><ci id="S5.SS2.SSS1.6.p6.5.m5.1.2.3.2.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.2.3.2">𝑒</ci><apply id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1"><minus id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.2.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1"></minus><apply id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3"><times id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3.1.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3.1"></times><ci id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3.2.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3.2">Θ</ci><apply id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3.3.2"><divide id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.3.3.2"></divide><apply id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2"><times id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.1.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.1"></times><ci id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.2.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.2">𝑛</ci><apply id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.3.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.3.1.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.3">superscript</csymbol><ci id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.3.2.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.3.2">𝑘</ci><ci id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.3.3.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.2.3.3">′</ci></apply></apply><ci id="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.3.cmml" xref="S5.SS2.SSS1.6.p6.5.m5.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.6.p6.5.m5.1c">k^{\prime}e^{-\Theta\left(\frac{nk^{\prime}}{k}\right)}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.6.p6.5.m5.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS2.SSS1.7.p7"> <p class="ltx_p" id="S5.SS2.SSS1.7.p7.4">Thus, the probability of exiting the first while loop during the <math alttext="i" class="ltx_Math" display="inline" id="S5.SS2.SSS1.7.p7.1.m1.1"><semantics id="S5.SS2.SSS1.7.p7.1.m1.1a"><mi id="S5.SS2.SSS1.7.p7.1.m1.1.1" xref="S5.SS2.SSS1.7.p7.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.7.p7.1.m1.1b"><ci id="S5.SS2.SSS1.7.p7.1.m1.1.1.cmml" xref="S5.SS2.SSS1.7.p7.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.7.p7.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.7.p7.1.m1.1d">italic_i</annotation></semantics></math>th iteration, where <math alttext="i\leq\alpha\frac{n}{k}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.7.p7.2.m2.1"><semantics id="S5.SS2.SSS1.7.p7.2.m2.1a"><mrow id="S5.SS2.SSS1.7.p7.2.m2.1.1" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.cmml"><mi id="S5.SS2.SSS1.7.p7.2.m2.1.1.2" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.2.cmml">i</mi><mo id="S5.SS2.SSS1.7.p7.2.m2.1.1.1" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.1.cmml">≤</mo><mrow id="S5.SS2.SSS1.7.p7.2.m2.1.1.3" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.3.cmml"><mi id="S5.SS2.SSS1.7.p7.2.m2.1.1.3.2" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.3.2.cmml">α</mi><mo id="S5.SS2.SSS1.7.p7.2.m2.1.1.3.1" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.3.1.cmml">⁢</mo><mfrac id="S5.SS2.SSS1.7.p7.2.m2.1.1.3.3" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.3.3.cmml"><mi id="S5.SS2.SSS1.7.p7.2.m2.1.1.3.3.2" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.3.3.2.cmml">n</mi><mi id="S5.SS2.SSS1.7.p7.2.m2.1.1.3.3.3" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.3.3.3.cmml">k</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.7.p7.2.m2.1b"><apply id="S5.SS2.SSS1.7.p7.2.m2.1.1.cmml" xref="S5.SS2.SSS1.7.p7.2.m2.1.1"><leq id="S5.SS2.SSS1.7.p7.2.m2.1.1.1.cmml" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.1"></leq><ci id="S5.SS2.SSS1.7.p7.2.m2.1.1.2.cmml" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.2">𝑖</ci><apply id="S5.SS2.SSS1.7.p7.2.m2.1.1.3.cmml" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.3"><times id="S5.SS2.SSS1.7.p7.2.m2.1.1.3.1.cmml" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.3.1"></times><ci id="S5.SS2.SSS1.7.p7.2.m2.1.1.3.2.cmml" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.3.2">𝛼</ci><apply id="S5.SS2.SSS1.7.p7.2.m2.1.1.3.3.cmml" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.3.3"><divide id="S5.SS2.SSS1.7.p7.2.m2.1.1.3.3.1.cmml" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.3.3"></divide><ci id="S5.SS2.SSS1.7.p7.2.m2.1.1.3.3.2.cmml" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.3.3.2">𝑛</ci><ci id="S5.SS2.SSS1.7.p7.2.m2.1.1.3.3.3.cmml" xref="S5.SS2.SSS1.7.p7.2.m2.1.1.3.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.7.p7.2.m2.1c">i\leq\alpha\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.7.p7.2.m2.1d">italic_i ≤ italic_α divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math>, is also at most <math alttext="k^{\prime}e^{-\Theta\left(\frac{nk^{\prime}}{k}\right)}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.7.p7.3.m3.1"><semantics id="S5.SS2.SSS1.7.p7.3.m3.1a"><mrow id="S5.SS2.SSS1.7.p7.3.m3.1.2" xref="S5.SS2.SSS1.7.p7.3.m3.1.2.cmml"><msup id="S5.SS2.SSS1.7.p7.3.m3.1.2.2" xref="S5.SS2.SSS1.7.p7.3.m3.1.2.2.cmml"><mi id="S5.SS2.SSS1.7.p7.3.m3.1.2.2.2" xref="S5.SS2.SSS1.7.p7.3.m3.1.2.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.7.p7.3.m3.1.2.2.3" xref="S5.SS2.SSS1.7.p7.3.m3.1.2.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS1.7.p7.3.m3.1.2.1" xref="S5.SS2.SSS1.7.p7.3.m3.1.2.1.cmml">⁢</mo><msup id="S5.SS2.SSS1.7.p7.3.m3.1.2.3" xref="S5.SS2.SSS1.7.p7.3.m3.1.2.3.cmml"><mi id="S5.SS2.SSS1.7.p7.3.m3.1.2.3.2" xref="S5.SS2.SSS1.7.p7.3.m3.1.2.3.2.cmml">e</mi><mrow id="S5.SS2.SSS1.7.p7.3.m3.1.1.1" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.cmml"><mo id="S5.SS2.SSS1.7.p7.3.m3.1.1.1a" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.cmml">−</mo><mrow id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3.cmml"><mi id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3.2" mathvariant="normal" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3.2.cmml">Θ</mi><mo id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3.1" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3.3.2" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.cmml"><mo id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3.3.2.1" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.cmml"><mrow id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.cmml"><mi id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.2" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.2.cmml">n</mi><mo id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.1" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.1.cmml">⁢</mo><msup id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.3" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.3.cmml"><mi id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.3.2" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.3.2.cmml">k</mi><mo id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.3.3" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.3.3.cmml">′</mo></msup></mrow><mi id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.3" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3.3.2.2" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.7.p7.3.m3.1b"><apply id="S5.SS2.SSS1.7.p7.3.m3.1.2.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.2"><times id="S5.SS2.SSS1.7.p7.3.m3.1.2.1.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.2.1"></times><apply id="S5.SS2.SSS1.7.p7.3.m3.1.2.2.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.7.p7.3.m3.1.2.2.1.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.2.2">superscript</csymbol><ci id="S5.SS2.SSS1.7.p7.3.m3.1.2.2.2.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.2.2.2">𝑘</ci><ci id="S5.SS2.SSS1.7.p7.3.m3.1.2.2.3.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.2.2.3">′</ci></apply><apply id="S5.SS2.SSS1.7.p7.3.m3.1.2.3.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.2.3"><csymbol cd="ambiguous" id="S5.SS2.SSS1.7.p7.3.m3.1.2.3.1.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.2.3">superscript</csymbol><ci id="S5.SS2.SSS1.7.p7.3.m3.1.2.3.2.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.2.3.2">𝑒</ci><apply id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1"><minus id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.2.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1"></minus><apply id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3"><times id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3.1.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3.1"></times><ci id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3.2.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3.2">Θ</ci><apply id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3.3.2"><divide id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.3.3.2"></divide><apply id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2"><times id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.1.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.1"></times><ci id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.2.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.2">𝑛</ci><apply id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.3.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.3.1.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.3">superscript</csymbol><ci id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.3.2.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.3.2">𝑘</ci><ci id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.3.3.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.2.3.3">′</ci></apply></apply><ci id="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.3.cmml" xref="S5.SS2.SSS1.7.p7.3.m3.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.7.p7.3.m3.1c">k^{\prime}e^{-\Theta\left(\frac{nk^{\prime}}{k}\right)}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.7.p7.3.m3.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT</annotation></semantics></math>. By a union bound, the probability that the algorithm exits the while loop before <math alttext="i=\alpha\frac{n}{k}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.7.p7.4.m4.1"><semantics id="S5.SS2.SSS1.7.p7.4.m4.1a"><mrow id="S5.SS2.SSS1.7.p7.4.m4.1.1" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.cmml"><mi id="S5.SS2.SSS1.7.p7.4.m4.1.1.2" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.2.cmml">i</mi><mo id="S5.SS2.SSS1.7.p7.4.m4.1.1.1" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.1.cmml">=</mo><mrow id="S5.SS2.SSS1.7.p7.4.m4.1.1.3" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.3.cmml"><mi id="S5.SS2.SSS1.7.p7.4.m4.1.1.3.2" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.3.2.cmml">α</mi><mo id="S5.SS2.SSS1.7.p7.4.m4.1.1.3.1" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.3.1.cmml">⁢</mo><mfrac id="S5.SS2.SSS1.7.p7.4.m4.1.1.3.3" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.3.3.cmml"><mi id="S5.SS2.SSS1.7.p7.4.m4.1.1.3.3.2" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.3.3.2.cmml">n</mi><mi id="S5.SS2.SSS1.7.p7.4.m4.1.1.3.3.3" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.3.3.3.cmml">k</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.7.p7.4.m4.1b"><apply id="S5.SS2.SSS1.7.p7.4.m4.1.1.cmml" xref="S5.SS2.SSS1.7.p7.4.m4.1.1"><eq id="S5.SS2.SSS1.7.p7.4.m4.1.1.1.cmml" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.1"></eq><ci id="S5.SS2.SSS1.7.p7.4.m4.1.1.2.cmml" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.2">𝑖</ci><apply id="S5.SS2.SSS1.7.p7.4.m4.1.1.3.cmml" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.3"><times id="S5.SS2.SSS1.7.p7.4.m4.1.1.3.1.cmml" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.3.1"></times><ci id="S5.SS2.SSS1.7.p7.4.m4.1.1.3.2.cmml" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.3.2">𝛼</ci><apply id="S5.SS2.SSS1.7.p7.4.m4.1.1.3.3.cmml" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.3.3"><divide id="S5.SS2.SSS1.7.p7.4.m4.1.1.3.3.1.cmml" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.3.3"></divide><ci id="S5.SS2.SSS1.7.p7.4.m4.1.1.3.3.2.cmml" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.3.3.2">𝑛</ci><ci id="S5.SS2.SSS1.7.p7.4.m4.1.1.3.3.3.cmml" xref="S5.SS2.SSS1.7.p7.4.m4.1.1.3.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.7.p7.4.m4.1c">i=\alpha\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.7.p7.4.m4.1d">italic_i = italic_α divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math> is bounded by</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex31"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\alpha\frac{n}{k}k^{\prime}e^{-\Theta\left(\frac{nk^{\prime}}{k}\right)}=e^{-% \Theta\left(\frac{nk^{\prime}}{k}\right)}." class="ltx_Math" display="block" id="S5.Ex31.m1.3"><semantics id="S5.Ex31.m1.3a"><mrow id="S5.Ex31.m1.3.3.1" xref="S5.Ex31.m1.3.3.1.1.cmml"><mrow id="S5.Ex31.m1.3.3.1.1" xref="S5.Ex31.m1.3.3.1.1.cmml"><mrow id="S5.Ex31.m1.3.3.1.1.2" xref="S5.Ex31.m1.3.3.1.1.2.cmml"><mi id="S5.Ex31.m1.3.3.1.1.2.2" xref="S5.Ex31.m1.3.3.1.1.2.2.cmml">α</mi><mo id="S5.Ex31.m1.3.3.1.1.2.1" xref="S5.Ex31.m1.3.3.1.1.2.1.cmml">⁢</mo><mfrac id="S5.Ex31.m1.3.3.1.1.2.3" xref="S5.Ex31.m1.3.3.1.1.2.3.cmml"><mi id="S5.Ex31.m1.3.3.1.1.2.3.2" xref="S5.Ex31.m1.3.3.1.1.2.3.2.cmml">n</mi><mi id="S5.Ex31.m1.3.3.1.1.2.3.3" xref="S5.Ex31.m1.3.3.1.1.2.3.3.cmml">k</mi></mfrac><mo id="S5.Ex31.m1.3.3.1.1.2.1a" xref="S5.Ex31.m1.3.3.1.1.2.1.cmml">⁢</mo><msup id="S5.Ex31.m1.3.3.1.1.2.4" xref="S5.Ex31.m1.3.3.1.1.2.4.cmml"><mi id="S5.Ex31.m1.3.3.1.1.2.4.2" xref="S5.Ex31.m1.3.3.1.1.2.4.2.cmml">k</mi><mo id="S5.Ex31.m1.3.3.1.1.2.4.3" xref="S5.Ex31.m1.3.3.1.1.2.4.3.cmml">′</mo></msup><mo id="S5.Ex31.m1.3.3.1.1.2.1b" xref="S5.Ex31.m1.3.3.1.1.2.1.cmml">⁢</mo><msup id="S5.Ex31.m1.3.3.1.1.2.5" xref="S5.Ex31.m1.3.3.1.1.2.5.cmml"><mi id="S5.Ex31.m1.3.3.1.1.2.5.2" xref="S5.Ex31.m1.3.3.1.1.2.5.2.cmml">e</mi><mrow id="S5.Ex31.m1.1.1.1" xref="S5.Ex31.m1.1.1.1.cmml"><mo id="S5.Ex31.m1.1.1.1a" xref="S5.Ex31.m1.1.1.1.cmml">−</mo><mrow id="S5.Ex31.m1.1.1.1.3" xref="S5.Ex31.m1.1.1.1.3.cmml"><mi id="S5.Ex31.m1.1.1.1.3.2" mathvariant="normal" xref="S5.Ex31.m1.1.1.1.3.2.cmml">Θ</mi><mo id="S5.Ex31.m1.1.1.1.3.1" xref="S5.Ex31.m1.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.Ex31.m1.1.1.1.3.3.2" xref="S5.Ex31.m1.1.1.1.1.cmml"><mo id="S5.Ex31.m1.1.1.1.3.3.2.1" xref="S5.Ex31.m1.1.1.1.1.cmml">(</mo><mfrac id="S5.Ex31.m1.1.1.1.1" xref="S5.Ex31.m1.1.1.1.1.cmml"><mrow id="S5.Ex31.m1.1.1.1.1.2" xref="S5.Ex31.m1.1.1.1.1.2.cmml"><mi id="S5.Ex31.m1.1.1.1.1.2.2" xref="S5.Ex31.m1.1.1.1.1.2.2.cmml">n</mi><mo id="S5.Ex31.m1.1.1.1.1.2.1" xref="S5.Ex31.m1.1.1.1.1.2.1.cmml">⁢</mo><msup id="S5.Ex31.m1.1.1.1.1.2.3" xref="S5.Ex31.m1.1.1.1.1.2.3.cmml"><mi id="S5.Ex31.m1.1.1.1.1.2.3.2" xref="S5.Ex31.m1.1.1.1.1.2.3.2.cmml">k</mi><mo id="S5.Ex31.m1.1.1.1.1.2.3.3" xref="S5.Ex31.m1.1.1.1.1.2.3.3.cmml">′</mo></msup></mrow><mi id="S5.Ex31.m1.1.1.1.1.3" xref="S5.Ex31.m1.1.1.1.1.3.cmml">k</mi></mfrac><mo id="S5.Ex31.m1.1.1.1.3.3.2.2" xref="S5.Ex31.m1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><mo id="S5.Ex31.m1.3.3.1.1.1" xref="S5.Ex31.m1.3.3.1.1.1.cmml">=</mo><msup id="S5.Ex31.m1.3.3.1.1.3" xref="S5.Ex31.m1.3.3.1.1.3.cmml"><mi id="S5.Ex31.m1.3.3.1.1.3.2" xref="S5.Ex31.m1.3.3.1.1.3.2.cmml">e</mi><mrow id="S5.Ex31.m1.2.2.1" xref="S5.Ex31.m1.2.2.1.cmml"><mo id="S5.Ex31.m1.2.2.1a" xref="S5.Ex31.m1.2.2.1.cmml">−</mo><mrow id="S5.Ex31.m1.2.2.1.3" xref="S5.Ex31.m1.2.2.1.3.cmml"><mi id="S5.Ex31.m1.2.2.1.3.2" mathvariant="normal" xref="S5.Ex31.m1.2.2.1.3.2.cmml">Θ</mi><mo id="S5.Ex31.m1.2.2.1.3.1" xref="S5.Ex31.m1.2.2.1.3.1.cmml">⁢</mo><mrow id="S5.Ex31.m1.2.2.1.3.3.2" xref="S5.Ex31.m1.2.2.1.1.cmml"><mo id="S5.Ex31.m1.2.2.1.3.3.2.1" xref="S5.Ex31.m1.2.2.1.1.cmml">(</mo><mfrac id="S5.Ex31.m1.2.2.1.1" xref="S5.Ex31.m1.2.2.1.1.cmml"><mrow id="S5.Ex31.m1.2.2.1.1.2" xref="S5.Ex31.m1.2.2.1.1.2.cmml"><mi id="S5.Ex31.m1.2.2.1.1.2.2" xref="S5.Ex31.m1.2.2.1.1.2.2.cmml">n</mi><mo id="S5.Ex31.m1.2.2.1.1.2.1" xref="S5.Ex31.m1.2.2.1.1.2.1.cmml">⁢</mo><msup id="S5.Ex31.m1.2.2.1.1.2.3" xref="S5.Ex31.m1.2.2.1.1.2.3.cmml"><mi id="S5.Ex31.m1.2.2.1.1.2.3.2" xref="S5.Ex31.m1.2.2.1.1.2.3.2.cmml">k</mi><mo id="S5.Ex31.m1.2.2.1.1.2.3.3" xref="S5.Ex31.m1.2.2.1.1.2.3.3.cmml">′</mo></msup></mrow><mi id="S5.Ex31.m1.2.2.1.1.3" xref="S5.Ex31.m1.2.2.1.1.3.cmml">k</mi></mfrac><mo id="S5.Ex31.m1.2.2.1.3.3.2.2" xref="S5.Ex31.m1.2.2.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><mo id="S5.Ex31.m1.3.3.1.2" lspace="0em" xref="S5.Ex31.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex31.m1.3b"><apply id="S5.Ex31.m1.3.3.1.1.cmml" xref="S5.Ex31.m1.3.3.1"><eq id="S5.Ex31.m1.3.3.1.1.1.cmml" xref="S5.Ex31.m1.3.3.1.1.1"></eq><apply id="S5.Ex31.m1.3.3.1.1.2.cmml" xref="S5.Ex31.m1.3.3.1.1.2"><times id="S5.Ex31.m1.3.3.1.1.2.1.cmml" xref="S5.Ex31.m1.3.3.1.1.2.1"></times><ci id="S5.Ex31.m1.3.3.1.1.2.2.cmml" xref="S5.Ex31.m1.3.3.1.1.2.2">𝛼</ci><apply id="S5.Ex31.m1.3.3.1.1.2.3.cmml" xref="S5.Ex31.m1.3.3.1.1.2.3"><divide id="S5.Ex31.m1.3.3.1.1.2.3.1.cmml" xref="S5.Ex31.m1.3.3.1.1.2.3"></divide><ci id="S5.Ex31.m1.3.3.1.1.2.3.2.cmml" xref="S5.Ex31.m1.3.3.1.1.2.3.2">𝑛</ci><ci id="S5.Ex31.m1.3.3.1.1.2.3.3.cmml" xref="S5.Ex31.m1.3.3.1.1.2.3.3">𝑘</ci></apply><apply id="S5.Ex31.m1.3.3.1.1.2.4.cmml" xref="S5.Ex31.m1.3.3.1.1.2.4"><csymbol cd="ambiguous" id="S5.Ex31.m1.3.3.1.1.2.4.1.cmml" xref="S5.Ex31.m1.3.3.1.1.2.4">superscript</csymbol><ci id="S5.Ex31.m1.3.3.1.1.2.4.2.cmml" xref="S5.Ex31.m1.3.3.1.1.2.4.2">𝑘</ci><ci id="S5.Ex31.m1.3.3.1.1.2.4.3.cmml" xref="S5.Ex31.m1.3.3.1.1.2.4.3">′</ci></apply><apply id="S5.Ex31.m1.3.3.1.1.2.5.cmml" xref="S5.Ex31.m1.3.3.1.1.2.5"><csymbol cd="ambiguous" id="S5.Ex31.m1.3.3.1.1.2.5.1.cmml" xref="S5.Ex31.m1.3.3.1.1.2.5">superscript</csymbol><ci id="S5.Ex31.m1.3.3.1.1.2.5.2.cmml" xref="S5.Ex31.m1.3.3.1.1.2.5.2">𝑒</ci><apply id="S5.Ex31.m1.1.1.1.cmml" xref="S5.Ex31.m1.1.1.1"><minus id="S5.Ex31.m1.1.1.1.2.cmml" xref="S5.Ex31.m1.1.1.1"></minus><apply id="S5.Ex31.m1.1.1.1.3.cmml" xref="S5.Ex31.m1.1.1.1.3"><times id="S5.Ex31.m1.1.1.1.3.1.cmml" xref="S5.Ex31.m1.1.1.1.3.1"></times><ci id="S5.Ex31.m1.1.1.1.3.2.cmml" xref="S5.Ex31.m1.1.1.1.3.2">Θ</ci><apply id="S5.Ex31.m1.1.1.1.1.cmml" xref="S5.Ex31.m1.1.1.1.3.3.2"><divide id="S5.Ex31.m1.1.1.1.1.1.cmml" xref="S5.Ex31.m1.1.1.1.3.3.2"></divide><apply id="S5.Ex31.m1.1.1.1.1.2.cmml" xref="S5.Ex31.m1.1.1.1.1.2"><times id="S5.Ex31.m1.1.1.1.1.2.1.cmml" xref="S5.Ex31.m1.1.1.1.1.2.1"></times><ci id="S5.Ex31.m1.1.1.1.1.2.2.cmml" xref="S5.Ex31.m1.1.1.1.1.2.2">𝑛</ci><apply id="S5.Ex31.m1.1.1.1.1.2.3.cmml" xref="S5.Ex31.m1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S5.Ex31.m1.1.1.1.1.2.3.1.cmml" xref="S5.Ex31.m1.1.1.1.1.2.3">superscript</csymbol><ci id="S5.Ex31.m1.1.1.1.1.2.3.2.cmml" xref="S5.Ex31.m1.1.1.1.1.2.3.2">𝑘</ci><ci id="S5.Ex31.m1.1.1.1.1.2.3.3.cmml" xref="S5.Ex31.m1.1.1.1.1.2.3.3">′</ci></apply></apply><ci id="S5.Ex31.m1.1.1.1.1.3.cmml" xref="S5.Ex31.m1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply><apply id="S5.Ex31.m1.3.3.1.1.3.cmml" xref="S5.Ex31.m1.3.3.1.1.3"><csymbol cd="ambiguous" id="S5.Ex31.m1.3.3.1.1.3.1.cmml" xref="S5.Ex31.m1.3.3.1.1.3">superscript</csymbol><ci id="S5.Ex31.m1.3.3.1.1.3.2.cmml" xref="S5.Ex31.m1.3.3.1.1.3.2">𝑒</ci><apply id="S5.Ex31.m1.2.2.1.cmml" xref="S5.Ex31.m1.2.2.1"><minus id="S5.Ex31.m1.2.2.1.2.cmml" xref="S5.Ex31.m1.2.2.1"></minus><apply id="S5.Ex31.m1.2.2.1.3.cmml" xref="S5.Ex31.m1.2.2.1.3"><times id="S5.Ex31.m1.2.2.1.3.1.cmml" xref="S5.Ex31.m1.2.2.1.3.1"></times><ci id="S5.Ex31.m1.2.2.1.3.2.cmml" xref="S5.Ex31.m1.2.2.1.3.2">Θ</ci><apply id="S5.Ex31.m1.2.2.1.1.cmml" xref="S5.Ex31.m1.2.2.1.3.3.2"><divide id="S5.Ex31.m1.2.2.1.1.1.cmml" xref="S5.Ex31.m1.2.2.1.3.3.2"></divide><apply id="S5.Ex31.m1.2.2.1.1.2.cmml" xref="S5.Ex31.m1.2.2.1.1.2"><times id="S5.Ex31.m1.2.2.1.1.2.1.cmml" xref="S5.Ex31.m1.2.2.1.1.2.1"></times><ci id="S5.Ex31.m1.2.2.1.1.2.2.cmml" xref="S5.Ex31.m1.2.2.1.1.2.2">𝑛</ci><apply id="S5.Ex31.m1.2.2.1.1.2.3.cmml" xref="S5.Ex31.m1.2.2.1.1.2.3"><csymbol cd="ambiguous" id="S5.Ex31.m1.2.2.1.1.2.3.1.cmml" xref="S5.Ex31.m1.2.2.1.1.2.3">superscript</csymbol><ci id="S5.Ex31.m1.2.2.1.1.2.3.2.cmml" xref="S5.Ex31.m1.2.2.1.1.2.3.2">𝑘</ci><ci id="S5.Ex31.m1.2.2.1.1.2.3.3.cmml" xref="S5.Ex31.m1.2.2.1.1.2.3.3">′</ci></apply></apply><ci id="S5.Ex31.m1.2.2.1.1.3.cmml" xref="S5.Ex31.m1.2.2.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex31.m1.3c">\alpha\frac{n}{k}k^{\prime}e^{-\Theta\left(\frac{nk^{\prime}}{k}\right)}=e^{-% \Theta\left(\frac{nk^{\prime}}{k}\right)}.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex31.m1.3d">italic_α divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT = italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S5.SS2.SSS1.8.p8"> <p class="ltx_p" id="S5.SS2.SSS1.8.p8.3">Therefore, with probability <math alttext="1-e^{-\Theta\left(\frac{nk^{\prime}}{k}\right)}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.8.p8.1.m1.1"><semantics id="S5.SS2.SSS1.8.p8.1.m1.1a"><mrow id="S5.SS2.SSS1.8.p8.1.m1.1.2" xref="S5.SS2.SSS1.8.p8.1.m1.1.2.cmml"><mn id="S5.SS2.SSS1.8.p8.1.m1.1.2.2" xref="S5.SS2.SSS1.8.p8.1.m1.1.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.8.p8.1.m1.1.2.1" xref="S5.SS2.SSS1.8.p8.1.m1.1.2.1.cmml">−</mo><msup id="S5.SS2.SSS1.8.p8.1.m1.1.2.3" xref="S5.SS2.SSS1.8.p8.1.m1.1.2.3.cmml"><mi id="S5.SS2.SSS1.8.p8.1.m1.1.2.3.2" xref="S5.SS2.SSS1.8.p8.1.m1.1.2.3.2.cmml">e</mi><mrow id="S5.SS2.SSS1.8.p8.1.m1.1.1.1" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.cmml"><mo id="S5.SS2.SSS1.8.p8.1.m1.1.1.1a" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.cmml">−</mo><mrow id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3.cmml"><mi id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3.2" mathvariant="normal" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3.2.cmml">Θ</mi><mo id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3.1" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3.3.2" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.cmml"><mo id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3.3.2.1" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.cmml"><mrow id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.cmml"><mi id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.2" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.2.cmml">n</mi><mo id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.1" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.1.cmml">⁢</mo><msup id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.3" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.3.cmml"><mi id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.3.2" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.3.2.cmml">k</mi><mo id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.3.3" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.3.3.cmml">′</mo></msup></mrow><mi id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.3" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3.3.2.2" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.8.p8.1.m1.1b"><apply id="S5.SS2.SSS1.8.p8.1.m1.1.2.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.2"><minus id="S5.SS2.SSS1.8.p8.1.m1.1.2.1.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.2.1"></minus><cn id="S5.SS2.SSS1.8.p8.1.m1.1.2.2.cmml" type="integer" xref="S5.SS2.SSS1.8.p8.1.m1.1.2.2">1</cn><apply id="S5.SS2.SSS1.8.p8.1.m1.1.2.3.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.2.3"><csymbol cd="ambiguous" id="S5.SS2.SSS1.8.p8.1.m1.1.2.3.1.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.2.3">superscript</csymbol><ci id="S5.SS2.SSS1.8.p8.1.m1.1.2.3.2.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.2.3.2">𝑒</ci><apply id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1"><minus id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.2.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1"></minus><apply id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3"><times id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3.1.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3.1"></times><ci id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3.2.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3.2">Θ</ci><apply id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3.3.2"><divide id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.3.3.2"></divide><apply id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2"><times id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.1.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.1"></times><ci id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.2.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.2">𝑛</ci><apply id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.3.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.3.1.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.3">superscript</csymbol><ci id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.3.2.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.3.2">𝑘</ci><ci id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.3.3.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.2.3.3">′</ci></apply></apply><ci id="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.3.cmml" xref="S5.SS2.SSS1.8.p8.1.m1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.8.p8.1.m1.1c">1-e^{-\Theta\left(\frac{nk^{\prime}}{k}\right)}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.8.p8.1.m1.1d">1 - italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT</annotation></semantics></math>, the algorithm returns <math alttext="\alpha\frac{n}{k}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.8.p8.2.m2.1"><semantics id="S5.SS2.SSS1.8.p8.2.m2.1a"><mrow id="S5.SS2.SSS1.8.p8.2.m2.1.1" xref="S5.SS2.SSS1.8.p8.2.m2.1.1.cmml"><mi id="S5.SS2.SSS1.8.p8.2.m2.1.1.2" xref="S5.SS2.SSS1.8.p8.2.m2.1.1.2.cmml">α</mi><mo id="S5.SS2.SSS1.8.p8.2.m2.1.1.1" xref="S5.SS2.SSS1.8.p8.2.m2.1.1.1.cmml">⁢</mo><mfrac id="S5.SS2.SSS1.8.p8.2.m2.1.1.3" xref="S5.SS2.SSS1.8.p8.2.m2.1.1.3.cmml"><mi id="S5.SS2.SSS1.8.p8.2.m2.1.1.3.2" xref="S5.SS2.SSS1.8.p8.2.m2.1.1.3.2.cmml">n</mi><mi id="S5.SS2.SSS1.8.p8.2.m2.1.1.3.3" xref="S5.SS2.SSS1.8.p8.2.m2.1.1.3.3.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.8.p8.2.m2.1b"><apply id="S5.SS2.SSS1.8.p8.2.m2.1.1.cmml" xref="S5.SS2.SSS1.8.p8.2.m2.1.1"><times id="S5.SS2.SSS1.8.p8.2.m2.1.1.1.cmml" xref="S5.SS2.SSS1.8.p8.2.m2.1.1.1"></times><ci id="S5.SS2.SSS1.8.p8.2.m2.1.1.2.cmml" xref="S5.SS2.SSS1.8.p8.2.m2.1.1.2">𝛼</ci><apply id="S5.SS2.SSS1.8.p8.2.m2.1.1.3.cmml" xref="S5.SS2.SSS1.8.p8.2.m2.1.1.3"><divide id="S5.SS2.SSS1.8.p8.2.m2.1.1.3.1.cmml" xref="S5.SS2.SSS1.8.p8.2.m2.1.1.3"></divide><ci id="S5.SS2.SSS1.8.p8.2.m2.1.1.3.2.cmml" xref="S5.SS2.SSS1.8.p8.2.m2.1.1.3.2">𝑛</ci><ci id="S5.SS2.SSS1.8.p8.2.m2.1.1.3.3.cmml" xref="S5.SS2.SSS1.8.p8.2.m2.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.8.p8.2.m2.1c">\alpha\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.8.p8.2.m2.1d">italic_α divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math> cliques of size <math alttext="k^{\prime}(\sqrt{1-\varepsilon})" class="ltx_Math" display="inline" id="S5.SS2.SSS1.8.p8.3.m3.1"><semantics id="S5.SS2.SSS1.8.p8.3.m3.1a"><mrow id="S5.SS2.SSS1.8.p8.3.m3.1.2" xref="S5.SS2.SSS1.8.p8.3.m3.1.2.cmml"><msup id="S5.SS2.SSS1.8.p8.3.m3.1.2.2" xref="S5.SS2.SSS1.8.p8.3.m3.1.2.2.cmml"><mi id="S5.SS2.SSS1.8.p8.3.m3.1.2.2.2" xref="S5.SS2.SSS1.8.p8.3.m3.1.2.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.8.p8.3.m3.1.2.2.3" xref="S5.SS2.SSS1.8.p8.3.m3.1.2.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS1.8.p8.3.m3.1.2.1" xref="S5.SS2.SSS1.8.p8.3.m3.1.2.1.cmml">⁢</mo><mrow id="S5.SS2.SSS1.8.p8.3.m3.1.2.3.2" xref="S5.SS2.SSS1.8.p8.3.m3.1.1.cmml"><mo id="S5.SS2.SSS1.8.p8.3.m3.1.2.3.2.1" stretchy="false" xref="S5.SS2.SSS1.8.p8.3.m3.1.1.cmml">(</mo><msqrt id="S5.SS2.SSS1.8.p8.3.m3.1.1" xref="S5.SS2.SSS1.8.p8.3.m3.1.1.cmml"><mrow id="S5.SS2.SSS1.8.p8.3.m3.1.1.2" xref="S5.SS2.SSS1.8.p8.3.m3.1.1.2.cmml"><mn id="S5.SS2.SSS1.8.p8.3.m3.1.1.2.2" xref="S5.SS2.SSS1.8.p8.3.m3.1.1.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.8.p8.3.m3.1.1.2.1" xref="S5.SS2.SSS1.8.p8.3.m3.1.1.2.1.cmml">−</mo><mi id="S5.SS2.SSS1.8.p8.3.m3.1.1.2.3" xref="S5.SS2.SSS1.8.p8.3.m3.1.1.2.3.cmml">ε</mi></mrow></msqrt><mo id="S5.SS2.SSS1.8.p8.3.m3.1.2.3.2.2" stretchy="false" xref="S5.SS2.SSS1.8.p8.3.m3.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.8.p8.3.m3.1b"><apply id="S5.SS2.SSS1.8.p8.3.m3.1.2.cmml" xref="S5.SS2.SSS1.8.p8.3.m3.1.2"><times id="S5.SS2.SSS1.8.p8.3.m3.1.2.1.cmml" xref="S5.SS2.SSS1.8.p8.3.m3.1.2.1"></times><apply id="S5.SS2.SSS1.8.p8.3.m3.1.2.2.cmml" xref="S5.SS2.SSS1.8.p8.3.m3.1.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.8.p8.3.m3.1.2.2.1.cmml" xref="S5.SS2.SSS1.8.p8.3.m3.1.2.2">superscript</csymbol><ci id="S5.SS2.SSS1.8.p8.3.m3.1.2.2.2.cmml" xref="S5.SS2.SSS1.8.p8.3.m3.1.2.2.2">𝑘</ci><ci id="S5.SS2.SSS1.8.p8.3.m3.1.2.2.3.cmml" xref="S5.SS2.SSS1.8.p8.3.m3.1.2.2.3">′</ci></apply><apply id="S5.SS2.SSS1.8.p8.3.m3.1.1.cmml" xref="S5.SS2.SSS1.8.p8.3.m3.1.2.3.2"><root id="S5.SS2.SSS1.8.p8.3.m3.1.1a.cmml" xref="S5.SS2.SSS1.8.p8.3.m3.1.2.3.2"></root><apply id="S5.SS2.SSS1.8.p8.3.m3.1.1.2.cmml" xref="S5.SS2.SSS1.8.p8.3.m3.1.1.2"><minus id="S5.SS2.SSS1.8.p8.3.m3.1.1.2.1.cmml" xref="S5.SS2.SSS1.8.p8.3.m3.1.1.2.1"></minus><cn id="S5.SS2.SSS1.8.p8.3.m3.1.1.2.2.cmml" type="integer" xref="S5.SS2.SSS1.8.p8.3.m3.1.1.2.2">1</cn><ci id="S5.SS2.SSS1.8.p8.3.m3.1.1.2.3.cmml" xref="S5.SS2.SSS1.8.p8.3.m3.1.1.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.8.p8.3.m3.1c">k^{\prime}(\sqrt{1-\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.8.p8.3.m3.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( square-root start_ARG 1 - italic_ε end_ARG )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S5.SS2.SSS1.p4"> <p class="ltx_p" id="S5.SS2.SSS1.p4.1">We now prove our main theorem for the low perturbation regime.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S5.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem8.1.1.1">Theorem 5.8</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem8.p1"> <p class="ltx_p" id="S5.Thmtheorem8.p1.4"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem8.p1.4.4">Consider aversion-to-enemies games given by random Turán graphs <math alttext="G=(n,k,p)" class="ltx_Math" display="inline" id="S5.Thmtheorem8.p1.1.1.m1.3"><semantics id="S5.Thmtheorem8.p1.1.1.m1.3a"><mrow id="S5.Thmtheorem8.p1.1.1.m1.3.4" xref="S5.Thmtheorem8.p1.1.1.m1.3.4.cmml"><mi id="S5.Thmtheorem8.p1.1.1.m1.3.4.2" xref="S5.Thmtheorem8.p1.1.1.m1.3.4.2.cmml">G</mi><mo id="S5.Thmtheorem8.p1.1.1.m1.3.4.1" xref="S5.Thmtheorem8.p1.1.1.m1.3.4.1.cmml">=</mo><mrow id="S5.Thmtheorem8.p1.1.1.m1.3.4.3.2" xref="S5.Thmtheorem8.p1.1.1.m1.3.4.3.1.cmml"><mo id="S5.Thmtheorem8.p1.1.1.m1.3.4.3.2.1" stretchy="false" xref="S5.Thmtheorem8.p1.1.1.m1.3.4.3.1.cmml">(</mo><mi id="S5.Thmtheorem8.p1.1.1.m1.1.1" xref="S5.Thmtheorem8.p1.1.1.m1.1.1.cmml">n</mi><mo id="S5.Thmtheorem8.p1.1.1.m1.3.4.3.2.2" xref="S5.Thmtheorem8.p1.1.1.m1.3.4.3.1.cmml">,</mo><mi id="S5.Thmtheorem8.p1.1.1.m1.2.2" xref="S5.Thmtheorem8.p1.1.1.m1.2.2.cmml">k</mi><mo id="S5.Thmtheorem8.p1.1.1.m1.3.4.3.2.3" xref="S5.Thmtheorem8.p1.1.1.m1.3.4.3.1.cmml">,</mo><mi id="S5.Thmtheorem8.p1.1.1.m1.3.3" xref="S5.Thmtheorem8.p1.1.1.m1.3.3.cmml">p</mi><mo id="S5.Thmtheorem8.p1.1.1.m1.3.4.3.2.4" stretchy="false" xref="S5.Thmtheorem8.p1.1.1.m1.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem8.p1.1.1.m1.3b"><apply id="S5.Thmtheorem8.p1.1.1.m1.3.4.cmml" xref="S5.Thmtheorem8.p1.1.1.m1.3.4"><eq id="S5.Thmtheorem8.p1.1.1.m1.3.4.1.cmml" xref="S5.Thmtheorem8.p1.1.1.m1.3.4.1"></eq><ci id="S5.Thmtheorem8.p1.1.1.m1.3.4.2.cmml" xref="S5.Thmtheorem8.p1.1.1.m1.3.4.2">𝐺</ci><vector id="S5.Thmtheorem8.p1.1.1.m1.3.4.3.1.cmml" xref="S5.Thmtheorem8.p1.1.1.m1.3.4.3.2"><ci id="S5.Thmtheorem8.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem8.p1.1.1.m1.1.1">𝑛</ci><ci id="S5.Thmtheorem8.p1.1.1.m1.2.2.cmml" xref="S5.Thmtheorem8.p1.1.1.m1.2.2">𝑘</ci><ci id="S5.Thmtheorem8.p1.1.1.m1.3.3.cmml" xref="S5.Thmtheorem8.p1.1.1.m1.3.3">𝑝</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem8.p1.1.1.m1.3c">G=(n,k,p)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem8.p1.1.1.m1.3d">italic_G = ( italic_n , italic_k , italic_p )</annotation></semantics></math>, where <math alttext="k\geq 2" class="ltx_Math" display="inline" id="S5.Thmtheorem8.p1.2.2.m2.1"><semantics id="S5.Thmtheorem8.p1.2.2.m2.1a"><mrow id="S5.Thmtheorem8.p1.2.2.m2.1.1" xref="S5.Thmtheorem8.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem8.p1.2.2.m2.1.1.2" xref="S5.Thmtheorem8.p1.2.2.m2.1.1.2.cmml">k</mi><mo id="S5.Thmtheorem8.p1.2.2.m2.1.1.1" xref="S5.Thmtheorem8.p1.2.2.m2.1.1.1.cmml">≥</mo><mn id="S5.Thmtheorem8.p1.2.2.m2.1.1.3" xref="S5.Thmtheorem8.p1.2.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem8.p1.2.2.m2.1b"><apply id="S5.Thmtheorem8.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem8.p1.2.2.m2.1.1"><geq id="S5.Thmtheorem8.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem8.p1.2.2.m2.1.1.1"></geq><ci id="S5.Thmtheorem8.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem8.p1.2.2.m2.1.1.2">𝑘</ci><cn id="S5.Thmtheorem8.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S5.Thmtheorem8.p1.2.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem8.p1.2.2.m2.1c">k\geq 2</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem8.p1.2.2.m2.1d">italic_k ≥ 2</annotation></semantics></math> and <math alttext="p=\mathcal{O}(\frac{1}{k})" class="ltx_Math" display="inline" id="S5.Thmtheorem8.p1.3.3.m3.1"><semantics id="S5.Thmtheorem8.p1.3.3.m3.1a"><mrow id="S5.Thmtheorem8.p1.3.3.m3.1.2" xref="S5.Thmtheorem8.p1.3.3.m3.1.2.cmml"><mi id="S5.Thmtheorem8.p1.3.3.m3.1.2.2" xref="S5.Thmtheorem8.p1.3.3.m3.1.2.2.cmml">p</mi><mo id="S5.Thmtheorem8.p1.3.3.m3.1.2.1" xref="S5.Thmtheorem8.p1.3.3.m3.1.2.1.cmml">=</mo><mrow id="S5.Thmtheorem8.p1.3.3.m3.1.2.3" xref="S5.Thmtheorem8.p1.3.3.m3.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem8.p1.3.3.m3.1.2.3.2" xref="S5.Thmtheorem8.p1.3.3.m3.1.2.3.2.cmml">𝒪</mi><mo id="S5.Thmtheorem8.p1.3.3.m3.1.2.3.1" xref="S5.Thmtheorem8.p1.3.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem8.p1.3.3.m3.1.2.3.3.2" xref="S5.Thmtheorem8.p1.3.3.m3.1.1.cmml"><mo id="S5.Thmtheorem8.p1.3.3.m3.1.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem8.p1.3.3.m3.1.1.cmml">(</mo><mfrac id="S5.Thmtheorem8.p1.3.3.m3.1.1" xref="S5.Thmtheorem8.p1.3.3.m3.1.1.cmml"><mn id="S5.Thmtheorem8.p1.3.3.m3.1.1.2" xref="S5.Thmtheorem8.p1.3.3.m3.1.1.2.cmml">1</mn><mi id="S5.Thmtheorem8.p1.3.3.m3.1.1.3" xref="S5.Thmtheorem8.p1.3.3.m3.1.1.3.cmml">k</mi></mfrac><mo id="S5.Thmtheorem8.p1.3.3.m3.1.2.3.3.2.2" stretchy="false" xref="S5.Thmtheorem8.p1.3.3.m3.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem8.p1.3.3.m3.1b"><apply id="S5.Thmtheorem8.p1.3.3.m3.1.2.cmml" xref="S5.Thmtheorem8.p1.3.3.m3.1.2"><eq id="S5.Thmtheorem8.p1.3.3.m3.1.2.1.cmml" xref="S5.Thmtheorem8.p1.3.3.m3.1.2.1"></eq><ci id="S5.Thmtheorem8.p1.3.3.m3.1.2.2.cmml" xref="S5.Thmtheorem8.p1.3.3.m3.1.2.2">𝑝</ci><apply id="S5.Thmtheorem8.p1.3.3.m3.1.2.3.cmml" xref="S5.Thmtheorem8.p1.3.3.m3.1.2.3"><times id="S5.Thmtheorem8.p1.3.3.m3.1.2.3.1.cmml" xref="S5.Thmtheorem8.p1.3.3.m3.1.2.3.1"></times><ci id="S5.Thmtheorem8.p1.3.3.m3.1.2.3.2.cmml" xref="S5.Thmtheorem8.p1.3.3.m3.1.2.3.2">𝒪</ci><apply id="S5.Thmtheorem8.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem8.p1.3.3.m3.1.2.3.3.2"><divide id="S5.Thmtheorem8.p1.3.3.m3.1.1.1.cmml" xref="S5.Thmtheorem8.p1.3.3.m3.1.2.3.3.2"></divide><cn id="S5.Thmtheorem8.p1.3.3.m3.1.1.2.cmml" type="integer" xref="S5.Thmtheorem8.p1.3.3.m3.1.1.2">1</cn><ci id="S5.Thmtheorem8.p1.3.3.m3.1.1.3.cmml" xref="S5.Thmtheorem8.p1.3.3.m3.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem8.p1.3.3.m3.1c">p=\mathcal{O}(\frac{1}{k})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem8.p1.3.3.m3.1d">italic_p = caligraphic_O ( divide start_ARG 1 end_ARG start_ARG italic_k end_ARG )</annotation></semantics></math>. Then there is a polynomial-time algorithm that returns a constant-factor approximation to maximum welfare with probability <math alttext="1-e^{-\Theta(n)}" class="ltx_Math" display="inline" id="S5.Thmtheorem8.p1.4.4.m4.1"><semantics id="S5.Thmtheorem8.p1.4.4.m4.1a"><mrow id="S5.Thmtheorem8.p1.4.4.m4.1.2" xref="S5.Thmtheorem8.p1.4.4.m4.1.2.cmml"><mn id="S5.Thmtheorem8.p1.4.4.m4.1.2.2" xref="S5.Thmtheorem8.p1.4.4.m4.1.2.2.cmml">1</mn><mo id="S5.Thmtheorem8.p1.4.4.m4.1.2.1" xref="S5.Thmtheorem8.p1.4.4.m4.1.2.1.cmml">−</mo><msup id="S5.Thmtheorem8.p1.4.4.m4.1.2.3" xref="S5.Thmtheorem8.p1.4.4.m4.1.2.3.cmml"><mi id="S5.Thmtheorem8.p1.4.4.m4.1.2.3.2" xref="S5.Thmtheorem8.p1.4.4.m4.1.2.3.2.cmml">e</mi><mrow id="S5.Thmtheorem8.p1.4.4.m4.1.1.1" xref="S5.Thmtheorem8.p1.4.4.m4.1.1.1.cmml"><mo id="S5.Thmtheorem8.p1.4.4.m4.1.1.1a" xref="S5.Thmtheorem8.p1.4.4.m4.1.1.1.cmml">−</mo><mrow id="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3" xref="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.cmml"><mi id="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.2" mathvariant="normal" xref="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.2.cmml">Θ</mi><mo id="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.1" xref="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.3.2" xref="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.cmml"><mo id="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.3.2.1" stretchy="false" xref="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.cmml">(</mo><mi id="S5.Thmtheorem8.p1.4.4.m4.1.1.1.1" xref="S5.Thmtheorem8.p1.4.4.m4.1.1.1.1.cmml">n</mi><mo id="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.3.2.2" stretchy="false" xref="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.cmml">)</mo></mrow></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem8.p1.4.4.m4.1b"><apply id="S5.Thmtheorem8.p1.4.4.m4.1.2.cmml" xref="S5.Thmtheorem8.p1.4.4.m4.1.2"><minus id="S5.Thmtheorem8.p1.4.4.m4.1.2.1.cmml" xref="S5.Thmtheorem8.p1.4.4.m4.1.2.1"></minus><cn id="S5.Thmtheorem8.p1.4.4.m4.1.2.2.cmml" type="integer" xref="S5.Thmtheorem8.p1.4.4.m4.1.2.2">1</cn><apply id="S5.Thmtheorem8.p1.4.4.m4.1.2.3.cmml" xref="S5.Thmtheorem8.p1.4.4.m4.1.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem8.p1.4.4.m4.1.2.3.1.cmml" xref="S5.Thmtheorem8.p1.4.4.m4.1.2.3">superscript</csymbol><ci id="S5.Thmtheorem8.p1.4.4.m4.1.2.3.2.cmml" xref="S5.Thmtheorem8.p1.4.4.m4.1.2.3.2">𝑒</ci><apply id="S5.Thmtheorem8.p1.4.4.m4.1.1.1.cmml" xref="S5.Thmtheorem8.p1.4.4.m4.1.1.1"><minus id="S5.Thmtheorem8.p1.4.4.m4.1.1.1.2.cmml" xref="S5.Thmtheorem8.p1.4.4.m4.1.1.1"></minus><apply id="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.cmml" xref="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3"><times id="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.1.cmml" xref="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.1"></times><ci id="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.2.cmml" xref="S5.Thmtheorem8.p1.4.4.m4.1.1.1.3.2">Θ</ci><ci id="S5.Thmtheorem8.p1.4.4.m4.1.1.1.1.cmml" xref="S5.Thmtheorem8.p1.4.4.m4.1.1.1.1">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem8.p1.4.4.m4.1c">1-e^{-\Theta(n)}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem8.p1.4.4.m4.1d">1 - italic_e start_POSTSUPERSCRIPT - roman_Θ ( italic_n ) end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S5.SS2.SSS1.11"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.SS2.SSS1.9.p1"> <p class="ltx_p" id="S5.SS2.SSS1.9.p1.10">Fix <math alttext="\varepsilon\in(0,1)" class="ltx_Math" display="inline" id="S5.SS2.SSS1.9.p1.1.m1.2"><semantics id="S5.SS2.SSS1.9.p1.1.m1.2a"><mrow id="S5.SS2.SSS1.9.p1.1.m1.2.3" xref="S5.SS2.SSS1.9.p1.1.m1.2.3.cmml"><mi id="S5.SS2.SSS1.9.p1.1.m1.2.3.2" xref="S5.SS2.SSS1.9.p1.1.m1.2.3.2.cmml">ε</mi><mo id="S5.SS2.SSS1.9.p1.1.m1.2.3.1" xref="S5.SS2.SSS1.9.p1.1.m1.2.3.1.cmml">∈</mo><mrow id="S5.SS2.SSS1.9.p1.1.m1.2.3.3.2" xref="S5.SS2.SSS1.9.p1.1.m1.2.3.3.1.cmml"><mo id="S5.SS2.SSS1.9.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S5.SS2.SSS1.9.p1.1.m1.2.3.3.1.cmml">(</mo><mn id="S5.SS2.SSS1.9.p1.1.m1.1.1" xref="S5.SS2.SSS1.9.p1.1.m1.1.1.cmml">0</mn><mo id="S5.SS2.SSS1.9.p1.1.m1.2.3.3.2.2" xref="S5.SS2.SSS1.9.p1.1.m1.2.3.3.1.cmml">,</mo><mn id="S5.SS2.SSS1.9.p1.1.m1.2.2" xref="S5.SS2.SSS1.9.p1.1.m1.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.9.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S5.SS2.SSS1.9.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.9.p1.1.m1.2b"><apply id="S5.SS2.SSS1.9.p1.1.m1.2.3.cmml" xref="S5.SS2.SSS1.9.p1.1.m1.2.3"><in id="S5.SS2.SSS1.9.p1.1.m1.2.3.1.cmml" xref="S5.SS2.SSS1.9.p1.1.m1.2.3.1"></in><ci id="S5.SS2.SSS1.9.p1.1.m1.2.3.2.cmml" xref="S5.SS2.SSS1.9.p1.1.m1.2.3.2">𝜀</ci><interval closure="open" id="S5.SS2.SSS1.9.p1.1.m1.2.3.3.1.cmml" xref="S5.SS2.SSS1.9.p1.1.m1.2.3.3.2"><cn id="S5.SS2.SSS1.9.p1.1.m1.1.1.cmml" type="integer" xref="S5.SS2.SSS1.9.p1.1.m1.1.1">0</cn><cn id="S5.SS2.SSS1.9.p1.1.m1.2.2.cmml" type="integer" xref="S5.SS2.SSS1.9.p1.1.m1.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.9.p1.1.m1.2c">\varepsilon\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.9.p1.1.m1.2d">italic_ε ∈ ( 0 , 1 )</annotation></semantics></math>, and consider <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg1" title="In 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a> for input <math alttext="\langle G,S=\{V_{1},\dots,V_{k}\},\varepsilon\rangle" class="ltx_Math" display="inline" id="S5.SS2.SSS1.9.p1.2.m2.5"><semantics id="S5.SS2.SSS1.9.p1.2.m2.5a"><mrow id="S5.SS2.SSS1.9.p1.2.m2.5.5.1" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.2.cmml"><mo id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.2" stretchy="false" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.2.1.cmml">⟨</mo><mrow id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.2.cmml"><mrow id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.cmml"><mrow id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.4.2" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.4.1.cmml"><mi id="S5.SS2.SSS1.9.p1.2.m2.2.2" xref="S5.SS2.SSS1.9.p1.2.m2.2.2.cmml">G</mi><mo id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.4.2.1" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.4.1.cmml">,</mo><mi id="S5.SS2.SSS1.9.p1.2.m2.3.3" xref="S5.SS2.SSS1.9.p1.2.m2.3.3.cmml">S</mi></mrow><mo id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.3" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.3.cmml">=</mo><mrow id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.3.cmml"><mo id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.3" stretchy="false" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.3.cmml">{</mo><msub id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.1.1.1" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.1.1.1.cmml"><mi id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.1.1.1.2" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.1.1.1.2.cmml">V</mi><mn id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.1.1.1.3" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.4" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.3.cmml">,</mo><mi id="S5.SS2.SSS1.9.p1.2.m2.1.1" mathvariant="normal" xref="S5.SS2.SSS1.9.p1.2.m2.1.1.cmml">…</mi><mo id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.5" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.3.cmml">,</mo><msub id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.2" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.2.cmml"><mi id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.2.2" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.2.2.cmml">V</mi><mi id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.2.3" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.2.3.cmml">k</mi></msub><mo id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.6" stretchy="false" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.3.cmml">}</mo></mrow></mrow><mo id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.2" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.2a.cmml">,</mo><mi id="S5.SS2.SSS1.9.p1.2.m2.4.4" xref="S5.SS2.SSS1.9.p1.2.m2.4.4.cmml">ε</mi></mrow><mo id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.3" stretchy="false" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.9.p1.2.m2.5b"><apply id="S5.SS2.SSS1.9.p1.2.m2.5.5.2.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1"><csymbol cd="latexml" id="S5.SS2.SSS1.9.p1.2.m2.5.5.2.1.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.2">delimited-⟨⟩</csymbol><apply id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.2.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.2a.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.2">formulae-sequence</csymbol><apply id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1"><eq id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.3.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.3"></eq><list id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.4.1.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.4.2"><ci id="S5.SS2.SSS1.9.p1.2.m2.2.2.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.2.2">𝐺</ci><ci id="S5.SS2.SSS1.9.p1.2.m2.3.3.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.3.3">𝑆</ci></list><set id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.3.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2"><apply id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.1.1.1">subscript</csymbol><ci id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.1.1.1.2.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.1.1.1.2">𝑉</ci><cn id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.1.1.1.3">1</cn></apply><ci id="S5.SS2.SSS1.9.p1.2.m2.1.1.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.1.1">…</ci><apply id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.2.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.2.1.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.2">subscript</csymbol><ci id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.2.2.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.2.2">𝑉</ci><ci id="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.2.3.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.5.5.1.1.1.1.2.2.2.3">𝑘</ci></apply></set></apply><ci id="S5.SS2.SSS1.9.p1.2.m2.4.4.cmml" xref="S5.SS2.SSS1.9.p1.2.m2.4.4">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.9.p1.2.m2.5c">\langle G,S=\{V_{1},\dots,V_{k}\},\varepsilon\rangle</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.9.p1.2.m2.5d">⟨ italic_G , italic_S = { italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } , italic_ε ⟩</annotation></semantics></math>. Since <math alttext="p=\mathcal{O}\left(\frac{1}{k}\right)" class="ltx_Math" display="inline" id="S5.SS2.SSS1.9.p1.3.m3.1"><semantics id="S5.SS2.SSS1.9.p1.3.m3.1a"><mrow id="S5.SS2.SSS1.9.p1.3.m3.1.2" xref="S5.SS2.SSS1.9.p1.3.m3.1.2.cmml"><mi id="S5.SS2.SSS1.9.p1.3.m3.1.2.2" xref="S5.SS2.SSS1.9.p1.3.m3.1.2.2.cmml">p</mi><mo id="S5.SS2.SSS1.9.p1.3.m3.1.2.1" xref="S5.SS2.SSS1.9.p1.3.m3.1.2.1.cmml">=</mo><mrow id="S5.SS2.SSS1.9.p1.3.m3.1.2.3" xref="S5.SS2.SSS1.9.p1.3.m3.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS1.9.p1.3.m3.1.2.3.2" xref="S5.SS2.SSS1.9.p1.3.m3.1.2.3.2.cmml">𝒪</mi><mo id="S5.SS2.SSS1.9.p1.3.m3.1.2.3.1" xref="S5.SS2.SSS1.9.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS1.9.p1.3.m3.1.2.3.3.2" xref="S5.SS2.SSS1.9.p1.3.m3.1.1.cmml"><mo id="S5.SS2.SSS1.9.p1.3.m3.1.2.3.3.2.1" xref="S5.SS2.SSS1.9.p1.3.m3.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS1.9.p1.3.m3.1.1" xref="S5.SS2.SSS1.9.p1.3.m3.1.1.cmml"><mn id="S5.SS2.SSS1.9.p1.3.m3.1.1.2" xref="S5.SS2.SSS1.9.p1.3.m3.1.1.2.cmml">1</mn><mi id="S5.SS2.SSS1.9.p1.3.m3.1.1.3" xref="S5.SS2.SSS1.9.p1.3.m3.1.1.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS1.9.p1.3.m3.1.2.3.3.2.2" xref="S5.SS2.SSS1.9.p1.3.m3.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.9.p1.3.m3.1b"><apply id="S5.SS2.SSS1.9.p1.3.m3.1.2.cmml" xref="S5.SS2.SSS1.9.p1.3.m3.1.2"><eq id="S5.SS2.SSS1.9.p1.3.m3.1.2.1.cmml" xref="S5.SS2.SSS1.9.p1.3.m3.1.2.1"></eq><ci id="S5.SS2.SSS1.9.p1.3.m3.1.2.2.cmml" xref="S5.SS2.SSS1.9.p1.3.m3.1.2.2">𝑝</ci><apply id="S5.SS2.SSS1.9.p1.3.m3.1.2.3.cmml" xref="S5.SS2.SSS1.9.p1.3.m3.1.2.3"><times id="S5.SS2.SSS1.9.p1.3.m3.1.2.3.1.cmml" xref="S5.SS2.SSS1.9.p1.3.m3.1.2.3.1"></times><ci id="S5.SS2.SSS1.9.p1.3.m3.1.2.3.2.cmml" xref="S5.SS2.SSS1.9.p1.3.m3.1.2.3.2">𝒪</ci><apply id="S5.SS2.SSS1.9.p1.3.m3.1.1.cmml" xref="S5.SS2.SSS1.9.p1.3.m3.1.2.3.3.2"><divide id="S5.SS2.SSS1.9.p1.3.m3.1.1.1.cmml" xref="S5.SS2.SSS1.9.p1.3.m3.1.2.3.3.2"></divide><cn id="S5.SS2.SSS1.9.p1.3.m3.1.1.2.cmml" type="integer" xref="S5.SS2.SSS1.9.p1.3.m3.1.1.2">1</cn><ci id="S5.SS2.SSS1.9.p1.3.m3.1.1.3.cmml" xref="S5.SS2.SSS1.9.p1.3.m3.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.9.p1.3.m3.1c">p=\mathcal{O}\left(\frac{1}{k}\right)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.9.p1.3.m3.1d">italic_p = caligraphic_O ( divide start_ARG 1 end_ARG start_ARG italic_k end_ARG )</annotation></semantics></math>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem7" title="Lemma 5.7. ‣ 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5.7</span></a> implies that the algorithm returns <math alttext="\alpha\frac{n}{k}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.9.p1.4.m4.1"><semantics id="S5.SS2.SSS1.9.p1.4.m4.1a"><mrow id="S5.SS2.SSS1.9.p1.4.m4.1.1" xref="S5.SS2.SSS1.9.p1.4.m4.1.1.cmml"><mi id="S5.SS2.SSS1.9.p1.4.m4.1.1.2" xref="S5.SS2.SSS1.9.p1.4.m4.1.1.2.cmml">α</mi><mo id="S5.SS2.SSS1.9.p1.4.m4.1.1.1" xref="S5.SS2.SSS1.9.p1.4.m4.1.1.1.cmml">⁢</mo><mfrac id="S5.SS2.SSS1.9.p1.4.m4.1.1.3" xref="S5.SS2.SSS1.9.p1.4.m4.1.1.3.cmml"><mi id="S5.SS2.SSS1.9.p1.4.m4.1.1.3.2" xref="S5.SS2.SSS1.9.p1.4.m4.1.1.3.2.cmml">n</mi><mi id="S5.SS2.SSS1.9.p1.4.m4.1.1.3.3" xref="S5.SS2.SSS1.9.p1.4.m4.1.1.3.3.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.9.p1.4.m4.1b"><apply id="S5.SS2.SSS1.9.p1.4.m4.1.1.cmml" xref="S5.SS2.SSS1.9.p1.4.m4.1.1"><times id="S5.SS2.SSS1.9.p1.4.m4.1.1.1.cmml" xref="S5.SS2.SSS1.9.p1.4.m4.1.1.1"></times><ci id="S5.SS2.SSS1.9.p1.4.m4.1.1.2.cmml" xref="S5.SS2.SSS1.9.p1.4.m4.1.1.2">𝛼</ci><apply id="S5.SS2.SSS1.9.p1.4.m4.1.1.3.cmml" xref="S5.SS2.SSS1.9.p1.4.m4.1.1.3"><divide id="S5.SS2.SSS1.9.p1.4.m4.1.1.3.1.cmml" xref="S5.SS2.SSS1.9.p1.4.m4.1.1.3"></divide><ci id="S5.SS2.SSS1.9.p1.4.m4.1.1.3.2.cmml" xref="S5.SS2.SSS1.9.p1.4.m4.1.1.3.2">𝑛</ci><ci id="S5.SS2.SSS1.9.p1.4.m4.1.1.3.3.cmml" xref="S5.SS2.SSS1.9.p1.4.m4.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.9.p1.4.m4.1c">\alpha\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.9.p1.4.m4.1d">italic_α divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math> cliques of size at least <math alttext="k(\sqrt{1-\varepsilon})" class="ltx_Math" display="inline" id="S5.SS2.SSS1.9.p1.5.m5.1"><semantics id="S5.SS2.SSS1.9.p1.5.m5.1a"><mrow id="S5.SS2.SSS1.9.p1.5.m5.1.2" xref="S5.SS2.SSS1.9.p1.5.m5.1.2.cmml"><mi id="S5.SS2.SSS1.9.p1.5.m5.1.2.2" xref="S5.SS2.SSS1.9.p1.5.m5.1.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.9.p1.5.m5.1.2.1" xref="S5.SS2.SSS1.9.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S5.SS2.SSS1.9.p1.5.m5.1.2.3.2" xref="S5.SS2.SSS1.9.p1.5.m5.1.1.cmml"><mo id="S5.SS2.SSS1.9.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S5.SS2.SSS1.9.p1.5.m5.1.1.cmml">(</mo><msqrt id="S5.SS2.SSS1.9.p1.5.m5.1.1" xref="S5.SS2.SSS1.9.p1.5.m5.1.1.cmml"><mrow id="S5.SS2.SSS1.9.p1.5.m5.1.1.2" xref="S5.SS2.SSS1.9.p1.5.m5.1.1.2.cmml"><mn id="S5.SS2.SSS1.9.p1.5.m5.1.1.2.2" xref="S5.SS2.SSS1.9.p1.5.m5.1.1.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.9.p1.5.m5.1.1.2.1" xref="S5.SS2.SSS1.9.p1.5.m5.1.1.2.1.cmml">−</mo><mi id="S5.SS2.SSS1.9.p1.5.m5.1.1.2.3" xref="S5.SS2.SSS1.9.p1.5.m5.1.1.2.3.cmml">ε</mi></mrow></msqrt><mo id="S5.SS2.SSS1.9.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S5.SS2.SSS1.9.p1.5.m5.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.9.p1.5.m5.1b"><apply id="S5.SS2.SSS1.9.p1.5.m5.1.2.cmml" xref="S5.SS2.SSS1.9.p1.5.m5.1.2"><times id="S5.SS2.SSS1.9.p1.5.m5.1.2.1.cmml" xref="S5.SS2.SSS1.9.p1.5.m5.1.2.1"></times><ci id="S5.SS2.SSS1.9.p1.5.m5.1.2.2.cmml" xref="S5.SS2.SSS1.9.p1.5.m5.1.2.2">𝑘</ci><apply id="S5.SS2.SSS1.9.p1.5.m5.1.1.cmml" xref="S5.SS2.SSS1.9.p1.5.m5.1.2.3.2"><root id="S5.SS2.SSS1.9.p1.5.m5.1.1a.cmml" xref="S5.SS2.SSS1.9.p1.5.m5.1.2.3.2"></root><apply id="S5.SS2.SSS1.9.p1.5.m5.1.1.2.cmml" xref="S5.SS2.SSS1.9.p1.5.m5.1.1.2"><minus id="S5.SS2.SSS1.9.p1.5.m5.1.1.2.1.cmml" xref="S5.SS2.SSS1.9.p1.5.m5.1.1.2.1"></minus><cn id="S5.SS2.SSS1.9.p1.5.m5.1.1.2.2.cmml" type="integer" xref="S5.SS2.SSS1.9.p1.5.m5.1.1.2.2">1</cn><ci id="S5.SS2.SSS1.9.p1.5.m5.1.1.2.3.cmml" xref="S5.SS2.SSS1.9.p1.5.m5.1.1.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.9.p1.5.m5.1c">k(\sqrt{1-\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.9.p1.5.m5.1d">italic_k ( square-root start_ARG 1 - italic_ε end_ARG )</annotation></semantics></math> with probability <math alttext="1-e^{-\Theta(n)}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.9.p1.6.m6.1"><semantics id="S5.SS2.SSS1.9.p1.6.m6.1a"><mrow id="S5.SS2.SSS1.9.p1.6.m6.1.2" xref="S5.SS2.SSS1.9.p1.6.m6.1.2.cmml"><mn id="S5.SS2.SSS1.9.p1.6.m6.1.2.2" xref="S5.SS2.SSS1.9.p1.6.m6.1.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.9.p1.6.m6.1.2.1" xref="S5.SS2.SSS1.9.p1.6.m6.1.2.1.cmml">−</mo><msup id="S5.SS2.SSS1.9.p1.6.m6.1.2.3" xref="S5.SS2.SSS1.9.p1.6.m6.1.2.3.cmml"><mi id="S5.SS2.SSS1.9.p1.6.m6.1.2.3.2" xref="S5.SS2.SSS1.9.p1.6.m6.1.2.3.2.cmml">e</mi><mrow id="S5.SS2.SSS1.9.p1.6.m6.1.1.1" xref="S5.SS2.SSS1.9.p1.6.m6.1.1.1.cmml"><mo id="S5.SS2.SSS1.9.p1.6.m6.1.1.1a" xref="S5.SS2.SSS1.9.p1.6.m6.1.1.1.cmml">−</mo><mrow id="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3" xref="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.cmml"><mi id="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.2" mathvariant="normal" xref="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.2.cmml">Θ</mi><mo id="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.1" xref="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.3.2" xref="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.cmml"><mo id="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.3.2.1" stretchy="false" xref="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.cmml">(</mo><mi id="S5.SS2.SSS1.9.p1.6.m6.1.1.1.1" xref="S5.SS2.SSS1.9.p1.6.m6.1.1.1.1.cmml">n</mi><mo id="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.3.2.2" stretchy="false" xref="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.cmml">)</mo></mrow></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.9.p1.6.m6.1b"><apply id="S5.SS2.SSS1.9.p1.6.m6.1.2.cmml" xref="S5.SS2.SSS1.9.p1.6.m6.1.2"><minus id="S5.SS2.SSS1.9.p1.6.m6.1.2.1.cmml" xref="S5.SS2.SSS1.9.p1.6.m6.1.2.1"></minus><cn id="S5.SS2.SSS1.9.p1.6.m6.1.2.2.cmml" type="integer" xref="S5.SS2.SSS1.9.p1.6.m6.1.2.2">1</cn><apply id="S5.SS2.SSS1.9.p1.6.m6.1.2.3.cmml" xref="S5.SS2.SSS1.9.p1.6.m6.1.2.3"><csymbol cd="ambiguous" id="S5.SS2.SSS1.9.p1.6.m6.1.2.3.1.cmml" xref="S5.SS2.SSS1.9.p1.6.m6.1.2.3">superscript</csymbol><ci id="S5.SS2.SSS1.9.p1.6.m6.1.2.3.2.cmml" xref="S5.SS2.SSS1.9.p1.6.m6.1.2.3.2">𝑒</ci><apply id="S5.SS2.SSS1.9.p1.6.m6.1.1.1.cmml" xref="S5.SS2.SSS1.9.p1.6.m6.1.1.1"><minus id="S5.SS2.SSS1.9.p1.6.m6.1.1.1.2.cmml" xref="S5.SS2.SSS1.9.p1.6.m6.1.1.1"></minus><apply id="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.cmml" xref="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3"><times id="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.1.cmml" xref="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.1"></times><ci id="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.2.cmml" xref="S5.SS2.SSS1.9.p1.6.m6.1.1.1.3.2">Θ</ci><ci id="S5.SS2.SSS1.9.p1.6.m6.1.1.1.1.cmml" xref="S5.SS2.SSS1.9.p1.6.m6.1.1.1.1">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.9.p1.6.m6.1c">1-e^{-\Theta(n)}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.9.p1.6.m6.1d">1 - italic_e start_POSTSUPERSCRIPT - roman_Θ ( italic_n ) end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="\alpha" class="ltx_Math" display="inline" id="S5.SS2.SSS1.9.p1.7.m7.1"><semantics id="S5.SS2.SSS1.9.p1.7.m7.1a"><mi id="S5.SS2.SSS1.9.p1.7.m7.1.1" xref="S5.SS2.SSS1.9.p1.7.m7.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.9.p1.7.m7.1b"><ci id="S5.SS2.SSS1.9.p1.7.m7.1.1.cmml" xref="S5.SS2.SSS1.9.p1.7.m7.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.9.p1.7.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.9.p1.7.m7.1d">italic_α</annotation></semantics></math> is any constant in the range <math alttext="(0,1)" class="ltx_Math" display="inline" id="S5.SS2.SSS1.9.p1.8.m8.2"><semantics id="S5.SS2.SSS1.9.p1.8.m8.2a"><mrow id="S5.SS2.SSS1.9.p1.8.m8.2.3.2" xref="S5.SS2.SSS1.9.p1.8.m8.2.3.1.cmml"><mo id="S5.SS2.SSS1.9.p1.8.m8.2.3.2.1" stretchy="false" xref="S5.SS2.SSS1.9.p1.8.m8.2.3.1.cmml">(</mo><mn id="S5.SS2.SSS1.9.p1.8.m8.1.1" xref="S5.SS2.SSS1.9.p1.8.m8.1.1.cmml">0</mn><mo id="S5.SS2.SSS1.9.p1.8.m8.2.3.2.2" xref="S5.SS2.SSS1.9.p1.8.m8.2.3.1.cmml">,</mo><mn id="S5.SS2.SSS1.9.p1.8.m8.2.2" xref="S5.SS2.SSS1.9.p1.8.m8.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.9.p1.8.m8.2.3.2.3" stretchy="false" xref="S5.SS2.SSS1.9.p1.8.m8.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.9.p1.8.m8.2b"><interval closure="open" id="S5.SS2.SSS1.9.p1.8.m8.2.3.1.cmml" xref="S5.SS2.SSS1.9.p1.8.m8.2.3.2"><cn id="S5.SS2.SSS1.9.p1.8.m8.1.1.cmml" type="integer" xref="S5.SS2.SSS1.9.p1.8.m8.1.1">0</cn><cn id="S5.SS2.SSS1.9.p1.8.m8.2.2.cmml" type="integer" xref="S5.SS2.SSS1.9.p1.8.m8.2.2">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.9.p1.8.m8.2c">(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.9.p1.8.m8.2d">( 0 , 1 )</annotation></semantics></math>. Each clique contains at least <math alttext="k\sqrt{1-\varepsilon}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.9.p1.9.m9.1"><semantics id="S5.SS2.SSS1.9.p1.9.m9.1a"><mrow id="S5.SS2.SSS1.9.p1.9.m9.1.1" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.cmml"><mi id="S5.SS2.SSS1.9.p1.9.m9.1.1.2" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.2.cmml">k</mi><mo id="S5.SS2.SSS1.9.p1.9.m9.1.1.1" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.1.cmml">⁢</mo><msqrt id="S5.SS2.SSS1.9.p1.9.m9.1.1.3" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.3.cmml"><mrow id="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2.cmml"><mn id="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2.2" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2.1" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2.1.cmml">−</mo><mi id="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2.3" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.9.p1.9.m9.1b"><apply id="S5.SS2.SSS1.9.p1.9.m9.1.1.cmml" xref="S5.SS2.SSS1.9.p1.9.m9.1.1"><times id="S5.SS2.SSS1.9.p1.9.m9.1.1.1.cmml" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.1"></times><ci id="S5.SS2.SSS1.9.p1.9.m9.1.1.2.cmml" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.2">𝑘</ci><apply id="S5.SS2.SSS1.9.p1.9.m9.1.1.3.cmml" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.3"><root id="S5.SS2.SSS1.9.p1.9.m9.1.1.3a.cmml" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.3"></root><apply id="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2.cmml" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2"><minus id="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2.1.cmml" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2.1"></minus><cn id="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2.2.cmml" type="integer" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2.2">1</cn><ci id="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2.3.cmml" xref="S5.SS2.SSS1.9.p1.9.m9.1.1.3.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.9.p1.9.m9.1c">k\sqrt{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.9.p1.9.m9.1d">italic_k square-root start_ARG 1 - italic_ε end_ARG</annotation></semantics></math> agents, and the utility of every agent in such cliques is at least <math alttext="k(\sqrt{1-\varepsilon})-1" class="ltx_Math" display="inline" id="S5.SS2.SSS1.9.p1.10.m10.1"><semantics id="S5.SS2.SSS1.9.p1.10.m10.1a"><mrow id="S5.SS2.SSS1.9.p1.10.m10.1.2" xref="S5.SS2.SSS1.9.p1.10.m10.1.2.cmml"><mrow id="S5.SS2.SSS1.9.p1.10.m10.1.2.2" xref="S5.SS2.SSS1.9.p1.10.m10.1.2.2.cmml"><mi id="S5.SS2.SSS1.9.p1.10.m10.1.2.2.2" xref="S5.SS2.SSS1.9.p1.10.m10.1.2.2.2.cmml">k</mi><mo id="S5.SS2.SSS1.9.p1.10.m10.1.2.2.1" xref="S5.SS2.SSS1.9.p1.10.m10.1.2.2.1.cmml">⁢</mo><mrow id="S5.SS2.SSS1.9.p1.10.m10.1.2.2.3.2" xref="S5.SS2.SSS1.9.p1.10.m10.1.1.cmml"><mo id="S5.SS2.SSS1.9.p1.10.m10.1.2.2.3.2.1" stretchy="false" xref="S5.SS2.SSS1.9.p1.10.m10.1.1.cmml">(</mo><msqrt id="S5.SS2.SSS1.9.p1.10.m10.1.1" xref="S5.SS2.SSS1.9.p1.10.m10.1.1.cmml"><mrow id="S5.SS2.SSS1.9.p1.10.m10.1.1.2" xref="S5.SS2.SSS1.9.p1.10.m10.1.1.2.cmml"><mn id="S5.SS2.SSS1.9.p1.10.m10.1.1.2.2" xref="S5.SS2.SSS1.9.p1.10.m10.1.1.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.9.p1.10.m10.1.1.2.1" xref="S5.SS2.SSS1.9.p1.10.m10.1.1.2.1.cmml">−</mo><mi id="S5.SS2.SSS1.9.p1.10.m10.1.1.2.3" xref="S5.SS2.SSS1.9.p1.10.m10.1.1.2.3.cmml">ε</mi></mrow></msqrt><mo id="S5.SS2.SSS1.9.p1.10.m10.1.2.2.3.2.2" stretchy="false" xref="S5.SS2.SSS1.9.p1.10.m10.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS2.SSS1.9.p1.10.m10.1.2.1" xref="S5.SS2.SSS1.9.p1.10.m10.1.2.1.cmml">−</mo><mn id="S5.SS2.SSS1.9.p1.10.m10.1.2.3" xref="S5.SS2.SSS1.9.p1.10.m10.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.9.p1.10.m10.1b"><apply id="S5.SS2.SSS1.9.p1.10.m10.1.2.cmml" xref="S5.SS2.SSS1.9.p1.10.m10.1.2"><minus id="S5.SS2.SSS1.9.p1.10.m10.1.2.1.cmml" xref="S5.SS2.SSS1.9.p1.10.m10.1.2.1"></minus><apply id="S5.SS2.SSS1.9.p1.10.m10.1.2.2.cmml" xref="S5.SS2.SSS1.9.p1.10.m10.1.2.2"><times id="S5.SS2.SSS1.9.p1.10.m10.1.2.2.1.cmml" xref="S5.SS2.SSS1.9.p1.10.m10.1.2.2.1"></times><ci id="S5.SS2.SSS1.9.p1.10.m10.1.2.2.2.cmml" xref="S5.SS2.SSS1.9.p1.10.m10.1.2.2.2">𝑘</ci><apply id="S5.SS2.SSS1.9.p1.10.m10.1.1.cmml" xref="S5.SS2.SSS1.9.p1.10.m10.1.2.2.3.2"><root id="S5.SS2.SSS1.9.p1.10.m10.1.1a.cmml" xref="S5.SS2.SSS1.9.p1.10.m10.1.2.2.3.2"></root><apply id="S5.SS2.SSS1.9.p1.10.m10.1.1.2.cmml" xref="S5.SS2.SSS1.9.p1.10.m10.1.1.2"><minus id="S5.SS2.SSS1.9.p1.10.m10.1.1.2.1.cmml" xref="S5.SS2.SSS1.9.p1.10.m10.1.1.2.1"></minus><cn id="S5.SS2.SSS1.9.p1.10.m10.1.1.2.2.cmml" type="integer" xref="S5.SS2.SSS1.9.p1.10.m10.1.1.2.2">1</cn><ci id="S5.SS2.SSS1.9.p1.10.m10.1.1.2.3.cmml" xref="S5.SS2.SSS1.9.p1.10.m10.1.1.2.3">𝜀</ci></apply></apply></apply><cn id="S5.SS2.SSS1.9.p1.10.m10.1.2.3.cmml" type="integer" xref="S5.SS2.SSS1.9.p1.10.m10.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.9.p1.10.m10.1c">k(\sqrt{1-\varepsilon})-1</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.9.p1.10.m10.1d">italic_k ( square-root start_ARG 1 - italic_ε end_ARG ) - 1</annotation></semantics></math>. Therefore, the social welfare of the partition returned by the algorithm is bounded as</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex32"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{SW}(\pi)\geq\alpha\frac{n}{k}k\sqrt{1-\varepsilon}(k(\sqrt{1-% \varepsilon})-1)=\alpha nk(1-\varepsilon)-\alpha n\sqrt{1-\varepsilon}\text{.}" class="ltx_Math" display="block" id="S5.Ex32.m1.4"><semantics id="S5.Ex32.m1.4a"><mrow id="S5.Ex32.m1.4.4" xref="S5.Ex32.m1.4.4.cmml"><mrow id="S5.Ex32.m1.4.4.4" xref="S5.Ex32.m1.4.4.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex32.m1.4.4.4.2" xref="S5.Ex32.m1.4.4.4.2.cmml">𝒮</mi><mo id="S5.Ex32.m1.4.4.4.1" xref="S5.Ex32.m1.4.4.4.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex32.m1.4.4.4.3" xref="S5.Ex32.m1.4.4.4.3.cmml">𝒲</mi><mo id="S5.Ex32.m1.4.4.4.1a" xref="S5.Ex32.m1.4.4.4.1.cmml">⁢</mo><mrow id="S5.Ex32.m1.4.4.4.4.2" xref="S5.Ex32.m1.4.4.4.cmml"><mo id="S5.Ex32.m1.4.4.4.4.2.1" stretchy="false" xref="S5.Ex32.m1.4.4.4.cmml">(</mo><mi id="S5.Ex32.m1.1.1" xref="S5.Ex32.m1.1.1.cmml">π</mi><mo id="S5.Ex32.m1.4.4.4.4.2.2" stretchy="false" xref="S5.Ex32.m1.4.4.4.cmml">)</mo></mrow></mrow><mo id="S5.Ex32.m1.4.4.5" xref="S5.Ex32.m1.4.4.5.cmml">≥</mo><mrow id="S5.Ex32.m1.3.3.1" xref="S5.Ex32.m1.3.3.1.cmml"><mi id="S5.Ex32.m1.3.3.1.3" xref="S5.Ex32.m1.3.3.1.3.cmml">α</mi><mo id="S5.Ex32.m1.3.3.1.2" xref="S5.Ex32.m1.3.3.1.2.cmml">⁢</mo><mfrac id="S5.Ex32.m1.3.3.1.4" xref="S5.Ex32.m1.3.3.1.4.cmml"><mi id="S5.Ex32.m1.3.3.1.4.2" xref="S5.Ex32.m1.3.3.1.4.2.cmml">n</mi><mi id="S5.Ex32.m1.3.3.1.4.3" xref="S5.Ex32.m1.3.3.1.4.3.cmml">k</mi></mfrac><mo id="S5.Ex32.m1.3.3.1.2a" xref="S5.Ex32.m1.3.3.1.2.cmml">⁢</mo><mi id="S5.Ex32.m1.3.3.1.5" xref="S5.Ex32.m1.3.3.1.5.cmml">k</mi><mo id="S5.Ex32.m1.3.3.1.2b" xref="S5.Ex32.m1.3.3.1.2.cmml">⁢</mo><msqrt id="S5.Ex32.m1.3.3.1.6" xref="S5.Ex32.m1.3.3.1.6.cmml"><mrow id="S5.Ex32.m1.3.3.1.6.2" xref="S5.Ex32.m1.3.3.1.6.2.cmml"><mn id="S5.Ex32.m1.3.3.1.6.2.2" xref="S5.Ex32.m1.3.3.1.6.2.2.cmml">1</mn><mo id="S5.Ex32.m1.3.3.1.6.2.1" xref="S5.Ex32.m1.3.3.1.6.2.1.cmml">−</mo><mi id="S5.Ex32.m1.3.3.1.6.2.3" xref="S5.Ex32.m1.3.3.1.6.2.3.cmml">ε</mi></mrow></msqrt><mo id="S5.Ex32.m1.3.3.1.2c" xref="S5.Ex32.m1.3.3.1.2.cmml">⁢</mo><mrow id="S5.Ex32.m1.3.3.1.1.1" xref="S5.Ex32.m1.3.3.1.1.1.1.cmml"><mo id="S5.Ex32.m1.3.3.1.1.1.2" stretchy="false" xref="S5.Ex32.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S5.Ex32.m1.3.3.1.1.1.1" xref="S5.Ex32.m1.3.3.1.1.1.1.cmml"><mrow id="S5.Ex32.m1.3.3.1.1.1.1.2" xref="S5.Ex32.m1.3.3.1.1.1.1.2.cmml"><mi id="S5.Ex32.m1.3.3.1.1.1.1.2.2" xref="S5.Ex32.m1.3.3.1.1.1.1.2.2.cmml">k</mi><mo id="S5.Ex32.m1.3.3.1.1.1.1.2.1" xref="S5.Ex32.m1.3.3.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S5.Ex32.m1.3.3.1.1.1.1.2.3.2" xref="S5.Ex32.m1.2.2.cmml"><mo id="S5.Ex32.m1.3.3.1.1.1.1.2.3.2.1" stretchy="false" xref="S5.Ex32.m1.2.2.cmml">(</mo><msqrt id="S5.Ex32.m1.2.2" xref="S5.Ex32.m1.2.2.cmml"><mrow id="S5.Ex32.m1.2.2.2" xref="S5.Ex32.m1.2.2.2.cmml"><mn id="S5.Ex32.m1.2.2.2.2" xref="S5.Ex32.m1.2.2.2.2.cmml">1</mn><mo id="S5.Ex32.m1.2.2.2.1" xref="S5.Ex32.m1.2.2.2.1.cmml">−</mo><mi id="S5.Ex32.m1.2.2.2.3" xref="S5.Ex32.m1.2.2.2.3.cmml">ε</mi></mrow></msqrt><mo id="S5.Ex32.m1.3.3.1.1.1.1.2.3.2.2" stretchy="false" xref="S5.Ex32.m1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.Ex32.m1.3.3.1.1.1.1.1" xref="S5.Ex32.m1.3.3.1.1.1.1.1.cmml">−</mo><mn id="S5.Ex32.m1.3.3.1.1.1.1.3" xref="S5.Ex32.m1.3.3.1.1.1.1.3.cmml">1</mn></mrow><mo id="S5.Ex32.m1.3.3.1.1.1.3" stretchy="false" xref="S5.Ex32.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex32.m1.4.4.6" xref="S5.Ex32.m1.4.4.6.cmml">=</mo><mrow id="S5.Ex32.m1.4.4.2" xref="S5.Ex32.m1.4.4.2.cmml"><mrow id="S5.Ex32.m1.4.4.2.1" xref="S5.Ex32.m1.4.4.2.1.cmml"><mi id="S5.Ex32.m1.4.4.2.1.3" xref="S5.Ex32.m1.4.4.2.1.3.cmml">α</mi><mo id="S5.Ex32.m1.4.4.2.1.2" xref="S5.Ex32.m1.4.4.2.1.2.cmml">⁢</mo><mi id="S5.Ex32.m1.4.4.2.1.4" xref="S5.Ex32.m1.4.4.2.1.4.cmml">n</mi><mo id="S5.Ex32.m1.4.4.2.1.2a" xref="S5.Ex32.m1.4.4.2.1.2.cmml">⁢</mo><mi id="S5.Ex32.m1.4.4.2.1.5" xref="S5.Ex32.m1.4.4.2.1.5.cmml">k</mi><mo id="S5.Ex32.m1.4.4.2.1.2b" xref="S5.Ex32.m1.4.4.2.1.2.cmml">⁢</mo><mrow id="S5.Ex32.m1.4.4.2.1.1.1" xref="S5.Ex32.m1.4.4.2.1.1.1.1.cmml"><mo id="S5.Ex32.m1.4.4.2.1.1.1.2" stretchy="false" xref="S5.Ex32.m1.4.4.2.1.1.1.1.cmml">(</mo><mrow id="S5.Ex32.m1.4.4.2.1.1.1.1" xref="S5.Ex32.m1.4.4.2.1.1.1.1.cmml"><mn id="S5.Ex32.m1.4.4.2.1.1.1.1.2" xref="S5.Ex32.m1.4.4.2.1.1.1.1.2.cmml">1</mn><mo id="S5.Ex32.m1.4.4.2.1.1.1.1.1" xref="S5.Ex32.m1.4.4.2.1.1.1.1.1.cmml">−</mo><mi id="S5.Ex32.m1.4.4.2.1.1.1.1.3" xref="S5.Ex32.m1.4.4.2.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.Ex32.m1.4.4.2.1.1.1.3" stretchy="false" xref="S5.Ex32.m1.4.4.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex32.m1.4.4.2.2" xref="S5.Ex32.m1.4.4.2.2.cmml">−</mo><mrow id="S5.Ex32.m1.4.4.2.3" xref="S5.Ex32.m1.4.4.2.3.cmml"><mi id="S5.Ex32.m1.4.4.2.3.2" xref="S5.Ex32.m1.4.4.2.3.2.cmml">α</mi><mo id="S5.Ex32.m1.4.4.2.3.1" xref="S5.Ex32.m1.4.4.2.3.1.cmml">⁢</mo><mi id="S5.Ex32.m1.4.4.2.3.3" xref="S5.Ex32.m1.4.4.2.3.3.cmml">n</mi><mo id="S5.Ex32.m1.4.4.2.3.1a" xref="S5.Ex32.m1.4.4.2.3.1.cmml">⁢</mo><msqrt id="S5.Ex32.m1.4.4.2.3.4" xref="S5.Ex32.m1.4.4.2.3.4.cmml"><mrow id="S5.Ex32.m1.4.4.2.3.4.2" xref="S5.Ex32.m1.4.4.2.3.4.2.cmml"><mn id="S5.Ex32.m1.4.4.2.3.4.2.2" xref="S5.Ex32.m1.4.4.2.3.4.2.2.cmml">1</mn><mo id="S5.Ex32.m1.4.4.2.3.4.2.1" xref="S5.Ex32.m1.4.4.2.3.4.2.1.cmml">−</mo><mi id="S5.Ex32.m1.4.4.2.3.4.2.3" xref="S5.Ex32.m1.4.4.2.3.4.2.3.cmml">ε</mi></mrow></msqrt><mo id="S5.Ex32.m1.4.4.2.3.1b" xref="S5.Ex32.m1.4.4.2.3.1.cmml">⁢</mo><mtext id="S5.Ex32.m1.4.4.2.3.5" xref="S5.Ex32.m1.4.4.2.3.5a.cmml">.</mtext></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex32.m1.4b"><apply id="S5.Ex32.m1.4.4.cmml" xref="S5.Ex32.m1.4.4"><and id="S5.Ex32.m1.4.4a.cmml" xref="S5.Ex32.m1.4.4"></and><apply id="S5.Ex32.m1.4.4b.cmml" xref="S5.Ex32.m1.4.4"><geq id="S5.Ex32.m1.4.4.5.cmml" xref="S5.Ex32.m1.4.4.5"></geq><apply id="S5.Ex32.m1.4.4.4.cmml" xref="S5.Ex32.m1.4.4.4"><times id="S5.Ex32.m1.4.4.4.1.cmml" xref="S5.Ex32.m1.4.4.4.1"></times><ci id="S5.Ex32.m1.4.4.4.2.cmml" xref="S5.Ex32.m1.4.4.4.2">𝒮</ci><ci id="S5.Ex32.m1.4.4.4.3.cmml" xref="S5.Ex32.m1.4.4.4.3">𝒲</ci><ci id="S5.Ex32.m1.1.1.cmml" xref="S5.Ex32.m1.1.1">𝜋</ci></apply><apply id="S5.Ex32.m1.3.3.1.cmml" xref="S5.Ex32.m1.3.3.1"><times id="S5.Ex32.m1.3.3.1.2.cmml" xref="S5.Ex32.m1.3.3.1.2"></times><ci id="S5.Ex32.m1.3.3.1.3.cmml" xref="S5.Ex32.m1.3.3.1.3">𝛼</ci><apply id="S5.Ex32.m1.3.3.1.4.cmml" xref="S5.Ex32.m1.3.3.1.4"><divide id="S5.Ex32.m1.3.3.1.4.1.cmml" xref="S5.Ex32.m1.3.3.1.4"></divide><ci id="S5.Ex32.m1.3.3.1.4.2.cmml" xref="S5.Ex32.m1.3.3.1.4.2">𝑛</ci><ci id="S5.Ex32.m1.3.3.1.4.3.cmml" xref="S5.Ex32.m1.3.3.1.4.3">𝑘</ci></apply><ci id="S5.Ex32.m1.3.3.1.5.cmml" xref="S5.Ex32.m1.3.3.1.5">𝑘</ci><apply id="S5.Ex32.m1.3.3.1.6.cmml" xref="S5.Ex32.m1.3.3.1.6"><root id="S5.Ex32.m1.3.3.1.6a.cmml" xref="S5.Ex32.m1.3.3.1.6"></root><apply id="S5.Ex32.m1.3.3.1.6.2.cmml" xref="S5.Ex32.m1.3.3.1.6.2"><minus id="S5.Ex32.m1.3.3.1.6.2.1.cmml" xref="S5.Ex32.m1.3.3.1.6.2.1"></minus><cn id="S5.Ex32.m1.3.3.1.6.2.2.cmml" type="integer" xref="S5.Ex32.m1.3.3.1.6.2.2">1</cn><ci id="S5.Ex32.m1.3.3.1.6.2.3.cmml" xref="S5.Ex32.m1.3.3.1.6.2.3">𝜀</ci></apply></apply><apply id="S5.Ex32.m1.3.3.1.1.1.1.cmml" xref="S5.Ex32.m1.3.3.1.1.1"><minus id="S5.Ex32.m1.3.3.1.1.1.1.1.cmml" xref="S5.Ex32.m1.3.3.1.1.1.1.1"></minus><apply id="S5.Ex32.m1.3.3.1.1.1.1.2.cmml" xref="S5.Ex32.m1.3.3.1.1.1.1.2"><times id="S5.Ex32.m1.3.3.1.1.1.1.2.1.cmml" xref="S5.Ex32.m1.3.3.1.1.1.1.2.1"></times><ci id="S5.Ex32.m1.3.3.1.1.1.1.2.2.cmml" xref="S5.Ex32.m1.3.3.1.1.1.1.2.2">𝑘</ci><apply id="S5.Ex32.m1.2.2.cmml" xref="S5.Ex32.m1.3.3.1.1.1.1.2.3.2"><root id="S5.Ex32.m1.2.2a.cmml" xref="S5.Ex32.m1.3.3.1.1.1.1.2.3.2"></root><apply id="S5.Ex32.m1.2.2.2.cmml" xref="S5.Ex32.m1.2.2.2"><minus id="S5.Ex32.m1.2.2.2.1.cmml" xref="S5.Ex32.m1.2.2.2.1"></minus><cn id="S5.Ex32.m1.2.2.2.2.cmml" type="integer" xref="S5.Ex32.m1.2.2.2.2">1</cn><ci id="S5.Ex32.m1.2.2.2.3.cmml" xref="S5.Ex32.m1.2.2.2.3">𝜀</ci></apply></apply></apply><cn id="S5.Ex32.m1.3.3.1.1.1.1.3.cmml" type="integer" xref="S5.Ex32.m1.3.3.1.1.1.1.3">1</cn></apply></apply></apply><apply id="S5.Ex32.m1.4.4c.cmml" xref="S5.Ex32.m1.4.4"><eq id="S5.Ex32.m1.4.4.6.cmml" xref="S5.Ex32.m1.4.4.6"></eq><share href="https://arxiv.org/html/2503.06017v1#S5.Ex32.m1.3.3.1.cmml" id="S5.Ex32.m1.4.4d.cmml" xref="S5.Ex32.m1.4.4"></share><apply id="S5.Ex32.m1.4.4.2.cmml" xref="S5.Ex32.m1.4.4.2"><minus id="S5.Ex32.m1.4.4.2.2.cmml" xref="S5.Ex32.m1.4.4.2.2"></minus><apply id="S5.Ex32.m1.4.4.2.1.cmml" xref="S5.Ex32.m1.4.4.2.1"><times id="S5.Ex32.m1.4.4.2.1.2.cmml" xref="S5.Ex32.m1.4.4.2.1.2"></times><ci id="S5.Ex32.m1.4.4.2.1.3.cmml" xref="S5.Ex32.m1.4.4.2.1.3">𝛼</ci><ci id="S5.Ex32.m1.4.4.2.1.4.cmml" xref="S5.Ex32.m1.4.4.2.1.4">𝑛</ci><ci id="S5.Ex32.m1.4.4.2.1.5.cmml" xref="S5.Ex32.m1.4.4.2.1.5">𝑘</ci><apply id="S5.Ex32.m1.4.4.2.1.1.1.1.cmml" xref="S5.Ex32.m1.4.4.2.1.1.1"><minus id="S5.Ex32.m1.4.4.2.1.1.1.1.1.cmml" xref="S5.Ex32.m1.4.4.2.1.1.1.1.1"></minus><cn id="S5.Ex32.m1.4.4.2.1.1.1.1.2.cmml" type="integer" xref="S5.Ex32.m1.4.4.2.1.1.1.1.2">1</cn><ci id="S5.Ex32.m1.4.4.2.1.1.1.1.3.cmml" xref="S5.Ex32.m1.4.4.2.1.1.1.1.3">𝜀</ci></apply></apply><apply id="S5.Ex32.m1.4.4.2.3.cmml" xref="S5.Ex32.m1.4.4.2.3"><times id="S5.Ex32.m1.4.4.2.3.1.cmml" xref="S5.Ex32.m1.4.4.2.3.1"></times><ci id="S5.Ex32.m1.4.4.2.3.2.cmml" xref="S5.Ex32.m1.4.4.2.3.2">𝛼</ci><ci id="S5.Ex32.m1.4.4.2.3.3.cmml" xref="S5.Ex32.m1.4.4.2.3.3">𝑛</ci><apply id="S5.Ex32.m1.4.4.2.3.4.cmml" xref="S5.Ex32.m1.4.4.2.3.4"><root id="S5.Ex32.m1.4.4.2.3.4a.cmml" xref="S5.Ex32.m1.4.4.2.3.4"></root><apply id="S5.Ex32.m1.4.4.2.3.4.2.cmml" xref="S5.Ex32.m1.4.4.2.3.4.2"><minus id="S5.Ex32.m1.4.4.2.3.4.2.1.cmml" xref="S5.Ex32.m1.4.4.2.3.4.2.1"></minus><cn id="S5.Ex32.m1.4.4.2.3.4.2.2.cmml" type="integer" xref="S5.Ex32.m1.4.4.2.3.4.2.2">1</cn><ci id="S5.Ex32.m1.4.4.2.3.4.2.3.cmml" xref="S5.Ex32.m1.4.4.2.3.4.2.3">𝜀</ci></apply></apply><ci id="S5.Ex32.m1.4.4.2.3.5a.cmml" xref="S5.Ex32.m1.4.4.2.3.5"><mtext id="S5.Ex32.m1.4.4.2.3.5.cmml" xref="S5.Ex32.m1.4.4.2.3.5">.</mtext></ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex32.m1.4c">\mathcal{SW}(\pi)\geq\alpha\frac{n}{k}k\sqrt{1-\varepsilon}(k(\sqrt{1-% \varepsilon})-1)=\alpha nk(1-\varepsilon)-\alpha n\sqrt{1-\varepsilon}\text{.}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex32.m1.4d">caligraphic_S caligraphic_W ( italic_π ) ≥ italic_α divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG italic_k square-root start_ARG 1 - italic_ε end_ARG ( italic_k ( square-root start_ARG 1 - italic_ε end_ARG ) - 1 ) = italic_α italic_n italic_k ( 1 - italic_ε ) - italic_α italic_n square-root start_ARG 1 - italic_ε end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S5.SS2.SSS1.10.p2"> <p class="ltx_p" id="S5.SS2.SSS1.10.p2.1">Moreover, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem6" title="Proposition 5.6. ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">5.6</span></a> implies that <math alttext="\mathcal{SW}(\pi^{*})\leq n(k-1)<nk" class="ltx_Math" display="inline" id="S5.SS2.SSS1.10.p2.1.m1.2"><semantics id="S5.SS2.SSS1.10.p2.1.m1.2a"><mrow id="S5.SS2.SSS1.10.p2.1.m1.2.2" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.cmml"><mrow id="S5.SS2.SSS1.10.p2.1.m1.1.1.1" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.3" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.3.cmml">𝒮</mi><mo id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.2" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.4" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.4.cmml">𝒲</mi><mo id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.2a" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.1.cmml"><mo id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.1" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.1.cmml"><mi id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.1.2" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.1.2.cmml">π</mi><mo id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.1.3" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS2.SSS1.10.p2.1.m1.2.2.4" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.4.cmml">≤</mo><mrow id="S5.SS2.SSS1.10.p2.1.m1.2.2.2" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.cmml"><mi id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.3" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.3.cmml">n</mi><mo id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.2" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.2.cmml">⁢</mo><mrow id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.cmml"><mo id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.2" stretchy="false" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.cmml"><mi id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.2" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.2.cmml">k</mi><mo id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.1" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.1.cmml">−</mo><mn id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.3" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.3" stretchy="false" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS2.SSS1.10.p2.1.m1.2.2.5" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.5.cmml"><</mo><mrow id="S5.SS2.SSS1.10.p2.1.m1.2.2.6" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.6.cmml"><mi id="S5.SS2.SSS1.10.p2.1.m1.2.2.6.2" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.6.2.cmml">n</mi><mo id="S5.SS2.SSS1.10.p2.1.m1.2.2.6.1" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.6.1.cmml">⁢</mo><mi id="S5.SS2.SSS1.10.p2.1.m1.2.2.6.3" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.6.3.cmml">k</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.10.p2.1.m1.2b"><apply id="S5.SS2.SSS1.10.p2.1.m1.2.2.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2"><and id="S5.SS2.SSS1.10.p2.1.m1.2.2a.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2"></and><apply id="S5.SS2.SSS1.10.p2.1.m1.2.2b.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2"><leq id="S5.SS2.SSS1.10.p2.1.m1.2.2.4.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.4"></leq><apply id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1"><times id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.2.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.2"></times><ci id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.3.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.3">𝒮</ci><ci id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.4.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.4">𝒲</ci><apply id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1">superscript</csymbol><ci id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.1.2.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.1.2">𝜋</ci><times id="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.1.3.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.1.1.1.1.1.1.3"></times></apply></apply><apply id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2"><times id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.2.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.2"></times><ci id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.3.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.3">𝑛</ci><apply id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1"><minus id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.1.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.1"></minus><ci id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.2.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.2">𝑘</ci><cn id="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.3.cmml" type="integer" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.2.1.1.1.3">1</cn></apply></apply></apply><apply id="S5.SS2.SSS1.10.p2.1.m1.2.2c.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2"><lt id="S5.SS2.SSS1.10.p2.1.m1.2.2.5.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.5"></lt><share href="https://arxiv.org/html/2503.06017v1#S5.SS2.SSS1.10.p2.1.m1.2.2.2.cmml" id="S5.SS2.SSS1.10.p2.1.m1.2.2d.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2"></share><apply id="S5.SS2.SSS1.10.p2.1.m1.2.2.6.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.6"><times id="S5.SS2.SSS1.10.p2.1.m1.2.2.6.1.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.6.1"></times><ci id="S5.SS2.SSS1.10.p2.1.m1.2.2.6.2.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.6.2">𝑛</ci><ci id="S5.SS2.SSS1.10.p2.1.m1.2.2.6.3.cmml" xref="S5.SS2.SSS1.10.p2.1.m1.2.2.6.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.10.p2.1.m1.2c">\mathcal{SW}(\pi^{*})\leq n(k-1)<nk</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.10.p2.1.m1.2d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ≤ italic_n ( italic_k - 1 ) < italic_n italic_k</annotation></semantics></math>. Hence,</p> <table class="ltx_equation ltx_eqn_table" id="S5.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{\mathcal{SW}(\pi)}{\mathcal{SW}(\pi^{*})}\geq\alpha(1-\varepsilon)-\frac% {\alpha\sqrt{1-\varepsilon}}{k}\text{.}" class="ltx_Math" display="block" id="S5.E3.m1.3"><semantics id="S5.E3.m1.3a"><mrow id="S5.E3.m1.3.3" xref="S5.E3.m1.3.3.cmml"><mfrac id="S5.E3.m1.2.2" xref="S5.E3.m1.2.2.cmml"><mrow id="S5.E3.m1.1.1.1" xref="S5.E3.m1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.E3.m1.1.1.1.3" xref="S5.E3.m1.1.1.1.3.cmml">𝒮</mi><mo id="S5.E3.m1.1.1.1.2" xref="S5.E3.m1.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.E3.m1.1.1.1.4" xref="S5.E3.m1.1.1.1.4.cmml">𝒲</mi><mo id="S5.E3.m1.1.1.1.2a" xref="S5.E3.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.E3.m1.1.1.1.5.2" xref="S5.E3.m1.1.1.1.cmml"><mo id="S5.E3.m1.1.1.1.5.2.1" stretchy="false" xref="S5.E3.m1.1.1.1.cmml">(</mo><mi id="S5.E3.m1.1.1.1.1" xref="S5.E3.m1.1.1.1.1.cmml">π</mi><mo id="S5.E3.m1.1.1.1.5.2.2" stretchy="false" xref="S5.E3.m1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S5.E3.m1.2.2.2" xref="S5.E3.m1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.E3.m1.2.2.2.3" xref="S5.E3.m1.2.2.2.3.cmml">𝒮</mi><mo id="S5.E3.m1.2.2.2.2" xref="S5.E3.m1.2.2.2.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.E3.m1.2.2.2.4" xref="S5.E3.m1.2.2.2.4.cmml">𝒲</mi><mo id="S5.E3.m1.2.2.2.2a" xref="S5.E3.m1.2.2.2.2.cmml">⁢</mo><mrow id="S5.E3.m1.2.2.2.1.1" xref="S5.E3.m1.2.2.2.1.1.1.cmml"><mo id="S5.E3.m1.2.2.2.1.1.2" stretchy="false" xref="S5.E3.m1.2.2.2.1.1.1.cmml">(</mo><msup id="S5.E3.m1.2.2.2.1.1.1" xref="S5.E3.m1.2.2.2.1.1.1.cmml"><mi id="S5.E3.m1.2.2.2.1.1.1.2" xref="S5.E3.m1.2.2.2.1.1.1.2.cmml">π</mi><mo id="S5.E3.m1.2.2.2.1.1.1.3" xref="S5.E3.m1.2.2.2.1.1.1.3.cmml">∗</mo></msup><mo id="S5.E3.m1.2.2.2.1.1.3" stretchy="false" xref="S5.E3.m1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mfrac><mo id="S5.E3.m1.3.3.2" xref="S5.E3.m1.3.3.2.cmml">≥</mo><mrow id="S5.E3.m1.3.3.1" xref="S5.E3.m1.3.3.1.cmml"><mrow id="S5.E3.m1.3.3.1.1" xref="S5.E3.m1.3.3.1.1.cmml"><mi id="S5.E3.m1.3.3.1.1.3" xref="S5.E3.m1.3.3.1.1.3.cmml">α</mi><mo id="S5.E3.m1.3.3.1.1.2" xref="S5.E3.m1.3.3.1.1.2.cmml">⁢</mo><mrow id="S5.E3.m1.3.3.1.1.1.1" xref="S5.E3.m1.3.3.1.1.1.1.1.cmml"><mo id="S5.E3.m1.3.3.1.1.1.1.2" stretchy="false" xref="S5.E3.m1.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S5.E3.m1.3.3.1.1.1.1.1" xref="S5.E3.m1.3.3.1.1.1.1.1.cmml"><mn id="S5.E3.m1.3.3.1.1.1.1.1.2" xref="S5.E3.m1.3.3.1.1.1.1.1.2.cmml">1</mn><mo id="S5.E3.m1.3.3.1.1.1.1.1.1" xref="S5.E3.m1.3.3.1.1.1.1.1.1.cmml">−</mo><mi id="S5.E3.m1.3.3.1.1.1.1.1.3" xref="S5.E3.m1.3.3.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.E3.m1.3.3.1.1.1.1.3" stretchy="false" xref="S5.E3.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.E3.m1.3.3.1.2" xref="S5.E3.m1.3.3.1.2.cmml">−</mo><mrow id="S5.E3.m1.3.3.1.3" xref="S5.E3.m1.3.3.1.3.cmml"><mfrac id="S5.E3.m1.3.3.1.3.2" xref="S5.E3.m1.3.3.1.3.2.cmml"><mrow id="S5.E3.m1.3.3.1.3.2.2" xref="S5.E3.m1.3.3.1.3.2.2.cmml"><mi id="S5.E3.m1.3.3.1.3.2.2.2" xref="S5.E3.m1.3.3.1.3.2.2.2.cmml">α</mi><mo id="S5.E3.m1.3.3.1.3.2.2.1" xref="S5.E3.m1.3.3.1.3.2.2.1.cmml">⁢</mo><msqrt id="S5.E3.m1.3.3.1.3.2.2.3" xref="S5.E3.m1.3.3.1.3.2.2.3.cmml"><mrow id="S5.E3.m1.3.3.1.3.2.2.3.2" xref="S5.E3.m1.3.3.1.3.2.2.3.2.cmml"><mn id="S5.E3.m1.3.3.1.3.2.2.3.2.2" xref="S5.E3.m1.3.3.1.3.2.2.3.2.2.cmml">1</mn><mo id="S5.E3.m1.3.3.1.3.2.2.3.2.1" xref="S5.E3.m1.3.3.1.3.2.2.3.2.1.cmml">−</mo><mi id="S5.E3.m1.3.3.1.3.2.2.3.2.3" xref="S5.E3.m1.3.3.1.3.2.2.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><mi id="S5.E3.m1.3.3.1.3.2.3" xref="S5.E3.m1.3.3.1.3.2.3.cmml">k</mi></mfrac><mo id="S5.E3.m1.3.3.1.3.1" xref="S5.E3.m1.3.3.1.3.1.cmml">⁢</mo><mtext id="S5.E3.m1.3.3.1.3.3" xref="S5.E3.m1.3.3.1.3.3a.cmml">.</mtext></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.E3.m1.3b"><apply id="S5.E3.m1.3.3.cmml" xref="S5.E3.m1.3.3"><geq id="S5.E3.m1.3.3.2.cmml" xref="S5.E3.m1.3.3.2"></geq><apply id="S5.E3.m1.2.2.cmml" xref="S5.E3.m1.2.2"><divide id="S5.E3.m1.2.2.3.cmml" xref="S5.E3.m1.2.2"></divide><apply id="S5.E3.m1.1.1.1.cmml" xref="S5.E3.m1.1.1.1"><times id="S5.E3.m1.1.1.1.2.cmml" xref="S5.E3.m1.1.1.1.2"></times><ci id="S5.E3.m1.1.1.1.3.cmml" xref="S5.E3.m1.1.1.1.3">𝒮</ci><ci id="S5.E3.m1.1.1.1.4.cmml" xref="S5.E3.m1.1.1.1.4">𝒲</ci><ci id="S5.E3.m1.1.1.1.1.cmml" xref="S5.E3.m1.1.1.1.1">𝜋</ci></apply><apply id="S5.E3.m1.2.2.2.cmml" xref="S5.E3.m1.2.2.2"><times id="S5.E3.m1.2.2.2.2.cmml" xref="S5.E3.m1.2.2.2.2"></times><ci id="S5.E3.m1.2.2.2.3.cmml" xref="S5.E3.m1.2.2.2.3">𝒮</ci><ci id="S5.E3.m1.2.2.2.4.cmml" xref="S5.E3.m1.2.2.2.4">𝒲</ci><apply id="S5.E3.m1.2.2.2.1.1.1.cmml" xref="S5.E3.m1.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.E3.m1.2.2.2.1.1.1.1.cmml" xref="S5.E3.m1.2.2.2.1.1">superscript</csymbol><ci id="S5.E3.m1.2.2.2.1.1.1.2.cmml" xref="S5.E3.m1.2.2.2.1.1.1.2">𝜋</ci><times id="S5.E3.m1.2.2.2.1.1.1.3.cmml" xref="S5.E3.m1.2.2.2.1.1.1.3"></times></apply></apply></apply><apply id="S5.E3.m1.3.3.1.cmml" xref="S5.E3.m1.3.3.1"><minus id="S5.E3.m1.3.3.1.2.cmml" xref="S5.E3.m1.3.3.1.2"></minus><apply id="S5.E3.m1.3.3.1.1.cmml" xref="S5.E3.m1.3.3.1.1"><times id="S5.E3.m1.3.3.1.1.2.cmml" xref="S5.E3.m1.3.3.1.1.2"></times><ci id="S5.E3.m1.3.3.1.1.3.cmml" xref="S5.E3.m1.3.3.1.1.3">𝛼</ci><apply id="S5.E3.m1.3.3.1.1.1.1.1.cmml" xref="S5.E3.m1.3.3.1.1.1.1"><minus id="S5.E3.m1.3.3.1.1.1.1.1.1.cmml" xref="S5.E3.m1.3.3.1.1.1.1.1.1"></minus><cn id="S5.E3.m1.3.3.1.1.1.1.1.2.cmml" type="integer" xref="S5.E3.m1.3.3.1.1.1.1.1.2">1</cn><ci id="S5.E3.m1.3.3.1.1.1.1.1.3.cmml" xref="S5.E3.m1.3.3.1.1.1.1.1.3">𝜀</ci></apply></apply><apply id="S5.E3.m1.3.3.1.3.cmml" xref="S5.E3.m1.3.3.1.3"><times id="S5.E3.m1.3.3.1.3.1.cmml" xref="S5.E3.m1.3.3.1.3.1"></times><apply id="S5.E3.m1.3.3.1.3.2.cmml" xref="S5.E3.m1.3.3.1.3.2"><divide id="S5.E3.m1.3.3.1.3.2.1.cmml" xref="S5.E3.m1.3.3.1.3.2"></divide><apply id="S5.E3.m1.3.3.1.3.2.2.cmml" xref="S5.E3.m1.3.3.1.3.2.2"><times id="S5.E3.m1.3.3.1.3.2.2.1.cmml" xref="S5.E3.m1.3.3.1.3.2.2.1"></times><ci id="S5.E3.m1.3.3.1.3.2.2.2.cmml" xref="S5.E3.m1.3.3.1.3.2.2.2">𝛼</ci><apply id="S5.E3.m1.3.3.1.3.2.2.3.cmml" xref="S5.E3.m1.3.3.1.3.2.2.3"><root id="S5.E3.m1.3.3.1.3.2.2.3a.cmml" xref="S5.E3.m1.3.3.1.3.2.2.3"></root><apply id="S5.E3.m1.3.3.1.3.2.2.3.2.cmml" xref="S5.E3.m1.3.3.1.3.2.2.3.2"><minus id="S5.E3.m1.3.3.1.3.2.2.3.2.1.cmml" xref="S5.E3.m1.3.3.1.3.2.2.3.2.1"></minus><cn id="S5.E3.m1.3.3.1.3.2.2.3.2.2.cmml" type="integer" xref="S5.E3.m1.3.3.1.3.2.2.3.2.2">1</cn><ci id="S5.E3.m1.3.3.1.3.2.2.3.2.3.cmml" xref="S5.E3.m1.3.3.1.3.2.2.3.2.3">𝜀</ci></apply></apply></apply><ci id="S5.E3.m1.3.3.1.3.2.3.cmml" xref="S5.E3.m1.3.3.1.3.2.3">𝑘</ci></apply><ci id="S5.E3.m1.3.3.1.3.3a.cmml" xref="S5.E3.m1.3.3.1.3.3"><mtext id="S5.E3.m1.3.3.1.3.3.cmml" xref="S5.E3.m1.3.3.1.3.3">.</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E3.m1.3c">\frac{\mathcal{SW}(\pi)}{\mathcal{SW}(\pi^{*})}\geq\alpha(1-\varepsilon)-\frac% {\alpha\sqrt{1-\varepsilon}}{k}\text{.}</annotation><annotation encoding="application/x-llamapun" id="S5.E3.m1.3d">divide start_ARG caligraphic_S caligraphic_W ( italic_π ) end_ARG start_ARG caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) end_ARG ≥ italic_α ( 1 - italic_ε ) - divide start_ARG italic_α square-root start_ARG 1 - italic_ε end_ARG end_ARG start_ARG italic_k end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S5.SS2.SSS1.11.p3"> <p class="ltx_p" id="S5.SS2.SSS1.11.p3.7">Note that <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S5.SS2.SSS1.11.p3.1.m1.1"><semantics id="S5.SS2.SSS1.11.p3.1.m1.1a"><mi id="S5.SS2.SSS1.11.p3.1.m1.1.1" xref="S5.SS2.SSS1.11.p3.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.11.p3.1.m1.1b"><ci id="S5.SS2.SSS1.11.p3.1.m1.1.1.cmml" xref="S5.SS2.SSS1.11.p3.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.11.p3.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.11.p3.1.m1.1d">italic_ε</annotation></semantics></math> is a parameter of our choice, and it can be chosen arbitrarily close to zero. In addition, <math alttext="\alpha" class="ltx_Math" display="inline" id="S5.SS2.SSS1.11.p3.2.m2.1"><semantics id="S5.SS2.SSS1.11.p3.2.m2.1a"><mi id="S5.SS2.SSS1.11.p3.2.m2.1.1" xref="S5.SS2.SSS1.11.p3.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.11.p3.2.m2.1b"><ci id="S5.SS2.SSS1.11.p3.2.m2.1.1.cmml" xref="S5.SS2.SSS1.11.p3.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.11.p3.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.11.p3.2.m2.1d">italic_α</annotation></semantics></math> is a constant with <math alttext="\alpha\in(0,1)" class="ltx_Math" display="inline" id="S5.SS2.SSS1.11.p3.3.m3.2"><semantics id="S5.SS2.SSS1.11.p3.3.m3.2a"><mrow id="S5.SS2.SSS1.11.p3.3.m3.2.3" xref="S5.SS2.SSS1.11.p3.3.m3.2.3.cmml"><mi id="S5.SS2.SSS1.11.p3.3.m3.2.3.2" xref="S5.SS2.SSS1.11.p3.3.m3.2.3.2.cmml">α</mi><mo id="S5.SS2.SSS1.11.p3.3.m3.2.3.1" xref="S5.SS2.SSS1.11.p3.3.m3.2.3.1.cmml">∈</mo><mrow id="S5.SS2.SSS1.11.p3.3.m3.2.3.3.2" xref="S5.SS2.SSS1.11.p3.3.m3.2.3.3.1.cmml"><mo id="S5.SS2.SSS1.11.p3.3.m3.2.3.3.2.1" stretchy="false" xref="S5.SS2.SSS1.11.p3.3.m3.2.3.3.1.cmml">(</mo><mn id="S5.SS2.SSS1.11.p3.3.m3.1.1" xref="S5.SS2.SSS1.11.p3.3.m3.1.1.cmml">0</mn><mo id="S5.SS2.SSS1.11.p3.3.m3.2.3.3.2.2" xref="S5.SS2.SSS1.11.p3.3.m3.2.3.3.1.cmml">,</mo><mn id="S5.SS2.SSS1.11.p3.3.m3.2.2" xref="S5.SS2.SSS1.11.p3.3.m3.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.11.p3.3.m3.2.3.3.2.3" stretchy="false" xref="S5.SS2.SSS1.11.p3.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.11.p3.3.m3.2b"><apply id="S5.SS2.SSS1.11.p3.3.m3.2.3.cmml" xref="S5.SS2.SSS1.11.p3.3.m3.2.3"><in id="S5.SS2.SSS1.11.p3.3.m3.2.3.1.cmml" xref="S5.SS2.SSS1.11.p3.3.m3.2.3.1"></in><ci id="S5.SS2.SSS1.11.p3.3.m3.2.3.2.cmml" xref="S5.SS2.SSS1.11.p3.3.m3.2.3.2">𝛼</ci><interval closure="open" id="S5.SS2.SSS1.11.p3.3.m3.2.3.3.1.cmml" xref="S5.SS2.SSS1.11.p3.3.m3.2.3.3.2"><cn id="S5.SS2.SSS1.11.p3.3.m3.1.1.cmml" type="integer" xref="S5.SS2.SSS1.11.p3.3.m3.1.1">0</cn><cn id="S5.SS2.SSS1.11.p3.3.m3.2.2.cmml" type="integer" xref="S5.SS2.SSS1.11.p3.3.m3.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.11.p3.3.m3.2c">\alpha\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.11.p3.3.m3.2d">italic_α ∈ ( 0 , 1 )</annotation></semantics></math>, meaning that <math alttext="\alpha" class="ltx_Math" display="inline" id="S5.SS2.SSS1.11.p3.4.m4.1"><semantics id="S5.SS2.SSS1.11.p3.4.m4.1a"><mi id="S5.SS2.SSS1.11.p3.4.m4.1.1" xref="S5.SS2.SSS1.11.p3.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.11.p3.4.m4.1b"><ci id="S5.SS2.SSS1.11.p3.4.m4.1.1.cmml" xref="S5.SS2.SSS1.11.p3.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.11.p3.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.11.p3.4.m4.1d">italic_α</annotation></semantics></math> can be made arbitrarily close to <math alttext="1" class="ltx_Math" display="inline" id="S5.SS2.SSS1.11.p3.5.m5.1"><semantics id="S5.SS2.SSS1.11.p3.5.m5.1a"><mn id="S5.SS2.SSS1.11.p3.5.m5.1.1" xref="S5.SS2.SSS1.11.p3.5.m5.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.11.p3.5.m5.1b"><cn id="S5.SS2.SSS1.11.p3.5.m5.1.1.cmml" type="integer" xref="S5.SS2.SSS1.11.p3.5.m5.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.11.p3.5.m5.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.11.p3.5.m5.1d">1</annotation></semantics></math>. This implies that the approximation factor as bounded in <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.E3" title="In Proof. ‣ 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">3</span></a> can be arbitrarily close to <math alttext="1-\frac{1}{k}\geq\frac{1}{2}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.11.p3.6.m6.1"><semantics id="S5.SS2.SSS1.11.p3.6.m6.1a"><mrow id="S5.SS2.SSS1.11.p3.6.m6.1.1" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.cmml"><mrow id="S5.SS2.SSS1.11.p3.6.m6.1.1.2" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.2.cmml"><mn id="S5.SS2.SSS1.11.p3.6.m6.1.1.2.2" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.2.2.cmml">1</mn><mo id="S5.SS2.SSS1.11.p3.6.m6.1.1.2.1" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.2.1.cmml">−</mo><mfrac id="S5.SS2.SSS1.11.p3.6.m6.1.1.2.3" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.2.3.cmml"><mn id="S5.SS2.SSS1.11.p3.6.m6.1.1.2.3.2" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.2.3.2.cmml">1</mn><mi id="S5.SS2.SSS1.11.p3.6.m6.1.1.2.3.3" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.2.3.3.cmml">k</mi></mfrac></mrow><mo id="S5.SS2.SSS1.11.p3.6.m6.1.1.1" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.1.cmml">≥</mo><mfrac id="S5.SS2.SSS1.11.p3.6.m6.1.1.3" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.3.cmml"><mn id="S5.SS2.SSS1.11.p3.6.m6.1.1.3.2" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.3.2.cmml">1</mn><mn id="S5.SS2.SSS1.11.p3.6.m6.1.1.3.3" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.3.3.cmml">2</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.11.p3.6.m6.1b"><apply id="S5.SS2.SSS1.11.p3.6.m6.1.1.cmml" xref="S5.SS2.SSS1.11.p3.6.m6.1.1"><geq id="S5.SS2.SSS1.11.p3.6.m6.1.1.1.cmml" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.1"></geq><apply id="S5.SS2.SSS1.11.p3.6.m6.1.1.2.cmml" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.2"><minus id="S5.SS2.SSS1.11.p3.6.m6.1.1.2.1.cmml" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.2.1"></minus><cn id="S5.SS2.SSS1.11.p3.6.m6.1.1.2.2.cmml" type="integer" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.2.2">1</cn><apply id="S5.SS2.SSS1.11.p3.6.m6.1.1.2.3.cmml" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.2.3"><divide id="S5.SS2.SSS1.11.p3.6.m6.1.1.2.3.1.cmml" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.2.3"></divide><cn id="S5.SS2.SSS1.11.p3.6.m6.1.1.2.3.2.cmml" type="integer" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.2.3.2">1</cn><ci id="S5.SS2.SSS1.11.p3.6.m6.1.1.2.3.3.cmml" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.2.3.3">𝑘</ci></apply></apply><apply id="S5.SS2.SSS1.11.p3.6.m6.1.1.3.cmml" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.3"><divide id="S5.SS2.SSS1.11.p3.6.m6.1.1.3.1.cmml" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.3"></divide><cn id="S5.SS2.SSS1.11.p3.6.m6.1.1.3.2.cmml" type="integer" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.3.2">1</cn><cn id="S5.SS2.SSS1.11.p3.6.m6.1.1.3.3.cmml" type="integer" xref="S5.SS2.SSS1.11.p3.6.m6.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.11.p3.6.m6.1c">1-\frac{1}{k}\geq\frac{1}{2}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.11.p3.6.m6.1d">1 - divide start_ARG 1 end_ARG start_ARG italic_k end_ARG ≥ divide start_ARG 1 end_ARG start_ARG 2 end_ARG</annotation></semantics></math>, where we use that <math alttext="k\geq 2" class="ltx_Math" display="inline" id="S5.SS2.SSS1.11.p3.7.m7.1"><semantics id="S5.SS2.SSS1.11.p3.7.m7.1a"><mrow id="S5.SS2.SSS1.11.p3.7.m7.1.1" xref="S5.SS2.SSS1.11.p3.7.m7.1.1.cmml"><mi id="S5.SS2.SSS1.11.p3.7.m7.1.1.2" xref="S5.SS2.SSS1.11.p3.7.m7.1.1.2.cmml">k</mi><mo id="S5.SS2.SSS1.11.p3.7.m7.1.1.1" xref="S5.SS2.SSS1.11.p3.7.m7.1.1.1.cmml">≥</mo><mn id="S5.SS2.SSS1.11.p3.7.m7.1.1.3" xref="S5.SS2.SSS1.11.p3.7.m7.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.11.p3.7.m7.1b"><apply id="S5.SS2.SSS1.11.p3.7.m7.1.1.cmml" xref="S5.SS2.SSS1.11.p3.7.m7.1.1"><geq id="S5.SS2.SSS1.11.p3.7.m7.1.1.1.cmml" xref="S5.SS2.SSS1.11.p3.7.m7.1.1.1"></geq><ci id="S5.SS2.SSS1.11.p3.7.m7.1.1.2.cmml" xref="S5.SS2.SSS1.11.p3.7.m7.1.1.2">𝑘</ci><cn id="S5.SS2.SSS1.11.p3.7.m7.1.1.3.cmml" type="integer" xref="S5.SS2.SSS1.11.p3.7.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.11.p3.7.m7.1c">k\geq 2</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.11.p3.7.m7.1d">italic_k ≥ 2</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S5.SS2.SSS1.p5"> <p class="ltx_p" id="S5.SS2.SSS1.p5.4">As we just showed, the approximation factor can be arbitrarily close to <math alttext="1-\frac{1}{k}" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p5.1.m1.1"><semantics id="S5.SS2.SSS1.p5.1.m1.1a"><mrow id="S5.SS2.SSS1.p5.1.m1.1.1" xref="S5.SS2.SSS1.p5.1.m1.1.1.cmml"><mn id="S5.SS2.SSS1.p5.1.m1.1.1.2" xref="S5.SS2.SSS1.p5.1.m1.1.1.2.cmml">1</mn><mo id="S5.SS2.SSS1.p5.1.m1.1.1.1" xref="S5.SS2.SSS1.p5.1.m1.1.1.1.cmml">−</mo><mfrac id="S5.SS2.SSS1.p5.1.m1.1.1.3" xref="S5.SS2.SSS1.p5.1.m1.1.1.3.cmml"><mn id="S5.SS2.SSS1.p5.1.m1.1.1.3.2" xref="S5.SS2.SSS1.p5.1.m1.1.1.3.2.cmml">1</mn><mi id="S5.SS2.SSS1.p5.1.m1.1.1.3.3" xref="S5.SS2.SSS1.p5.1.m1.1.1.3.3.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p5.1.m1.1b"><apply id="S5.SS2.SSS1.p5.1.m1.1.1.cmml" xref="S5.SS2.SSS1.p5.1.m1.1.1"><minus id="S5.SS2.SSS1.p5.1.m1.1.1.1.cmml" xref="S5.SS2.SSS1.p5.1.m1.1.1.1"></minus><cn id="S5.SS2.SSS1.p5.1.m1.1.1.2.cmml" type="integer" xref="S5.SS2.SSS1.p5.1.m1.1.1.2">1</cn><apply id="S5.SS2.SSS1.p5.1.m1.1.1.3.cmml" xref="S5.SS2.SSS1.p5.1.m1.1.1.3"><divide id="S5.SS2.SSS1.p5.1.m1.1.1.3.1.cmml" xref="S5.SS2.SSS1.p5.1.m1.1.1.3"></divide><cn id="S5.SS2.SSS1.p5.1.m1.1.1.3.2.cmml" type="integer" xref="S5.SS2.SSS1.p5.1.m1.1.1.3.2">1</cn><ci id="S5.SS2.SSS1.p5.1.m1.1.1.3.3.cmml" xref="S5.SS2.SSS1.p5.1.m1.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p5.1.m1.1c">1-\frac{1}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p5.1.m1.1d">1 - divide start_ARG 1 end_ARG start_ARG italic_k end_ARG</annotation></semantics></math>. Hence, in case that <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p5.2.m2.1"><semantics id="S5.SS2.SSS1.p5.2.m2.1a"><mi id="S5.SS2.SSS1.p5.2.m2.1.1" xref="S5.SS2.SSS1.p5.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p5.2.m2.1b"><ci id="S5.SS2.SSS1.p5.2.m2.1.1.cmml" xref="S5.SS2.SSS1.p5.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p5.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p5.2.m2.1d">italic_k</annotation></semantics></math> tends to infinity as <math alttext="n" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p5.3.m3.1"><semantics id="S5.SS2.SSS1.p5.3.m3.1a"><mi id="S5.SS2.SSS1.p5.3.m3.1.1" xref="S5.SS2.SSS1.p5.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p5.3.m3.1b"><ci id="S5.SS2.SSS1.p5.3.m3.1.1.cmml" xref="S5.SS2.SSS1.p5.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p5.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p5.3.m3.1d">italic_n</annotation></semantics></math> tends to infinity, we obtain nearly optimal partitions for large <math alttext="n" class="ltx_Math" display="inline" id="S5.SS2.SSS1.p5.4.m4.1"><semantics id="S5.SS2.SSS1.p5.4.m4.1a"><mi id="S5.SS2.SSS1.p5.4.m4.1.1" xref="S5.SS2.SSS1.p5.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS1.p5.4.m4.1b"><ci id="S5.SS2.SSS1.p5.4.m4.1.1.cmml" xref="S5.SS2.SSS1.p5.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS1.p5.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS1.p5.4.m4.1d">italic_n</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsubsection" id="S5.SS2.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">5.2.2 </span>High Perturbation Regime for Random Turán Graphs</h4> <div class="ltx_para" id="S5.SS2.SSS2.p1"> <p class="ltx_p" id="S5.SS2.SSS2.p1.5">We now present a second algorithm, which uses <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg1" title="In 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a> as a subroutine, for the case when the perturbation probability is constant, i.e., <math alttext="p=c" class="ltx_Math" display="inline" id="S5.SS2.SSS2.p1.1.m1.1"><semantics id="S5.SS2.SSS2.p1.1.m1.1a"><mrow id="S5.SS2.SSS2.p1.1.m1.1.1" xref="S5.SS2.SSS2.p1.1.m1.1.1.cmml"><mi id="S5.SS2.SSS2.p1.1.m1.1.1.2" xref="S5.SS2.SSS2.p1.1.m1.1.1.2.cmml">p</mi><mo id="S5.SS2.SSS2.p1.1.m1.1.1.1" xref="S5.SS2.SSS2.p1.1.m1.1.1.1.cmml">=</mo><mi id="S5.SS2.SSS2.p1.1.m1.1.1.3" xref="S5.SS2.SSS2.p1.1.m1.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.p1.1.m1.1b"><apply id="S5.SS2.SSS2.p1.1.m1.1.1.cmml" xref="S5.SS2.SSS2.p1.1.m1.1.1"><eq id="S5.SS2.SSS2.p1.1.m1.1.1.1.cmml" xref="S5.SS2.SSS2.p1.1.m1.1.1.1"></eq><ci id="S5.SS2.SSS2.p1.1.m1.1.1.2.cmml" xref="S5.SS2.SSS2.p1.1.m1.1.1.2">𝑝</ci><ci id="S5.SS2.SSS2.p1.1.m1.1.1.3.cmml" xref="S5.SS2.SSS2.p1.1.m1.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.p1.1.m1.1c">p=c</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.p1.1.m1.1d">italic_p = italic_c</annotation></semantics></math> for some constant <math alttext="c\in(0,1)" class="ltx_Math" display="inline" id="S5.SS2.SSS2.p1.2.m2.2"><semantics id="S5.SS2.SSS2.p1.2.m2.2a"><mrow id="S5.SS2.SSS2.p1.2.m2.2.3" xref="S5.SS2.SSS2.p1.2.m2.2.3.cmml"><mi id="S5.SS2.SSS2.p1.2.m2.2.3.2" xref="S5.SS2.SSS2.p1.2.m2.2.3.2.cmml">c</mi><mo id="S5.SS2.SSS2.p1.2.m2.2.3.1" xref="S5.SS2.SSS2.p1.2.m2.2.3.1.cmml">∈</mo><mrow id="S5.SS2.SSS2.p1.2.m2.2.3.3.2" xref="S5.SS2.SSS2.p1.2.m2.2.3.3.1.cmml"><mo id="S5.SS2.SSS2.p1.2.m2.2.3.3.2.1" stretchy="false" xref="S5.SS2.SSS2.p1.2.m2.2.3.3.1.cmml">(</mo><mn id="S5.SS2.SSS2.p1.2.m2.1.1" xref="S5.SS2.SSS2.p1.2.m2.1.1.cmml">0</mn><mo id="S5.SS2.SSS2.p1.2.m2.2.3.3.2.2" xref="S5.SS2.SSS2.p1.2.m2.2.3.3.1.cmml">,</mo><mn id="S5.SS2.SSS2.p1.2.m2.2.2" xref="S5.SS2.SSS2.p1.2.m2.2.2.cmml">1</mn><mo id="S5.SS2.SSS2.p1.2.m2.2.3.3.2.3" stretchy="false" xref="S5.SS2.SSS2.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.p1.2.m2.2b"><apply id="S5.SS2.SSS2.p1.2.m2.2.3.cmml" xref="S5.SS2.SSS2.p1.2.m2.2.3"><in id="S5.SS2.SSS2.p1.2.m2.2.3.1.cmml" xref="S5.SS2.SSS2.p1.2.m2.2.3.1"></in><ci id="S5.SS2.SSS2.p1.2.m2.2.3.2.cmml" xref="S5.SS2.SSS2.p1.2.m2.2.3.2">𝑐</ci><interval closure="open" id="S5.SS2.SSS2.p1.2.m2.2.3.3.1.cmml" xref="S5.SS2.SSS2.p1.2.m2.2.3.3.2"><cn id="S5.SS2.SSS2.p1.2.m2.1.1.cmml" type="integer" xref="S5.SS2.SSS2.p1.2.m2.1.1">0</cn><cn id="S5.SS2.SSS2.p1.2.m2.2.2.cmml" type="integer" xref="S5.SS2.SSS2.p1.2.m2.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.p1.2.m2.2c">c\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.p1.2.m2.2d">italic_c ∈ ( 0 , 1 )</annotation></semantics></math>. <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg2" title="In 5.2.2 High Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">2</span></a> partitions the set of color classes <math alttext="\{V_{1},\cdots,V_{k}\}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.p1.3.m3.3"><semantics id="S5.SS2.SSS2.p1.3.m3.3a"><mrow id="S5.SS2.SSS2.p1.3.m3.3.3.2" xref="S5.SS2.SSS2.p1.3.m3.3.3.3.cmml"><mo id="S5.SS2.SSS2.p1.3.m3.3.3.2.3" stretchy="false" xref="S5.SS2.SSS2.p1.3.m3.3.3.3.cmml">{</mo><msub id="S5.SS2.SSS2.p1.3.m3.2.2.1.1" xref="S5.SS2.SSS2.p1.3.m3.2.2.1.1.cmml"><mi id="S5.SS2.SSS2.p1.3.m3.2.2.1.1.2" xref="S5.SS2.SSS2.p1.3.m3.2.2.1.1.2.cmml">V</mi><mn id="S5.SS2.SSS2.p1.3.m3.2.2.1.1.3" xref="S5.SS2.SSS2.p1.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S5.SS2.SSS2.p1.3.m3.3.3.2.4" xref="S5.SS2.SSS2.p1.3.m3.3.3.3.cmml">,</mo><mi id="S5.SS2.SSS2.p1.3.m3.1.1" mathvariant="normal" xref="S5.SS2.SSS2.p1.3.m3.1.1.cmml">⋯</mi><mo id="S5.SS2.SSS2.p1.3.m3.3.3.2.5" xref="S5.SS2.SSS2.p1.3.m3.3.3.3.cmml">,</mo><msub id="S5.SS2.SSS2.p1.3.m3.3.3.2.2" xref="S5.SS2.SSS2.p1.3.m3.3.3.2.2.cmml"><mi id="S5.SS2.SSS2.p1.3.m3.3.3.2.2.2" xref="S5.SS2.SSS2.p1.3.m3.3.3.2.2.2.cmml">V</mi><mi id="S5.SS2.SSS2.p1.3.m3.3.3.2.2.3" xref="S5.SS2.SSS2.p1.3.m3.3.3.2.2.3.cmml">k</mi></msub><mo id="S5.SS2.SSS2.p1.3.m3.3.3.2.6" stretchy="false" xref="S5.SS2.SSS2.p1.3.m3.3.3.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.p1.3.m3.3b"><set id="S5.SS2.SSS2.p1.3.m3.3.3.3.cmml" xref="S5.SS2.SSS2.p1.3.m3.3.3.2"><apply id="S5.SS2.SSS2.p1.3.m3.2.2.1.1.cmml" xref="S5.SS2.SSS2.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.p1.3.m3.2.2.1.1.1.cmml" xref="S5.SS2.SSS2.p1.3.m3.2.2.1.1">subscript</csymbol><ci id="S5.SS2.SSS2.p1.3.m3.2.2.1.1.2.cmml" xref="S5.SS2.SSS2.p1.3.m3.2.2.1.1.2">𝑉</ci><cn id="S5.SS2.SSS2.p1.3.m3.2.2.1.1.3.cmml" type="integer" xref="S5.SS2.SSS2.p1.3.m3.2.2.1.1.3">1</cn></apply><ci id="S5.SS2.SSS2.p1.3.m3.1.1.cmml" xref="S5.SS2.SSS2.p1.3.m3.1.1">⋯</ci><apply id="S5.SS2.SSS2.p1.3.m3.3.3.2.2.cmml" xref="S5.SS2.SSS2.p1.3.m3.3.3.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS2.p1.3.m3.3.3.2.2.1.cmml" xref="S5.SS2.SSS2.p1.3.m3.3.3.2.2">subscript</csymbol><ci id="S5.SS2.SSS2.p1.3.m3.3.3.2.2.2.cmml" xref="S5.SS2.SSS2.p1.3.m3.3.3.2.2.2">𝑉</ci><ci id="S5.SS2.SSS2.p1.3.m3.3.3.2.2.3.cmml" xref="S5.SS2.SSS2.p1.3.m3.3.3.2.2.3">𝑘</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.p1.3.m3.3c">\{V_{1},\cdots,V_{k}\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.p1.3.m3.3d">{ italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }</annotation></semantics></math> into <math alttext="\lceil ck\rceil" class="ltx_Math" display="inline" id="S5.SS2.SSS2.p1.4.m4.1"><semantics id="S5.SS2.SSS2.p1.4.m4.1a"><mrow id="S5.SS2.SSS2.p1.4.m4.1.1.1" xref="S5.SS2.SSS2.p1.4.m4.1.1.2.cmml"><mo id="S5.SS2.SSS2.p1.4.m4.1.1.1.2" stretchy="false" xref="S5.SS2.SSS2.p1.4.m4.1.1.2.1.cmml">⌈</mo><mrow id="S5.SS2.SSS2.p1.4.m4.1.1.1.1" xref="S5.SS2.SSS2.p1.4.m4.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.p1.4.m4.1.1.1.1.2" xref="S5.SS2.SSS2.p1.4.m4.1.1.1.1.2.cmml">c</mi><mo id="S5.SS2.SSS2.p1.4.m4.1.1.1.1.1" xref="S5.SS2.SSS2.p1.4.m4.1.1.1.1.1.cmml">⁢</mo><mi id="S5.SS2.SSS2.p1.4.m4.1.1.1.1.3" xref="S5.SS2.SSS2.p1.4.m4.1.1.1.1.3.cmml">k</mi></mrow><mo id="S5.SS2.SSS2.p1.4.m4.1.1.1.3" stretchy="false" xref="S5.SS2.SSS2.p1.4.m4.1.1.2.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.p1.4.m4.1b"><apply id="S5.SS2.SSS2.p1.4.m4.1.1.2.cmml" xref="S5.SS2.SSS2.p1.4.m4.1.1.1"><ceiling id="S5.SS2.SSS2.p1.4.m4.1.1.2.1.cmml" xref="S5.SS2.SSS2.p1.4.m4.1.1.1.2"></ceiling><apply id="S5.SS2.SSS2.p1.4.m4.1.1.1.1.cmml" xref="S5.SS2.SSS2.p1.4.m4.1.1.1.1"><times id="S5.SS2.SSS2.p1.4.m4.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.p1.4.m4.1.1.1.1.1"></times><ci id="S5.SS2.SSS2.p1.4.m4.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.p1.4.m4.1.1.1.1.2">𝑐</ci><ci id="S5.SS2.SSS2.p1.4.m4.1.1.1.1.3.cmml" xref="S5.SS2.SSS2.p1.4.m4.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.p1.4.m4.1c">\lceil ck\rceil</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.p1.4.m4.1d">⌈ italic_c italic_k ⌉</annotation></semantics></math> disjoint sets, so that the sizes of the sets differ by at most one. Denote these disjoint sets by <math alttext="\{S_{1},\dots,S_{\lceil ck\rceil}\}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.p1.5.m5.4"><semantics id="S5.SS2.SSS2.p1.5.m5.4a"><mrow id="S5.SS2.SSS2.p1.5.m5.4.4.2" xref="S5.SS2.SSS2.p1.5.m5.4.4.3.cmml"><mo id="S5.SS2.SSS2.p1.5.m5.4.4.2.3" stretchy="false" xref="S5.SS2.SSS2.p1.5.m5.4.4.3.cmml">{</mo><msub id="S5.SS2.SSS2.p1.5.m5.3.3.1.1" xref="S5.SS2.SSS2.p1.5.m5.3.3.1.1.cmml"><mi id="S5.SS2.SSS2.p1.5.m5.3.3.1.1.2" xref="S5.SS2.SSS2.p1.5.m5.3.3.1.1.2.cmml">S</mi><mn id="S5.SS2.SSS2.p1.5.m5.3.3.1.1.3" xref="S5.SS2.SSS2.p1.5.m5.3.3.1.1.3.cmml">1</mn></msub><mo id="S5.SS2.SSS2.p1.5.m5.4.4.2.4" xref="S5.SS2.SSS2.p1.5.m5.4.4.3.cmml">,</mo><mi id="S5.SS2.SSS2.p1.5.m5.2.2" mathvariant="normal" xref="S5.SS2.SSS2.p1.5.m5.2.2.cmml">…</mi><mo id="S5.SS2.SSS2.p1.5.m5.4.4.2.5" xref="S5.SS2.SSS2.p1.5.m5.4.4.3.cmml">,</mo><msub id="S5.SS2.SSS2.p1.5.m5.4.4.2.2" xref="S5.SS2.SSS2.p1.5.m5.4.4.2.2.cmml"><mi id="S5.SS2.SSS2.p1.5.m5.4.4.2.2.2" xref="S5.SS2.SSS2.p1.5.m5.4.4.2.2.2.cmml">S</mi><mrow id="S5.SS2.SSS2.p1.5.m5.1.1.1.1" xref="S5.SS2.SSS2.p1.5.m5.1.1.1.2.cmml"><mo id="S5.SS2.SSS2.p1.5.m5.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSS2.p1.5.m5.1.1.1.2.1.cmml">⌈</mo><mrow id="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1" xref="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1.2" xref="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1.2.cmml">c</mi><mo id="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1.1" xref="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1.3" xref="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1.3.cmml">k</mi></mrow><mo id="S5.SS2.SSS2.p1.5.m5.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSS2.p1.5.m5.1.1.1.2.1.cmml">⌉</mo></mrow></msub><mo id="S5.SS2.SSS2.p1.5.m5.4.4.2.6" stretchy="false" xref="S5.SS2.SSS2.p1.5.m5.4.4.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.p1.5.m5.4b"><set id="S5.SS2.SSS2.p1.5.m5.4.4.3.cmml" xref="S5.SS2.SSS2.p1.5.m5.4.4.2"><apply id="S5.SS2.SSS2.p1.5.m5.3.3.1.1.cmml" xref="S5.SS2.SSS2.p1.5.m5.3.3.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.p1.5.m5.3.3.1.1.1.cmml" xref="S5.SS2.SSS2.p1.5.m5.3.3.1.1">subscript</csymbol><ci id="S5.SS2.SSS2.p1.5.m5.3.3.1.1.2.cmml" xref="S5.SS2.SSS2.p1.5.m5.3.3.1.1.2">𝑆</ci><cn id="S5.SS2.SSS2.p1.5.m5.3.3.1.1.3.cmml" type="integer" xref="S5.SS2.SSS2.p1.5.m5.3.3.1.1.3">1</cn></apply><ci id="S5.SS2.SSS2.p1.5.m5.2.2.cmml" xref="S5.SS2.SSS2.p1.5.m5.2.2">…</ci><apply id="S5.SS2.SSS2.p1.5.m5.4.4.2.2.cmml" xref="S5.SS2.SSS2.p1.5.m5.4.4.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS2.p1.5.m5.4.4.2.2.1.cmml" xref="S5.SS2.SSS2.p1.5.m5.4.4.2.2">subscript</csymbol><ci id="S5.SS2.SSS2.p1.5.m5.4.4.2.2.2.cmml" xref="S5.SS2.SSS2.p1.5.m5.4.4.2.2.2">𝑆</ci><apply id="S5.SS2.SSS2.p1.5.m5.1.1.1.2.cmml" xref="S5.SS2.SSS2.p1.5.m5.1.1.1.1"><ceiling id="S5.SS2.SSS2.p1.5.m5.1.1.1.2.1.cmml" xref="S5.SS2.SSS2.p1.5.m5.1.1.1.1.2"></ceiling><apply id="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1"><times id="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1.1"></times><ci id="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1.2">𝑐</ci><ci id="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1.3.cmml" xref="S5.SS2.SSS2.p1.5.m5.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.p1.5.m5.4c">\{S_{1},\dots,S_{\lceil ck\rceil}\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.p1.5.m5.4d">{ italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_S start_POSTSUBSCRIPT ⌈ italic_c italic_k ⌉ end_POSTSUBSCRIPT }</annotation></semantics></math>.<span class="ltx_note ltx_role_footnote" id="footnote6"><sup class="ltx_note_mark">6</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">6</sup><span class="ltx_tag ltx_tag_note">6</span> For example, if <math alttext="k=11" class="ltx_Math" display="inline" id="footnote6.m1.1"><semantics id="footnote6.m1.1b"><mrow id="footnote6.m1.1.1" xref="footnote6.m1.1.1.cmml"><mi id="footnote6.m1.1.1.2" xref="footnote6.m1.1.1.2.cmml">k</mi><mo id="footnote6.m1.1.1.1" xref="footnote6.m1.1.1.1.cmml">=</mo><mn id="footnote6.m1.1.1.3" xref="footnote6.m1.1.1.3.cmml">11</mn></mrow><annotation-xml encoding="MathML-Content" id="footnote6.m1.1c"><apply id="footnote6.m1.1.1.cmml" xref="footnote6.m1.1.1"><eq id="footnote6.m1.1.1.1.cmml" xref="footnote6.m1.1.1.1"></eq><ci id="footnote6.m1.1.1.2.cmml" xref="footnote6.m1.1.1.2">𝑘</ci><cn id="footnote6.m1.1.1.3.cmml" type="integer" xref="footnote6.m1.1.1.3">11</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m1.1d">k=11</annotation><annotation encoding="application/x-llamapun" id="footnote6.m1.1e">italic_k = 11</annotation></semantics></math> and <math alttext="c=\frac{1}{3}" class="ltx_Math" display="inline" id="footnote6.m2.1"><semantics id="footnote6.m2.1b"><mrow id="footnote6.m2.1.1" xref="footnote6.m2.1.1.cmml"><mi id="footnote6.m2.1.1.2" xref="footnote6.m2.1.1.2.cmml">c</mi><mo id="footnote6.m2.1.1.1" xref="footnote6.m2.1.1.1.cmml">=</mo><mfrac id="footnote6.m2.1.1.3" xref="footnote6.m2.1.1.3.cmml"><mn id="footnote6.m2.1.1.3.2" xref="footnote6.m2.1.1.3.2.cmml">1</mn><mn id="footnote6.m2.1.1.3.3" xref="footnote6.m2.1.1.3.3.cmml">3</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="footnote6.m2.1c"><apply id="footnote6.m2.1.1.cmml" xref="footnote6.m2.1.1"><eq id="footnote6.m2.1.1.1.cmml" xref="footnote6.m2.1.1.1"></eq><ci id="footnote6.m2.1.1.2.cmml" xref="footnote6.m2.1.1.2">𝑐</ci><apply id="footnote6.m2.1.1.3.cmml" xref="footnote6.m2.1.1.3"><divide id="footnote6.m2.1.1.3.1.cmml" xref="footnote6.m2.1.1.3"></divide><cn id="footnote6.m2.1.1.3.2.cmml" type="integer" xref="footnote6.m2.1.1.3.2">1</cn><cn id="footnote6.m2.1.1.3.3.cmml" type="integer" xref="footnote6.m2.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m2.1d">c=\frac{1}{3}</annotation><annotation encoding="application/x-llamapun" id="footnote6.m2.1e">italic_c = divide start_ARG 1 end_ARG start_ARG 3 end_ARG</annotation></semantics></math>, one can take <math alttext="S_{1}=\{V_{1},V_{2},V_{3}\}" class="ltx_Math" display="inline" id="footnote6.m3.3"><semantics id="footnote6.m3.3b"><mrow id="footnote6.m3.3.3" xref="footnote6.m3.3.3.cmml"><msub id="footnote6.m3.3.3.5" xref="footnote6.m3.3.3.5.cmml"><mi id="footnote6.m3.3.3.5.2" xref="footnote6.m3.3.3.5.2.cmml">S</mi><mn id="footnote6.m3.3.3.5.3" xref="footnote6.m3.3.3.5.3.cmml">1</mn></msub><mo id="footnote6.m3.3.3.4" xref="footnote6.m3.3.3.4.cmml">=</mo><mrow id="footnote6.m3.3.3.3.3" xref="footnote6.m3.3.3.3.4.cmml"><mo id="footnote6.m3.3.3.3.3.4" stretchy="false" xref="footnote6.m3.3.3.3.4.cmml">{</mo><msub id="footnote6.m3.1.1.1.1.1" xref="footnote6.m3.1.1.1.1.1.cmml"><mi id="footnote6.m3.1.1.1.1.1.2" xref="footnote6.m3.1.1.1.1.1.2.cmml">V</mi><mn id="footnote6.m3.1.1.1.1.1.3" xref="footnote6.m3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="footnote6.m3.3.3.3.3.5" xref="footnote6.m3.3.3.3.4.cmml">,</mo><msub id="footnote6.m3.2.2.2.2.2" xref="footnote6.m3.2.2.2.2.2.cmml"><mi id="footnote6.m3.2.2.2.2.2.2" xref="footnote6.m3.2.2.2.2.2.2.cmml">V</mi><mn id="footnote6.m3.2.2.2.2.2.3" xref="footnote6.m3.2.2.2.2.2.3.cmml">2</mn></msub><mo id="footnote6.m3.3.3.3.3.6" xref="footnote6.m3.3.3.3.4.cmml">,</mo><msub id="footnote6.m3.3.3.3.3.3" xref="footnote6.m3.3.3.3.3.3.cmml"><mi id="footnote6.m3.3.3.3.3.3.2" xref="footnote6.m3.3.3.3.3.3.2.cmml">V</mi><mn id="footnote6.m3.3.3.3.3.3.3" xref="footnote6.m3.3.3.3.3.3.3.cmml">3</mn></msub><mo id="footnote6.m3.3.3.3.3.7" stretchy="false" xref="footnote6.m3.3.3.3.4.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote6.m3.3c"><apply id="footnote6.m3.3.3.cmml" xref="footnote6.m3.3.3"><eq id="footnote6.m3.3.3.4.cmml" xref="footnote6.m3.3.3.4"></eq><apply id="footnote6.m3.3.3.5.cmml" xref="footnote6.m3.3.3.5"><csymbol cd="ambiguous" id="footnote6.m3.3.3.5.1.cmml" xref="footnote6.m3.3.3.5">subscript</csymbol><ci id="footnote6.m3.3.3.5.2.cmml" xref="footnote6.m3.3.3.5.2">𝑆</ci><cn id="footnote6.m3.3.3.5.3.cmml" type="integer" xref="footnote6.m3.3.3.5.3">1</cn></apply><set id="footnote6.m3.3.3.3.4.cmml" xref="footnote6.m3.3.3.3.3"><apply id="footnote6.m3.1.1.1.1.1.cmml" xref="footnote6.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="footnote6.m3.1.1.1.1.1.1.cmml" xref="footnote6.m3.1.1.1.1.1">subscript</csymbol><ci id="footnote6.m3.1.1.1.1.1.2.cmml" xref="footnote6.m3.1.1.1.1.1.2">𝑉</ci><cn id="footnote6.m3.1.1.1.1.1.3.cmml" type="integer" xref="footnote6.m3.1.1.1.1.1.3">1</cn></apply><apply id="footnote6.m3.2.2.2.2.2.cmml" xref="footnote6.m3.2.2.2.2.2"><csymbol cd="ambiguous" id="footnote6.m3.2.2.2.2.2.1.cmml" xref="footnote6.m3.2.2.2.2.2">subscript</csymbol><ci id="footnote6.m3.2.2.2.2.2.2.cmml" xref="footnote6.m3.2.2.2.2.2.2">𝑉</ci><cn id="footnote6.m3.2.2.2.2.2.3.cmml" type="integer" xref="footnote6.m3.2.2.2.2.2.3">2</cn></apply><apply id="footnote6.m3.3.3.3.3.3.cmml" xref="footnote6.m3.3.3.3.3.3"><csymbol cd="ambiguous" id="footnote6.m3.3.3.3.3.3.1.cmml" xref="footnote6.m3.3.3.3.3.3">subscript</csymbol><ci id="footnote6.m3.3.3.3.3.3.2.cmml" xref="footnote6.m3.3.3.3.3.3.2">𝑉</ci><cn id="footnote6.m3.3.3.3.3.3.3.cmml" type="integer" xref="footnote6.m3.3.3.3.3.3.3">3</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m3.3d">S_{1}=\{V_{1},V_{2},V_{3}\}</annotation><annotation encoding="application/x-llamapun" id="footnote6.m3.3e">italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = { italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT }</annotation></semantics></math>, <math alttext="S_{2}=\{V_{4},V_{5},V_{6}\},S_{3}=\{V_{7},V_{8},V_{9}\}," class="ltx_Math" display="inline" id="footnote6.m4.1"><semantics id="footnote6.m4.1b"><mrow id="footnote6.m4.1.1.1"><mrow id="footnote6.m4.1.1.1.1.2" xref="footnote6.m4.1.1.1.1.3.cmml"><mrow id="footnote6.m4.1.1.1.1.1.1" xref="footnote6.m4.1.1.1.1.1.1.cmml"><msub id="footnote6.m4.1.1.1.1.1.1.5" xref="footnote6.m4.1.1.1.1.1.1.5.cmml"><mi id="footnote6.m4.1.1.1.1.1.1.5.2" xref="footnote6.m4.1.1.1.1.1.1.5.2.cmml">S</mi><mn id="footnote6.m4.1.1.1.1.1.1.5.3" xref="footnote6.m4.1.1.1.1.1.1.5.3.cmml">2</mn></msub><mo id="footnote6.m4.1.1.1.1.1.1.4" xref="footnote6.m4.1.1.1.1.1.1.4.cmml">=</mo><mrow id="footnote6.m4.1.1.1.1.1.1.3.3" xref="footnote6.m4.1.1.1.1.1.1.3.4.cmml"><mo id="footnote6.m4.1.1.1.1.1.1.3.3.4" stretchy="false" xref="footnote6.m4.1.1.1.1.1.1.3.4.cmml">{</mo><msub id="footnote6.m4.1.1.1.1.1.1.1.1.1" xref="footnote6.m4.1.1.1.1.1.1.1.1.1.cmml"><mi id="footnote6.m4.1.1.1.1.1.1.1.1.1.2" xref="footnote6.m4.1.1.1.1.1.1.1.1.1.2.cmml">V</mi><mn id="footnote6.m4.1.1.1.1.1.1.1.1.1.3" xref="footnote6.m4.1.1.1.1.1.1.1.1.1.3.cmml">4</mn></msub><mo id="footnote6.m4.1.1.1.1.1.1.3.3.5" xref="footnote6.m4.1.1.1.1.1.1.3.4.cmml">,</mo><msub id="footnote6.m4.1.1.1.1.1.1.2.2.2" xref="footnote6.m4.1.1.1.1.1.1.2.2.2.cmml"><mi id="footnote6.m4.1.1.1.1.1.1.2.2.2.2" xref="footnote6.m4.1.1.1.1.1.1.2.2.2.2.cmml">V</mi><mn id="footnote6.m4.1.1.1.1.1.1.2.2.2.3" xref="footnote6.m4.1.1.1.1.1.1.2.2.2.3.cmml">5</mn></msub><mo id="footnote6.m4.1.1.1.1.1.1.3.3.6" xref="footnote6.m4.1.1.1.1.1.1.3.4.cmml">,</mo><msub id="footnote6.m4.1.1.1.1.1.1.3.3.3" xref="footnote6.m4.1.1.1.1.1.1.3.3.3.cmml"><mi id="footnote6.m4.1.1.1.1.1.1.3.3.3.2" xref="footnote6.m4.1.1.1.1.1.1.3.3.3.2.cmml">V</mi><mn id="footnote6.m4.1.1.1.1.1.1.3.3.3.3" xref="footnote6.m4.1.1.1.1.1.1.3.3.3.3.cmml">6</mn></msub><mo id="footnote6.m4.1.1.1.1.1.1.3.3.7" stretchy="false" xref="footnote6.m4.1.1.1.1.1.1.3.4.cmml">}</mo></mrow></mrow><mo id="footnote6.m4.1.1.1.1.2.3" xref="footnote6.m4.1.1.1.1.3a.cmml">,</mo><mrow id="footnote6.m4.1.1.1.1.2.2" xref="footnote6.m4.1.1.1.1.2.2.cmml"><msub id="footnote6.m4.1.1.1.1.2.2.5" xref="footnote6.m4.1.1.1.1.2.2.5.cmml"><mi id="footnote6.m4.1.1.1.1.2.2.5.2" xref="footnote6.m4.1.1.1.1.2.2.5.2.cmml">S</mi><mn id="footnote6.m4.1.1.1.1.2.2.5.3" xref="footnote6.m4.1.1.1.1.2.2.5.3.cmml">3</mn></msub><mo id="footnote6.m4.1.1.1.1.2.2.4" xref="footnote6.m4.1.1.1.1.2.2.4.cmml">=</mo><mrow id="footnote6.m4.1.1.1.1.2.2.3.3" xref="footnote6.m4.1.1.1.1.2.2.3.4.cmml"><mo id="footnote6.m4.1.1.1.1.2.2.3.3.4" stretchy="false" xref="footnote6.m4.1.1.1.1.2.2.3.4.cmml">{</mo><msub id="footnote6.m4.1.1.1.1.2.2.1.1.1" xref="footnote6.m4.1.1.1.1.2.2.1.1.1.cmml"><mi id="footnote6.m4.1.1.1.1.2.2.1.1.1.2" xref="footnote6.m4.1.1.1.1.2.2.1.1.1.2.cmml">V</mi><mn id="footnote6.m4.1.1.1.1.2.2.1.1.1.3" xref="footnote6.m4.1.1.1.1.2.2.1.1.1.3.cmml">7</mn></msub><mo id="footnote6.m4.1.1.1.1.2.2.3.3.5" xref="footnote6.m4.1.1.1.1.2.2.3.4.cmml">,</mo><msub id="footnote6.m4.1.1.1.1.2.2.2.2.2" xref="footnote6.m4.1.1.1.1.2.2.2.2.2.cmml"><mi id="footnote6.m4.1.1.1.1.2.2.2.2.2.2" xref="footnote6.m4.1.1.1.1.2.2.2.2.2.2.cmml">V</mi><mn id="footnote6.m4.1.1.1.1.2.2.2.2.2.3" xref="footnote6.m4.1.1.1.1.2.2.2.2.2.3.cmml">8</mn></msub><mo id="footnote6.m4.1.1.1.1.2.2.3.3.6" xref="footnote6.m4.1.1.1.1.2.2.3.4.cmml">,</mo><msub id="footnote6.m4.1.1.1.1.2.2.3.3.3" xref="footnote6.m4.1.1.1.1.2.2.3.3.3.cmml"><mi id="footnote6.m4.1.1.1.1.2.2.3.3.3.2" xref="footnote6.m4.1.1.1.1.2.2.3.3.3.2.cmml">V</mi><mn id="footnote6.m4.1.1.1.1.2.2.3.3.3.3" xref="footnote6.m4.1.1.1.1.2.2.3.3.3.3.cmml">9</mn></msub><mo id="footnote6.m4.1.1.1.1.2.2.3.3.7" stretchy="false" xref="footnote6.m4.1.1.1.1.2.2.3.4.cmml">}</mo></mrow></mrow></mrow><mo id="footnote6.m4.1.1.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="footnote6.m4.1c"><apply id="footnote6.m4.1.1.1.1.3.cmml" xref="footnote6.m4.1.1.1.1.2"><csymbol cd="ambiguous" id="footnote6.m4.1.1.1.1.3a.cmml" xref="footnote6.m4.1.1.1.1.2.3">formulae-sequence</csymbol><apply id="footnote6.m4.1.1.1.1.1.1.cmml" xref="footnote6.m4.1.1.1.1.1.1"><eq id="footnote6.m4.1.1.1.1.1.1.4.cmml" xref="footnote6.m4.1.1.1.1.1.1.4"></eq><apply id="footnote6.m4.1.1.1.1.1.1.5.cmml" xref="footnote6.m4.1.1.1.1.1.1.5"><csymbol cd="ambiguous" id="footnote6.m4.1.1.1.1.1.1.5.1.cmml" xref="footnote6.m4.1.1.1.1.1.1.5">subscript</csymbol><ci id="footnote6.m4.1.1.1.1.1.1.5.2.cmml" xref="footnote6.m4.1.1.1.1.1.1.5.2">𝑆</ci><cn id="footnote6.m4.1.1.1.1.1.1.5.3.cmml" type="integer" xref="footnote6.m4.1.1.1.1.1.1.5.3">2</cn></apply><set id="footnote6.m4.1.1.1.1.1.1.3.4.cmml" xref="footnote6.m4.1.1.1.1.1.1.3.3"><apply id="footnote6.m4.1.1.1.1.1.1.1.1.1.cmml" xref="footnote6.m4.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="footnote6.m4.1.1.1.1.1.1.1.1.1.1.cmml" xref="footnote6.m4.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="footnote6.m4.1.1.1.1.1.1.1.1.1.2.cmml" xref="footnote6.m4.1.1.1.1.1.1.1.1.1.2">𝑉</ci><cn id="footnote6.m4.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="footnote6.m4.1.1.1.1.1.1.1.1.1.3">4</cn></apply><apply id="footnote6.m4.1.1.1.1.1.1.2.2.2.cmml" xref="footnote6.m4.1.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="footnote6.m4.1.1.1.1.1.1.2.2.2.1.cmml" xref="footnote6.m4.1.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="footnote6.m4.1.1.1.1.1.1.2.2.2.2.cmml" xref="footnote6.m4.1.1.1.1.1.1.2.2.2.2">𝑉</ci><cn id="footnote6.m4.1.1.1.1.1.1.2.2.2.3.cmml" type="integer" xref="footnote6.m4.1.1.1.1.1.1.2.2.2.3">5</cn></apply><apply id="footnote6.m4.1.1.1.1.1.1.3.3.3.cmml" xref="footnote6.m4.1.1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="footnote6.m4.1.1.1.1.1.1.3.3.3.1.cmml" xref="footnote6.m4.1.1.1.1.1.1.3.3.3">subscript</csymbol><ci id="footnote6.m4.1.1.1.1.1.1.3.3.3.2.cmml" xref="footnote6.m4.1.1.1.1.1.1.3.3.3.2">𝑉</ci><cn id="footnote6.m4.1.1.1.1.1.1.3.3.3.3.cmml" type="integer" xref="footnote6.m4.1.1.1.1.1.1.3.3.3.3">6</cn></apply></set></apply><apply id="footnote6.m4.1.1.1.1.2.2.cmml" xref="footnote6.m4.1.1.1.1.2.2"><eq id="footnote6.m4.1.1.1.1.2.2.4.cmml" xref="footnote6.m4.1.1.1.1.2.2.4"></eq><apply id="footnote6.m4.1.1.1.1.2.2.5.cmml" xref="footnote6.m4.1.1.1.1.2.2.5"><csymbol cd="ambiguous" id="footnote6.m4.1.1.1.1.2.2.5.1.cmml" xref="footnote6.m4.1.1.1.1.2.2.5">subscript</csymbol><ci id="footnote6.m4.1.1.1.1.2.2.5.2.cmml" xref="footnote6.m4.1.1.1.1.2.2.5.2">𝑆</ci><cn id="footnote6.m4.1.1.1.1.2.2.5.3.cmml" type="integer" xref="footnote6.m4.1.1.1.1.2.2.5.3">3</cn></apply><set id="footnote6.m4.1.1.1.1.2.2.3.4.cmml" xref="footnote6.m4.1.1.1.1.2.2.3.3"><apply id="footnote6.m4.1.1.1.1.2.2.1.1.1.cmml" xref="footnote6.m4.1.1.1.1.2.2.1.1.1"><csymbol cd="ambiguous" id="footnote6.m4.1.1.1.1.2.2.1.1.1.1.cmml" xref="footnote6.m4.1.1.1.1.2.2.1.1.1">subscript</csymbol><ci id="footnote6.m4.1.1.1.1.2.2.1.1.1.2.cmml" xref="footnote6.m4.1.1.1.1.2.2.1.1.1.2">𝑉</ci><cn id="footnote6.m4.1.1.1.1.2.2.1.1.1.3.cmml" type="integer" xref="footnote6.m4.1.1.1.1.2.2.1.1.1.3">7</cn></apply><apply id="footnote6.m4.1.1.1.1.2.2.2.2.2.cmml" xref="footnote6.m4.1.1.1.1.2.2.2.2.2"><csymbol cd="ambiguous" id="footnote6.m4.1.1.1.1.2.2.2.2.2.1.cmml" xref="footnote6.m4.1.1.1.1.2.2.2.2.2">subscript</csymbol><ci id="footnote6.m4.1.1.1.1.2.2.2.2.2.2.cmml" xref="footnote6.m4.1.1.1.1.2.2.2.2.2.2">𝑉</ci><cn id="footnote6.m4.1.1.1.1.2.2.2.2.2.3.cmml" type="integer" xref="footnote6.m4.1.1.1.1.2.2.2.2.2.3">8</cn></apply><apply id="footnote6.m4.1.1.1.1.2.2.3.3.3.cmml" xref="footnote6.m4.1.1.1.1.2.2.3.3.3"><csymbol cd="ambiguous" id="footnote6.m4.1.1.1.1.2.2.3.3.3.1.cmml" xref="footnote6.m4.1.1.1.1.2.2.3.3.3">subscript</csymbol><ci id="footnote6.m4.1.1.1.1.2.2.3.3.3.2.cmml" xref="footnote6.m4.1.1.1.1.2.2.3.3.3.2">𝑉</ci><cn id="footnote6.m4.1.1.1.1.2.2.3.3.3.3.cmml" type="integer" xref="footnote6.m4.1.1.1.1.2.2.3.3.3.3">9</cn></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m4.1d">S_{2}=\{V_{4},V_{5},V_{6}\},S_{3}=\{V_{7},V_{8},V_{9}\},</annotation><annotation encoding="application/x-llamapun" id="footnote6.m4.1e">italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = { italic_V start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT } , italic_S start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = { italic_V start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT } ,</annotation></semantics></math> and <math alttext="S_{4}=\{V_{10},V_{11}\}" class="ltx_Math" display="inline" id="footnote6.m5.2"><semantics id="footnote6.m5.2b"><mrow id="footnote6.m5.2.2" xref="footnote6.m5.2.2.cmml"><msub id="footnote6.m5.2.2.4" xref="footnote6.m5.2.2.4.cmml"><mi id="footnote6.m5.2.2.4.2" xref="footnote6.m5.2.2.4.2.cmml">S</mi><mn id="footnote6.m5.2.2.4.3" xref="footnote6.m5.2.2.4.3.cmml">4</mn></msub><mo id="footnote6.m5.2.2.3" xref="footnote6.m5.2.2.3.cmml">=</mo><mrow id="footnote6.m5.2.2.2.2" xref="footnote6.m5.2.2.2.3.cmml"><mo id="footnote6.m5.2.2.2.2.3" stretchy="false" xref="footnote6.m5.2.2.2.3.cmml">{</mo><msub id="footnote6.m5.1.1.1.1.1" xref="footnote6.m5.1.1.1.1.1.cmml"><mi id="footnote6.m5.1.1.1.1.1.2" xref="footnote6.m5.1.1.1.1.1.2.cmml">V</mi><mn id="footnote6.m5.1.1.1.1.1.3" xref="footnote6.m5.1.1.1.1.1.3.cmml">10</mn></msub><mo id="footnote6.m5.2.2.2.2.4" xref="footnote6.m5.2.2.2.3.cmml">,</mo><msub id="footnote6.m5.2.2.2.2.2" xref="footnote6.m5.2.2.2.2.2.cmml"><mi id="footnote6.m5.2.2.2.2.2.2" xref="footnote6.m5.2.2.2.2.2.2.cmml">V</mi><mn id="footnote6.m5.2.2.2.2.2.3" xref="footnote6.m5.2.2.2.2.2.3.cmml">11</mn></msub><mo id="footnote6.m5.2.2.2.2.5" stretchy="false" xref="footnote6.m5.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote6.m5.2c"><apply id="footnote6.m5.2.2.cmml" xref="footnote6.m5.2.2"><eq id="footnote6.m5.2.2.3.cmml" xref="footnote6.m5.2.2.3"></eq><apply id="footnote6.m5.2.2.4.cmml" xref="footnote6.m5.2.2.4"><csymbol cd="ambiguous" id="footnote6.m5.2.2.4.1.cmml" xref="footnote6.m5.2.2.4">subscript</csymbol><ci id="footnote6.m5.2.2.4.2.cmml" xref="footnote6.m5.2.2.4.2">𝑆</ci><cn id="footnote6.m5.2.2.4.3.cmml" type="integer" xref="footnote6.m5.2.2.4.3">4</cn></apply><set id="footnote6.m5.2.2.2.3.cmml" xref="footnote6.m5.2.2.2.2"><apply id="footnote6.m5.1.1.1.1.1.cmml" xref="footnote6.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="footnote6.m5.1.1.1.1.1.1.cmml" xref="footnote6.m5.1.1.1.1.1">subscript</csymbol><ci id="footnote6.m5.1.1.1.1.1.2.cmml" xref="footnote6.m5.1.1.1.1.1.2">𝑉</ci><cn id="footnote6.m5.1.1.1.1.1.3.cmml" type="integer" xref="footnote6.m5.1.1.1.1.1.3">10</cn></apply><apply id="footnote6.m5.2.2.2.2.2.cmml" xref="footnote6.m5.2.2.2.2.2"><csymbol cd="ambiguous" id="footnote6.m5.2.2.2.2.2.1.cmml" xref="footnote6.m5.2.2.2.2.2">subscript</csymbol><ci id="footnote6.m5.2.2.2.2.2.2.cmml" xref="footnote6.m5.2.2.2.2.2.2">𝑉</ci><cn id="footnote6.m5.2.2.2.2.2.3.cmml" type="integer" xref="footnote6.m5.2.2.2.2.2.3">11</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m5.2d">S_{4}=\{V_{10},V_{11}\}</annotation><annotation encoding="application/x-llamapun" id="footnote6.m5.2e">italic_S start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = { italic_V start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT }</annotation></semantics></math>. </span></span></span></p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S5.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem9.1.1.1">Lemma 5.9</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem9.p1"> <p class="ltx_p" id="S5.Thmtheorem9.p1.6"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem9.p1.6.6">Consider a random Turán graph <math alttext="G=(n,k,p)" class="ltx_Math" display="inline" id="S5.Thmtheorem9.p1.1.1.m1.3"><semantics id="S5.Thmtheorem9.p1.1.1.m1.3a"><mrow id="S5.Thmtheorem9.p1.1.1.m1.3.4" xref="S5.Thmtheorem9.p1.1.1.m1.3.4.cmml"><mi id="S5.Thmtheorem9.p1.1.1.m1.3.4.2" xref="S5.Thmtheorem9.p1.1.1.m1.3.4.2.cmml">G</mi><mo id="S5.Thmtheorem9.p1.1.1.m1.3.4.1" xref="S5.Thmtheorem9.p1.1.1.m1.3.4.1.cmml">=</mo><mrow id="S5.Thmtheorem9.p1.1.1.m1.3.4.3.2" xref="S5.Thmtheorem9.p1.1.1.m1.3.4.3.1.cmml"><mo id="S5.Thmtheorem9.p1.1.1.m1.3.4.3.2.1" stretchy="false" xref="S5.Thmtheorem9.p1.1.1.m1.3.4.3.1.cmml">(</mo><mi id="S5.Thmtheorem9.p1.1.1.m1.1.1" xref="S5.Thmtheorem9.p1.1.1.m1.1.1.cmml">n</mi><mo id="S5.Thmtheorem9.p1.1.1.m1.3.4.3.2.2" xref="S5.Thmtheorem9.p1.1.1.m1.3.4.3.1.cmml">,</mo><mi id="S5.Thmtheorem9.p1.1.1.m1.2.2" xref="S5.Thmtheorem9.p1.1.1.m1.2.2.cmml">k</mi><mo id="S5.Thmtheorem9.p1.1.1.m1.3.4.3.2.3" xref="S5.Thmtheorem9.p1.1.1.m1.3.4.3.1.cmml">,</mo><mi id="S5.Thmtheorem9.p1.1.1.m1.3.3" xref="S5.Thmtheorem9.p1.1.1.m1.3.3.cmml">p</mi><mo id="S5.Thmtheorem9.p1.1.1.m1.3.4.3.2.4" stretchy="false" xref="S5.Thmtheorem9.p1.1.1.m1.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem9.p1.1.1.m1.3b"><apply id="S5.Thmtheorem9.p1.1.1.m1.3.4.cmml" xref="S5.Thmtheorem9.p1.1.1.m1.3.4"><eq id="S5.Thmtheorem9.p1.1.1.m1.3.4.1.cmml" xref="S5.Thmtheorem9.p1.1.1.m1.3.4.1"></eq><ci id="S5.Thmtheorem9.p1.1.1.m1.3.4.2.cmml" xref="S5.Thmtheorem9.p1.1.1.m1.3.4.2">𝐺</ci><vector id="S5.Thmtheorem9.p1.1.1.m1.3.4.3.1.cmml" xref="S5.Thmtheorem9.p1.1.1.m1.3.4.3.2"><ci id="S5.Thmtheorem9.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem9.p1.1.1.m1.1.1">𝑛</ci><ci id="S5.Thmtheorem9.p1.1.1.m1.2.2.cmml" xref="S5.Thmtheorem9.p1.1.1.m1.2.2">𝑘</ci><ci id="S5.Thmtheorem9.p1.1.1.m1.3.3.cmml" xref="S5.Thmtheorem9.p1.1.1.m1.3.3">𝑝</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem9.p1.1.1.m1.3c">G=(n,k,p)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem9.p1.1.1.m1.3d">italic_G = ( italic_n , italic_k , italic_p )</annotation></semantics></math> where <math alttext="p=c" class="ltx_Math" display="inline" id="S5.Thmtheorem9.p1.2.2.m2.1"><semantics id="S5.Thmtheorem9.p1.2.2.m2.1a"><mrow id="S5.Thmtheorem9.p1.2.2.m2.1.1" xref="S5.Thmtheorem9.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem9.p1.2.2.m2.1.1.2" xref="S5.Thmtheorem9.p1.2.2.m2.1.1.2.cmml">p</mi><mo id="S5.Thmtheorem9.p1.2.2.m2.1.1.1" xref="S5.Thmtheorem9.p1.2.2.m2.1.1.1.cmml">=</mo><mi id="S5.Thmtheorem9.p1.2.2.m2.1.1.3" xref="S5.Thmtheorem9.p1.2.2.m2.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem9.p1.2.2.m2.1b"><apply id="S5.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem9.p1.2.2.m2.1.1"><eq id="S5.Thmtheorem9.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem9.p1.2.2.m2.1.1.1"></eq><ci id="S5.Thmtheorem9.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem9.p1.2.2.m2.1.1.2">𝑝</ci><ci id="S5.Thmtheorem9.p1.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem9.p1.2.2.m2.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem9.p1.2.2.m2.1c">p=c</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem9.p1.2.2.m2.1d">italic_p = italic_c</annotation></semantics></math> for some constant <math alttext="c" class="ltx_Math" display="inline" id="S5.Thmtheorem9.p1.3.3.m3.1"><semantics id="S5.Thmtheorem9.p1.3.3.m3.1a"><mi id="S5.Thmtheorem9.p1.3.3.m3.1.1" xref="S5.Thmtheorem9.p1.3.3.m3.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem9.p1.3.3.m3.1b"><ci id="S5.Thmtheorem9.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem9.p1.3.3.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem9.p1.3.3.m3.1c">c</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem9.p1.3.3.m3.1d">italic_c</annotation></semantics></math>. Let <math alttext="\pi" class="ltx_Math" display="inline" id="S5.Thmtheorem9.p1.4.4.m4.1"><semantics id="S5.Thmtheorem9.p1.4.4.m4.1a"><mi id="S5.Thmtheorem9.p1.4.4.m4.1.1" xref="S5.Thmtheorem9.p1.4.4.m4.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem9.p1.4.4.m4.1b"><ci id="S5.Thmtheorem9.p1.4.4.m4.1.1.cmml" xref="S5.Thmtheorem9.p1.4.4.m4.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem9.p1.4.4.m4.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem9.p1.4.4.m4.1d">italic_π</annotation></semantics></math> be the partition returned by <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg2" title="In 5.2.2 High Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">2</span></a>. Then <math alttext="\mathcal{SW}(\pi)=\Omega(n)" class="ltx_Math" display="inline" id="S5.Thmtheorem9.p1.5.5.m5.2"><semantics id="S5.Thmtheorem9.p1.5.5.m5.2a"><mrow id="S5.Thmtheorem9.p1.5.5.m5.2.3" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.cmml"><mrow id="S5.Thmtheorem9.p1.5.5.m5.2.3.2" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem9.p1.5.5.m5.2.3.2.2" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.2.2.cmml">𝒮</mi><mo id="S5.Thmtheorem9.p1.5.5.m5.2.3.2.1" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem9.p1.5.5.m5.2.3.2.3" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.2.3.cmml">𝒲</mi><mo id="S5.Thmtheorem9.p1.5.5.m5.2.3.2.1a" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem9.p1.5.5.m5.2.3.2.4.2" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.2.cmml"><mo id="S5.Thmtheorem9.p1.5.5.m5.2.3.2.4.2.1" stretchy="false" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.2.cmml">(</mo><mi id="S5.Thmtheorem9.p1.5.5.m5.1.1" xref="S5.Thmtheorem9.p1.5.5.m5.1.1.cmml">π</mi><mo id="S5.Thmtheorem9.p1.5.5.m5.2.3.2.4.2.2" stretchy="false" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem9.p1.5.5.m5.2.3.1" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.1.cmml">=</mo><mrow id="S5.Thmtheorem9.p1.5.5.m5.2.3.3" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.3.cmml"><mi id="S5.Thmtheorem9.p1.5.5.m5.2.3.3.2" mathvariant="normal" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.3.2.cmml">Ω</mi><mo id="S5.Thmtheorem9.p1.5.5.m5.2.3.3.1" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem9.p1.5.5.m5.2.3.3.3.2" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.3.cmml"><mo id="S5.Thmtheorem9.p1.5.5.m5.2.3.3.3.2.1" stretchy="false" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.3.cmml">(</mo><mi id="S5.Thmtheorem9.p1.5.5.m5.2.2" xref="S5.Thmtheorem9.p1.5.5.m5.2.2.cmml">n</mi><mo id="S5.Thmtheorem9.p1.5.5.m5.2.3.3.3.2.2" stretchy="false" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem9.p1.5.5.m5.2b"><apply id="S5.Thmtheorem9.p1.5.5.m5.2.3.cmml" xref="S5.Thmtheorem9.p1.5.5.m5.2.3"><eq id="S5.Thmtheorem9.p1.5.5.m5.2.3.1.cmml" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.1"></eq><apply id="S5.Thmtheorem9.p1.5.5.m5.2.3.2.cmml" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.2"><times id="S5.Thmtheorem9.p1.5.5.m5.2.3.2.1.cmml" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.2.1"></times><ci id="S5.Thmtheorem9.p1.5.5.m5.2.3.2.2.cmml" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.2.2">𝒮</ci><ci id="S5.Thmtheorem9.p1.5.5.m5.2.3.2.3.cmml" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.2.3">𝒲</ci><ci id="S5.Thmtheorem9.p1.5.5.m5.1.1.cmml" xref="S5.Thmtheorem9.p1.5.5.m5.1.1">𝜋</ci></apply><apply id="S5.Thmtheorem9.p1.5.5.m5.2.3.3.cmml" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.3"><times id="S5.Thmtheorem9.p1.5.5.m5.2.3.3.1.cmml" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.3.1"></times><ci id="S5.Thmtheorem9.p1.5.5.m5.2.3.3.2.cmml" xref="S5.Thmtheorem9.p1.5.5.m5.2.3.3.2">Ω</ci><ci id="S5.Thmtheorem9.p1.5.5.m5.2.2.cmml" xref="S5.Thmtheorem9.p1.5.5.m5.2.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem9.p1.5.5.m5.2c">\mathcal{SW}(\pi)=\Omega(n)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem9.p1.5.5.m5.2d">caligraphic_S caligraphic_W ( italic_π ) = roman_Ω ( italic_n )</annotation></semantics></math> with probability <math alttext="1-ne^{-\Theta(\frac{n}{k})}" class="ltx_Math" display="inline" id="S5.Thmtheorem9.p1.6.6.m6.1"><semantics id="S5.Thmtheorem9.p1.6.6.m6.1a"><mrow id="S5.Thmtheorem9.p1.6.6.m6.1.2" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.cmml"><mn id="S5.Thmtheorem9.p1.6.6.m6.1.2.2" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.2.cmml">1</mn><mo id="S5.Thmtheorem9.p1.6.6.m6.1.2.1" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.1.cmml">−</mo><mrow id="S5.Thmtheorem9.p1.6.6.m6.1.2.3" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.3.cmml"><mi id="S5.Thmtheorem9.p1.6.6.m6.1.2.3.2" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.3.2.cmml">n</mi><mo id="S5.Thmtheorem9.p1.6.6.m6.1.2.3.1" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.3.1.cmml">⁢</mo><msup id="S5.Thmtheorem9.p1.6.6.m6.1.2.3.3" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.3.3.cmml"><mi id="S5.Thmtheorem9.p1.6.6.m6.1.2.3.3.2" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.3.3.2.cmml">e</mi><mrow id="S5.Thmtheorem9.p1.6.6.m6.1.1.1" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.cmml"><mo id="S5.Thmtheorem9.p1.6.6.m6.1.1.1a" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.cmml">−</mo><mrow id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3.cmml"><mi id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3.2" mathvariant="normal" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3.2.cmml">Θ</mi><mo id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3.1" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3.3.2" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.1.cmml"><mo id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3.3.2.1" stretchy="false" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.1.cmml">(</mo><mfrac id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.1" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.1.cmml"><mi id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.1.2" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.1.2.cmml">n</mi><mi id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.1.3" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.1.3.cmml">k</mi></mfrac><mo id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3.3.2.2" stretchy="false" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem9.p1.6.6.m6.1b"><apply id="S5.Thmtheorem9.p1.6.6.m6.1.2.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.2"><minus id="S5.Thmtheorem9.p1.6.6.m6.1.2.1.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.1"></minus><cn id="S5.Thmtheorem9.p1.6.6.m6.1.2.2.cmml" type="integer" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.2">1</cn><apply id="S5.Thmtheorem9.p1.6.6.m6.1.2.3.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.3"><times id="S5.Thmtheorem9.p1.6.6.m6.1.2.3.1.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.3.1"></times><ci id="S5.Thmtheorem9.p1.6.6.m6.1.2.3.2.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.3.2">𝑛</ci><apply id="S5.Thmtheorem9.p1.6.6.m6.1.2.3.3.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem9.p1.6.6.m6.1.2.3.3.1.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.3.3">superscript</csymbol><ci id="S5.Thmtheorem9.p1.6.6.m6.1.2.3.3.2.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.2.3.3.2">𝑒</ci><apply id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1"><minus id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.2.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1"></minus><apply id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3"><times id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3.1.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3.1"></times><ci id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3.2.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3.2">Θ</ci><apply id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.1.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3.3.2"><divide id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.1.1.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.3.3.2"></divide><ci id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.1.2.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.1.2">𝑛</ci><ci id="S5.Thmtheorem9.p1.6.6.m6.1.1.1.1.3.cmml" xref="S5.Thmtheorem9.p1.6.6.m6.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem9.p1.6.6.m6.1c">1-ne^{-\Theta(\frac{n}{k})}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem9.p1.6.6.m6.1d">1 - italic_n italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S5.SS2.SSS2.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.SS2.SSS2.1.p1"> <p class="ltx_p" id="S5.SS2.SSS2.1.p1.19">At least <math alttext="\lfloor ck\rfloor" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.1.m1.1"><semantics id="S5.SS2.SSS2.1.p1.1.m1.1a"><mrow id="S5.SS2.SSS2.1.p1.1.m1.1.1.1" xref="S5.SS2.SSS2.1.p1.1.m1.1.1.2.cmml"><mo id="S5.SS2.SSS2.1.p1.1.m1.1.1.1.2" stretchy="false" xref="S5.SS2.SSS2.1.p1.1.m1.1.1.2.1.cmml">⌊</mo><mrow id="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1" xref="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1.2" xref="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1.2.cmml">c</mi><mo id="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1.1" xref="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1.3" xref="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1.3.cmml">k</mi></mrow><mo id="S5.SS2.SSS2.1.p1.1.m1.1.1.1.3" stretchy="false" xref="S5.SS2.SSS2.1.p1.1.m1.1.1.2.1.cmml">⌋</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.1.m1.1b"><apply id="S5.SS2.SSS2.1.p1.1.m1.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.1.m1.1.1.1"><floor id="S5.SS2.SSS2.1.p1.1.m1.1.1.2.1.cmml" xref="S5.SS2.SSS2.1.p1.1.m1.1.1.1.2"></floor><apply id="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1"><times id="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1.1"></times><ci id="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1.2">𝑐</ci><ci id="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.1.m1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.1.m1.1c">\lfloor ck\rfloor</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.1.m1.1d">⌊ italic_c italic_k ⌋</annotation></semantics></math> of the sets in <math alttext="\{S_{1},\dots,S_{\lceil ck\rceil}\}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.2.m2.4"><semantics id="S5.SS2.SSS2.1.p1.2.m2.4a"><mrow id="S5.SS2.SSS2.1.p1.2.m2.4.4.2" xref="S5.SS2.SSS2.1.p1.2.m2.4.4.3.cmml"><mo id="S5.SS2.SSS2.1.p1.2.m2.4.4.2.3" stretchy="false" xref="S5.SS2.SSS2.1.p1.2.m2.4.4.3.cmml">{</mo><msub id="S5.SS2.SSS2.1.p1.2.m2.3.3.1.1" xref="S5.SS2.SSS2.1.p1.2.m2.3.3.1.1.cmml"><mi id="S5.SS2.SSS2.1.p1.2.m2.3.3.1.1.2" xref="S5.SS2.SSS2.1.p1.2.m2.3.3.1.1.2.cmml">S</mi><mn id="S5.SS2.SSS2.1.p1.2.m2.3.3.1.1.3" xref="S5.SS2.SSS2.1.p1.2.m2.3.3.1.1.3.cmml">1</mn></msub><mo id="S5.SS2.SSS2.1.p1.2.m2.4.4.2.4" xref="S5.SS2.SSS2.1.p1.2.m2.4.4.3.cmml">,</mo><mi id="S5.SS2.SSS2.1.p1.2.m2.2.2" mathvariant="normal" xref="S5.SS2.SSS2.1.p1.2.m2.2.2.cmml">…</mi><mo id="S5.SS2.SSS2.1.p1.2.m2.4.4.2.5" xref="S5.SS2.SSS2.1.p1.2.m2.4.4.3.cmml">,</mo><msub id="S5.SS2.SSS2.1.p1.2.m2.4.4.2.2" xref="S5.SS2.SSS2.1.p1.2.m2.4.4.2.2.cmml"><mi id="S5.SS2.SSS2.1.p1.2.m2.4.4.2.2.2" xref="S5.SS2.SSS2.1.p1.2.m2.4.4.2.2.2.cmml">S</mi><mrow id="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1" xref="S5.SS2.SSS2.1.p1.2.m2.1.1.1.2.cmml"><mo id="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSS2.1.p1.2.m2.1.1.1.2.1.cmml">⌈</mo><mrow id="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1" xref="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1.2" xref="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1.2.cmml">c</mi><mo id="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1.1" xref="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1.3" xref="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1.3.cmml">k</mi></mrow><mo id="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSS2.1.p1.2.m2.1.1.1.2.1.cmml">⌉</mo></mrow></msub><mo id="S5.SS2.SSS2.1.p1.2.m2.4.4.2.6" stretchy="false" xref="S5.SS2.SSS2.1.p1.2.m2.4.4.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.2.m2.4b"><set id="S5.SS2.SSS2.1.p1.2.m2.4.4.3.cmml" xref="S5.SS2.SSS2.1.p1.2.m2.4.4.2"><apply id="S5.SS2.SSS2.1.p1.2.m2.3.3.1.1.cmml" xref="S5.SS2.SSS2.1.p1.2.m2.3.3.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.2.m2.3.3.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.2.m2.3.3.1.1">subscript</csymbol><ci id="S5.SS2.SSS2.1.p1.2.m2.3.3.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.2.m2.3.3.1.1.2">𝑆</ci><cn id="S5.SS2.SSS2.1.p1.2.m2.3.3.1.1.3.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.2.m2.3.3.1.1.3">1</cn></apply><ci id="S5.SS2.SSS2.1.p1.2.m2.2.2.cmml" xref="S5.SS2.SSS2.1.p1.2.m2.2.2">…</ci><apply id="S5.SS2.SSS2.1.p1.2.m2.4.4.2.2.cmml" xref="S5.SS2.SSS2.1.p1.2.m2.4.4.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.2.m2.4.4.2.2.1.cmml" xref="S5.SS2.SSS2.1.p1.2.m2.4.4.2.2">subscript</csymbol><ci id="S5.SS2.SSS2.1.p1.2.m2.4.4.2.2.2.cmml" xref="S5.SS2.SSS2.1.p1.2.m2.4.4.2.2.2">𝑆</ci><apply id="S5.SS2.SSS2.1.p1.2.m2.1.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1"><ceiling id="S5.SS2.SSS2.1.p1.2.m2.1.1.1.2.1.cmml" xref="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.2"></ceiling><apply id="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1"><times id="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1.1"></times><ci id="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1.2">𝑐</ci><ci id="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.2.m2.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.2.m2.4c">\{S_{1},\dots,S_{\lceil ck\rceil}\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.2.m2.4d">{ italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_S start_POSTSUBSCRIPT ⌈ italic_c italic_k ⌉ end_POSTSUBSCRIPT }</annotation></semantics></math> contain <math alttext="k^{\prime}=\left\lfloor\frac{1}{c}\right\rfloor" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.3.m3.1"><semantics id="S5.SS2.SSS2.1.p1.3.m3.1a"><mrow id="S5.SS2.SSS2.1.p1.3.m3.1.2" xref="S5.SS2.SSS2.1.p1.3.m3.1.2.cmml"><msup id="S5.SS2.SSS2.1.p1.3.m3.1.2.2" xref="S5.SS2.SSS2.1.p1.3.m3.1.2.2.cmml"><mi id="S5.SS2.SSS2.1.p1.3.m3.1.2.2.2" xref="S5.SS2.SSS2.1.p1.3.m3.1.2.2.2.cmml">k</mi><mo id="S5.SS2.SSS2.1.p1.3.m3.1.2.2.3" xref="S5.SS2.SSS2.1.p1.3.m3.1.2.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS2.1.p1.3.m3.1.2.1" xref="S5.SS2.SSS2.1.p1.3.m3.1.2.1.cmml">=</mo><mrow id="S5.SS2.SSS2.1.p1.3.m3.1.2.3.2" xref="S5.SS2.SSS2.1.p1.3.m3.1.2.3.1.cmml"><mo id="S5.SS2.SSS2.1.p1.3.m3.1.2.3.2.1" xref="S5.SS2.SSS2.1.p1.3.m3.1.2.3.1.1.cmml">⌊</mo><mfrac id="S5.SS2.SSS2.1.p1.3.m3.1.1" xref="S5.SS2.SSS2.1.p1.3.m3.1.1.cmml"><mn id="S5.SS2.SSS2.1.p1.3.m3.1.1.2" xref="S5.SS2.SSS2.1.p1.3.m3.1.1.2.cmml">1</mn><mi id="S5.SS2.SSS2.1.p1.3.m3.1.1.3" xref="S5.SS2.SSS2.1.p1.3.m3.1.1.3.cmml">c</mi></mfrac><mo id="S5.SS2.SSS2.1.p1.3.m3.1.2.3.2.2" xref="S5.SS2.SSS2.1.p1.3.m3.1.2.3.1.1.cmml">⌋</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.3.m3.1b"><apply id="S5.SS2.SSS2.1.p1.3.m3.1.2.cmml" xref="S5.SS2.SSS2.1.p1.3.m3.1.2"><eq id="S5.SS2.SSS2.1.p1.3.m3.1.2.1.cmml" xref="S5.SS2.SSS2.1.p1.3.m3.1.2.1"></eq><apply id="S5.SS2.SSS2.1.p1.3.m3.1.2.2.cmml" xref="S5.SS2.SSS2.1.p1.3.m3.1.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.3.m3.1.2.2.1.cmml" xref="S5.SS2.SSS2.1.p1.3.m3.1.2.2">superscript</csymbol><ci id="S5.SS2.SSS2.1.p1.3.m3.1.2.2.2.cmml" xref="S5.SS2.SSS2.1.p1.3.m3.1.2.2.2">𝑘</ci><ci id="S5.SS2.SSS2.1.p1.3.m3.1.2.2.3.cmml" xref="S5.SS2.SSS2.1.p1.3.m3.1.2.2.3">′</ci></apply><apply id="S5.SS2.SSS2.1.p1.3.m3.1.2.3.1.cmml" xref="S5.SS2.SSS2.1.p1.3.m3.1.2.3.2"><floor id="S5.SS2.SSS2.1.p1.3.m3.1.2.3.1.1.cmml" xref="S5.SS2.SSS2.1.p1.3.m3.1.2.3.2.1"></floor><apply id="S5.SS2.SSS2.1.p1.3.m3.1.1.cmml" xref="S5.SS2.SSS2.1.p1.3.m3.1.1"><divide id="S5.SS2.SSS2.1.p1.3.m3.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.3.m3.1.1"></divide><cn id="S5.SS2.SSS2.1.p1.3.m3.1.1.2.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.3.m3.1.1.2">1</cn><ci id="S5.SS2.SSS2.1.p1.3.m3.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.3.m3.1.1.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.3.m3.1c">k^{\prime}=\left\lfloor\frac{1}{c}\right\rfloor</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.3.m3.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ⌊ divide start_ARG 1 end_ARG start_ARG italic_c end_ARG ⌋</annotation></semantics></math> color classes. Since <math alttext="k^{\prime}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.4.m4.1"><semantics id="S5.SS2.SSS2.1.p1.4.m4.1a"><msup id="S5.SS2.SSS2.1.p1.4.m4.1.1" xref="S5.SS2.SSS2.1.p1.4.m4.1.1.cmml"><mi id="S5.SS2.SSS2.1.p1.4.m4.1.1.2" xref="S5.SS2.SSS2.1.p1.4.m4.1.1.2.cmml">k</mi><mo id="S5.SS2.SSS2.1.p1.4.m4.1.1.3" xref="S5.SS2.SSS2.1.p1.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.4.m4.1b"><apply id="S5.SS2.SSS2.1.p1.4.m4.1.1.cmml" xref="S5.SS2.SSS2.1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.4.m4.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.4.m4.1.1">superscript</csymbol><ci id="S5.SS2.SSS2.1.p1.4.m4.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.4.m4.1.1.2">𝑘</ci><ci id="S5.SS2.SSS2.1.p1.4.m4.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.4.m4.1c">k^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.4.m4.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is a constant, it follows that <math alttext="p=\mathcal{O}\left(\frac{1}{k^{\prime}}\right)" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.5.m5.1"><semantics id="S5.SS2.SSS2.1.p1.5.m5.1a"><mrow id="S5.SS2.SSS2.1.p1.5.m5.1.2" xref="S5.SS2.SSS2.1.p1.5.m5.1.2.cmml"><mi id="S5.SS2.SSS2.1.p1.5.m5.1.2.2" xref="S5.SS2.SSS2.1.p1.5.m5.1.2.2.cmml">p</mi><mo id="S5.SS2.SSS2.1.p1.5.m5.1.2.1" xref="S5.SS2.SSS2.1.p1.5.m5.1.2.1.cmml">=</mo><mrow id="S5.SS2.SSS2.1.p1.5.m5.1.2.3" xref="S5.SS2.SSS2.1.p1.5.m5.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS2.1.p1.5.m5.1.2.3.2" xref="S5.SS2.SSS2.1.p1.5.m5.1.2.3.2.cmml">𝒪</mi><mo id="S5.SS2.SSS2.1.p1.5.m5.1.2.3.1" xref="S5.SS2.SSS2.1.p1.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.1.p1.5.m5.1.2.3.3.2" xref="S5.SS2.SSS2.1.p1.5.m5.1.1.cmml"><mo id="S5.SS2.SSS2.1.p1.5.m5.1.2.3.3.2.1" xref="S5.SS2.SSS2.1.p1.5.m5.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS2.1.p1.5.m5.1.1" xref="S5.SS2.SSS2.1.p1.5.m5.1.1.cmml"><mn id="S5.SS2.SSS2.1.p1.5.m5.1.1.2" xref="S5.SS2.SSS2.1.p1.5.m5.1.1.2.cmml">1</mn><msup id="S5.SS2.SSS2.1.p1.5.m5.1.1.3" xref="S5.SS2.SSS2.1.p1.5.m5.1.1.3.cmml"><mi id="S5.SS2.SSS2.1.p1.5.m5.1.1.3.2" xref="S5.SS2.SSS2.1.p1.5.m5.1.1.3.2.cmml">k</mi><mo id="S5.SS2.SSS2.1.p1.5.m5.1.1.3.3" xref="S5.SS2.SSS2.1.p1.5.m5.1.1.3.3.cmml">′</mo></msup></mfrac><mo id="S5.SS2.SSS2.1.p1.5.m5.1.2.3.3.2.2" xref="S5.SS2.SSS2.1.p1.5.m5.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.5.m5.1b"><apply id="S5.SS2.SSS2.1.p1.5.m5.1.2.cmml" xref="S5.SS2.SSS2.1.p1.5.m5.1.2"><eq id="S5.SS2.SSS2.1.p1.5.m5.1.2.1.cmml" xref="S5.SS2.SSS2.1.p1.5.m5.1.2.1"></eq><ci id="S5.SS2.SSS2.1.p1.5.m5.1.2.2.cmml" xref="S5.SS2.SSS2.1.p1.5.m5.1.2.2">𝑝</ci><apply id="S5.SS2.SSS2.1.p1.5.m5.1.2.3.cmml" xref="S5.SS2.SSS2.1.p1.5.m5.1.2.3"><times id="S5.SS2.SSS2.1.p1.5.m5.1.2.3.1.cmml" xref="S5.SS2.SSS2.1.p1.5.m5.1.2.3.1"></times><ci id="S5.SS2.SSS2.1.p1.5.m5.1.2.3.2.cmml" xref="S5.SS2.SSS2.1.p1.5.m5.1.2.3.2">𝒪</ci><apply id="S5.SS2.SSS2.1.p1.5.m5.1.1.cmml" xref="S5.SS2.SSS2.1.p1.5.m5.1.2.3.3.2"><divide id="S5.SS2.SSS2.1.p1.5.m5.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.5.m5.1.2.3.3.2"></divide><cn id="S5.SS2.SSS2.1.p1.5.m5.1.1.2.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.5.m5.1.1.2">1</cn><apply id="S5.SS2.SSS2.1.p1.5.m5.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.5.m5.1.1.3.1.cmml" xref="S5.SS2.SSS2.1.p1.5.m5.1.1.3">superscript</csymbol><ci id="S5.SS2.SSS2.1.p1.5.m5.1.1.3.2.cmml" xref="S5.SS2.SSS2.1.p1.5.m5.1.1.3.2">𝑘</ci><ci id="S5.SS2.SSS2.1.p1.5.m5.1.1.3.3.cmml" xref="S5.SS2.SSS2.1.p1.5.m5.1.1.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.5.m5.1c">p=\mathcal{O}\left(\frac{1}{k^{\prime}}\right)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.5.m5.1d">italic_p = caligraphic_O ( divide start_ARG 1 end_ARG start_ARG italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG )</annotation></semantics></math>. Each set of colors forms a subproblem, and by applying <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg1" title="In 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a> to each subproblem, we obtain <math alttext="\alpha\frac{n}{k}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.6.m6.1"><semantics id="S5.SS2.SSS2.1.p1.6.m6.1a"><mrow id="S5.SS2.SSS2.1.p1.6.m6.1.1" xref="S5.SS2.SSS2.1.p1.6.m6.1.1.cmml"><mi id="S5.SS2.SSS2.1.p1.6.m6.1.1.2" xref="S5.SS2.SSS2.1.p1.6.m6.1.1.2.cmml">α</mi><mo id="S5.SS2.SSS2.1.p1.6.m6.1.1.1" xref="S5.SS2.SSS2.1.p1.6.m6.1.1.1.cmml">⁢</mo><mfrac id="S5.SS2.SSS2.1.p1.6.m6.1.1.3" xref="S5.SS2.SSS2.1.p1.6.m6.1.1.3.cmml"><mi id="S5.SS2.SSS2.1.p1.6.m6.1.1.3.2" xref="S5.SS2.SSS2.1.p1.6.m6.1.1.3.2.cmml">n</mi><mi id="S5.SS2.SSS2.1.p1.6.m6.1.1.3.3" xref="S5.SS2.SSS2.1.p1.6.m6.1.1.3.3.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.6.m6.1b"><apply id="S5.SS2.SSS2.1.p1.6.m6.1.1.cmml" xref="S5.SS2.SSS2.1.p1.6.m6.1.1"><times id="S5.SS2.SSS2.1.p1.6.m6.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.6.m6.1.1.1"></times><ci id="S5.SS2.SSS2.1.p1.6.m6.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.6.m6.1.1.2">𝛼</ci><apply id="S5.SS2.SSS2.1.p1.6.m6.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.6.m6.1.1.3"><divide id="S5.SS2.SSS2.1.p1.6.m6.1.1.3.1.cmml" xref="S5.SS2.SSS2.1.p1.6.m6.1.1.3"></divide><ci id="S5.SS2.SSS2.1.p1.6.m6.1.1.3.2.cmml" xref="S5.SS2.SSS2.1.p1.6.m6.1.1.3.2">𝑛</ci><ci id="S5.SS2.SSS2.1.p1.6.m6.1.1.3.3.cmml" xref="S5.SS2.SSS2.1.p1.6.m6.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.6.m6.1c">\alpha\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.6.m6.1d">italic_α divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math> coalitions of size <math alttext="k^{\prime}\sqrt{1-\varepsilon}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.7.m7.1"><semantics id="S5.SS2.SSS2.1.p1.7.m7.1a"><mrow id="S5.SS2.SSS2.1.p1.7.m7.1.1" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.cmml"><msup id="S5.SS2.SSS2.1.p1.7.m7.1.1.2" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.2.cmml"><mi id="S5.SS2.SSS2.1.p1.7.m7.1.1.2.2" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.2.2.cmml">k</mi><mo id="S5.SS2.SSS2.1.p1.7.m7.1.1.2.3" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS2.1.p1.7.m7.1.1.1" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.1.cmml">⁢</mo><msqrt id="S5.SS2.SSS2.1.p1.7.m7.1.1.3" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.3.cmml"><mrow id="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2.cmml"><mn id="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2.2" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2.2.cmml">1</mn><mo id="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2.1" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2.1.cmml">−</mo><mi id="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2.3" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.7.m7.1b"><apply id="S5.SS2.SSS2.1.p1.7.m7.1.1.cmml" xref="S5.SS2.SSS2.1.p1.7.m7.1.1"><times id="S5.SS2.SSS2.1.p1.7.m7.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.1"></times><apply id="S5.SS2.SSS2.1.p1.7.m7.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.7.m7.1.1.2.1.cmml" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.2">superscript</csymbol><ci id="S5.SS2.SSS2.1.p1.7.m7.1.1.2.2.cmml" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.2.2">𝑘</ci><ci id="S5.SS2.SSS2.1.p1.7.m7.1.1.2.3.cmml" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.2.3">′</ci></apply><apply id="S5.SS2.SSS2.1.p1.7.m7.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.3"><root id="S5.SS2.SSS2.1.p1.7.m7.1.1.3a.cmml" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.3"></root><apply id="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2.cmml" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2"><minus id="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2.1.cmml" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2.1"></minus><cn id="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2.2.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2.2">1</cn><ci id="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2.3.cmml" xref="S5.SS2.SSS2.1.p1.7.m7.1.1.3.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.7.m7.1c">k^{\prime}\sqrt{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.7.m7.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_ε end_ARG</annotation></semantics></math> with probability <math alttext="1-e^{-\Theta\left(\frac{n}{k}\right)}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.8.m8.1"><semantics id="S5.SS2.SSS2.1.p1.8.m8.1a"><mrow id="S5.SS2.SSS2.1.p1.8.m8.1.2" xref="S5.SS2.SSS2.1.p1.8.m8.1.2.cmml"><mn id="S5.SS2.SSS2.1.p1.8.m8.1.2.2" xref="S5.SS2.SSS2.1.p1.8.m8.1.2.2.cmml">1</mn><mo id="S5.SS2.SSS2.1.p1.8.m8.1.2.1" xref="S5.SS2.SSS2.1.p1.8.m8.1.2.1.cmml">−</mo><msup id="S5.SS2.SSS2.1.p1.8.m8.1.2.3" xref="S5.SS2.SSS2.1.p1.8.m8.1.2.3.cmml"><mi id="S5.SS2.SSS2.1.p1.8.m8.1.2.3.2" xref="S5.SS2.SSS2.1.p1.8.m8.1.2.3.2.cmml">e</mi><mrow id="S5.SS2.SSS2.1.p1.8.m8.1.1.1" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.cmml"><mo id="S5.SS2.SSS2.1.p1.8.m8.1.1.1a" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.cmml">−</mo><mrow id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3.cmml"><mi id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3.2" mathvariant="normal" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3.2.cmml">Θ</mi><mo id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3.1" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3.3.2" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.1.cmml"><mo id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3.3.2.1" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.1" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.1.2" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.1.2.cmml">n</mi><mi id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.1.3" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.1.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3.3.2.2" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.8.m8.1b"><apply id="S5.SS2.SSS2.1.p1.8.m8.1.2.cmml" xref="S5.SS2.SSS2.1.p1.8.m8.1.2"><minus id="S5.SS2.SSS2.1.p1.8.m8.1.2.1.cmml" xref="S5.SS2.SSS2.1.p1.8.m8.1.2.1"></minus><cn id="S5.SS2.SSS2.1.p1.8.m8.1.2.2.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.8.m8.1.2.2">1</cn><apply id="S5.SS2.SSS2.1.p1.8.m8.1.2.3.cmml" xref="S5.SS2.SSS2.1.p1.8.m8.1.2.3"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.8.m8.1.2.3.1.cmml" xref="S5.SS2.SSS2.1.p1.8.m8.1.2.3">superscript</csymbol><ci id="S5.SS2.SSS2.1.p1.8.m8.1.2.3.2.cmml" xref="S5.SS2.SSS2.1.p1.8.m8.1.2.3.2">𝑒</ci><apply id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1"><minus id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1"></minus><apply id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3"><times id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3.1.cmml" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3.1"></times><ci id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3.2.cmml" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3.2">Θ</ci><apply id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3.3.2"><divide id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.3.3.2"></divide><ci id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.1.2">𝑛</ci><ci id="S5.SS2.SSS2.1.p1.8.m8.1.1.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.8.m8.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.8.m8.1c">1-e^{-\Theta\left(\frac{n}{k}\right)}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.8.m8.1d">1 - italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT</annotation></semantics></math>, for some fixed constants <math alttext="\varepsilon\in(0,1)" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.9.m9.2"><semantics id="S5.SS2.SSS2.1.p1.9.m9.2a"><mrow id="S5.SS2.SSS2.1.p1.9.m9.2.3" xref="S5.SS2.SSS2.1.p1.9.m9.2.3.cmml"><mi id="S5.SS2.SSS2.1.p1.9.m9.2.3.2" xref="S5.SS2.SSS2.1.p1.9.m9.2.3.2.cmml">ε</mi><mo id="S5.SS2.SSS2.1.p1.9.m9.2.3.1" xref="S5.SS2.SSS2.1.p1.9.m9.2.3.1.cmml">∈</mo><mrow id="S5.SS2.SSS2.1.p1.9.m9.2.3.3.2" xref="S5.SS2.SSS2.1.p1.9.m9.2.3.3.1.cmml"><mo id="S5.SS2.SSS2.1.p1.9.m9.2.3.3.2.1" stretchy="false" xref="S5.SS2.SSS2.1.p1.9.m9.2.3.3.1.cmml">(</mo><mn id="S5.SS2.SSS2.1.p1.9.m9.1.1" xref="S5.SS2.SSS2.1.p1.9.m9.1.1.cmml">0</mn><mo id="S5.SS2.SSS2.1.p1.9.m9.2.3.3.2.2" xref="S5.SS2.SSS2.1.p1.9.m9.2.3.3.1.cmml">,</mo><mn id="S5.SS2.SSS2.1.p1.9.m9.2.2" xref="S5.SS2.SSS2.1.p1.9.m9.2.2.cmml">1</mn><mo id="S5.SS2.SSS2.1.p1.9.m9.2.3.3.2.3" stretchy="false" xref="S5.SS2.SSS2.1.p1.9.m9.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.9.m9.2b"><apply id="S5.SS2.SSS2.1.p1.9.m9.2.3.cmml" xref="S5.SS2.SSS2.1.p1.9.m9.2.3"><in id="S5.SS2.SSS2.1.p1.9.m9.2.3.1.cmml" xref="S5.SS2.SSS2.1.p1.9.m9.2.3.1"></in><ci id="S5.SS2.SSS2.1.p1.9.m9.2.3.2.cmml" xref="S5.SS2.SSS2.1.p1.9.m9.2.3.2">𝜀</ci><interval closure="open" id="S5.SS2.SSS2.1.p1.9.m9.2.3.3.1.cmml" xref="S5.SS2.SSS2.1.p1.9.m9.2.3.3.2"><cn id="S5.SS2.SSS2.1.p1.9.m9.1.1.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.9.m9.1.1">0</cn><cn id="S5.SS2.SSS2.1.p1.9.m9.2.2.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.9.m9.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.9.m9.2c">\varepsilon\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.9.m9.2d">italic_ε ∈ ( 0 , 1 )</annotation></semantics></math> and <math alttext="\alpha\in(0,1)" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.10.m10.2"><semantics id="S5.SS2.SSS2.1.p1.10.m10.2a"><mrow id="S5.SS2.SSS2.1.p1.10.m10.2.3" xref="S5.SS2.SSS2.1.p1.10.m10.2.3.cmml"><mi id="S5.SS2.SSS2.1.p1.10.m10.2.3.2" xref="S5.SS2.SSS2.1.p1.10.m10.2.3.2.cmml">α</mi><mo id="S5.SS2.SSS2.1.p1.10.m10.2.3.1" xref="S5.SS2.SSS2.1.p1.10.m10.2.3.1.cmml">∈</mo><mrow id="S5.SS2.SSS2.1.p1.10.m10.2.3.3.2" xref="S5.SS2.SSS2.1.p1.10.m10.2.3.3.1.cmml"><mo id="S5.SS2.SSS2.1.p1.10.m10.2.3.3.2.1" stretchy="false" xref="S5.SS2.SSS2.1.p1.10.m10.2.3.3.1.cmml">(</mo><mn id="S5.SS2.SSS2.1.p1.10.m10.1.1" xref="S5.SS2.SSS2.1.p1.10.m10.1.1.cmml">0</mn><mo id="S5.SS2.SSS2.1.p1.10.m10.2.3.3.2.2" xref="S5.SS2.SSS2.1.p1.10.m10.2.3.3.1.cmml">,</mo><mn id="S5.SS2.SSS2.1.p1.10.m10.2.2" xref="S5.SS2.SSS2.1.p1.10.m10.2.2.cmml">1</mn><mo id="S5.SS2.SSS2.1.p1.10.m10.2.3.3.2.3" stretchy="false" xref="S5.SS2.SSS2.1.p1.10.m10.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.10.m10.2b"><apply id="S5.SS2.SSS2.1.p1.10.m10.2.3.cmml" xref="S5.SS2.SSS2.1.p1.10.m10.2.3"><in id="S5.SS2.SSS2.1.p1.10.m10.2.3.1.cmml" xref="S5.SS2.SSS2.1.p1.10.m10.2.3.1"></in><ci id="S5.SS2.SSS2.1.p1.10.m10.2.3.2.cmml" xref="S5.SS2.SSS2.1.p1.10.m10.2.3.2">𝛼</ci><interval closure="open" id="S5.SS2.SSS2.1.p1.10.m10.2.3.3.1.cmml" xref="S5.SS2.SSS2.1.p1.10.m10.2.3.3.2"><cn id="S5.SS2.SSS2.1.p1.10.m10.1.1.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.10.m10.1.1">0</cn><cn id="S5.SS2.SSS2.1.p1.10.m10.2.2.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.10.m10.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.10.m10.2c">\alpha\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.10.m10.2d">italic_α ∈ ( 0 , 1 )</annotation></semantics></math>, as established by <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem7" title="Lemma 5.7. ‣ 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5.7</span></a>. By a union bound, the probability that <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg1" title="In 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a> returns fewer than <math alttext="\alpha\frac{n}{k}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.11.m11.1"><semantics id="S5.SS2.SSS2.1.p1.11.m11.1a"><mrow id="S5.SS2.SSS2.1.p1.11.m11.1.1" xref="S5.SS2.SSS2.1.p1.11.m11.1.1.cmml"><mi id="S5.SS2.SSS2.1.p1.11.m11.1.1.2" xref="S5.SS2.SSS2.1.p1.11.m11.1.1.2.cmml">α</mi><mo id="S5.SS2.SSS2.1.p1.11.m11.1.1.1" xref="S5.SS2.SSS2.1.p1.11.m11.1.1.1.cmml">⁢</mo><mfrac id="S5.SS2.SSS2.1.p1.11.m11.1.1.3" xref="S5.SS2.SSS2.1.p1.11.m11.1.1.3.cmml"><mi id="S5.SS2.SSS2.1.p1.11.m11.1.1.3.2" xref="S5.SS2.SSS2.1.p1.11.m11.1.1.3.2.cmml">n</mi><mi id="S5.SS2.SSS2.1.p1.11.m11.1.1.3.3" xref="S5.SS2.SSS2.1.p1.11.m11.1.1.3.3.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.11.m11.1b"><apply id="S5.SS2.SSS2.1.p1.11.m11.1.1.cmml" xref="S5.SS2.SSS2.1.p1.11.m11.1.1"><times id="S5.SS2.SSS2.1.p1.11.m11.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.11.m11.1.1.1"></times><ci id="S5.SS2.SSS2.1.p1.11.m11.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.11.m11.1.1.2">𝛼</ci><apply id="S5.SS2.SSS2.1.p1.11.m11.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.11.m11.1.1.3"><divide id="S5.SS2.SSS2.1.p1.11.m11.1.1.3.1.cmml" xref="S5.SS2.SSS2.1.p1.11.m11.1.1.3"></divide><ci id="S5.SS2.SSS2.1.p1.11.m11.1.1.3.2.cmml" xref="S5.SS2.SSS2.1.p1.11.m11.1.1.3.2">𝑛</ci><ci id="S5.SS2.SSS2.1.p1.11.m11.1.1.3.3.cmml" xref="S5.SS2.SSS2.1.p1.11.m11.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.11.m11.1c">\alpha\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.11.m11.1d">italic_α divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math> coalitions of size <math alttext="k^{\prime}\sqrt{1-\varepsilon}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.12.m12.1"><semantics id="S5.SS2.SSS2.1.p1.12.m12.1a"><mrow id="S5.SS2.SSS2.1.p1.12.m12.1.1" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.cmml"><msup id="S5.SS2.SSS2.1.p1.12.m12.1.1.2" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.2.cmml"><mi id="S5.SS2.SSS2.1.p1.12.m12.1.1.2.2" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.2.2.cmml">k</mi><mo id="S5.SS2.SSS2.1.p1.12.m12.1.1.2.3" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS2.1.p1.12.m12.1.1.1" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.1.cmml">⁢</mo><msqrt id="S5.SS2.SSS2.1.p1.12.m12.1.1.3" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.3.cmml"><mrow id="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2.cmml"><mn id="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2.2" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2.2.cmml">1</mn><mo id="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2.1" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2.1.cmml">−</mo><mi id="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2.3" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.12.m12.1b"><apply id="S5.SS2.SSS2.1.p1.12.m12.1.1.cmml" xref="S5.SS2.SSS2.1.p1.12.m12.1.1"><times id="S5.SS2.SSS2.1.p1.12.m12.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.1"></times><apply id="S5.SS2.SSS2.1.p1.12.m12.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.12.m12.1.1.2.1.cmml" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.2">superscript</csymbol><ci id="S5.SS2.SSS2.1.p1.12.m12.1.1.2.2.cmml" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.2.2">𝑘</ci><ci id="S5.SS2.SSS2.1.p1.12.m12.1.1.2.3.cmml" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.2.3">′</ci></apply><apply id="S5.SS2.SSS2.1.p1.12.m12.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.3"><root id="S5.SS2.SSS2.1.p1.12.m12.1.1.3a.cmml" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.3"></root><apply id="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2.cmml" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2"><minus id="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2.1.cmml" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2.1"></minus><cn id="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2.2.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2.2">1</cn><ci id="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2.3.cmml" xref="S5.SS2.SSS2.1.p1.12.m12.1.1.3.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.12.m12.1c">k^{\prime}\sqrt{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.12.m12.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_ε end_ARG</annotation></semantics></math> in at least one subproblem is at most <math alttext="\lceil ck\rceil e^{-\Theta\left(\frac{n}{k}\right)}\leq ne^{-\Theta\left(\frac% {n}{k}\right)}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.13.m13.3"><semantics id="S5.SS2.SSS2.1.p1.13.m13.3a"><mrow id="S5.SS2.SSS2.1.p1.13.m13.3.3" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.cmml"><mrow id="S5.SS2.SSS2.1.p1.13.m13.3.3.1" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.cmml"><mrow id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.2.cmml"><mo id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.2" stretchy="false" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.2.1.cmml">⌈</mo><mrow id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1.2" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1.2.cmml">c</mi><mo id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1.1" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1.1.cmml">⁢</mo><mi id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1.3" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1.3.cmml">k</mi></mrow><mo id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.3" stretchy="false" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.2.1.cmml">⌉</mo></mrow><mo id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.2" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.2.cmml">⁢</mo><msup id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.3" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.3.cmml"><mi id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.3.2" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.3.2.cmml">e</mi><mrow id="S5.SS2.SSS2.1.p1.13.m13.1.1.1" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.cmml"><mo id="S5.SS2.SSS2.1.p1.13.m13.1.1.1a" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.cmml">−</mo><mrow id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3.cmml"><mi id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3.2" mathvariant="normal" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3.2.cmml">Θ</mi><mo id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3.1" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3.3.2" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.1.cmml"><mo id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3.3.2.1" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.1" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.1.2" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.1.2.cmml">n</mi><mi id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.1.3" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.1.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3.3.2.2" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><mo id="S5.SS2.SSS2.1.p1.13.m13.3.3.2" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.2.cmml">≤</mo><mrow id="S5.SS2.SSS2.1.p1.13.m13.3.3.3" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.3.cmml"><mi id="S5.SS2.SSS2.1.p1.13.m13.3.3.3.2" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.3.2.cmml">n</mi><mo id="S5.SS2.SSS2.1.p1.13.m13.3.3.3.1" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.3.1.cmml">⁢</mo><msup id="S5.SS2.SSS2.1.p1.13.m13.3.3.3.3" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.3.3.cmml"><mi id="S5.SS2.SSS2.1.p1.13.m13.3.3.3.3.2" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.3.3.2.cmml">e</mi><mrow id="S5.SS2.SSS2.1.p1.13.m13.2.2.1" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.cmml"><mo id="S5.SS2.SSS2.1.p1.13.m13.2.2.1a" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.cmml">−</mo><mrow id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3.cmml"><mi id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3.2" mathvariant="normal" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3.2.cmml">Θ</mi><mo id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3.1" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3.3.2" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.1.cmml"><mo id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3.3.2.1" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.1" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.1.cmml"><mi id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.1.2" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.1.2.cmml">n</mi><mi id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.1.3" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.1.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3.3.2.2" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.13.m13.3b"><apply id="S5.SS2.SSS2.1.p1.13.m13.3.3.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3"><leq id="S5.SS2.SSS2.1.p1.13.m13.3.3.2.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.2"></leq><apply id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1"><times id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.2.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.2"></times><apply id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1"><ceiling id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.2.1.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.2"></ceiling><apply id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1"><times id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1.1"></times><ci id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1.2">𝑐</ci><ci id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.1.1.1.3">𝑘</ci></apply></apply><apply id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.3.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.3"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.3.1.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.3">superscript</csymbol><ci id="S5.SS2.SSS2.1.p1.13.m13.3.3.1.3.2.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.1.3.2">𝑒</ci><apply id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1"><minus id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1"></minus><apply id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3"><times id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3.1.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3.1"></times><ci id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3.2.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3.2">Θ</ci><apply id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3.3.2"><divide id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.3.3.2"></divide><ci id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.1.2">𝑛</ci><ci id="S5.SS2.SSS2.1.p1.13.m13.1.1.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply><apply id="S5.SS2.SSS2.1.p1.13.m13.3.3.3.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.3"><times id="S5.SS2.SSS2.1.p1.13.m13.3.3.3.1.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.3.1"></times><ci id="S5.SS2.SSS2.1.p1.13.m13.3.3.3.2.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.3.2">𝑛</ci><apply id="S5.SS2.SSS2.1.p1.13.m13.3.3.3.3.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.3.3"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.13.m13.3.3.3.3.1.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.3.3">superscript</csymbol><ci id="S5.SS2.SSS2.1.p1.13.m13.3.3.3.3.2.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.3.3.3.3.2">𝑒</ci><apply id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1"><minus id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.2.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1"></minus><apply id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3"><times id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3.1.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3.1"></times><ci id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3.2.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3.2">Θ</ci><apply id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.1.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3.3.2"><divide id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.3.3.2"></divide><ci id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.1.2">𝑛</ci><ci id="S5.SS2.SSS2.1.p1.13.m13.2.2.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.13.m13.2.2.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.13.m13.3c">\lceil ck\rceil e^{-\Theta\left(\frac{n}{k}\right)}\leq ne^{-\Theta\left(\frac% {n}{k}\right)}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.13.m13.3d">⌈ italic_c italic_k ⌉ italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT ≤ italic_n italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT</annotation></semantics></math>. Since <math alttext="k=o\left(\frac{n}{\log n}\right)" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.14.m14.1"><semantics id="S5.SS2.SSS2.1.p1.14.m14.1a"><mrow id="S5.SS2.SSS2.1.p1.14.m14.1.2" xref="S5.SS2.SSS2.1.p1.14.m14.1.2.cmml"><mi id="S5.SS2.SSS2.1.p1.14.m14.1.2.2" xref="S5.SS2.SSS2.1.p1.14.m14.1.2.2.cmml">k</mi><mo id="S5.SS2.SSS2.1.p1.14.m14.1.2.1" xref="S5.SS2.SSS2.1.p1.14.m14.1.2.1.cmml">=</mo><mrow id="S5.SS2.SSS2.1.p1.14.m14.1.2.3" xref="S5.SS2.SSS2.1.p1.14.m14.1.2.3.cmml"><mi id="S5.SS2.SSS2.1.p1.14.m14.1.2.3.2" xref="S5.SS2.SSS2.1.p1.14.m14.1.2.3.2.cmml">o</mi><mo id="S5.SS2.SSS2.1.p1.14.m14.1.2.3.1" xref="S5.SS2.SSS2.1.p1.14.m14.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.1.p1.14.m14.1.2.3.3.2" xref="S5.SS2.SSS2.1.p1.14.m14.1.1.cmml"><mo id="S5.SS2.SSS2.1.p1.14.m14.1.2.3.3.2.1" xref="S5.SS2.SSS2.1.p1.14.m14.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS2.1.p1.14.m14.1.1" xref="S5.SS2.SSS2.1.p1.14.m14.1.1.cmml"><mi id="S5.SS2.SSS2.1.p1.14.m14.1.1.2" xref="S5.SS2.SSS2.1.p1.14.m14.1.1.2.cmml">n</mi><mrow id="S5.SS2.SSS2.1.p1.14.m14.1.1.3" xref="S5.SS2.SSS2.1.p1.14.m14.1.1.3.cmml"><mi id="S5.SS2.SSS2.1.p1.14.m14.1.1.3.1" xref="S5.SS2.SSS2.1.p1.14.m14.1.1.3.1.cmml">log</mi><mo id="S5.SS2.SSS2.1.p1.14.m14.1.1.3a" lspace="0.167em" xref="S5.SS2.SSS2.1.p1.14.m14.1.1.3.cmml">⁡</mo><mi id="S5.SS2.SSS2.1.p1.14.m14.1.1.3.2" xref="S5.SS2.SSS2.1.p1.14.m14.1.1.3.2.cmml">n</mi></mrow></mfrac><mo id="S5.SS2.SSS2.1.p1.14.m14.1.2.3.3.2.2" xref="S5.SS2.SSS2.1.p1.14.m14.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.14.m14.1b"><apply id="S5.SS2.SSS2.1.p1.14.m14.1.2.cmml" xref="S5.SS2.SSS2.1.p1.14.m14.1.2"><eq id="S5.SS2.SSS2.1.p1.14.m14.1.2.1.cmml" xref="S5.SS2.SSS2.1.p1.14.m14.1.2.1"></eq><ci id="S5.SS2.SSS2.1.p1.14.m14.1.2.2.cmml" xref="S5.SS2.SSS2.1.p1.14.m14.1.2.2">𝑘</ci><apply id="S5.SS2.SSS2.1.p1.14.m14.1.2.3.cmml" xref="S5.SS2.SSS2.1.p1.14.m14.1.2.3"><times id="S5.SS2.SSS2.1.p1.14.m14.1.2.3.1.cmml" xref="S5.SS2.SSS2.1.p1.14.m14.1.2.3.1"></times><ci id="S5.SS2.SSS2.1.p1.14.m14.1.2.3.2.cmml" xref="S5.SS2.SSS2.1.p1.14.m14.1.2.3.2">𝑜</ci><apply id="S5.SS2.SSS2.1.p1.14.m14.1.1.cmml" xref="S5.SS2.SSS2.1.p1.14.m14.1.2.3.3.2"><divide id="S5.SS2.SSS2.1.p1.14.m14.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.14.m14.1.2.3.3.2"></divide><ci id="S5.SS2.SSS2.1.p1.14.m14.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.14.m14.1.1.2">𝑛</ci><apply id="S5.SS2.SSS2.1.p1.14.m14.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.14.m14.1.1.3"><log id="S5.SS2.SSS2.1.p1.14.m14.1.1.3.1.cmml" xref="S5.SS2.SSS2.1.p1.14.m14.1.1.3.1"></log><ci id="S5.SS2.SSS2.1.p1.14.m14.1.1.3.2.cmml" xref="S5.SS2.SSS2.1.p1.14.m14.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.14.m14.1c">k=o\left(\frac{n}{\log n}\right)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.14.m14.1d">italic_k = italic_o ( divide start_ARG italic_n end_ARG start_ARG roman_log italic_n end_ARG )</annotation></semantics></math>, this probability approaches zero as <math alttext="n" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.15.m15.1"><semantics id="S5.SS2.SSS2.1.p1.15.m15.1a"><mi id="S5.SS2.SSS2.1.p1.15.m15.1.1" xref="S5.SS2.SSS2.1.p1.15.m15.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.15.m15.1b"><ci id="S5.SS2.SSS2.1.p1.15.m15.1.1.cmml" xref="S5.SS2.SSS2.1.p1.15.m15.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.15.m15.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.15.m15.1d">italic_n</annotation></semantics></math> increases. Therefore, with probability <math alttext="1-ne^{-\Theta\left(\frac{n}{k}\right)}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.16.m16.1"><semantics id="S5.SS2.SSS2.1.p1.16.m16.1a"><mrow id="S5.SS2.SSS2.1.p1.16.m16.1.2" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.cmml"><mn id="S5.SS2.SSS2.1.p1.16.m16.1.2.2" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.2.cmml">1</mn><mo id="S5.SS2.SSS2.1.p1.16.m16.1.2.1" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.1.cmml">−</mo><mrow id="S5.SS2.SSS2.1.p1.16.m16.1.2.3" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.3.cmml"><mi id="S5.SS2.SSS2.1.p1.16.m16.1.2.3.2" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.3.2.cmml">n</mi><mo id="S5.SS2.SSS2.1.p1.16.m16.1.2.3.1" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.3.1.cmml">⁢</mo><msup id="S5.SS2.SSS2.1.p1.16.m16.1.2.3.3" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.3.3.cmml"><mi id="S5.SS2.SSS2.1.p1.16.m16.1.2.3.3.2" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.3.3.2.cmml">e</mi><mrow id="S5.SS2.SSS2.1.p1.16.m16.1.1.1" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.cmml"><mo id="S5.SS2.SSS2.1.p1.16.m16.1.1.1a" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.cmml">−</mo><mrow id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3.cmml"><mi id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3.2" mathvariant="normal" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3.2.cmml">Θ</mi><mo id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3.1" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3.3.2" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.1.cmml"><mo id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3.3.2.1" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.1" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.1.2" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.1.2.cmml">n</mi><mi id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.1.3" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.1.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3.3.2.2" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.16.m16.1b"><apply id="S5.SS2.SSS2.1.p1.16.m16.1.2.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.2"><minus id="S5.SS2.SSS2.1.p1.16.m16.1.2.1.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.1"></minus><cn id="S5.SS2.SSS2.1.p1.16.m16.1.2.2.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.2">1</cn><apply id="S5.SS2.SSS2.1.p1.16.m16.1.2.3.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.3"><times id="S5.SS2.SSS2.1.p1.16.m16.1.2.3.1.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.3.1"></times><ci id="S5.SS2.SSS2.1.p1.16.m16.1.2.3.2.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.3.2">𝑛</ci><apply id="S5.SS2.SSS2.1.p1.16.m16.1.2.3.3.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.3.3"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.16.m16.1.2.3.3.1.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.3.3">superscript</csymbol><ci id="S5.SS2.SSS2.1.p1.16.m16.1.2.3.3.2.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.2.3.3.2">𝑒</ci><apply id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1"><minus id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1"></minus><apply id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3"><times id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3.1.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3.1"></times><ci id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3.2.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3.2">Θ</ci><apply id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3.3.2"><divide id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.3.3.2"></divide><ci id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.1.2">𝑛</ci><ci id="S5.SS2.SSS2.1.p1.16.m16.1.1.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.16.m16.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.16.m16.1c">1-ne^{-\Theta\left(\frac{n}{k}\right)}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.16.m16.1d">1 - italic_n italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT</annotation></semantics></math>, all subproblems return at least <math alttext="\alpha\frac{n}{k}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.17.m17.1"><semantics id="S5.SS2.SSS2.1.p1.17.m17.1a"><mrow id="S5.SS2.SSS2.1.p1.17.m17.1.1" xref="S5.SS2.SSS2.1.p1.17.m17.1.1.cmml"><mi id="S5.SS2.SSS2.1.p1.17.m17.1.1.2" xref="S5.SS2.SSS2.1.p1.17.m17.1.1.2.cmml">α</mi><mo id="S5.SS2.SSS2.1.p1.17.m17.1.1.1" xref="S5.SS2.SSS2.1.p1.17.m17.1.1.1.cmml">⁢</mo><mfrac id="S5.SS2.SSS2.1.p1.17.m17.1.1.3" xref="S5.SS2.SSS2.1.p1.17.m17.1.1.3.cmml"><mi id="S5.SS2.SSS2.1.p1.17.m17.1.1.3.2" xref="S5.SS2.SSS2.1.p1.17.m17.1.1.3.2.cmml">n</mi><mi id="S5.SS2.SSS2.1.p1.17.m17.1.1.3.3" xref="S5.SS2.SSS2.1.p1.17.m17.1.1.3.3.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.17.m17.1b"><apply id="S5.SS2.SSS2.1.p1.17.m17.1.1.cmml" xref="S5.SS2.SSS2.1.p1.17.m17.1.1"><times id="S5.SS2.SSS2.1.p1.17.m17.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.17.m17.1.1.1"></times><ci id="S5.SS2.SSS2.1.p1.17.m17.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.17.m17.1.1.2">𝛼</ci><apply id="S5.SS2.SSS2.1.p1.17.m17.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.17.m17.1.1.3"><divide id="S5.SS2.SSS2.1.p1.17.m17.1.1.3.1.cmml" xref="S5.SS2.SSS2.1.p1.17.m17.1.1.3"></divide><ci id="S5.SS2.SSS2.1.p1.17.m17.1.1.3.2.cmml" xref="S5.SS2.SSS2.1.p1.17.m17.1.1.3.2">𝑛</ci><ci id="S5.SS2.SSS2.1.p1.17.m17.1.1.3.3.cmml" xref="S5.SS2.SSS2.1.p1.17.m17.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.17.m17.1c">\alpha\frac{n}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.17.m17.1d">italic_α divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG</annotation></semantics></math> coalitions of size <math alttext="k^{\prime}\sqrt{1-\varepsilon}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.18.m18.1"><semantics id="S5.SS2.SSS2.1.p1.18.m18.1a"><mrow id="S5.SS2.SSS2.1.p1.18.m18.1.1" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.cmml"><msup id="S5.SS2.SSS2.1.p1.18.m18.1.1.2" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.2.cmml"><mi id="S5.SS2.SSS2.1.p1.18.m18.1.1.2.2" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.2.2.cmml">k</mi><mo id="S5.SS2.SSS2.1.p1.18.m18.1.1.2.3" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS2.1.p1.18.m18.1.1.1" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.1.cmml">⁢</mo><msqrt id="S5.SS2.SSS2.1.p1.18.m18.1.1.3" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.3.cmml"><mrow id="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2.cmml"><mn id="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2.2" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2.2.cmml">1</mn><mo id="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2.1" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2.1.cmml">−</mo><mi id="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2.3" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.18.m18.1b"><apply id="S5.SS2.SSS2.1.p1.18.m18.1.1.cmml" xref="S5.SS2.SSS2.1.p1.18.m18.1.1"><times id="S5.SS2.SSS2.1.p1.18.m18.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.1"></times><apply id="S5.SS2.SSS2.1.p1.18.m18.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.18.m18.1.1.2.1.cmml" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.2">superscript</csymbol><ci id="S5.SS2.SSS2.1.p1.18.m18.1.1.2.2.cmml" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.2.2">𝑘</ci><ci id="S5.SS2.SSS2.1.p1.18.m18.1.1.2.3.cmml" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.2.3">′</ci></apply><apply id="S5.SS2.SSS2.1.p1.18.m18.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.3"><root id="S5.SS2.SSS2.1.p1.18.m18.1.1.3a.cmml" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.3"></root><apply id="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2.cmml" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2"><minus id="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2.1.cmml" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2.1"></minus><cn id="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2.2.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2.2">1</cn><ci id="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2.3.cmml" xref="S5.SS2.SSS2.1.p1.18.m18.1.1.3.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.18.m18.1c">k^{\prime}\sqrt{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.18.m18.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_ε end_ARG</annotation></semantics></math>. The utility of agents in these coalitions is least <math alttext="k^{\prime}\sqrt{1-\varepsilon}-1" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.19.m19.1"><semantics id="S5.SS2.SSS2.1.p1.19.m19.1a"><mrow id="S5.SS2.SSS2.1.p1.19.m19.1.1" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.cmml"><mrow id="S5.SS2.SSS2.1.p1.19.m19.1.1.2" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.cmml"><msup id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.2" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.2.cmml"><mi id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.2.2" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.2.2.cmml">k</mi><mo id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.2.3" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.1" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.1.cmml">⁢</mo><msqrt id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.cmml"><mrow id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2.cmml"><mn id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2.2" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2.2.cmml">1</mn><mo id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2.1" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2.1.cmml">−</mo><mi id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2.3" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><mo id="S5.SS2.SSS2.1.p1.19.m19.1.1.1" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.1.cmml">−</mo><mn id="S5.SS2.SSS2.1.p1.19.m19.1.1.3" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.19.m19.1b"><apply id="S5.SS2.SSS2.1.p1.19.m19.1.1.cmml" xref="S5.SS2.SSS2.1.p1.19.m19.1.1"><minus id="S5.SS2.SSS2.1.p1.19.m19.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.1"></minus><apply id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2"><times id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.1.cmml" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.1"></times><apply id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.2.cmml" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.2.1.cmml" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.2">superscript</csymbol><ci id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.2.2.cmml" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.2.2">𝑘</ci><ci id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.2.3.cmml" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.2.3">′</ci></apply><apply id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.cmml" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3"><root id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3a.cmml" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3"></root><apply id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2.cmml" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2"><minus id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2.1.cmml" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2.1"></minus><cn id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2.2.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2.2">1</cn><ci id="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2.3.cmml" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.2.3.2.3">𝜀</ci></apply></apply></apply><cn id="S5.SS2.SSS2.1.p1.19.m19.1.1.3.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.19.m19.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.19.m19.1c">k^{\prime}\sqrt{1-\varepsilon}-1</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.19.m19.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_ε end_ARG - 1</annotation></semantics></math>. Therefore,</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex33"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{SW}(\pi)\geq\lfloor kc\rfloor\alpha\frac{n}{k}(k^{\prime}\sqrt{1-% \varepsilon})(k^{\prime}\sqrt{1-\varepsilon}-1)\text{.}" class="ltx_Math" display="block" id="S5.Ex33.m1.4"><semantics id="S5.Ex33.m1.4a"><mrow id="S5.Ex33.m1.4.4" xref="S5.Ex33.m1.4.4.cmml"><mrow id="S5.Ex33.m1.4.4.5" xref="S5.Ex33.m1.4.4.5.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex33.m1.4.4.5.2" xref="S5.Ex33.m1.4.4.5.2.cmml">𝒮</mi><mo id="S5.Ex33.m1.4.4.5.1" xref="S5.Ex33.m1.4.4.5.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex33.m1.4.4.5.3" xref="S5.Ex33.m1.4.4.5.3.cmml">𝒲</mi><mo id="S5.Ex33.m1.4.4.5.1a" xref="S5.Ex33.m1.4.4.5.1.cmml">⁢</mo><mrow id="S5.Ex33.m1.4.4.5.4.2" xref="S5.Ex33.m1.4.4.5.cmml"><mo id="S5.Ex33.m1.4.4.5.4.2.1" stretchy="false" xref="S5.Ex33.m1.4.4.5.cmml">(</mo><mi id="S5.Ex33.m1.1.1" xref="S5.Ex33.m1.1.1.cmml">π</mi><mo id="S5.Ex33.m1.4.4.5.4.2.2" stretchy="false" xref="S5.Ex33.m1.4.4.5.cmml">)</mo></mrow></mrow><mo id="S5.Ex33.m1.4.4.4" xref="S5.Ex33.m1.4.4.4.cmml">≥</mo><mrow id="S5.Ex33.m1.4.4.3" xref="S5.Ex33.m1.4.4.3.cmml"><mrow id="S5.Ex33.m1.2.2.1.1.1" xref="S5.Ex33.m1.2.2.1.1.2.cmml"><mo id="S5.Ex33.m1.2.2.1.1.1.2" stretchy="false" xref="S5.Ex33.m1.2.2.1.1.2.1.cmml">⌊</mo><mrow id="S5.Ex33.m1.2.2.1.1.1.1" xref="S5.Ex33.m1.2.2.1.1.1.1.cmml"><mi id="S5.Ex33.m1.2.2.1.1.1.1.2" xref="S5.Ex33.m1.2.2.1.1.1.1.2.cmml">k</mi><mo id="S5.Ex33.m1.2.2.1.1.1.1.1" xref="S5.Ex33.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex33.m1.2.2.1.1.1.1.3" xref="S5.Ex33.m1.2.2.1.1.1.1.3.cmml">c</mi></mrow><mo id="S5.Ex33.m1.2.2.1.1.1.3" stretchy="false" xref="S5.Ex33.m1.2.2.1.1.2.1.cmml">⌋</mo></mrow><mo id="S5.Ex33.m1.4.4.3.4" xref="S5.Ex33.m1.4.4.3.4.cmml">⁢</mo><mi id="S5.Ex33.m1.4.4.3.5" xref="S5.Ex33.m1.4.4.3.5.cmml">α</mi><mo id="S5.Ex33.m1.4.4.3.4a" xref="S5.Ex33.m1.4.4.3.4.cmml">⁢</mo><mfrac id="S5.Ex33.m1.4.4.3.6" xref="S5.Ex33.m1.4.4.3.6.cmml"><mi id="S5.Ex33.m1.4.4.3.6.2" xref="S5.Ex33.m1.4.4.3.6.2.cmml">n</mi><mi id="S5.Ex33.m1.4.4.3.6.3" xref="S5.Ex33.m1.4.4.3.6.3.cmml">k</mi></mfrac><mo id="S5.Ex33.m1.4.4.3.4b" xref="S5.Ex33.m1.4.4.3.4.cmml">⁢</mo><mrow id="S5.Ex33.m1.3.3.2.2.1" xref="S5.Ex33.m1.3.3.2.2.1.1.cmml"><mo id="S5.Ex33.m1.3.3.2.2.1.2" stretchy="false" xref="S5.Ex33.m1.3.3.2.2.1.1.cmml">(</mo><mrow id="S5.Ex33.m1.3.3.2.2.1.1" xref="S5.Ex33.m1.3.3.2.2.1.1.cmml"><msup id="S5.Ex33.m1.3.3.2.2.1.1.2" xref="S5.Ex33.m1.3.3.2.2.1.1.2.cmml"><mi id="S5.Ex33.m1.3.3.2.2.1.1.2.2" xref="S5.Ex33.m1.3.3.2.2.1.1.2.2.cmml">k</mi><mo id="S5.Ex33.m1.3.3.2.2.1.1.2.3" xref="S5.Ex33.m1.3.3.2.2.1.1.2.3.cmml">′</mo></msup><mo id="S5.Ex33.m1.3.3.2.2.1.1.1" xref="S5.Ex33.m1.3.3.2.2.1.1.1.cmml">⁢</mo><msqrt id="S5.Ex33.m1.3.3.2.2.1.1.3" xref="S5.Ex33.m1.3.3.2.2.1.1.3.cmml"><mrow id="S5.Ex33.m1.3.3.2.2.1.1.3.2" xref="S5.Ex33.m1.3.3.2.2.1.1.3.2.cmml"><mn id="S5.Ex33.m1.3.3.2.2.1.1.3.2.2" xref="S5.Ex33.m1.3.3.2.2.1.1.3.2.2.cmml">1</mn><mo id="S5.Ex33.m1.3.3.2.2.1.1.3.2.1" xref="S5.Ex33.m1.3.3.2.2.1.1.3.2.1.cmml">−</mo><mi id="S5.Ex33.m1.3.3.2.2.1.1.3.2.3" xref="S5.Ex33.m1.3.3.2.2.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><mo id="S5.Ex33.m1.3.3.2.2.1.3" stretchy="false" xref="S5.Ex33.m1.3.3.2.2.1.1.cmml">)</mo></mrow><mo id="S5.Ex33.m1.4.4.3.4c" xref="S5.Ex33.m1.4.4.3.4.cmml">⁢</mo><mrow id="S5.Ex33.m1.4.4.3.3.1" xref="S5.Ex33.m1.4.4.3.3.1.1.cmml"><mo id="S5.Ex33.m1.4.4.3.3.1.2" stretchy="false" xref="S5.Ex33.m1.4.4.3.3.1.1.cmml">(</mo><mrow id="S5.Ex33.m1.4.4.3.3.1.1" xref="S5.Ex33.m1.4.4.3.3.1.1.cmml"><mrow id="S5.Ex33.m1.4.4.3.3.1.1.2" xref="S5.Ex33.m1.4.4.3.3.1.1.2.cmml"><msup id="S5.Ex33.m1.4.4.3.3.1.1.2.2" xref="S5.Ex33.m1.4.4.3.3.1.1.2.2.cmml"><mi id="S5.Ex33.m1.4.4.3.3.1.1.2.2.2" xref="S5.Ex33.m1.4.4.3.3.1.1.2.2.2.cmml">k</mi><mo id="S5.Ex33.m1.4.4.3.3.1.1.2.2.3" xref="S5.Ex33.m1.4.4.3.3.1.1.2.2.3.cmml">′</mo></msup><mo id="S5.Ex33.m1.4.4.3.3.1.1.2.1" xref="S5.Ex33.m1.4.4.3.3.1.1.2.1.cmml">⁢</mo><msqrt id="S5.Ex33.m1.4.4.3.3.1.1.2.3" xref="S5.Ex33.m1.4.4.3.3.1.1.2.3.cmml"><mrow id="S5.Ex33.m1.4.4.3.3.1.1.2.3.2" xref="S5.Ex33.m1.4.4.3.3.1.1.2.3.2.cmml"><mn id="S5.Ex33.m1.4.4.3.3.1.1.2.3.2.2" xref="S5.Ex33.m1.4.4.3.3.1.1.2.3.2.2.cmml">1</mn><mo id="S5.Ex33.m1.4.4.3.3.1.1.2.3.2.1" xref="S5.Ex33.m1.4.4.3.3.1.1.2.3.2.1.cmml">−</mo><mi id="S5.Ex33.m1.4.4.3.3.1.1.2.3.2.3" xref="S5.Ex33.m1.4.4.3.3.1.1.2.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><mo id="S5.Ex33.m1.4.4.3.3.1.1.1" xref="S5.Ex33.m1.4.4.3.3.1.1.1.cmml">−</mo><mn id="S5.Ex33.m1.4.4.3.3.1.1.3" xref="S5.Ex33.m1.4.4.3.3.1.1.3.cmml">1</mn></mrow><mo id="S5.Ex33.m1.4.4.3.3.1.3" stretchy="false" xref="S5.Ex33.m1.4.4.3.3.1.1.cmml">)</mo></mrow><mo id="S5.Ex33.m1.4.4.3.4d" xref="S5.Ex33.m1.4.4.3.4.cmml">⁢</mo><mtext id="S5.Ex33.m1.4.4.3.7" xref="S5.Ex33.m1.4.4.3.7a.cmml">.</mtext></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex33.m1.4b"><apply id="S5.Ex33.m1.4.4.cmml" xref="S5.Ex33.m1.4.4"><geq id="S5.Ex33.m1.4.4.4.cmml" xref="S5.Ex33.m1.4.4.4"></geq><apply id="S5.Ex33.m1.4.4.5.cmml" xref="S5.Ex33.m1.4.4.5"><times id="S5.Ex33.m1.4.4.5.1.cmml" xref="S5.Ex33.m1.4.4.5.1"></times><ci id="S5.Ex33.m1.4.4.5.2.cmml" xref="S5.Ex33.m1.4.4.5.2">𝒮</ci><ci id="S5.Ex33.m1.4.4.5.3.cmml" xref="S5.Ex33.m1.4.4.5.3">𝒲</ci><ci id="S5.Ex33.m1.1.1.cmml" xref="S5.Ex33.m1.1.1">𝜋</ci></apply><apply id="S5.Ex33.m1.4.4.3.cmml" xref="S5.Ex33.m1.4.4.3"><times id="S5.Ex33.m1.4.4.3.4.cmml" xref="S5.Ex33.m1.4.4.3.4"></times><apply id="S5.Ex33.m1.2.2.1.1.2.cmml" xref="S5.Ex33.m1.2.2.1.1.1"><floor id="S5.Ex33.m1.2.2.1.1.2.1.cmml" xref="S5.Ex33.m1.2.2.1.1.1.2"></floor><apply id="S5.Ex33.m1.2.2.1.1.1.1.cmml" xref="S5.Ex33.m1.2.2.1.1.1.1"><times id="S5.Ex33.m1.2.2.1.1.1.1.1.cmml" xref="S5.Ex33.m1.2.2.1.1.1.1.1"></times><ci id="S5.Ex33.m1.2.2.1.1.1.1.2.cmml" xref="S5.Ex33.m1.2.2.1.1.1.1.2">𝑘</ci><ci id="S5.Ex33.m1.2.2.1.1.1.1.3.cmml" xref="S5.Ex33.m1.2.2.1.1.1.1.3">𝑐</ci></apply></apply><ci id="S5.Ex33.m1.4.4.3.5.cmml" xref="S5.Ex33.m1.4.4.3.5">𝛼</ci><apply id="S5.Ex33.m1.4.4.3.6.cmml" xref="S5.Ex33.m1.4.4.3.6"><divide id="S5.Ex33.m1.4.4.3.6.1.cmml" xref="S5.Ex33.m1.4.4.3.6"></divide><ci id="S5.Ex33.m1.4.4.3.6.2.cmml" xref="S5.Ex33.m1.4.4.3.6.2">𝑛</ci><ci id="S5.Ex33.m1.4.4.3.6.3.cmml" xref="S5.Ex33.m1.4.4.3.6.3">𝑘</ci></apply><apply id="S5.Ex33.m1.3.3.2.2.1.1.cmml" xref="S5.Ex33.m1.3.3.2.2.1"><times id="S5.Ex33.m1.3.3.2.2.1.1.1.cmml" xref="S5.Ex33.m1.3.3.2.2.1.1.1"></times><apply id="S5.Ex33.m1.3.3.2.2.1.1.2.cmml" xref="S5.Ex33.m1.3.3.2.2.1.1.2"><csymbol cd="ambiguous" id="S5.Ex33.m1.3.3.2.2.1.1.2.1.cmml" xref="S5.Ex33.m1.3.3.2.2.1.1.2">superscript</csymbol><ci id="S5.Ex33.m1.3.3.2.2.1.1.2.2.cmml" xref="S5.Ex33.m1.3.3.2.2.1.1.2.2">𝑘</ci><ci id="S5.Ex33.m1.3.3.2.2.1.1.2.3.cmml" xref="S5.Ex33.m1.3.3.2.2.1.1.2.3">′</ci></apply><apply id="S5.Ex33.m1.3.3.2.2.1.1.3.cmml" xref="S5.Ex33.m1.3.3.2.2.1.1.3"><root id="S5.Ex33.m1.3.3.2.2.1.1.3a.cmml" xref="S5.Ex33.m1.3.3.2.2.1.1.3"></root><apply id="S5.Ex33.m1.3.3.2.2.1.1.3.2.cmml" xref="S5.Ex33.m1.3.3.2.2.1.1.3.2"><minus id="S5.Ex33.m1.3.3.2.2.1.1.3.2.1.cmml" xref="S5.Ex33.m1.3.3.2.2.1.1.3.2.1"></minus><cn id="S5.Ex33.m1.3.3.2.2.1.1.3.2.2.cmml" type="integer" xref="S5.Ex33.m1.3.3.2.2.1.1.3.2.2">1</cn><ci id="S5.Ex33.m1.3.3.2.2.1.1.3.2.3.cmml" xref="S5.Ex33.m1.3.3.2.2.1.1.3.2.3">𝜀</ci></apply></apply></apply><apply id="S5.Ex33.m1.4.4.3.3.1.1.cmml" xref="S5.Ex33.m1.4.4.3.3.1"><minus id="S5.Ex33.m1.4.4.3.3.1.1.1.cmml" xref="S5.Ex33.m1.4.4.3.3.1.1.1"></minus><apply id="S5.Ex33.m1.4.4.3.3.1.1.2.cmml" xref="S5.Ex33.m1.4.4.3.3.1.1.2"><times id="S5.Ex33.m1.4.4.3.3.1.1.2.1.cmml" xref="S5.Ex33.m1.4.4.3.3.1.1.2.1"></times><apply id="S5.Ex33.m1.4.4.3.3.1.1.2.2.cmml" xref="S5.Ex33.m1.4.4.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S5.Ex33.m1.4.4.3.3.1.1.2.2.1.cmml" xref="S5.Ex33.m1.4.4.3.3.1.1.2.2">superscript</csymbol><ci id="S5.Ex33.m1.4.4.3.3.1.1.2.2.2.cmml" xref="S5.Ex33.m1.4.4.3.3.1.1.2.2.2">𝑘</ci><ci id="S5.Ex33.m1.4.4.3.3.1.1.2.2.3.cmml" xref="S5.Ex33.m1.4.4.3.3.1.1.2.2.3">′</ci></apply><apply id="S5.Ex33.m1.4.4.3.3.1.1.2.3.cmml" xref="S5.Ex33.m1.4.4.3.3.1.1.2.3"><root id="S5.Ex33.m1.4.4.3.3.1.1.2.3a.cmml" xref="S5.Ex33.m1.4.4.3.3.1.1.2.3"></root><apply id="S5.Ex33.m1.4.4.3.3.1.1.2.3.2.cmml" xref="S5.Ex33.m1.4.4.3.3.1.1.2.3.2"><minus id="S5.Ex33.m1.4.4.3.3.1.1.2.3.2.1.cmml" xref="S5.Ex33.m1.4.4.3.3.1.1.2.3.2.1"></minus><cn id="S5.Ex33.m1.4.4.3.3.1.1.2.3.2.2.cmml" type="integer" xref="S5.Ex33.m1.4.4.3.3.1.1.2.3.2.2">1</cn><ci id="S5.Ex33.m1.4.4.3.3.1.1.2.3.2.3.cmml" xref="S5.Ex33.m1.4.4.3.3.1.1.2.3.2.3">𝜀</ci></apply></apply></apply><cn id="S5.Ex33.m1.4.4.3.3.1.1.3.cmml" type="integer" xref="S5.Ex33.m1.4.4.3.3.1.1.3">1</cn></apply><ci id="S5.Ex33.m1.4.4.3.7a.cmml" xref="S5.Ex33.m1.4.4.3.7"><mtext id="S5.Ex33.m1.4.4.3.7.cmml" xref="S5.Ex33.m1.4.4.3.7">.</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex33.m1.4c">\mathcal{SW}(\pi)\geq\lfloor kc\rfloor\alpha\frac{n}{k}(k^{\prime}\sqrt{1-% \varepsilon})(k^{\prime}\sqrt{1-\varepsilon}-1)\text{.}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex33.m1.4d">caligraphic_S caligraphic_W ( italic_π ) ≥ ⌊ italic_k italic_c ⌋ italic_α divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_ε end_ARG ) ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_ε end_ARG - 1 ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS2.SSS2.1.p1.23">Since <math alttext="k^{\prime}=\left\lfloor\frac{1}{c}\right\rfloor" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.20.m1.1"><semantics id="S5.SS2.SSS2.1.p1.20.m1.1a"><mrow id="S5.SS2.SSS2.1.p1.20.m1.1.2" xref="S5.SS2.SSS2.1.p1.20.m1.1.2.cmml"><msup id="S5.SS2.SSS2.1.p1.20.m1.1.2.2" xref="S5.SS2.SSS2.1.p1.20.m1.1.2.2.cmml"><mi id="S5.SS2.SSS2.1.p1.20.m1.1.2.2.2" xref="S5.SS2.SSS2.1.p1.20.m1.1.2.2.2.cmml">k</mi><mo id="S5.SS2.SSS2.1.p1.20.m1.1.2.2.3" xref="S5.SS2.SSS2.1.p1.20.m1.1.2.2.3.cmml">′</mo></msup><mo id="S5.SS2.SSS2.1.p1.20.m1.1.2.1" xref="S5.SS2.SSS2.1.p1.20.m1.1.2.1.cmml">=</mo><mrow id="S5.SS2.SSS2.1.p1.20.m1.1.2.3.2" xref="S5.SS2.SSS2.1.p1.20.m1.1.2.3.1.cmml"><mo id="S5.SS2.SSS2.1.p1.20.m1.1.2.3.2.1" xref="S5.SS2.SSS2.1.p1.20.m1.1.2.3.1.1.cmml">⌊</mo><mfrac id="S5.SS2.SSS2.1.p1.20.m1.1.1" xref="S5.SS2.SSS2.1.p1.20.m1.1.1.cmml"><mn id="S5.SS2.SSS2.1.p1.20.m1.1.1.2" xref="S5.SS2.SSS2.1.p1.20.m1.1.1.2.cmml">1</mn><mi id="S5.SS2.SSS2.1.p1.20.m1.1.1.3" xref="S5.SS2.SSS2.1.p1.20.m1.1.1.3.cmml">c</mi></mfrac><mo id="S5.SS2.SSS2.1.p1.20.m1.1.2.3.2.2" xref="S5.SS2.SSS2.1.p1.20.m1.1.2.3.1.1.cmml">⌋</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.20.m1.1b"><apply id="S5.SS2.SSS2.1.p1.20.m1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.20.m1.1.2"><eq id="S5.SS2.SSS2.1.p1.20.m1.1.2.1.cmml" xref="S5.SS2.SSS2.1.p1.20.m1.1.2.1"></eq><apply id="S5.SS2.SSS2.1.p1.20.m1.1.2.2.cmml" xref="S5.SS2.SSS2.1.p1.20.m1.1.2.2"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.20.m1.1.2.2.1.cmml" xref="S5.SS2.SSS2.1.p1.20.m1.1.2.2">superscript</csymbol><ci id="S5.SS2.SSS2.1.p1.20.m1.1.2.2.2.cmml" xref="S5.SS2.SSS2.1.p1.20.m1.1.2.2.2">𝑘</ci><ci id="S5.SS2.SSS2.1.p1.20.m1.1.2.2.3.cmml" xref="S5.SS2.SSS2.1.p1.20.m1.1.2.2.3">′</ci></apply><apply id="S5.SS2.SSS2.1.p1.20.m1.1.2.3.1.cmml" xref="S5.SS2.SSS2.1.p1.20.m1.1.2.3.2"><floor id="S5.SS2.SSS2.1.p1.20.m1.1.2.3.1.1.cmml" xref="S5.SS2.SSS2.1.p1.20.m1.1.2.3.2.1"></floor><apply id="S5.SS2.SSS2.1.p1.20.m1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.20.m1.1.1"><divide id="S5.SS2.SSS2.1.p1.20.m1.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.20.m1.1.1"></divide><cn id="S5.SS2.SSS2.1.p1.20.m1.1.1.2.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.20.m1.1.1.2">1</cn><ci id="S5.SS2.SSS2.1.p1.20.m1.1.1.3.cmml" xref="S5.SS2.SSS2.1.p1.20.m1.1.1.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.20.m1.1c">k^{\prime}=\left\lfloor\frac{1}{c}\right\rfloor</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.20.m1.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ⌊ divide start_ARG 1 end_ARG start_ARG italic_c end_ARG ⌋</annotation></semantics></math> and <math alttext="c" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.21.m2.1"><semantics id="S5.SS2.SSS2.1.p1.21.m2.1a"><mi id="S5.SS2.SSS2.1.p1.21.m2.1.1" xref="S5.SS2.SSS2.1.p1.21.m2.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.21.m2.1b"><ci id="S5.SS2.SSS2.1.p1.21.m2.1.1.cmml" xref="S5.SS2.SSS2.1.p1.21.m2.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.21.m2.1c">c</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.21.m2.1d">italic_c</annotation></semantics></math> is constant, it follows that <math alttext="\mathcal{SW}(\pi)\geq nc_{0}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.22.m3.1"><semantics id="S5.SS2.SSS2.1.p1.22.m3.1a"><mrow id="S5.SS2.SSS2.1.p1.22.m3.1.2" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.cmml"><mrow id="S5.SS2.SSS2.1.p1.22.m3.1.2.2" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS2.1.p1.22.m3.1.2.2.2" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.2.2.cmml">𝒮</mi><mo id="S5.SS2.SSS2.1.p1.22.m3.1.2.2.1" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS2.1.p1.22.m3.1.2.2.3" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.2.3.cmml">𝒲</mi><mo id="S5.SS2.SSS2.1.p1.22.m3.1.2.2.1a" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.2.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.1.p1.22.m3.1.2.2.4.2" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.2.cmml"><mo id="S5.SS2.SSS2.1.p1.22.m3.1.2.2.4.2.1" stretchy="false" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.2.cmml">(</mo><mi id="S5.SS2.SSS2.1.p1.22.m3.1.1" xref="S5.SS2.SSS2.1.p1.22.m3.1.1.cmml">π</mi><mo id="S5.SS2.SSS2.1.p1.22.m3.1.2.2.4.2.2" stretchy="false" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.SS2.SSS2.1.p1.22.m3.1.2.1" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.1.cmml">≥</mo><mrow id="S5.SS2.SSS2.1.p1.22.m3.1.2.3" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.3.cmml"><mi id="S5.SS2.SSS2.1.p1.22.m3.1.2.3.2" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.3.2.cmml">n</mi><mo id="S5.SS2.SSS2.1.p1.22.m3.1.2.3.1" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.3.1.cmml">⁢</mo><msub id="S5.SS2.SSS2.1.p1.22.m3.1.2.3.3" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.3.3.cmml"><mi id="S5.SS2.SSS2.1.p1.22.m3.1.2.3.3.2" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.3.3.2.cmml">c</mi><mn id="S5.SS2.SSS2.1.p1.22.m3.1.2.3.3.3" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.3.3.3.cmml">0</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.22.m3.1b"><apply id="S5.SS2.SSS2.1.p1.22.m3.1.2.cmml" xref="S5.SS2.SSS2.1.p1.22.m3.1.2"><geq id="S5.SS2.SSS2.1.p1.22.m3.1.2.1.cmml" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.1"></geq><apply id="S5.SS2.SSS2.1.p1.22.m3.1.2.2.cmml" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.2"><times id="S5.SS2.SSS2.1.p1.22.m3.1.2.2.1.cmml" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.2.1"></times><ci id="S5.SS2.SSS2.1.p1.22.m3.1.2.2.2.cmml" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.2.2">𝒮</ci><ci id="S5.SS2.SSS2.1.p1.22.m3.1.2.2.3.cmml" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.2.3">𝒲</ci><ci id="S5.SS2.SSS2.1.p1.22.m3.1.1.cmml" xref="S5.SS2.SSS2.1.p1.22.m3.1.1">𝜋</ci></apply><apply id="S5.SS2.SSS2.1.p1.22.m3.1.2.3.cmml" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.3"><times id="S5.SS2.SSS2.1.p1.22.m3.1.2.3.1.cmml" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.3.1"></times><ci id="S5.SS2.SSS2.1.p1.22.m3.1.2.3.2.cmml" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.3.2">𝑛</ci><apply id="S5.SS2.SSS2.1.p1.22.m3.1.2.3.3.cmml" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.3.3"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.22.m3.1.2.3.3.1.cmml" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.3.3">subscript</csymbol><ci id="S5.SS2.SSS2.1.p1.22.m3.1.2.3.3.2.cmml" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.3.3.2">𝑐</ci><cn id="S5.SS2.SSS2.1.p1.22.m3.1.2.3.3.3.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.22.m3.1.2.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.22.m3.1c">\mathcal{SW}(\pi)\geq nc_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.22.m3.1d">caligraphic_S caligraphic_W ( italic_π ) ≥ italic_n italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> for some constant <math alttext="c_{0}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.1.p1.23.m4.1"><semantics id="S5.SS2.SSS2.1.p1.23.m4.1a"><msub id="S5.SS2.SSS2.1.p1.23.m4.1.1" xref="S5.SS2.SSS2.1.p1.23.m4.1.1.cmml"><mi id="S5.SS2.SSS2.1.p1.23.m4.1.1.2" xref="S5.SS2.SSS2.1.p1.23.m4.1.1.2.cmml">c</mi><mn id="S5.SS2.SSS2.1.p1.23.m4.1.1.3" xref="S5.SS2.SSS2.1.p1.23.m4.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.1.p1.23.m4.1b"><apply id="S5.SS2.SSS2.1.p1.23.m4.1.1.cmml" xref="S5.SS2.SSS2.1.p1.23.m4.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.1.p1.23.m4.1.1.1.cmml" xref="S5.SS2.SSS2.1.p1.23.m4.1.1">subscript</csymbol><ci id="S5.SS2.SSS2.1.p1.23.m4.1.1.2.cmml" xref="S5.SS2.SSS2.1.p1.23.m4.1.1.2">𝑐</ci><cn id="S5.SS2.SSS2.1.p1.23.m4.1.1.3.cmml" type="integer" xref="S5.SS2.SSS2.1.p1.23.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.1.p1.23.m4.1c">c_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.1.p1.23.m4.1d">italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg2"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg2.7.1.1">Algorithm 2</span> </span> Dividing into smaller subproblems</figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_left" id="alg2.4.4"><span class="ltx_text ltx_font_bold" id="alg2.4.4.1">Input:</span> <math alttext="\langle G,\varepsilon\rangle" class="ltx_Math" display="inline" id="alg2.1.1.m1.2"><semantics id="alg2.1.1.m1.2a"><mrow id="alg2.1.1.m1.2.3.2" xref="alg2.1.1.m1.2.3.1.cmml"><mo id="alg2.1.1.m1.2.3.2.1" stretchy="false" xref="alg2.1.1.m1.2.3.1.cmml">⟨</mo><mi id="alg2.1.1.m1.1.1" xref="alg2.1.1.m1.1.1.cmml">G</mi><mo id="alg2.1.1.m1.2.3.2.2" xref="alg2.1.1.m1.2.3.1.cmml">,</mo><mi id="alg2.1.1.m1.2.2" xref="alg2.1.1.m1.2.2.cmml">ε</mi><mo id="alg2.1.1.m1.2.3.2.3" stretchy="false" xref="alg2.1.1.m1.2.3.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="alg2.1.1.m1.2b"><list id="alg2.1.1.m1.2.3.1.cmml" xref="alg2.1.1.m1.2.3.2"><ci id="alg2.1.1.m1.1.1.cmml" xref="alg2.1.1.m1.1.1">𝐺</ci><ci id="alg2.1.1.m1.2.2.cmml" xref="alg2.1.1.m1.2.2">𝜀</ci></list></annotation-xml><annotation encoding="application/x-tex" id="alg2.1.1.m1.2c">\langle G,\varepsilon\rangle</annotation><annotation encoding="application/x-llamapun" id="alg2.1.1.m1.2d">⟨ italic_G , italic_ε ⟩</annotation></semantics></math> where <math alttext="G=(n,k,p)" class="ltx_Math" display="inline" id="alg2.2.2.m2.3"><semantics id="alg2.2.2.m2.3a"><mrow id="alg2.2.2.m2.3.4" xref="alg2.2.2.m2.3.4.cmml"><mi id="alg2.2.2.m2.3.4.2" xref="alg2.2.2.m2.3.4.2.cmml">G</mi><mo id="alg2.2.2.m2.3.4.1" xref="alg2.2.2.m2.3.4.1.cmml">=</mo><mrow id="alg2.2.2.m2.3.4.3.2" xref="alg2.2.2.m2.3.4.3.1.cmml"><mo id="alg2.2.2.m2.3.4.3.2.1" stretchy="false" xref="alg2.2.2.m2.3.4.3.1.cmml">(</mo><mi id="alg2.2.2.m2.1.1" xref="alg2.2.2.m2.1.1.cmml">n</mi><mo id="alg2.2.2.m2.3.4.3.2.2" xref="alg2.2.2.m2.3.4.3.1.cmml">,</mo><mi id="alg2.2.2.m2.2.2" xref="alg2.2.2.m2.2.2.cmml">k</mi><mo id="alg2.2.2.m2.3.4.3.2.3" xref="alg2.2.2.m2.3.4.3.1.cmml">,</mo><mi id="alg2.2.2.m2.3.3" xref="alg2.2.2.m2.3.3.cmml">p</mi><mo id="alg2.2.2.m2.3.4.3.2.4" stretchy="false" xref="alg2.2.2.m2.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.2.2.m2.3b"><apply id="alg2.2.2.m2.3.4.cmml" xref="alg2.2.2.m2.3.4"><eq id="alg2.2.2.m2.3.4.1.cmml" xref="alg2.2.2.m2.3.4.1"></eq><ci id="alg2.2.2.m2.3.4.2.cmml" xref="alg2.2.2.m2.3.4.2">𝐺</ci><vector id="alg2.2.2.m2.3.4.3.1.cmml" xref="alg2.2.2.m2.3.4.3.2"><ci id="alg2.2.2.m2.1.1.cmml" xref="alg2.2.2.m2.1.1">𝑛</ci><ci id="alg2.2.2.m2.2.2.cmml" xref="alg2.2.2.m2.2.2">𝑘</ci><ci id="alg2.2.2.m2.3.3.cmml" xref="alg2.2.2.m2.3.3">𝑝</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.2.2.m2.3c">G=(n,k,p)</annotation><annotation encoding="application/x-llamapun" id="alg2.2.2.m2.3d">italic_G = ( italic_n , italic_k , italic_p )</annotation></semantics></math> is a random Turán graph and <math alttext="p=c" class="ltx_Math" display="inline" id="alg2.3.3.m3.1"><semantics id="alg2.3.3.m3.1a"><mrow id="alg2.3.3.m3.1.1" xref="alg2.3.3.m3.1.1.cmml"><mi id="alg2.3.3.m3.1.1.2" xref="alg2.3.3.m3.1.1.2.cmml">p</mi><mo id="alg2.3.3.m3.1.1.1" xref="alg2.3.3.m3.1.1.1.cmml">=</mo><mi id="alg2.3.3.m3.1.1.3" xref="alg2.3.3.m3.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="alg2.3.3.m3.1b"><apply id="alg2.3.3.m3.1.1.cmml" xref="alg2.3.3.m3.1.1"><eq id="alg2.3.3.m3.1.1.1.cmml" xref="alg2.3.3.m3.1.1.1"></eq><ci id="alg2.3.3.m3.1.1.2.cmml" xref="alg2.3.3.m3.1.1.2">𝑝</ci><ci id="alg2.3.3.m3.1.1.3.cmml" xref="alg2.3.3.m3.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.3.3.m3.1c">p=c</annotation><annotation encoding="application/x-llamapun" id="alg2.3.3.m3.1d">italic_p = italic_c</annotation></semantics></math> for some constant <math alttext="c\in(0,1)" class="ltx_Math" display="inline" id="alg2.4.4.m4.2"><semantics id="alg2.4.4.m4.2a"><mrow id="alg2.4.4.m4.2.3" xref="alg2.4.4.m4.2.3.cmml"><mi id="alg2.4.4.m4.2.3.2" xref="alg2.4.4.m4.2.3.2.cmml">c</mi><mo id="alg2.4.4.m4.2.3.1" xref="alg2.4.4.m4.2.3.1.cmml">∈</mo><mrow id="alg2.4.4.m4.2.3.3.2" xref="alg2.4.4.m4.2.3.3.1.cmml"><mo id="alg2.4.4.m4.2.3.3.2.1" stretchy="false" xref="alg2.4.4.m4.2.3.3.1.cmml">(</mo><mn id="alg2.4.4.m4.1.1" xref="alg2.4.4.m4.1.1.cmml">0</mn><mo id="alg2.4.4.m4.2.3.3.2.2" xref="alg2.4.4.m4.2.3.3.1.cmml">,</mo><mn id="alg2.4.4.m4.2.2" xref="alg2.4.4.m4.2.2.cmml">1</mn><mo id="alg2.4.4.m4.2.3.3.2.3" stretchy="false" xref="alg2.4.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.4.4.m4.2b"><apply id="alg2.4.4.m4.2.3.cmml" xref="alg2.4.4.m4.2.3"><in id="alg2.4.4.m4.2.3.1.cmml" xref="alg2.4.4.m4.2.3.1"></in><ci id="alg2.4.4.m4.2.3.2.cmml" xref="alg2.4.4.m4.2.3.2">𝑐</ci><interval closure="open" id="alg2.4.4.m4.2.3.3.1.cmml" xref="alg2.4.4.m4.2.3.3.2"><cn id="alg2.4.4.m4.1.1.cmml" type="integer" xref="alg2.4.4.m4.1.1">0</cn><cn id="alg2.4.4.m4.2.2.cmml" type="integer" xref="alg2.4.4.m4.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.4.4.m4.2c">c\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="alg2.4.4.m4.2d">italic_c ∈ ( 0 , 1 )</annotation></semantics></math></p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_left" id="alg2.5.5"><span class="ltx_text ltx_font_bold" id="alg2.5.5.1">Output:</span> Partition <math alttext="\pi" class="ltx_Math" display="inline" id="alg2.5.5.m1.1"><semantics id="alg2.5.5.m1.1a"><mi id="alg2.5.5.m1.1.1" xref="alg2.5.5.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="alg2.5.5.m1.1b"><ci id="alg2.5.5.m1.1.1.cmml" xref="alg2.5.5.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.5.5.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="alg2.5.5.m1.1d">italic_π</annotation></semantics></math></p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <div class="ltx_listing ltx_figure_panel ltx_listing" id="alg2.8"> <div class="ltx_listingline" id="alg2.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l1.1.1.1" style="font-size:80%;">1:</span></span><math alttext="\pi\leftarrow\emptyset" class="ltx_Math" display="inline" id="alg2.l1.m1.1"><semantics id="alg2.l1.m1.1a"><mrow id="alg2.l1.m1.1.1" xref="alg2.l1.m1.1.1.cmml"><mi id="alg2.l1.m1.1.1.2" xref="alg2.l1.m1.1.1.2.cmml">π</mi><mo id="alg2.l1.m1.1.1.1" stretchy="false" xref="alg2.l1.m1.1.1.1.cmml">←</mo><mi id="alg2.l1.m1.1.1.3" mathvariant="normal" xref="alg2.l1.m1.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="alg2.l1.m1.1b"><apply id="alg2.l1.m1.1.1.cmml" xref="alg2.l1.m1.1.1"><ci id="alg2.l1.m1.1.1.1.cmml" xref="alg2.l1.m1.1.1.1">←</ci><ci id="alg2.l1.m1.1.1.2.cmml" xref="alg2.l1.m1.1.1.2">𝜋</ci><emptyset id="alg2.l1.m1.1.1.3.cmml" xref="alg2.l1.m1.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m1.1c">\pi\leftarrow\emptyset</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m1.1d">italic_π ← ∅</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg2.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l2.1.1.1" style="font-size:80%;">2:</span></span>Partition <math alttext="\{V_{1},\dots,V_{k}\}" class="ltx_Math" display="inline" id="alg2.l2.m1.3"><semantics id="alg2.l2.m1.3a"><mrow id="alg2.l2.m1.3.3.2" xref="alg2.l2.m1.3.3.3.cmml"><mo id="alg2.l2.m1.3.3.2.3" stretchy="false" xref="alg2.l2.m1.3.3.3.cmml">{</mo><msub id="alg2.l2.m1.2.2.1.1" xref="alg2.l2.m1.2.2.1.1.cmml"><mi id="alg2.l2.m1.2.2.1.1.2" xref="alg2.l2.m1.2.2.1.1.2.cmml">V</mi><mn id="alg2.l2.m1.2.2.1.1.3" xref="alg2.l2.m1.2.2.1.1.3.cmml">1</mn></msub><mo id="alg2.l2.m1.3.3.2.4" xref="alg2.l2.m1.3.3.3.cmml">,</mo><mi id="alg2.l2.m1.1.1" mathvariant="normal" xref="alg2.l2.m1.1.1.cmml">…</mi><mo id="alg2.l2.m1.3.3.2.5" xref="alg2.l2.m1.3.3.3.cmml">,</mo><msub id="alg2.l2.m1.3.3.2.2" xref="alg2.l2.m1.3.3.2.2.cmml"><mi id="alg2.l2.m1.3.3.2.2.2" xref="alg2.l2.m1.3.3.2.2.2.cmml">V</mi><mi id="alg2.l2.m1.3.3.2.2.3" xref="alg2.l2.m1.3.3.2.2.3.cmml">k</mi></msub><mo id="alg2.l2.m1.3.3.2.6" stretchy="false" xref="alg2.l2.m1.3.3.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="alg2.l2.m1.3b"><set id="alg2.l2.m1.3.3.3.cmml" xref="alg2.l2.m1.3.3.2"><apply id="alg2.l2.m1.2.2.1.1.cmml" xref="alg2.l2.m1.2.2.1.1"><csymbol cd="ambiguous" id="alg2.l2.m1.2.2.1.1.1.cmml" xref="alg2.l2.m1.2.2.1.1">subscript</csymbol><ci id="alg2.l2.m1.2.2.1.1.2.cmml" xref="alg2.l2.m1.2.2.1.1.2">𝑉</ci><cn id="alg2.l2.m1.2.2.1.1.3.cmml" type="integer" xref="alg2.l2.m1.2.2.1.1.3">1</cn></apply><ci id="alg2.l2.m1.1.1.cmml" xref="alg2.l2.m1.1.1">…</ci><apply id="alg2.l2.m1.3.3.2.2.cmml" xref="alg2.l2.m1.3.3.2.2"><csymbol cd="ambiguous" id="alg2.l2.m1.3.3.2.2.1.cmml" xref="alg2.l2.m1.3.3.2.2">subscript</csymbol><ci id="alg2.l2.m1.3.3.2.2.2.cmml" xref="alg2.l2.m1.3.3.2.2.2">𝑉</ci><ci id="alg2.l2.m1.3.3.2.2.3.cmml" xref="alg2.l2.m1.3.3.2.2.3">𝑘</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="alg2.l2.m1.3c">\{V_{1},\dots,V_{k}\}</annotation><annotation encoding="application/x-llamapun" id="alg2.l2.m1.3d">{ italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }</annotation></semantics></math> into <math alttext="\lceil ck\rceil" class="ltx_Math" display="inline" id="alg2.l2.m2.1"><semantics id="alg2.l2.m2.1a"><mrow id="alg2.l2.m2.1.1.1" xref="alg2.l2.m2.1.1.2.cmml"><mo id="alg2.l2.m2.1.1.1.2" stretchy="false" xref="alg2.l2.m2.1.1.2.1.cmml">⌈</mo><mrow id="alg2.l2.m2.1.1.1.1" xref="alg2.l2.m2.1.1.1.1.cmml"><mi id="alg2.l2.m2.1.1.1.1.2" xref="alg2.l2.m2.1.1.1.1.2.cmml">c</mi><mo id="alg2.l2.m2.1.1.1.1.1" xref="alg2.l2.m2.1.1.1.1.1.cmml">⁢</mo><mi id="alg2.l2.m2.1.1.1.1.3" xref="alg2.l2.m2.1.1.1.1.3.cmml">k</mi></mrow><mo id="alg2.l2.m2.1.1.1.3" stretchy="false" xref="alg2.l2.m2.1.1.2.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="alg2.l2.m2.1b"><apply id="alg2.l2.m2.1.1.2.cmml" xref="alg2.l2.m2.1.1.1"><ceiling id="alg2.l2.m2.1.1.2.1.cmml" xref="alg2.l2.m2.1.1.1.2"></ceiling><apply id="alg2.l2.m2.1.1.1.1.cmml" xref="alg2.l2.m2.1.1.1.1"><times id="alg2.l2.m2.1.1.1.1.1.cmml" xref="alg2.l2.m2.1.1.1.1.1"></times><ci id="alg2.l2.m2.1.1.1.1.2.cmml" xref="alg2.l2.m2.1.1.1.1.2">𝑐</ci><ci id="alg2.l2.m2.1.1.1.1.3.cmml" xref="alg2.l2.m2.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l2.m2.1c">\lceil ck\rceil</annotation><annotation encoding="application/x-llamapun" id="alg2.l2.m2.1d">⌈ italic_c italic_k ⌉</annotation></semantics></math> disjoint sets <math alttext="S_{1},\dots,S_{\lceil ck\rceil}" class="ltx_Math" display="inline" id="alg2.l2.m3.4"><semantics id="alg2.l2.m3.4a"><mrow id="alg2.l2.m3.4.4.2" xref="alg2.l2.m3.4.4.3.cmml"><msub id="alg2.l2.m3.3.3.1.1" xref="alg2.l2.m3.3.3.1.1.cmml"><mi id="alg2.l2.m3.3.3.1.1.2" xref="alg2.l2.m3.3.3.1.1.2.cmml">S</mi><mn id="alg2.l2.m3.3.3.1.1.3" xref="alg2.l2.m3.3.3.1.1.3.cmml">1</mn></msub><mo id="alg2.l2.m3.4.4.2.3" xref="alg2.l2.m3.4.4.3.cmml">,</mo><mi id="alg2.l2.m3.2.2" mathvariant="normal" xref="alg2.l2.m3.2.2.cmml">…</mi><mo id="alg2.l2.m3.4.4.2.4" xref="alg2.l2.m3.4.4.3.cmml">,</mo><msub id="alg2.l2.m3.4.4.2.2" xref="alg2.l2.m3.4.4.2.2.cmml"><mi id="alg2.l2.m3.4.4.2.2.2" xref="alg2.l2.m3.4.4.2.2.2.cmml">S</mi><mrow id="alg2.l2.m3.1.1.1.1" xref="alg2.l2.m3.1.1.1.2.cmml"><mo id="alg2.l2.m3.1.1.1.1.2" stretchy="false" xref="alg2.l2.m3.1.1.1.2.1.cmml">⌈</mo><mrow id="alg2.l2.m3.1.1.1.1.1" xref="alg2.l2.m3.1.1.1.1.1.cmml"><mi id="alg2.l2.m3.1.1.1.1.1.2" xref="alg2.l2.m3.1.1.1.1.1.2.cmml">c</mi><mo id="alg2.l2.m3.1.1.1.1.1.1" xref="alg2.l2.m3.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg2.l2.m3.1.1.1.1.1.3" xref="alg2.l2.m3.1.1.1.1.1.3.cmml">k</mi></mrow><mo id="alg2.l2.m3.1.1.1.1.3" stretchy="false" xref="alg2.l2.m3.1.1.1.2.1.cmml">⌉</mo></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="alg2.l2.m3.4b"><list id="alg2.l2.m3.4.4.3.cmml" xref="alg2.l2.m3.4.4.2"><apply id="alg2.l2.m3.3.3.1.1.cmml" xref="alg2.l2.m3.3.3.1.1"><csymbol cd="ambiguous" id="alg2.l2.m3.3.3.1.1.1.cmml" xref="alg2.l2.m3.3.3.1.1">subscript</csymbol><ci id="alg2.l2.m3.3.3.1.1.2.cmml" xref="alg2.l2.m3.3.3.1.1.2">𝑆</ci><cn id="alg2.l2.m3.3.3.1.1.3.cmml" type="integer" xref="alg2.l2.m3.3.3.1.1.3">1</cn></apply><ci id="alg2.l2.m3.2.2.cmml" xref="alg2.l2.m3.2.2">…</ci><apply id="alg2.l2.m3.4.4.2.2.cmml" xref="alg2.l2.m3.4.4.2.2"><csymbol cd="ambiguous" id="alg2.l2.m3.4.4.2.2.1.cmml" xref="alg2.l2.m3.4.4.2.2">subscript</csymbol><ci id="alg2.l2.m3.4.4.2.2.2.cmml" xref="alg2.l2.m3.4.4.2.2.2">𝑆</ci><apply id="alg2.l2.m3.1.1.1.2.cmml" xref="alg2.l2.m3.1.1.1.1"><ceiling id="alg2.l2.m3.1.1.1.2.1.cmml" xref="alg2.l2.m3.1.1.1.1.2"></ceiling><apply id="alg2.l2.m3.1.1.1.1.1.cmml" xref="alg2.l2.m3.1.1.1.1.1"><times id="alg2.l2.m3.1.1.1.1.1.1.cmml" xref="alg2.l2.m3.1.1.1.1.1.1"></times><ci id="alg2.l2.m3.1.1.1.1.1.2.cmml" xref="alg2.l2.m3.1.1.1.1.1.2">𝑐</ci><ci id="alg2.l2.m3.1.1.1.1.1.3.cmml" xref="alg2.l2.m3.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="alg2.l2.m3.4c">S_{1},\dots,S_{\lceil ck\rceil}</annotation><annotation encoding="application/x-llamapun" id="alg2.l2.m3.4d">italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_S start_POSTSUBSCRIPT ⌈ italic_c italic_k ⌉ end_POSTSUBSCRIPT</annotation></semantics></math> that differ in size by at most one </div> <div class="ltx_listingline" id="alg2.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l3.1.1.1" style="font-size:80%;">3:</span></span><span class="ltx_text ltx_font_bold" id="alg2.l3.2">for</span> each group of colors <math alttext="S\in\{S_{1},\cdots,S_{\lceil kc\rceil}\}" class="ltx_Math" display="inline" id="alg2.l3.m1.4"><semantics id="alg2.l3.m1.4a"><mrow id="alg2.l3.m1.4.4" xref="alg2.l3.m1.4.4.cmml"><mi id="alg2.l3.m1.4.4.4" xref="alg2.l3.m1.4.4.4.cmml">S</mi><mo id="alg2.l3.m1.4.4.3" xref="alg2.l3.m1.4.4.3.cmml">∈</mo><mrow id="alg2.l3.m1.4.4.2.2" xref="alg2.l3.m1.4.4.2.3.cmml"><mo id="alg2.l3.m1.4.4.2.2.3" stretchy="false" xref="alg2.l3.m1.4.4.2.3.cmml">{</mo><msub id="alg2.l3.m1.3.3.1.1.1" xref="alg2.l3.m1.3.3.1.1.1.cmml"><mi id="alg2.l3.m1.3.3.1.1.1.2" xref="alg2.l3.m1.3.3.1.1.1.2.cmml">S</mi><mn id="alg2.l3.m1.3.3.1.1.1.3" xref="alg2.l3.m1.3.3.1.1.1.3.cmml">1</mn></msub><mo id="alg2.l3.m1.4.4.2.2.4" xref="alg2.l3.m1.4.4.2.3.cmml">,</mo><mi id="alg2.l3.m1.2.2" mathvariant="normal" xref="alg2.l3.m1.2.2.cmml">⋯</mi><mo id="alg2.l3.m1.4.4.2.2.5" xref="alg2.l3.m1.4.4.2.3.cmml">,</mo><msub id="alg2.l3.m1.4.4.2.2.2" xref="alg2.l3.m1.4.4.2.2.2.cmml"><mi id="alg2.l3.m1.4.4.2.2.2.2" xref="alg2.l3.m1.4.4.2.2.2.2.cmml">S</mi><mrow id="alg2.l3.m1.1.1.1.1" xref="alg2.l3.m1.1.1.1.2.cmml"><mo id="alg2.l3.m1.1.1.1.1.2" stretchy="false" xref="alg2.l3.m1.1.1.1.2.1.cmml">⌈</mo><mrow id="alg2.l3.m1.1.1.1.1.1" xref="alg2.l3.m1.1.1.1.1.1.cmml"><mi id="alg2.l3.m1.1.1.1.1.1.2" xref="alg2.l3.m1.1.1.1.1.1.2.cmml">k</mi><mo id="alg2.l3.m1.1.1.1.1.1.1" xref="alg2.l3.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg2.l3.m1.1.1.1.1.1.3" xref="alg2.l3.m1.1.1.1.1.1.3.cmml">c</mi></mrow><mo id="alg2.l3.m1.1.1.1.1.3" stretchy="false" xref="alg2.l3.m1.1.1.1.2.1.cmml">⌉</mo></mrow></msub><mo id="alg2.l3.m1.4.4.2.2.6" stretchy="false" xref="alg2.l3.m1.4.4.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l3.m1.4b"><apply id="alg2.l3.m1.4.4.cmml" xref="alg2.l3.m1.4.4"><in id="alg2.l3.m1.4.4.3.cmml" xref="alg2.l3.m1.4.4.3"></in><ci id="alg2.l3.m1.4.4.4.cmml" xref="alg2.l3.m1.4.4.4">𝑆</ci><set id="alg2.l3.m1.4.4.2.3.cmml" xref="alg2.l3.m1.4.4.2.2"><apply id="alg2.l3.m1.3.3.1.1.1.cmml" xref="alg2.l3.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="alg2.l3.m1.3.3.1.1.1.1.cmml" xref="alg2.l3.m1.3.3.1.1.1">subscript</csymbol><ci id="alg2.l3.m1.3.3.1.1.1.2.cmml" xref="alg2.l3.m1.3.3.1.1.1.2">𝑆</ci><cn id="alg2.l3.m1.3.3.1.1.1.3.cmml" type="integer" xref="alg2.l3.m1.3.3.1.1.1.3">1</cn></apply><ci id="alg2.l3.m1.2.2.cmml" xref="alg2.l3.m1.2.2">⋯</ci><apply id="alg2.l3.m1.4.4.2.2.2.cmml" xref="alg2.l3.m1.4.4.2.2.2"><csymbol cd="ambiguous" id="alg2.l3.m1.4.4.2.2.2.1.cmml" xref="alg2.l3.m1.4.4.2.2.2">subscript</csymbol><ci id="alg2.l3.m1.4.4.2.2.2.2.cmml" xref="alg2.l3.m1.4.4.2.2.2.2">𝑆</ci><apply id="alg2.l3.m1.1.1.1.2.cmml" xref="alg2.l3.m1.1.1.1.1"><ceiling id="alg2.l3.m1.1.1.1.2.1.cmml" xref="alg2.l3.m1.1.1.1.1.2"></ceiling><apply id="alg2.l3.m1.1.1.1.1.1.cmml" xref="alg2.l3.m1.1.1.1.1.1"><times id="alg2.l3.m1.1.1.1.1.1.1.cmml" xref="alg2.l3.m1.1.1.1.1.1.1"></times><ci id="alg2.l3.m1.1.1.1.1.1.2.cmml" xref="alg2.l3.m1.1.1.1.1.1.2">𝑘</ci><ci id="alg2.l3.m1.1.1.1.1.1.3.cmml" xref="alg2.l3.m1.1.1.1.1.1.3">𝑐</ci></apply></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l3.m1.4c">S\in\{S_{1},\cdots,S_{\lceil kc\rceil}\}</annotation><annotation encoding="application/x-llamapun" id="alg2.l3.m1.4d">italic_S ∈ { italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_S start_POSTSUBSCRIPT ⌈ italic_k italic_c ⌉ end_POSTSUBSCRIPT }</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg2.l3.3">do</span> </div> <div class="ltx_listingline" id="alg2.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l4.1.1.1" style="font-size:80%;">4:</span></span> Let <math alttext="\pi_{S}" class="ltx_Math" display="inline" id="alg2.l4.m1.1"><semantics id="alg2.l4.m1.1a"><msub id="alg2.l4.m1.1.1" xref="alg2.l4.m1.1.1.cmml"><mi id="alg2.l4.m1.1.1.2" xref="alg2.l4.m1.1.1.2.cmml">π</mi><mi id="alg2.l4.m1.1.1.3" xref="alg2.l4.m1.1.1.3.cmml">S</mi></msub><annotation-xml encoding="MathML-Content" id="alg2.l4.m1.1b"><apply id="alg2.l4.m1.1.1.cmml" xref="alg2.l4.m1.1.1"><csymbol cd="ambiguous" id="alg2.l4.m1.1.1.1.cmml" xref="alg2.l4.m1.1.1">subscript</csymbol><ci id="alg2.l4.m1.1.1.2.cmml" xref="alg2.l4.m1.1.1.2">𝜋</ci><ci id="alg2.l4.m1.1.1.3.cmml" xref="alg2.l4.m1.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l4.m1.1c">\pi_{S}</annotation><annotation encoding="application/x-llamapun" id="alg2.l4.m1.1d">italic_π start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT</annotation></semantics></math> be the partition within vertices in color classes of <math alttext="S" class="ltx_Math" display="inline" id="alg2.l4.m2.1"><semantics id="alg2.l4.m2.1a"><mi id="alg2.l4.m2.1.1" xref="alg2.l4.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="alg2.l4.m2.1b"><ci id="alg2.l4.m2.1.1.cmml" xref="alg2.l4.m2.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l4.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="alg2.l4.m2.1d">italic_S</annotation></semantics></math>, after applying <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg1" title="In 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a> on input <math alttext="\langle G,S,\varepsilon\rangle" class="ltx_Math" display="inline" id="alg2.l4.m3.3"><semantics id="alg2.l4.m3.3a"><mrow id="alg2.l4.m3.3.4.2" xref="alg2.l4.m3.3.4.1.cmml"><mo id="alg2.l4.m3.3.4.2.1" stretchy="false" xref="alg2.l4.m3.3.4.1.cmml">⟨</mo><mi id="alg2.l4.m3.1.1" xref="alg2.l4.m3.1.1.cmml">G</mi><mo id="alg2.l4.m3.3.4.2.2" xref="alg2.l4.m3.3.4.1.cmml">,</mo><mi id="alg2.l4.m3.2.2" xref="alg2.l4.m3.2.2.cmml">S</mi><mo id="alg2.l4.m3.3.4.2.3" xref="alg2.l4.m3.3.4.1.cmml">,</mo><mi id="alg2.l4.m3.3.3" xref="alg2.l4.m3.3.3.cmml">ε</mi><mo id="alg2.l4.m3.3.4.2.4" stretchy="false" xref="alg2.l4.m3.3.4.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="alg2.l4.m3.3b"><list id="alg2.l4.m3.3.4.1.cmml" xref="alg2.l4.m3.3.4.2"><ci id="alg2.l4.m3.1.1.cmml" xref="alg2.l4.m3.1.1">𝐺</ci><ci id="alg2.l4.m3.2.2.cmml" xref="alg2.l4.m3.2.2">𝑆</ci><ci id="alg2.l4.m3.3.3.cmml" xref="alg2.l4.m3.3.3">𝜀</ci></list></annotation-xml><annotation encoding="application/x-tex" id="alg2.l4.m3.3c">\langle G,S,\varepsilon\rangle</annotation><annotation encoding="application/x-llamapun" id="alg2.l4.m3.3d">⟨ italic_G , italic_S , italic_ε ⟩</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg2.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l5.1.1.1" style="font-size:80%;">5:</span></span> <math alttext="\pi\leftarrow\pi\cup\pi_{S}" class="ltx_Math" display="inline" id="alg2.l5.m1.1"><semantics id="alg2.l5.m1.1a"><mrow id="alg2.l5.m1.1.1" xref="alg2.l5.m1.1.1.cmml"><mi id="alg2.l5.m1.1.1.2" xref="alg2.l5.m1.1.1.2.cmml">π</mi><mo id="alg2.l5.m1.1.1.1" stretchy="false" xref="alg2.l5.m1.1.1.1.cmml">←</mo><mrow id="alg2.l5.m1.1.1.3" xref="alg2.l5.m1.1.1.3.cmml"><mi id="alg2.l5.m1.1.1.3.2" xref="alg2.l5.m1.1.1.3.2.cmml">π</mi><mo id="alg2.l5.m1.1.1.3.1" xref="alg2.l5.m1.1.1.3.1.cmml">∪</mo><msub id="alg2.l5.m1.1.1.3.3" xref="alg2.l5.m1.1.1.3.3.cmml"><mi id="alg2.l5.m1.1.1.3.3.2" xref="alg2.l5.m1.1.1.3.3.2.cmml">π</mi><mi id="alg2.l5.m1.1.1.3.3.3" xref="alg2.l5.m1.1.1.3.3.3.cmml">S</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l5.m1.1b"><apply id="alg2.l5.m1.1.1.cmml" xref="alg2.l5.m1.1.1"><ci id="alg2.l5.m1.1.1.1.cmml" xref="alg2.l5.m1.1.1.1">←</ci><ci id="alg2.l5.m1.1.1.2.cmml" xref="alg2.l5.m1.1.1.2">𝜋</ci><apply id="alg2.l5.m1.1.1.3.cmml" xref="alg2.l5.m1.1.1.3"><union id="alg2.l5.m1.1.1.3.1.cmml" xref="alg2.l5.m1.1.1.3.1"></union><ci id="alg2.l5.m1.1.1.3.2.cmml" xref="alg2.l5.m1.1.1.3.2">𝜋</ci><apply id="alg2.l5.m1.1.1.3.3.cmml" xref="alg2.l5.m1.1.1.3.3"><csymbol cd="ambiguous" id="alg2.l5.m1.1.1.3.3.1.cmml" xref="alg2.l5.m1.1.1.3.3">subscript</csymbol><ci id="alg2.l5.m1.1.1.3.3.2.cmml" xref="alg2.l5.m1.1.1.3.3.2">𝜋</ci><ci id="alg2.l5.m1.1.1.3.3.3.cmml" xref="alg2.l5.m1.1.1.3.3.3">𝑆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l5.m1.1c">\pi\leftarrow\pi\cup\pi_{S}</annotation><annotation encoding="application/x-llamapun" id="alg2.l5.m1.1d">italic_π ← italic_π ∪ italic_π start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg2.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l6.1.1.1" style="font-size:80%;">6:</span></span><span class="ltx_text ltx_font_bold" id="alg2.l6.2">return</span> <math alttext="\pi" class="ltx_Math" display="inline" id="alg2.l6.m1.1"><semantics id="alg2.l6.m1.1a"><mi id="alg2.l6.m1.1.1" xref="alg2.l6.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="alg2.l6.m1.1b"><ci id="alg2.l6.m1.1.1.cmml" xref="alg2.l6.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l6.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="alg2.l6.m1.1d">italic_π</annotation></semantics></math> </div> </div> </div> </div> </figure> <div class="ltx_para" id="S5.SS2.SSS2.p2"> <p class="ltx_p" id="S5.SS2.SSS2.p2.1">We now bound the maximum welfare. The proof is similar to our arguments for Erdős-Rényi graphs.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S5.Thmtheorem10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem10.1.1.1">Lemma 5.10</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem10.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem10.p1"> <p class="ltx_p" id="S5.Thmtheorem10.p1.5"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem10.p1.5.5">If <math alttext="k=\Omega(\log n)" class="ltx_Math" display="inline" id="S5.Thmtheorem10.p1.1.1.m1.1"><semantics id="S5.Thmtheorem10.p1.1.1.m1.1a"><mrow id="S5.Thmtheorem10.p1.1.1.m1.1.1" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem10.p1.1.1.m1.1.1.3" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.3.cmml">k</mi><mo id="S5.Thmtheorem10.p1.1.1.m1.1.1.2" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.2.cmml">=</mo><mrow id="S5.Thmtheorem10.p1.1.1.m1.1.1.1" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1.cmml"><mi id="S5.Thmtheorem10.p1.1.1.m1.1.1.1.3" mathvariant="normal" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1.3.cmml">Ω</mi><mo id="S5.Thmtheorem10.p1.1.1.m1.1.1.1.2" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1.1" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1.1.cmml">log</mi><mo id="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1a" lspace="0.167em" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1.cmml">⁡</mo><mi id="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1.2" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem10.p1.1.1.m1.1b"><apply id="S5.Thmtheorem10.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem10.p1.1.1.m1.1.1"><eq id="S5.Thmtheorem10.p1.1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.2"></eq><ci id="S5.Thmtheorem10.p1.1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.3">𝑘</ci><apply id="S5.Thmtheorem10.p1.1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1"><times id="S5.Thmtheorem10.p1.1.1.m1.1.1.1.2.cmml" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1.2"></times><ci id="S5.Thmtheorem10.p1.1.1.m1.1.1.1.3.cmml" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1.3">Ω</ci><apply id="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1"><log id="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1.1"></log><ci id="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem10.p1.1.1.m1.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem10.p1.1.1.m1.1c">k=\Omega(\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem10.p1.1.1.m1.1d">italic_k = roman_Ω ( roman_log italic_n )</annotation></semantics></math> and <math alttext="p=c" class="ltx_Math" display="inline" id="S5.Thmtheorem10.p1.2.2.m2.1"><semantics id="S5.Thmtheorem10.p1.2.2.m2.1a"><mrow id="S5.Thmtheorem10.p1.2.2.m2.1.1" xref="S5.Thmtheorem10.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem10.p1.2.2.m2.1.1.2" xref="S5.Thmtheorem10.p1.2.2.m2.1.1.2.cmml">p</mi><mo id="S5.Thmtheorem10.p1.2.2.m2.1.1.1" xref="S5.Thmtheorem10.p1.2.2.m2.1.1.1.cmml">=</mo><mi id="S5.Thmtheorem10.p1.2.2.m2.1.1.3" xref="S5.Thmtheorem10.p1.2.2.m2.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem10.p1.2.2.m2.1b"><apply id="S5.Thmtheorem10.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem10.p1.2.2.m2.1.1"><eq id="S5.Thmtheorem10.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem10.p1.2.2.m2.1.1.1"></eq><ci id="S5.Thmtheorem10.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem10.p1.2.2.m2.1.1.2">𝑝</ci><ci id="S5.Thmtheorem10.p1.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem10.p1.2.2.m2.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem10.p1.2.2.m2.1c">p=c</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem10.p1.2.2.m2.1d">italic_p = italic_c</annotation></semantics></math> for some constant <math alttext="c\in(0,1)" class="ltx_Math" display="inline" id="S5.Thmtheorem10.p1.3.3.m3.2"><semantics id="S5.Thmtheorem10.p1.3.3.m3.2a"><mrow id="S5.Thmtheorem10.p1.3.3.m3.2.3" xref="S5.Thmtheorem10.p1.3.3.m3.2.3.cmml"><mi id="S5.Thmtheorem10.p1.3.3.m3.2.3.2" xref="S5.Thmtheorem10.p1.3.3.m3.2.3.2.cmml">c</mi><mo id="S5.Thmtheorem10.p1.3.3.m3.2.3.1" xref="S5.Thmtheorem10.p1.3.3.m3.2.3.1.cmml">∈</mo><mrow id="S5.Thmtheorem10.p1.3.3.m3.2.3.3.2" xref="S5.Thmtheorem10.p1.3.3.m3.2.3.3.1.cmml"><mo id="S5.Thmtheorem10.p1.3.3.m3.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem10.p1.3.3.m3.2.3.3.1.cmml">(</mo><mn id="S5.Thmtheorem10.p1.3.3.m3.1.1" xref="S5.Thmtheorem10.p1.3.3.m3.1.1.cmml">0</mn><mo id="S5.Thmtheorem10.p1.3.3.m3.2.3.3.2.2" xref="S5.Thmtheorem10.p1.3.3.m3.2.3.3.1.cmml">,</mo><mn id="S5.Thmtheorem10.p1.3.3.m3.2.2" xref="S5.Thmtheorem10.p1.3.3.m3.2.2.cmml">1</mn><mo id="S5.Thmtheorem10.p1.3.3.m3.2.3.3.2.3" stretchy="false" xref="S5.Thmtheorem10.p1.3.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem10.p1.3.3.m3.2b"><apply id="S5.Thmtheorem10.p1.3.3.m3.2.3.cmml" xref="S5.Thmtheorem10.p1.3.3.m3.2.3"><in id="S5.Thmtheorem10.p1.3.3.m3.2.3.1.cmml" xref="S5.Thmtheorem10.p1.3.3.m3.2.3.1"></in><ci id="S5.Thmtheorem10.p1.3.3.m3.2.3.2.cmml" xref="S5.Thmtheorem10.p1.3.3.m3.2.3.2">𝑐</ci><interval closure="open" id="S5.Thmtheorem10.p1.3.3.m3.2.3.3.1.cmml" xref="S5.Thmtheorem10.p1.3.3.m3.2.3.3.2"><cn id="S5.Thmtheorem10.p1.3.3.m3.1.1.cmml" type="integer" xref="S5.Thmtheorem10.p1.3.3.m3.1.1">0</cn><cn id="S5.Thmtheorem10.p1.3.3.m3.2.2.cmml" type="integer" xref="S5.Thmtheorem10.p1.3.3.m3.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem10.p1.3.3.m3.2c">c\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem10.p1.3.3.m3.2d">italic_c ∈ ( 0 , 1 )</annotation></semantics></math>, then the maximum welfare satisfies <math alttext="SW(\pi^{*})=\mathcal{O}(n\log n)" class="ltx_Math" display="inline" id="S5.Thmtheorem10.p1.4.4.m4.2"><semantics id="S5.Thmtheorem10.p1.4.4.m4.2a"><mrow id="S5.Thmtheorem10.p1.4.4.m4.2.2" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.cmml"><mrow id="S5.Thmtheorem10.p1.4.4.m4.1.1.1" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.cmml"><mi id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.3" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.3.cmml">S</mi><mo id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.2" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.2.cmml">⁢</mo><mi id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.4" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.4.cmml">W</mi><mo id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.2a" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.1" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.1.2" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.1.2.cmml">π</mi><mo id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.1.3" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem10.p1.4.4.m4.2.2.3" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.3.cmml">=</mo><mrow id="S5.Thmtheorem10.p1.4.4.m4.2.2.2" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.3" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.3.cmml">𝒪</mi><mo id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.2" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.2.cmml">⁢</mo><mrow id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.cmml"><mo id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.2" stretchy="false" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.cmml"><mi id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.2" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.2.cmml">n</mi><mo id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.1" lspace="0.167em" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.3" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.3.cmml"><mi id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.3.1" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.3.1.cmml">log</mi><mo id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.3a" lspace="0.167em" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.3.cmml">⁡</mo><mi id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.3.2" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.3" stretchy="false" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem10.p1.4.4.m4.2b"><apply id="S5.Thmtheorem10.p1.4.4.m4.2.2.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.2.2"><eq id="S5.Thmtheorem10.p1.4.4.m4.2.2.3.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.3"></eq><apply id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1"><times id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.2.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.2"></times><ci id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.3.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.3">𝑆</ci><ci id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.4.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.4">𝑊</ci><apply id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.1.2">𝜋</ci><times id="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.1.1.1.1.1.1.3"></times></apply></apply><apply id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2"><times id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.2.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.2"></times><ci id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.3.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.3">𝒪</ci><apply id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1"><times id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.1"></times><ci id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.2.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.2">𝑛</ci><apply id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.3.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.3"><log id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.3.1.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.3.1"></log><ci id="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.3.2.cmml" xref="S5.Thmtheorem10.p1.4.4.m4.2.2.2.1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem10.p1.4.4.m4.2c">SW(\pi^{*})=\mathcal{O}(n\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem10.p1.4.4.m4.2d">italic_S italic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = caligraphic_O ( italic_n roman_log italic_n )</annotation></semantics></math> with probability <math alttext="1-\left(\frac{ce}{2\log_{e}n}\right)^{\frac{2}{c}\log_{e}n}" class="ltx_Math" display="inline" id="S5.Thmtheorem10.p1.5.5.m5.1"><semantics id="S5.Thmtheorem10.p1.5.5.m5.1a"><mrow id="S5.Thmtheorem10.p1.5.5.m5.1.2" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.cmml"><mn id="S5.Thmtheorem10.p1.5.5.m5.1.2.2" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.2.cmml">1</mn><mo id="S5.Thmtheorem10.p1.5.5.m5.1.2.1" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.1.cmml">−</mo><msup id="S5.Thmtheorem10.p1.5.5.m5.1.2.3" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.cmml"><mrow id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.2.2" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.cmml"><mo id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.2.2.1" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.cmml">(</mo><mfrac id="S5.Thmtheorem10.p1.5.5.m5.1.1" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.cmml"><mrow id="S5.Thmtheorem10.p1.5.5.m5.1.1.2" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.2.cmml"><mi id="S5.Thmtheorem10.p1.5.5.m5.1.1.2.2" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.2.2.cmml">c</mi><mo id="S5.Thmtheorem10.p1.5.5.m5.1.1.2.1" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem10.p1.5.5.m5.1.1.2.3" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.2.3.cmml">e</mi></mrow><mrow id="S5.Thmtheorem10.p1.5.5.m5.1.1.3" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.cmml"><mn id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.2" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.2.cmml">2</mn><mo id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.1" lspace="0.167em" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.cmml"><msub id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.1" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.1.cmml"><mi id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.1.2" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.1.2.cmml">log</mi><mi id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.1.3" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.1.3.cmml">e</mi></msub><mo id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3a" lspace="0.167em" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.cmml">⁡</mo><mi id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.2" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.2.cmml">n</mi></mrow></mrow></mfrac><mo id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.2.2.2" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.cmml">)</mo></mrow><mrow id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.cmml"><mfrac id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.2" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.2.cmml"><mn id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.2.2" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.2.2.cmml">2</mn><mi id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.2.3" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.2.3.cmml">c</mi></mfrac><mo id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.1" lspace="0.167em" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.cmml"><msub id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.1" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.1.cmml"><mi id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.1.2" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.1.2.cmml">log</mi><mi id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.1.3" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.1.3.cmml">e</mi></msub><mo id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3a" lspace="0.167em" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.cmml">⁡</mo><mi id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.2" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.2.cmml">n</mi></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem10.p1.5.5.m5.1b"><apply id="S5.Thmtheorem10.p1.5.5.m5.1.2.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2"><minus id="S5.Thmtheorem10.p1.5.5.m5.1.2.1.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.1"></minus><cn id="S5.Thmtheorem10.p1.5.5.m5.1.2.2.cmml" type="integer" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.2">1</cn><apply id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.1.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3">superscript</csymbol><apply id="S5.Thmtheorem10.p1.5.5.m5.1.1.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.2.2"><divide id="S5.Thmtheorem10.p1.5.5.m5.1.1.1.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.2.2"></divide><apply id="S5.Thmtheorem10.p1.5.5.m5.1.1.2.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.2"><times id="S5.Thmtheorem10.p1.5.5.m5.1.1.2.1.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.2.1"></times><ci id="S5.Thmtheorem10.p1.5.5.m5.1.1.2.2.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.2.2">𝑐</ci><ci id="S5.Thmtheorem10.p1.5.5.m5.1.1.2.3.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.2.3">𝑒</ci></apply><apply id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3"><times id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.1.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.1"></times><cn id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.2.cmml" type="integer" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.2">2</cn><apply id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3"><apply id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.1.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.1"><csymbol cd="ambiguous" id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.1.1.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.1">subscript</csymbol><log id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.1.2.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.1.2"></log><ci id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.1.3.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.1.3">𝑒</ci></apply><ci id="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.2.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.1.3.3.2">𝑛</ci></apply></apply></apply><apply id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3"><times id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.1.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.1"></times><apply id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.2.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.2"><divide id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.2.1.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.2"></divide><cn id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.2.2.cmml" type="integer" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.2.2">2</cn><ci id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.2.3.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.2.3">𝑐</ci></apply><apply id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3"><apply id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.1.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.1"><csymbol cd="ambiguous" id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.1.1.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.1">subscript</csymbol><log id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.1.2.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.1.2"></log><ci id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.1.3.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.1.3">𝑒</ci></apply><ci id="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.2.cmml" xref="S5.Thmtheorem10.p1.5.5.m5.1.2.3.3.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem10.p1.5.5.m5.1c">1-\left(\frac{ce}{2\log_{e}n}\right)^{\frac{2}{c}\log_{e}n}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem10.p1.5.5.m5.1d">1 - ( divide start_ARG italic_c italic_e end_ARG start_ARG 2 roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 2 end_ARG start_ARG italic_c end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S5.SS2.SSS2.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.SS2.SSS2.2.p1"> <p class="ltx_p" id="S5.SS2.SSS2.2.p1.15">Recall that <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.2.p1.1.m1.1"><semantics id="S5.SS2.SSS2.2.p1.1.m1.1a"><msup id="S5.SS2.SSS2.2.p1.1.m1.1.1" xref="S5.SS2.SSS2.2.p1.1.m1.1.1.cmml"><mi id="S5.SS2.SSS2.2.p1.1.m1.1.1.2" xref="S5.SS2.SSS2.2.p1.1.m1.1.1.2.cmml">G</mi><mo id="S5.SS2.SSS2.2.p1.1.m1.1.1.3" xref="S5.SS2.SSS2.2.p1.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.2.p1.1.m1.1b"><apply id="S5.SS2.SSS2.2.p1.1.m1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.2.p1.1.m1.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.1.m1.1.1">superscript</csymbol><ci id="S5.SS2.SSS2.2.p1.1.m1.1.1.2.cmml" xref="S5.SS2.SSS2.2.p1.1.m1.1.1.2">𝐺</ci><ci id="S5.SS2.SSS2.2.p1.1.m1.1.1.3.cmml" xref="S5.SS2.SSS2.2.p1.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.2.p1.1.m1.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.2.p1.1.m1.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is the graph obtained from <math alttext="G" class="ltx_Math" display="inline" id="S5.SS2.SSS2.2.p1.2.m2.1"><semantics id="S5.SS2.SSS2.2.p1.2.m2.1a"><mi id="S5.SS2.SSS2.2.p1.2.m2.1.1" xref="S5.SS2.SSS2.2.p1.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.2.p1.2.m2.1b"><ci id="S5.SS2.SSS2.2.p1.2.m2.1.1.cmml" xref="S5.SS2.SSS2.2.p1.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.2.p1.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.2.p1.2.m2.1d">italic_G</annotation></semantics></math> by removing all edges with weight <math alttext="-n" class="ltx_Math" display="inline" id="S5.SS2.SSS2.2.p1.3.m3.1"><semantics id="S5.SS2.SSS2.2.p1.3.m3.1a"><mrow id="S5.SS2.SSS2.2.p1.3.m3.1.1" xref="S5.SS2.SSS2.2.p1.3.m3.1.1.cmml"><mo id="S5.SS2.SSS2.2.p1.3.m3.1.1a" xref="S5.SS2.SSS2.2.p1.3.m3.1.1.cmml">−</mo><mi id="S5.SS2.SSS2.2.p1.3.m3.1.1.2" xref="S5.SS2.SSS2.2.p1.3.m3.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.2.p1.3.m3.1b"><apply id="S5.SS2.SSS2.2.p1.3.m3.1.1.cmml" xref="S5.SS2.SSS2.2.p1.3.m3.1.1"><minus id="S5.SS2.SSS2.2.p1.3.m3.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.3.m3.1.1"></minus><ci id="S5.SS2.SSS2.2.p1.3.m3.1.1.2.cmml" xref="S5.SS2.SSS2.2.p1.3.m3.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.2.p1.3.m3.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.2.p1.3.m3.1d">- italic_n</annotation></semantics></math>. Define <math alttext="t=\frac{2}{c}\log_{e}n+1" class="ltx_Math" display="inline" id="S5.SS2.SSS2.2.p1.4.m4.1"><semantics id="S5.SS2.SSS2.2.p1.4.m4.1a"><mrow id="S5.SS2.SSS2.2.p1.4.m4.1.1" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.cmml"><mi id="S5.SS2.SSS2.2.p1.4.m4.1.1.2" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.2.cmml">t</mi><mo id="S5.SS2.SSS2.2.p1.4.m4.1.1.1" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.1.cmml">=</mo><mrow id="S5.SS2.SSS2.2.p1.4.m4.1.1.3" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.cmml"><mrow id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.cmml"><mfrac id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.2" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.2.cmml"><mn id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.2.2" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.2.2.cmml">2</mn><mi id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.2.3" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.2.3.cmml">c</mi></mfrac><mo id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.1" lspace="0.167em" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.cmml"><msub id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.1" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.1.cmml"><mi id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.1.2" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.1.2.cmml">log</mi><mi id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.1.3" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.1.3.cmml">e</mi></msub><mo id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3a" lspace="0.167em" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.cmml">⁡</mo><mi id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.2" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.2.cmml">n</mi></mrow></mrow><mo id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.1" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.1.cmml">+</mo><mn id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.3" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.2.p1.4.m4.1b"><apply id="S5.SS2.SSS2.2.p1.4.m4.1.1.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1"><eq id="S5.SS2.SSS2.2.p1.4.m4.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.1"></eq><ci id="S5.SS2.SSS2.2.p1.4.m4.1.1.2.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.2">𝑡</ci><apply id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3"><plus id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.1.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.1"></plus><apply id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2"><times id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.1.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.1"></times><apply id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.2.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.2"><divide id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.2.1.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.2"></divide><cn id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.2.2.cmml" type="integer" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.2.2">2</cn><ci id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.2.3.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.2.3">𝑐</ci></apply><apply id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3"><apply id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.1.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.1.1.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.1">subscript</csymbol><log id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.1.2.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.1.2"></log><ci id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.1.3.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.1.3">𝑒</ci></apply><ci id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.2.cmml" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.2.3.2">𝑛</ci></apply></apply><cn id="S5.SS2.SSS2.2.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S5.SS2.SSS2.2.p1.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.2.p1.4.m4.1c">t=\frac{2}{c}\log_{e}n+1</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.2.p1.4.m4.1d">italic_t = divide start_ARG 2 end_ARG start_ARG italic_c end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n + 1</annotation></semantics></math>, and let <math alttext="S^{t}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.2.p1.5.m5.1"><semantics id="S5.SS2.SSS2.2.p1.5.m5.1a"><msup id="S5.SS2.SSS2.2.p1.5.m5.1.1" xref="S5.SS2.SSS2.2.p1.5.m5.1.1.cmml"><mi id="S5.SS2.SSS2.2.p1.5.m5.1.1.2" xref="S5.SS2.SSS2.2.p1.5.m5.1.1.2.cmml">S</mi><mi id="S5.SS2.SSS2.2.p1.5.m5.1.1.3" xref="S5.SS2.SSS2.2.p1.5.m5.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.2.p1.5.m5.1b"><apply id="S5.SS2.SSS2.2.p1.5.m5.1.1.cmml" xref="S5.SS2.SSS2.2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.2.p1.5.m5.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.5.m5.1.1">superscript</csymbol><ci id="S5.SS2.SSS2.2.p1.5.m5.1.1.2.cmml" xref="S5.SS2.SSS2.2.p1.5.m5.1.1.2">𝑆</ci><ci id="S5.SS2.SSS2.2.p1.5.m5.1.1.3.cmml" xref="S5.SS2.SSS2.2.p1.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.2.p1.5.m5.1c">S^{t}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.2.p1.5.m5.1d">italic_S start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> represent the set of all groups of <math alttext="t" class="ltx_Math" display="inline" id="S5.SS2.SSS2.2.p1.6.m6.1"><semantics id="S5.SS2.SSS2.2.p1.6.m6.1a"><mi id="S5.SS2.SSS2.2.p1.6.m6.1.1" xref="S5.SS2.SSS2.2.p1.6.m6.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.2.p1.6.m6.1b"><ci id="S5.SS2.SSS2.2.p1.6.m6.1.1.cmml" xref="S5.SS2.SSS2.2.p1.6.m6.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.2.p1.6.m6.1c">t</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.2.p1.6.m6.1d">italic_t</annotation></semantics></math> agents, each from a different color class. It is easy to observe that <math alttext="|S^{t}|=\binom{k}{t}(\frac{n}{k})^{t}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.2.p1.7.m7.4"><semantics id="S5.SS2.SSS2.2.p1.7.m7.4a"><mrow id="S5.SS2.SSS2.2.p1.7.m7.4.4" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.cmml"><mrow id="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.1.2.cmml"><mo id="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.2" stretchy="false" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.1.2.1.cmml">|</mo><msup id="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.1" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.1.cmml"><mi id="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.1.2" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.1.2.cmml">S</mi><mi id="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.1.3" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.1.3.cmml">t</mi></msup><mo id="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.3" stretchy="false" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.1.2.1.cmml">|</mo></mrow><mo id="S5.SS2.SSS2.2.p1.7.m7.4.4.2" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.2.cmml">=</mo><mrow id="S5.SS2.SSS2.2.p1.7.m7.4.4.3" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.3.cmml"><mrow id="S5.SS2.SSS2.2.p1.7.m7.2.2.4" xref="S5.SS2.SSS2.2.p1.7.m7.2.2.3.cmml"><mo id="S5.SS2.SSS2.2.p1.7.m7.2.2.4.1" xref="S5.SS2.SSS2.2.p1.7.m7.2.2.3.1.cmml">(</mo><mfrac id="S5.SS2.SSS2.2.p1.7.m7.2.2.2.2" linethickness="0pt" xref="S5.SS2.SSS2.2.p1.7.m7.2.2.3.cmml"><mi id="S5.SS2.SSS2.2.p1.7.m7.1.1.1.1.1.1" xref="S5.SS2.SSS2.2.p1.7.m7.1.1.1.1.1.1.cmml">k</mi><mi id="S5.SS2.SSS2.2.p1.7.m7.2.2.2.2.2.1" xref="S5.SS2.SSS2.2.p1.7.m7.2.2.2.2.2.1.cmml">t</mi></mfrac><mo id="S5.SS2.SSS2.2.p1.7.m7.2.2.4.2" xref="S5.SS2.SSS2.2.p1.7.m7.2.2.3.1.cmml">)</mo></mrow><mo id="S5.SS2.SSS2.2.p1.7.m7.4.4.3.1" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.3.1.cmml">⁢</mo><msup id="S5.SS2.SSS2.2.p1.7.m7.4.4.3.2" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.3.2.cmml"><mrow id="S5.SS2.SSS2.2.p1.7.m7.4.4.3.2.2.2" xref="S5.SS2.SSS2.2.p1.7.m7.3.3.cmml"><mo id="S5.SS2.SSS2.2.p1.7.m7.4.4.3.2.2.2.1" stretchy="false" xref="S5.SS2.SSS2.2.p1.7.m7.3.3.cmml">(</mo><mfrac id="S5.SS2.SSS2.2.p1.7.m7.3.3" xref="S5.SS2.SSS2.2.p1.7.m7.3.3.cmml"><mi id="S5.SS2.SSS2.2.p1.7.m7.3.3.2" xref="S5.SS2.SSS2.2.p1.7.m7.3.3.2.cmml">n</mi><mi id="S5.SS2.SSS2.2.p1.7.m7.3.3.3" xref="S5.SS2.SSS2.2.p1.7.m7.3.3.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS2.2.p1.7.m7.4.4.3.2.2.2.2" stretchy="false" xref="S5.SS2.SSS2.2.p1.7.m7.3.3.cmml">)</mo></mrow><mi id="S5.SS2.SSS2.2.p1.7.m7.4.4.3.2.3" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.3.2.3.cmml">t</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.2.p1.7.m7.4b"><apply id="S5.SS2.SSS2.2.p1.7.m7.4.4.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.4.4"><eq id="S5.SS2.SSS2.2.p1.7.m7.4.4.2.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.2"></eq><apply id="S5.SS2.SSS2.2.p1.7.m7.4.4.1.2.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1"><abs id="S5.SS2.SSS2.2.p1.7.m7.4.4.1.2.1.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.2"></abs><apply id="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.1">superscript</csymbol><ci id="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.1.2.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.1.2">𝑆</ci><ci id="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.1.3.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.1.1.1.3">𝑡</ci></apply></apply><apply id="S5.SS2.SSS2.2.p1.7.m7.4.4.3.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.3"><times id="S5.SS2.SSS2.2.p1.7.m7.4.4.3.1.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.3.1"></times><apply id="S5.SS2.SSS2.2.p1.7.m7.2.2.3.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.2.2.4"><csymbol cd="latexml" id="S5.SS2.SSS2.2.p1.7.m7.2.2.3.1.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.2.2.4.1">binomial</csymbol><ci id="S5.SS2.SSS2.2.p1.7.m7.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.1.1.1.1.1.1">𝑘</ci><ci id="S5.SS2.SSS2.2.p1.7.m7.2.2.2.2.2.1.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.2.2.2.2.2.1">𝑡</ci></apply><apply id="S5.SS2.SSS2.2.p1.7.m7.4.4.3.2.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.3.2"><csymbol cd="ambiguous" id="S5.SS2.SSS2.2.p1.7.m7.4.4.3.2.1.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.3.2">superscript</csymbol><apply id="S5.SS2.SSS2.2.p1.7.m7.3.3.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.3.2.2.2"><divide id="S5.SS2.SSS2.2.p1.7.m7.3.3.1.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.3.2.2.2"></divide><ci id="S5.SS2.SSS2.2.p1.7.m7.3.3.2.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.3.3.2">𝑛</ci><ci id="S5.SS2.SSS2.2.p1.7.m7.3.3.3.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.3.3.3">𝑘</ci></apply><ci id="S5.SS2.SSS2.2.p1.7.m7.4.4.3.2.3.cmml" xref="S5.SS2.SSS2.2.p1.7.m7.4.4.3.2.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.2.p1.7.m7.4c">|S^{t}|=\binom{k}{t}(\frac{n}{k})^{t}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.2.p1.7.m7.4d">| italic_S start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT | = ( FRACOP start_ARG italic_k end_ARG start_ARG italic_t end_ARG ) ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>. For any <math alttext="s\in S^{t}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.2.p1.8.m8.1"><semantics id="S5.SS2.SSS2.2.p1.8.m8.1a"><mrow id="S5.SS2.SSS2.2.p1.8.m8.1.1" xref="S5.SS2.SSS2.2.p1.8.m8.1.1.cmml"><mi id="S5.SS2.SSS2.2.p1.8.m8.1.1.2" xref="S5.SS2.SSS2.2.p1.8.m8.1.1.2.cmml">s</mi><mo id="S5.SS2.SSS2.2.p1.8.m8.1.1.1" xref="S5.SS2.SSS2.2.p1.8.m8.1.1.1.cmml">∈</mo><msup id="S5.SS2.SSS2.2.p1.8.m8.1.1.3" xref="S5.SS2.SSS2.2.p1.8.m8.1.1.3.cmml"><mi id="S5.SS2.SSS2.2.p1.8.m8.1.1.3.2" xref="S5.SS2.SSS2.2.p1.8.m8.1.1.3.2.cmml">S</mi><mi id="S5.SS2.SSS2.2.p1.8.m8.1.1.3.3" xref="S5.SS2.SSS2.2.p1.8.m8.1.1.3.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.2.p1.8.m8.1b"><apply id="S5.SS2.SSS2.2.p1.8.m8.1.1.cmml" xref="S5.SS2.SSS2.2.p1.8.m8.1.1"><in id="S5.SS2.SSS2.2.p1.8.m8.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.8.m8.1.1.1"></in><ci id="S5.SS2.SSS2.2.p1.8.m8.1.1.2.cmml" xref="S5.SS2.SSS2.2.p1.8.m8.1.1.2">𝑠</ci><apply id="S5.SS2.SSS2.2.p1.8.m8.1.1.3.cmml" xref="S5.SS2.SSS2.2.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.SSS2.2.p1.8.m8.1.1.3.1.cmml" xref="S5.SS2.SSS2.2.p1.8.m8.1.1.3">superscript</csymbol><ci id="S5.SS2.SSS2.2.p1.8.m8.1.1.3.2.cmml" xref="S5.SS2.SSS2.2.p1.8.m8.1.1.3.2">𝑆</ci><ci id="S5.SS2.SSS2.2.p1.8.m8.1.1.3.3.cmml" xref="S5.SS2.SSS2.2.p1.8.m8.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.2.p1.8.m8.1c">s\in S^{t}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.2.p1.8.m8.1d">italic_s ∈ italic_S start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>, let <math alttext="E_{s}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.2.p1.9.m9.1"><semantics id="S5.SS2.SSS2.2.p1.9.m9.1a"><msub id="S5.SS2.SSS2.2.p1.9.m9.1.1" xref="S5.SS2.SSS2.2.p1.9.m9.1.1.cmml"><mi id="S5.SS2.SSS2.2.p1.9.m9.1.1.2" xref="S5.SS2.SSS2.2.p1.9.m9.1.1.2.cmml">E</mi><mi id="S5.SS2.SSS2.2.p1.9.m9.1.1.3" xref="S5.SS2.SSS2.2.p1.9.m9.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.2.p1.9.m9.1b"><apply id="S5.SS2.SSS2.2.p1.9.m9.1.1.cmml" xref="S5.SS2.SSS2.2.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.2.p1.9.m9.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.9.m9.1.1">subscript</csymbol><ci id="S5.SS2.SSS2.2.p1.9.m9.1.1.2.cmml" xref="S5.SS2.SSS2.2.p1.9.m9.1.1.2">𝐸</ci><ci id="S5.SS2.SSS2.2.p1.9.m9.1.1.3.cmml" xref="S5.SS2.SSS2.2.p1.9.m9.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.2.p1.9.m9.1c">E_{s}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.2.p1.9.m9.1d">italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> denote the event that all vertices in <math alttext="s" class="ltx_Math" display="inline" id="S5.SS2.SSS2.2.p1.10.m10.1"><semantics id="S5.SS2.SSS2.2.p1.10.m10.1a"><mi id="S5.SS2.SSS2.2.p1.10.m10.1.1" xref="S5.SS2.SSS2.2.p1.10.m10.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.2.p1.10.m10.1b"><ci id="S5.SS2.SSS2.2.p1.10.m10.1.1.cmml" xref="S5.SS2.SSS2.2.p1.10.m10.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.2.p1.10.m10.1c">s</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.2.p1.10.m10.1d">italic_s</annotation></semantics></math> form a clique in <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.2.p1.11.m11.1"><semantics id="S5.SS2.SSS2.2.p1.11.m11.1a"><msup id="S5.SS2.SSS2.2.p1.11.m11.1.1" xref="S5.SS2.SSS2.2.p1.11.m11.1.1.cmml"><mi id="S5.SS2.SSS2.2.p1.11.m11.1.1.2" xref="S5.SS2.SSS2.2.p1.11.m11.1.1.2.cmml">G</mi><mo id="S5.SS2.SSS2.2.p1.11.m11.1.1.3" xref="S5.SS2.SSS2.2.p1.11.m11.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.2.p1.11.m11.1b"><apply id="S5.SS2.SSS2.2.p1.11.m11.1.1.cmml" xref="S5.SS2.SSS2.2.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.2.p1.11.m11.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.11.m11.1.1">superscript</csymbol><ci id="S5.SS2.SSS2.2.p1.11.m11.1.1.2.cmml" xref="S5.SS2.SSS2.2.p1.11.m11.1.1.2">𝐺</ci><ci id="S5.SS2.SSS2.2.p1.11.m11.1.1.3.cmml" xref="S5.SS2.SSS2.2.p1.11.m11.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.2.p1.11.m11.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.2.p1.11.m11.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, so the probability of this event is <math alttext="P(E_{s})=(1-p)^{\binom{t}{2}}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.2.p1.12.m12.4"><semantics id="S5.SS2.SSS2.2.p1.12.m12.4a"><mrow id="S5.SS2.SSS2.2.p1.12.m12.4.4" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.cmml"><mrow id="S5.SS2.SSS2.2.p1.12.m12.3.3.1" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1.cmml"><mi id="S5.SS2.SSS2.2.p1.12.m12.3.3.1.3" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1.3.cmml">P</mi><mo id="S5.SS2.SSS2.2.p1.12.m12.3.3.1.2" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1.2.cmml">⁢</mo><mrow id="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.1.cmml"><mo id="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.2" stretchy="false" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.1.cmml">(</mo><msub id="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.1" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.1.2" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.1.2.cmml">E</mi><mi id="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.1.3" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.1.3.cmml">s</mi></msub><mo id="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.3" stretchy="false" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS2.SSS2.2.p1.12.m12.4.4.3" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.3.cmml">=</mo><msup id="S5.SS2.SSS2.2.p1.12.m12.4.4.2" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.2.cmml"><mrow id="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.cmml"><mo id="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.2" stretchy="false" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.cmml">(</mo><mrow id="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.cmml"><mn id="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.2" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.2.cmml">1</mn><mo id="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.1" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.1.cmml">−</mo><mi id="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.3" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.3.cmml">p</mi></mrow><mo id="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.3" stretchy="false" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.cmml">)</mo></mrow><mrow id="S5.SS2.SSS2.2.p1.12.m12.2.2.2.4" xref="S5.SS2.SSS2.2.p1.12.m12.2.2.2.3.cmml"><mo id="S5.SS2.SSS2.2.p1.12.m12.2.2.2.4.1" xref="S5.SS2.SSS2.2.p1.12.m12.2.2.2.3.1.cmml">(</mo><mfrac id="S5.SS2.SSS2.2.p1.12.m12.2.2.2.2.2.2" linethickness="0pt" xref="S5.SS2.SSS2.2.p1.12.m12.2.2.2.3.cmml"><mi id="S5.SS2.SSS2.2.p1.12.m12.1.1.1.1.1.1.1.1" xref="S5.SS2.SSS2.2.p1.12.m12.1.1.1.1.1.1.1.1.cmml">t</mi><mn id="S5.SS2.SSS2.2.p1.12.m12.2.2.2.2.2.2.2.1" xref="S5.SS2.SSS2.2.p1.12.m12.2.2.2.2.2.2.2.1.cmml">2</mn></mfrac><mo id="S5.SS2.SSS2.2.p1.12.m12.2.2.2.4.2" xref="S5.SS2.SSS2.2.p1.12.m12.2.2.2.3.1.cmml">)</mo></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.2.p1.12.m12.4b"><apply id="S5.SS2.SSS2.2.p1.12.m12.4.4.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.4.4"><eq id="S5.SS2.SSS2.2.p1.12.m12.4.4.3.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.3"></eq><apply id="S5.SS2.SSS2.2.p1.12.m12.3.3.1.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1"><times id="S5.SS2.SSS2.2.p1.12.m12.3.3.1.2.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1.2"></times><ci id="S5.SS2.SSS2.2.p1.12.m12.3.3.1.3.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1.3">𝑃</ci><apply id="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1">subscript</csymbol><ci id="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.1.2">𝐸</ci><ci id="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.1.3.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.3.3.1.1.1.1.3">𝑠</ci></apply></apply><apply id="S5.SS2.SSS2.2.p1.12.m12.4.4.2.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.2"><csymbol cd="ambiguous" id="S5.SS2.SSS2.2.p1.12.m12.4.4.2.2.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.2">superscript</csymbol><apply id="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1"><minus id="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.1"></minus><cn id="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.2.cmml" type="integer" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.2">1</cn><ci id="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.3.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.4.4.2.1.1.1.3">𝑝</ci></apply><apply id="S5.SS2.SSS2.2.p1.12.m12.2.2.2.3.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.2.2.2.4"><csymbol cd="latexml" id="S5.SS2.SSS2.2.p1.12.m12.2.2.2.3.1.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.2.2.2.4.1">binomial</csymbol><ci id="S5.SS2.SSS2.2.p1.12.m12.1.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.12.m12.1.1.1.1.1.1.1.1">𝑡</ci><cn id="S5.SS2.SSS2.2.p1.12.m12.2.2.2.2.2.2.2.1.cmml" type="integer" xref="S5.SS2.SSS2.2.p1.12.m12.2.2.2.2.2.2.2.1">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.2.p1.12.m12.4c">P(E_{s})=(1-p)^{\binom{t}{2}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.2.p1.12.m12.4d">italic_P ( italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) = ( 1 - italic_p ) start_POSTSUPERSCRIPT ( FRACOP start_ARG italic_t end_ARG start_ARG 2 end_ARG ) end_POSTSUPERSCRIPT</annotation></semantics></math>. Let <math alttext="X^{t}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.2.p1.13.m13.1"><semantics id="S5.SS2.SSS2.2.p1.13.m13.1a"><msup id="S5.SS2.SSS2.2.p1.13.m13.1.1" xref="S5.SS2.SSS2.2.p1.13.m13.1.1.cmml"><mi id="S5.SS2.SSS2.2.p1.13.m13.1.1.2" xref="S5.SS2.SSS2.2.p1.13.m13.1.1.2.cmml">X</mi><mi id="S5.SS2.SSS2.2.p1.13.m13.1.1.3" xref="S5.SS2.SSS2.2.p1.13.m13.1.1.3.cmml">t</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.2.p1.13.m13.1b"><apply id="S5.SS2.SSS2.2.p1.13.m13.1.1.cmml" xref="S5.SS2.SSS2.2.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.2.p1.13.m13.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.13.m13.1.1">superscript</csymbol><ci id="S5.SS2.SSS2.2.p1.13.m13.1.1.2.cmml" xref="S5.SS2.SSS2.2.p1.13.m13.1.1.2">𝑋</ci><ci id="S5.SS2.SSS2.2.p1.13.m13.1.1.3.cmml" xref="S5.SS2.SSS2.2.p1.13.m13.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.2.p1.13.m13.1c">X^{t}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.2.p1.13.m13.1d">italic_X start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math> denote the number of cliques of size <math alttext="t" class="ltx_Math" display="inline" id="S5.SS2.SSS2.2.p1.14.m14.1"><semantics id="S5.SS2.SSS2.2.p1.14.m14.1a"><mi id="S5.SS2.SSS2.2.p1.14.m14.1.1" xref="S5.SS2.SSS2.2.p1.14.m14.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.2.p1.14.m14.1b"><ci id="S5.SS2.SSS2.2.p1.14.m14.1.1.cmml" xref="S5.SS2.SSS2.2.p1.14.m14.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.2.p1.14.m14.1c">t</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.2.p1.14.m14.1d">italic_t</annotation></semantics></math> in <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.2.p1.15.m15.1"><semantics id="S5.SS2.SSS2.2.p1.15.m15.1a"><msup id="S5.SS2.SSS2.2.p1.15.m15.1.1" xref="S5.SS2.SSS2.2.p1.15.m15.1.1.cmml"><mi id="S5.SS2.SSS2.2.p1.15.m15.1.1.2" xref="S5.SS2.SSS2.2.p1.15.m15.1.1.2.cmml">G</mi><mo id="S5.SS2.SSS2.2.p1.15.m15.1.1.3" xref="S5.SS2.SSS2.2.p1.15.m15.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.2.p1.15.m15.1b"><apply id="S5.SS2.SSS2.2.p1.15.m15.1.1.cmml" xref="S5.SS2.SSS2.2.p1.15.m15.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.2.p1.15.m15.1.1.1.cmml" xref="S5.SS2.SSS2.2.p1.15.m15.1.1">superscript</csymbol><ci id="S5.SS2.SSS2.2.p1.15.m15.1.1.2.cmml" xref="S5.SS2.SSS2.2.p1.15.m15.1.1.2">𝐺</ci><ci id="S5.SS2.SSS2.2.p1.15.m15.1.1.3.cmml" xref="S5.SS2.SSS2.2.p1.15.m15.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.2.p1.15.m15.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.2.p1.15.m15.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS2.SSS2.3.p2"> <table class="ltx_equationgroup ltx_eqn_table" id="S5.E4"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S5.E4X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathbb{E}[X^{t}]" class="ltx_Math" display="inline" id="S5.E4X.2.1.1.m1.1"><semantics id="S5.E4X.2.1.1.m1.1a"><mrow id="S5.E4X.2.1.1.m1.1.1" xref="S5.E4X.2.1.1.m1.1.1.cmml"><mi id="S5.E4X.2.1.1.m1.1.1.3" xref="S5.E4X.2.1.1.m1.1.1.3.cmml">𝔼</mi><mo id="S5.E4X.2.1.1.m1.1.1.2" xref="S5.E4X.2.1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S5.E4X.2.1.1.m1.1.1.1.1" xref="S5.E4X.2.1.1.m1.1.1.1.2.cmml"><mo id="S5.E4X.2.1.1.m1.1.1.1.1.2" stretchy="false" xref="S5.E4X.2.1.1.m1.1.1.1.2.1.cmml">[</mo><msup id="S5.E4X.2.1.1.m1.1.1.1.1.1" xref="S5.E4X.2.1.1.m1.1.1.1.1.1.cmml"><mi id="S5.E4X.2.1.1.m1.1.1.1.1.1.2" xref="S5.E4X.2.1.1.m1.1.1.1.1.1.2.cmml">X</mi><mi id="S5.E4X.2.1.1.m1.1.1.1.1.1.3" xref="S5.E4X.2.1.1.m1.1.1.1.1.1.3.cmml">t</mi></msup><mo id="S5.E4X.2.1.1.m1.1.1.1.1.3" stretchy="false" xref="S5.E4X.2.1.1.m1.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.E4X.2.1.1.m1.1b"><apply id="S5.E4X.2.1.1.m1.1.1.cmml" xref="S5.E4X.2.1.1.m1.1.1"><times id="S5.E4X.2.1.1.m1.1.1.2.cmml" xref="S5.E4X.2.1.1.m1.1.1.2"></times><ci id="S5.E4X.2.1.1.m1.1.1.3.cmml" xref="S5.E4X.2.1.1.m1.1.1.3">𝔼</ci><apply id="S5.E4X.2.1.1.m1.1.1.1.2.cmml" xref="S5.E4X.2.1.1.m1.1.1.1.1"><csymbol cd="latexml" id="S5.E4X.2.1.1.m1.1.1.1.2.1.cmml" xref="S5.E4X.2.1.1.m1.1.1.1.1.2">delimited-[]</csymbol><apply id="S5.E4X.2.1.1.m1.1.1.1.1.1.cmml" xref="S5.E4X.2.1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.E4X.2.1.1.m1.1.1.1.1.1.1.cmml" xref="S5.E4X.2.1.1.m1.1.1.1.1.1">superscript</csymbol><ci id="S5.E4X.2.1.1.m1.1.1.1.1.1.2.cmml" xref="S5.E4X.2.1.1.m1.1.1.1.1.1.2">𝑋</ci><ci id="S5.E4X.2.1.1.m1.1.1.1.1.1.3.cmml" xref="S5.E4X.2.1.1.m1.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E4X.2.1.1.m1.1c">\displaystyle\mathbb{E}[X^{t}]</annotation><annotation encoding="application/x-llamapun" id="S5.E4X.2.1.1.m1.1d">blackboard_E [ italic_X start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ]</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{s\in S^{t}}P(E_{s})=\binom{k}{t}(\frac{n}{k})^{t}(1-p)^{% \binom{t}{2}}" class="ltx_Math" display="inline" id="S5.E4X.3.2.2.m1.3"><semantics id="S5.E4X.3.2.2.m1.3a"><mrow id="S5.E4X.3.2.2.m1.3.3" xref="S5.E4X.3.2.2.m1.3.3.cmml"><mi id="S5.E4X.3.2.2.m1.3.3.4" xref="S5.E4X.3.2.2.m1.3.3.4.cmml"></mi><mo id="S5.E4X.3.2.2.m1.3.3.5" xref="S5.E4X.3.2.2.m1.3.3.5.cmml">=</mo><mrow id="S5.E4X.3.2.2.m1.2.2.1" xref="S5.E4X.3.2.2.m1.2.2.1.cmml"><mstyle displaystyle="true" id="S5.E4X.3.2.2.m1.2.2.1.2" xref="S5.E4X.3.2.2.m1.2.2.1.2.cmml"><munder id="S5.E4X.3.2.2.m1.2.2.1.2a" xref="S5.E4X.3.2.2.m1.2.2.1.2.cmml"><mo id="S5.E4X.3.2.2.m1.2.2.1.2.2" movablelimits="false" xref="S5.E4X.3.2.2.m1.2.2.1.2.2.cmml">∑</mo><mrow id="S5.E4X.3.2.2.m1.2.2.1.2.3" xref="S5.E4X.3.2.2.m1.2.2.1.2.3.cmml"><mi id="S5.E4X.3.2.2.m1.2.2.1.2.3.2" xref="S5.E4X.3.2.2.m1.2.2.1.2.3.2.cmml">s</mi><mo id="S5.E4X.3.2.2.m1.2.2.1.2.3.1" xref="S5.E4X.3.2.2.m1.2.2.1.2.3.1.cmml">∈</mo><msup id="S5.E4X.3.2.2.m1.2.2.1.2.3.3" xref="S5.E4X.3.2.2.m1.2.2.1.2.3.3.cmml"><mi id="S5.E4X.3.2.2.m1.2.2.1.2.3.3.2" xref="S5.E4X.3.2.2.m1.2.2.1.2.3.3.2.cmml">S</mi><mi id="S5.E4X.3.2.2.m1.2.2.1.2.3.3.3" xref="S5.E4X.3.2.2.m1.2.2.1.2.3.3.3.cmml">t</mi></msup></mrow></munder></mstyle><mrow id="S5.E4X.3.2.2.m1.2.2.1.1" xref="S5.E4X.3.2.2.m1.2.2.1.1.cmml"><mi id="S5.E4X.3.2.2.m1.2.2.1.1.3" xref="S5.E4X.3.2.2.m1.2.2.1.1.3.cmml">P</mi><mo id="S5.E4X.3.2.2.m1.2.2.1.1.2" xref="S5.E4X.3.2.2.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="S5.E4X.3.2.2.m1.2.2.1.1.1.1" xref="S5.E4X.3.2.2.m1.2.2.1.1.1.1.1.cmml"><mo id="S5.E4X.3.2.2.m1.2.2.1.1.1.1.2" stretchy="false" xref="S5.E4X.3.2.2.m1.2.2.1.1.1.1.1.cmml">(</mo><msub id="S5.E4X.3.2.2.m1.2.2.1.1.1.1.1" xref="S5.E4X.3.2.2.m1.2.2.1.1.1.1.1.cmml"><mi id="S5.E4X.3.2.2.m1.2.2.1.1.1.1.1.2" xref="S5.E4X.3.2.2.m1.2.2.1.1.1.1.1.2.cmml">E</mi><mi id="S5.E4X.3.2.2.m1.2.2.1.1.1.1.1.3" xref="S5.E4X.3.2.2.m1.2.2.1.1.1.1.1.3.cmml">s</mi></msub><mo id="S5.E4X.3.2.2.m1.2.2.1.1.1.1.3" stretchy="false" xref="S5.E4X.3.2.2.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.E4X.3.2.2.m1.3.3.6" xref="S5.E4X.3.2.2.m1.3.3.6.cmml">=</mo><mrow id="S5.E4X.3.2.2.m1.3.3.2" xref="S5.E4X.3.2.2.m1.3.3.2.cmml"><mrow id="S5.E4.m1.2.2.2.2.2.2a.4" xref="S5.E4.m1.2.2.2.2.2.2a.3.cmml"><mo id="S5.E4.m1.2.2.2.2.2.2a.4.1" xref="S5.E4.m1.2.2.2.2.2.2a.3.1.cmml">(</mo><mstyle displaystyle="true" id="S5.E4.m1.2.2.2.2.2.2.2.2a" xref="S5.E4.m1.2.2.2.2.2.2a.3.cmml"><mfrac id="S5.E4.m1.2.2.2.2.2.2.2.2aa" linethickness="0pt" xref="S5.E4.m1.2.2.2.2.2.2a.3.cmml"><mi id="S5.E4.m1.1.1.1.1.1.1.1.1.1.1.mf" xref="S5.E4.m1.1.1.1.1.1.1.1.1.1.1.mf.cmml">k</mi><mi id="S5.E4.m1.2.2.2.2.2.2.2.2.2.1.mf" xref="S5.E4.m1.2.2.2.2.2.2.2.2.2.1.mf.cmml">t</mi></mfrac></mstyle><mo id="S5.E4.m1.2.2.2.2.2.2a.4.2" xref="S5.E4.m1.2.2.2.2.2.2a.3.1.cmml">)</mo></mrow><mo id="S5.E4X.3.2.2.m1.3.3.2.2" xref="S5.E4X.3.2.2.m1.3.3.2.2.cmml">⁢</mo><msup id="S5.E4X.3.2.2.m1.3.3.2.3" xref="S5.E4X.3.2.2.m1.3.3.2.3.cmml"><mrow id="S5.E4X.3.2.2.m1.3.3.2.3.2.2" xref="S5.E4X.3.2.2.m1.1.1.cmml"><mo id="S5.E4X.3.2.2.m1.3.3.2.3.2.2.1" stretchy="false" xref="S5.E4X.3.2.2.m1.1.1.cmml">(</mo><mstyle displaystyle="true" id="S5.E4X.3.2.2.m1.1.1" xref="S5.E4X.3.2.2.m1.1.1.cmml"><mfrac id="S5.E4X.3.2.2.m1.1.1a" xref="S5.E4X.3.2.2.m1.1.1.cmml"><mi id="S5.E4X.3.2.2.m1.1.1.2" xref="S5.E4X.3.2.2.m1.1.1.2.cmml">n</mi><mi id="S5.E4X.3.2.2.m1.1.1.3" xref="S5.E4X.3.2.2.m1.1.1.3.cmml">k</mi></mfrac></mstyle><mo id="S5.E4X.3.2.2.m1.3.3.2.3.2.2.2" stretchy="false" xref="S5.E4X.3.2.2.m1.1.1.cmml">)</mo></mrow><mi id="S5.E4X.3.2.2.m1.3.3.2.3.3" xref="S5.E4X.3.2.2.m1.3.3.2.3.3.cmml">t</mi></msup><mo id="S5.E4X.3.2.2.m1.3.3.2.2a" xref="S5.E4X.3.2.2.m1.3.3.2.2.cmml">⁢</mo><msup id="S5.E4X.3.2.2.m1.3.3.2.1" xref="S5.E4X.3.2.2.m1.3.3.2.1.cmml"><mrow id="S5.E4X.3.2.2.m1.3.3.2.1.1.1" xref="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.cmml"><mo id="S5.E4X.3.2.2.m1.3.3.2.1.1.1.2" stretchy="false" xref="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.cmml">(</mo><mrow id="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1" xref="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.cmml"><mn id="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.2" xref="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.2.cmml">1</mn><mo id="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.1" xref="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.1.cmml">−</mo><mi id="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.3" xref="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.3.cmml">p</mi></mrow><mo id="S5.E4X.3.2.2.m1.3.3.2.1.1.1.3" stretchy="false" xref="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.cmml">)</mo></mrow><mrow id="S5.E4.m1.4.4.4.4.4.4.2a.4" xref="S5.E4.m1.4.4.4.4.4.4.2a.3.cmml"><mo id="S5.E4.m1.4.4.4.4.4.4.2a.4.1" xref="S5.E4.m1.4.4.4.4.4.4.2a.3.1.cmml">(</mo><mfrac id="S5.E4.m1.4.4.4.4.4.4.2.2.2.2a" linethickness="0pt" xref="S5.E4.m1.4.4.4.4.4.4.2a.3.cmml"><mi id="S5.E4.m1.3.3.3.3.3.3.1.1.1.1.1.1.mf" xref="S5.E4.m1.3.3.3.3.3.3.1.1.1.1.1.1.mf.cmml">t</mi><mn id="S5.E4.m1.4.4.4.4.4.4.2.2.2.2.2.1.mf" xref="S5.E4.m1.4.4.4.4.4.4.2.2.2.2.2.1.mf.cmml">2</mn></mfrac><mo id="S5.E4.m1.4.4.4.4.4.4.2a.4.2" xref="S5.E4.m1.4.4.4.4.4.4.2a.3.1.cmml">)</mo></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.E4X.3.2.2.m1.3b"><apply id="S5.E4X.3.2.2.m1.3.3.cmml" xref="S5.E4X.3.2.2.m1.3.3"><and id="S5.E4X.3.2.2.m1.3.3a.cmml" xref="S5.E4X.3.2.2.m1.3.3"></and><apply id="S5.E4X.3.2.2.m1.3.3b.cmml" xref="S5.E4X.3.2.2.m1.3.3"><eq id="S5.E4X.3.2.2.m1.3.3.5.cmml" xref="S5.E4X.3.2.2.m1.3.3.5"></eq><csymbol cd="latexml" id="S5.E4X.3.2.2.m1.3.3.4.cmml" xref="S5.E4X.3.2.2.m1.3.3.4">absent</csymbol><apply id="S5.E4X.3.2.2.m1.2.2.1.cmml" xref="S5.E4X.3.2.2.m1.2.2.1"><apply id="S5.E4X.3.2.2.m1.2.2.1.2.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.2"><csymbol cd="ambiguous" id="S5.E4X.3.2.2.m1.2.2.1.2.1.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.2">subscript</csymbol><sum id="S5.E4X.3.2.2.m1.2.2.1.2.2.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.2.2"></sum><apply id="S5.E4X.3.2.2.m1.2.2.1.2.3.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.2.3"><in id="S5.E4X.3.2.2.m1.2.2.1.2.3.1.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.2.3.1"></in><ci id="S5.E4X.3.2.2.m1.2.2.1.2.3.2.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.2.3.2">𝑠</ci><apply id="S5.E4X.3.2.2.m1.2.2.1.2.3.3.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.2.3.3"><csymbol cd="ambiguous" id="S5.E4X.3.2.2.m1.2.2.1.2.3.3.1.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.2.3.3">superscript</csymbol><ci id="S5.E4X.3.2.2.m1.2.2.1.2.3.3.2.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.2.3.3.2">𝑆</ci><ci id="S5.E4X.3.2.2.m1.2.2.1.2.3.3.3.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S5.E4X.3.2.2.m1.2.2.1.1.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.1"><times id="S5.E4X.3.2.2.m1.2.2.1.1.2.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.1.2"></times><ci id="S5.E4X.3.2.2.m1.2.2.1.1.3.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.1.3">𝑃</ci><apply id="S5.E4X.3.2.2.m1.2.2.1.1.1.1.1.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.E4X.3.2.2.m1.2.2.1.1.1.1.1.1.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.1.1.1">subscript</csymbol><ci id="S5.E4X.3.2.2.m1.2.2.1.1.1.1.1.2.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.1.1.1.1.2">𝐸</ci><ci id="S5.E4X.3.2.2.m1.2.2.1.1.1.1.1.3.cmml" xref="S5.E4X.3.2.2.m1.2.2.1.1.1.1.1.3">𝑠</ci></apply></apply></apply></apply><apply id="S5.E4X.3.2.2.m1.3.3c.cmml" xref="S5.E4X.3.2.2.m1.3.3"><eq id="S5.E4X.3.2.2.m1.3.3.6.cmml" xref="S5.E4X.3.2.2.m1.3.3.6"></eq><share href="https://arxiv.org/html/2503.06017v1#S5.E4X.3.2.2.m1.2.2.1.cmml" id="S5.E4X.3.2.2.m1.3.3d.cmml" xref="S5.E4X.3.2.2.m1.3.3"></share><apply id="S5.E4X.3.2.2.m1.3.3.2.cmml" xref="S5.E4X.3.2.2.m1.3.3.2"><times id="S5.E4X.3.2.2.m1.3.3.2.2.cmml" xref="S5.E4X.3.2.2.m1.3.3.2.2"></times><apply id="S5.E4.m1.2.2.2.2.2.2a.3.cmml" xref="S5.E4.m1.2.2.2.2.2.2a.4"><csymbol cd="latexml" id="S5.E4.m1.2.2.2.2.2.2a.3.1.cmml" xref="S5.E4.m1.2.2.2.2.2.2a.4.1">binomial</csymbol><ci id="S5.E4.m1.1.1.1.1.1.1.1.1.1.1.mf.cmml" xref="S5.E4.m1.1.1.1.1.1.1.1.1.1.1.mf">𝑘</ci><ci id="S5.E4.m1.2.2.2.2.2.2.2.2.2.1.mf.cmml" xref="S5.E4.m1.2.2.2.2.2.2.2.2.2.1.mf">𝑡</ci></apply><apply id="S5.E4X.3.2.2.m1.3.3.2.3.cmml" xref="S5.E4X.3.2.2.m1.3.3.2.3"><csymbol cd="ambiguous" id="S5.E4X.3.2.2.m1.3.3.2.3.1.cmml" xref="S5.E4X.3.2.2.m1.3.3.2.3">superscript</csymbol><apply id="S5.E4X.3.2.2.m1.1.1.cmml" xref="S5.E4X.3.2.2.m1.3.3.2.3.2.2"><divide id="S5.E4X.3.2.2.m1.1.1.1.cmml" xref="S5.E4X.3.2.2.m1.3.3.2.3.2.2"></divide><ci id="S5.E4X.3.2.2.m1.1.1.2.cmml" xref="S5.E4X.3.2.2.m1.1.1.2">𝑛</ci><ci id="S5.E4X.3.2.2.m1.1.1.3.cmml" xref="S5.E4X.3.2.2.m1.1.1.3">𝑘</ci></apply><ci id="S5.E4X.3.2.2.m1.3.3.2.3.3.cmml" xref="S5.E4X.3.2.2.m1.3.3.2.3.3">𝑡</ci></apply><apply id="S5.E4X.3.2.2.m1.3.3.2.1.cmml" xref="S5.E4X.3.2.2.m1.3.3.2.1"><csymbol cd="ambiguous" id="S5.E4X.3.2.2.m1.3.3.2.1.2.cmml" xref="S5.E4X.3.2.2.m1.3.3.2.1">superscript</csymbol><apply id="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.cmml" xref="S5.E4X.3.2.2.m1.3.3.2.1.1.1"><minus id="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.1.cmml" xref="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.1"></minus><cn id="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.2.cmml" type="integer" xref="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.2">1</cn><ci id="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.3.cmml" xref="S5.E4X.3.2.2.m1.3.3.2.1.1.1.1.3">𝑝</ci></apply><apply id="S5.E4.m1.4.4.4.4.4.4.2a.3.cmml" xref="S5.E4.m1.4.4.4.4.4.4.2a.4"><csymbol cd="latexml" id="S5.E4.m1.4.4.4.4.4.4.2a.3.1.cmml" xref="S5.E4.m1.4.4.4.4.4.4.2a.4.1">binomial</csymbol><ci id="S5.E4.m1.3.3.3.3.3.3.1.1.1.1.1.1.mf.cmml" xref="S5.E4.m1.3.3.3.3.3.3.1.1.1.1.1.1.mf">𝑡</ci><cn id="S5.E4.m1.4.4.4.4.4.4.2.2.2.2.2.1.mf.cmml" type="integer" xref="S5.E4.m1.4.4.4.4.4.4.2.2.2.2.2.1.mf">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E4X.3.2.2.m1.3c">\displaystyle=\sum_{s\in S^{t}}P(E_{s})=\binom{k}{t}(\frac{n}{k})^{t}(1-p)^{% \binom{t}{2}}</annotation><annotation encoding="application/x-llamapun" id="S5.E4X.3.2.2.m1.3d">= ∑ start_POSTSUBSCRIPT italic_s ∈ italic_S start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_P ( italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) = ( FRACOP start_ARG italic_k end_ARG start_ARG italic_t end_ARG ) ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( 1 - italic_p ) start_POSTSUPERSCRIPT ( FRACOP start_ARG italic_t end_ARG start_ARG 2 end_ARG ) end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="4"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(4)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S5.E4Xa"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq(\frac{ek}{t})^{t}(\frac{n}{k})^{t}(1-p)^{\binom{t}{2}}" class="ltx_Math" display="inline" id="S5.E4Xa.2.1.1.m1.3"><semantics id="S5.E4Xa.2.1.1.m1.3a"><mrow id="S5.E4Xa.2.1.1.m1.3.3" xref="S5.E4Xa.2.1.1.m1.3.3.cmml"><mi id="S5.E4Xa.2.1.1.m1.3.3.3" xref="S5.E4Xa.2.1.1.m1.3.3.3.cmml"></mi><mo id="S5.E4Xa.2.1.1.m1.3.3.2" xref="S5.E4Xa.2.1.1.m1.3.3.2.cmml">≤</mo><mrow id="S5.E4Xa.2.1.1.m1.3.3.1" xref="S5.E4Xa.2.1.1.m1.3.3.1.cmml"><msup id="S5.E4Xa.2.1.1.m1.3.3.1.3" xref="S5.E4Xa.2.1.1.m1.3.3.1.3.cmml"><mrow id="S5.E4Xa.2.1.1.m1.3.3.1.3.2.2" xref="S5.E4Xa.2.1.1.m1.1.1.cmml"><mo id="S5.E4Xa.2.1.1.m1.3.3.1.3.2.2.1" stretchy="false" xref="S5.E4Xa.2.1.1.m1.1.1.cmml">(</mo><mstyle displaystyle="true" id="S5.E4Xa.2.1.1.m1.1.1" xref="S5.E4Xa.2.1.1.m1.1.1.cmml"><mfrac id="S5.E4Xa.2.1.1.m1.1.1a" xref="S5.E4Xa.2.1.1.m1.1.1.cmml"><mrow id="S5.E4Xa.2.1.1.m1.1.1.2" xref="S5.E4Xa.2.1.1.m1.1.1.2.cmml"><mi id="S5.E4Xa.2.1.1.m1.1.1.2.2" xref="S5.E4Xa.2.1.1.m1.1.1.2.2.cmml">e</mi><mo id="S5.E4Xa.2.1.1.m1.1.1.2.1" xref="S5.E4Xa.2.1.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S5.E4Xa.2.1.1.m1.1.1.2.3" xref="S5.E4Xa.2.1.1.m1.1.1.2.3.cmml">k</mi></mrow><mi id="S5.E4Xa.2.1.1.m1.1.1.3" xref="S5.E4Xa.2.1.1.m1.1.1.3.cmml">t</mi></mfrac></mstyle><mo id="S5.E4Xa.2.1.1.m1.3.3.1.3.2.2.2" stretchy="false" xref="S5.E4Xa.2.1.1.m1.1.1.cmml">)</mo></mrow><mi id="S5.E4Xa.2.1.1.m1.3.3.1.3.3" xref="S5.E4Xa.2.1.1.m1.3.3.1.3.3.cmml">t</mi></msup><mo id="S5.E4Xa.2.1.1.m1.3.3.1.2" xref="S5.E4Xa.2.1.1.m1.3.3.1.2.cmml">⁢</mo><msup id="S5.E4Xa.2.1.1.m1.3.3.1.4" xref="S5.E4Xa.2.1.1.m1.3.3.1.4.cmml"><mrow id="S5.E4Xa.2.1.1.m1.3.3.1.4.2.2" xref="S5.E4Xa.2.1.1.m1.2.2.cmml"><mo id="S5.E4Xa.2.1.1.m1.3.3.1.4.2.2.1" stretchy="false" xref="S5.E4Xa.2.1.1.m1.2.2.cmml">(</mo><mstyle displaystyle="true" id="S5.E4Xa.2.1.1.m1.2.2" xref="S5.E4Xa.2.1.1.m1.2.2.cmml"><mfrac id="S5.E4Xa.2.1.1.m1.2.2a" xref="S5.E4Xa.2.1.1.m1.2.2.cmml"><mi id="S5.E4Xa.2.1.1.m1.2.2.2" xref="S5.E4Xa.2.1.1.m1.2.2.2.cmml">n</mi><mi id="S5.E4Xa.2.1.1.m1.2.2.3" xref="S5.E4Xa.2.1.1.m1.2.2.3.cmml">k</mi></mfrac></mstyle><mo id="S5.E4Xa.2.1.1.m1.3.3.1.4.2.2.2" stretchy="false" xref="S5.E4Xa.2.1.1.m1.2.2.cmml">)</mo></mrow><mi id="S5.E4Xa.2.1.1.m1.3.3.1.4.3" xref="S5.E4Xa.2.1.1.m1.3.3.1.4.3.cmml">t</mi></msup><mo id="S5.E4Xa.2.1.1.m1.3.3.1.2a" xref="S5.E4Xa.2.1.1.m1.3.3.1.2.cmml">⁢</mo><msup id="S5.E4Xa.2.1.1.m1.3.3.1.1" xref="S5.E4Xa.2.1.1.m1.3.3.1.1.cmml"><mrow id="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1" xref="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.cmml"><mo id="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.2" stretchy="false" xref="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1" xref="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.cmml"><mn id="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.2" xref="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.2.cmml">1</mn><mo id="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.1" xref="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.1.cmml">−</mo><mi id="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.3" xref="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.3" stretchy="false" xref="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S5.E4.m1.6.6.6.2.2.2.2a.4" xref="S5.E4.m1.6.6.6.2.2.2.2a.3.cmml"><mo id="S5.E4.m1.6.6.6.2.2.2.2a.4.1" xref="S5.E4.m1.6.6.6.2.2.2.2a.3.1.cmml">(</mo><mfrac id="S5.E4.m1.6.6.6.2.2.2.2.2.2.2a" linethickness="0pt" xref="S5.E4.m1.6.6.6.2.2.2.2a.3.cmml"><mi id="S5.E4.m1.5.5.5.1.1.1.1.1.1.1.1.1.mf" xref="S5.E4.m1.5.5.5.1.1.1.1.1.1.1.1.1.mf.cmml">t</mi><mn id="S5.E4.m1.6.6.6.2.2.2.2.2.2.2.2.1.mf" xref="S5.E4.m1.6.6.6.2.2.2.2.2.2.2.2.1.mf.cmml">2</mn></mfrac><mo id="S5.E4.m1.6.6.6.2.2.2.2a.4.2" xref="S5.E4.m1.6.6.6.2.2.2.2a.3.1.cmml">)</mo></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.E4Xa.2.1.1.m1.3b"><apply id="S5.E4Xa.2.1.1.m1.3.3.cmml" xref="S5.E4Xa.2.1.1.m1.3.3"><leq id="S5.E4Xa.2.1.1.m1.3.3.2.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.2"></leq><csymbol cd="latexml" id="S5.E4Xa.2.1.1.m1.3.3.3.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.3">absent</csymbol><apply id="S5.E4Xa.2.1.1.m1.3.3.1.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1"><times id="S5.E4Xa.2.1.1.m1.3.3.1.2.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.2"></times><apply id="S5.E4Xa.2.1.1.m1.3.3.1.3.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.3"><csymbol cd="ambiguous" id="S5.E4Xa.2.1.1.m1.3.3.1.3.1.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.3">superscript</csymbol><apply id="S5.E4Xa.2.1.1.m1.1.1.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.3.2.2"><divide id="S5.E4Xa.2.1.1.m1.1.1.1.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.3.2.2"></divide><apply id="S5.E4Xa.2.1.1.m1.1.1.2.cmml" xref="S5.E4Xa.2.1.1.m1.1.1.2"><times id="S5.E4Xa.2.1.1.m1.1.1.2.1.cmml" xref="S5.E4Xa.2.1.1.m1.1.1.2.1"></times><ci id="S5.E4Xa.2.1.1.m1.1.1.2.2.cmml" xref="S5.E4Xa.2.1.1.m1.1.1.2.2">𝑒</ci><ci id="S5.E4Xa.2.1.1.m1.1.1.2.3.cmml" xref="S5.E4Xa.2.1.1.m1.1.1.2.3">𝑘</ci></apply><ci id="S5.E4Xa.2.1.1.m1.1.1.3.cmml" xref="S5.E4Xa.2.1.1.m1.1.1.3">𝑡</ci></apply><ci id="S5.E4Xa.2.1.1.m1.3.3.1.3.3.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.3.3">𝑡</ci></apply><apply id="S5.E4Xa.2.1.1.m1.3.3.1.4.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.4"><csymbol cd="ambiguous" id="S5.E4Xa.2.1.1.m1.3.3.1.4.1.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.4">superscript</csymbol><apply id="S5.E4Xa.2.1.1.m1.2.2.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.4.2.2"><divide id="S5.E4Xa.2.1.1.m1.2.2.1.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.4.2.2"></divide><ci id="S5.E4Xa.2.1.1.m1.2.2.2.cmml" xref="S5.E4Xa.2.1.1.m1.2.2.2">𝑛</ci><ci id="S5.E4Xa.2.1.1.m1.2.2.3.cmml" xref="S5.E4Xa.2.1.1.m1.2.2.3">𝑘</ci></apply><ci id="S5.E4Xa.2.1.1.m1.3.3.1.4.3.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.4.3">𝑡</ci></apply><apply id="S5.E4Xa.2.1.1.m1.3.3.1.1.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.1"><csymbol cd="ambiguous" id="S5.E4Xa.2.1.1.m1.3.3.1.1.2.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.1">superscript</csymbol><apply id="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1"><minus id="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.1.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.1"></minus><cn id="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.2.cmml" type="integer" xref="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.2">1</cn><ci id="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.3.cmml" xref="S5.E4Xa.2.1.1.m1.3.3.1.1.1.1.1.3">𝑝</ci></apply><apply id="S5.E4.m1.6.6.6.2.2.2.2a.3.cmml" xref="S5.E4.m1.6.6.6.2.2.2.2a.4"><csymbol cd="latexml" id="S5.E4.m1.6.6.6.2.2.2.2a.3.1.cmml" xref="S5.E4.m1.6.6.6.2.2.2.2a.4.1">binomial</csymbol><ci id="S5.E4.m1.5.5.5.1.1.1.1.1.1.1.1.1.mf.cmml" xref="S5.E4.m1.5.5.5.1.1.1.1.1.1.1.1.1.mf">𝑡</ci><cn id="S5.E4.m1.6.6.6.2.2.2.2.2.2.2.2.1.mf.cmml" type="integer" xref="S5.E4.m1.6.6.6.2.2.2.2.2.2.2.2.1.mf">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E4Xa.2.1.1.m1.3c">\displaystyle\leq(\frac{ek}{t})^{t}(\frac{n}{k})^{t}(1-p)^{\binom{t}{2}}</annotation><annotation encoding="application/x-llamapun" id="S5.E4Xa.2.1.1.m1.3d">≤ ( divide start_ARG italic_e italic_k end_ARG start_ARG italic_t end_ARG ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( 1 - italic_p ) start_POSTSUPERSCRIPT ( FRACOP start_ARG italic_t end_ARG start_ARG 2 end_ARG ) end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S5.E4Xb"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=(\frac{en}{t}(1-p)^{\frac{t-1}{2}})^{t}" class="ltx_Math" display="inline" id="S5.E4Xb.2.1.1.m1.1"><semantics id="S5.E4Xb.2.1.1.m1.1a"><mrow id="S5.E4Xb.2.1.1.m1.1.1" xref="S5.E4Xb.2.1.1.m1.1.1.cmml"><mi id="S5.E4Xb.2.1.1.m1.1.1.3" xref="S5.E4Xb.2.1.1.m1.1.1.3.cmml"></mi><mo id="S5.E4Xb.2.1.1.m1.1.1.2" xref="S5.E4Xb.2.1.1.m1.1.1.2.cmml">=</mo><msup id="S5.E4Xb.2.1.1.m1.1.1.1" xref="S5.E4Xb.2.1.1.m1.1.1.1.cmml"><mrow id="S5.E4Xb.2.1.1.m1.1.1.1.1.1" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.cmml"><mo id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.cmml"><mfrac id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3a" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.cmml"><mrow id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2.cmml"><mi id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2.2" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2.2.cmml">e</mi><mo id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2.1" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2.1.cmml">⁢</mo><mi id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2.3" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2.3.cmml">n</mi></mrow><mi id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.3" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.3.cmml">t</mi></mfrac></mstyle><mo id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.2" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.2.cmml">⁢</mo><msup id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.cmml"><mrow id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mn id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mfrac id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.cmml"><mrow id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2.cmml"><mi id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2.2" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2.2.cmml">t</mi><mo id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2.1" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2.1.cmml">−</mo><mn id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2.3" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2.3.cmml">1</mn></mrow><mn id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.3" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.3.cmml">2</mn></mfrac></msup></mrow><mo id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S5.E4Xb.2.1.1.m1.1.1.1.3" xref="S5.E4Xb.2.1.1.m1.1.1.1.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.E4Xb.2.1.1.m1.1b"><apply id="S5.E4Xb.2.1.1.m1.1.1.cmml" xref="S5.E4Xb.2.1.1.m1.1.1"><eq id="S5.E4Xb.2.1.1.m1.1.1.2.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.2"></eq><csymbol cd="latexml" id="S5.E4Xb.2.1.1.m1.1.1.3.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.3">absent</csymbol><apply id="S5.E4Xb.2.1.1.m1.1.1.1.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1"><csymbol cd="ambiguous" id="S5.E4Xb.2.1.1.m1.1.1.1.2.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1">superscript</csymbol><apply id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1"><times id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.2.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.2"></times><apply id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3"><divide id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.1.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3"></divide><apply id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2"><times id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2.1.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2.1"></times><ci id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2.2.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2.2">𝑒</ci><ci id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2.3.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.2.3">𝑛</ci></apply><ci id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.3.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.3.3">𝑡</ci></apply><apply id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1">superscript</csymbol><apply id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1"><minus id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.1"></minus><cn id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.2">1</cn><ci id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3"><divide id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.1.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3"></divide><apply id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2"><minus id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2.1.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2.1"></minus><ci id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2.2.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2.2">𝑡</ci><cn id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2.3.cmml" type="integer" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.2.3">1</cn></apply><cn id="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.E4Xb.2.1.1.m1.1.1.1.1.1.1.1.3.3">2</cn></apply></apply></apply><ci id="S5.E4Xb.2.1.1.m1.1.1.1.3.cmml" xref="S5.E4Xb.2.1.1.m1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E4Xb.2.1.1.m1.1c">\displaystyle=(\frac{en}{t}(1-p)^{\frac{t-1}{2}})^{t}</annotation><annotation encoding="application/x-llamapun" id="S5.E4Xb.2.1.1.m1.1d">= ( divide start_ARG italic_e italic_n end_ARG start_ARG italic_t end_ARG ( 1 - italic_p ) start_POSTSUPERSCRIPT divide start_ARG italic_t - 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S5.E4Xc"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\left(\frac{en}{t}e^{-p\frac{t-1}{2}}\right)^{t}" class="ltx_Math" display="inline" id="S5.E4Xc.2.1.1.m1.1"><semantics id="S5.E4Xc.2.1.1.m1.1a"><mrow id="S5.E4Xc.2.1.1.m1.1.1" xref="S5.E4Xc.2.1.1.m1.1.1.cmml"><mi id="S5.E4Xc.2.1.1.m1.1.1.3" xref="S5.E4Xc.2.1.1.m1.1.1.3.cmml"></mi><mo id="S5.E4Xc.2.1.1.m1.1.1.2" xref="S5.E4Xc.2.1.1.m1.1.1.2.cmml">≤</mo><msup id="S5.E4Xc.2.1.1.m1.1.1.1" xref="S5.E4Xc.2.1.1.m1.1.1.1.cmml"><mrow id="S5.E4Xc.2.1.1.m1.1.1.1.1.1" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.cmml"><mo id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.2" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.cmml"><mfrac id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2a" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.cmml"><mrow id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2.cmml"><mi id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2.2" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2.2.cmml">e</mi><mo id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2.1" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2.1.cmml">⁢</mo><mi id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2.3" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2.3.cmml">n</mi></mrow><mi id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.3" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.3.cmml">t</mi></mfrac></mstyle><mo id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.1" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.cmml"><mi id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.2" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.2.cmml">e</mi><mrow id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.cmml"><mo id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3a" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.cmml">−</mo><mrow id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.cmml"><mi id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.2" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.2.cmml">p</mi><mo id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.1" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.1.cmml">⁢</mo><mfrac id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.cmml"><mrow id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2.cmml"><mi id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2.2" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2.2.cmml">t</mi><mo id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2.1" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2.1.cmml">−</mo><mn id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2.3" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2.3.cmml">1</mn></mrow><mn id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.3" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.3.cmml">2</mn></mfrac></mrow></mrow></msup></mrow><mo id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.3" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S5.E4Xc.2.1.1.m1.1.1.1.3" xref="S5.E4Xc.2.1.1.m1.1.1.1.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.E4Xc.2.1.1.m1.1b"><apply id="S5.E4Xc.2.1.1.m1.1.1.cmml" xref="S5.E4Xc.2.1.1.m1.1.1"><leq id="S5.E4Xc.2.1.1.m1.1.1.2.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.2"></leq><csymbol cd="latexml" id="S5.E4Xc.2.1.1.m1.1.1.3.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.3">absent</csymbol><apply id="S5.E4Xc.2.1.1.m1.1.1.1.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1"><csymbol cd="ambiguous" id="S5.E4Xc.2.1.1.m1.1.1.1.2.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1">superscript</csymbol><apply id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1"><times id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.1"></times><apply id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2"><divide id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.1.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2"></divide><apply id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2"><times id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2.1.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2.1"></times><ci id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2.2.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2.2">𝑒</ci><ci id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2.3.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.2.3">𝑛</ci></apply><ci id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.3.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.2.3">𝑡</ci></apply><apply id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.1.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.2.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.2">𝑒</ci><apply id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3"><minus id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.1.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3"></minus><apply id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2"><times id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.1.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.1"></times><ci id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.2.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.2">𝑝</ci><apply id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3"><divide id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.1.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3"></divide><apply id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2"><minus id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2.1.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2.1"></minus><ci id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2.2.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2.2">𝑡</ci><cn id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2.3.cmml" type="integer" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.2.3">1</cn></apply><cn id="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.3.cmml" type="integer" xref="S5.E4Xc.2.1.1.m1.1.1.1.1.1.1.3.3.2.3.3">2</cn></apply></apply></apply></apply></apply><ci id="S5.E4Xc.2.1.1.m1.1.1.1.3.cmml" xref="S5.E4Xc.2.1.1.m1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E4Xc.2.1.1.m1.1c">\displaystyle\leq\left(\frac{en}{t}e^{-p\frac{t-1}{2}}\right)^{t}</annotation><annotation encoding="application/x-llamapun" id="S5.E4Xc.2.1.1.m1.1d">≤ ( divide start_ARG italic_e italic_n end_ARG start_ARG italic_t end_ARG italic_e start_POSTSUPERSCRIPT - italic_p divide start_ARG italic_t - 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S5.SS2.SSS2.3.p2.3">where we used <math alttext="1-p\leq e^{-p}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.3.p2.1.m1.1"><semantics id="S5.SS2.SSS2.3.p2.1.m1.1a"><mrow id="S5.SS2.SSS2.3.p2.1.m1.1.1" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.cmml"><mrow id="S5.SS2.SSS2.3.p2.1.m1.1.1.2" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.2.cmml"><mn id="S5.SS2.SSS2.3.p2.1.m1.1.1.2.2" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.2.2.cmml">1</mn><mo id="S5.SS2.SSS2.3.p2.1.m1.1.1.2.1" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.2.1.cmml">−</mo><mi id="S5.SS2.SSS2.3.p2.1.m1.1.1.2.3" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.2.3.cmml">p</mi></mrow><mo id="S5.SS2.SSS2.3.p2.1.m1.1.1.1" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.1.cmml">≤</mo><msup id="S5.SS2.SSS2.3.p2.1.m1.1.1.3" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.3.cmml"><mi id="S5.SS2.SSS2.3.p2.1.m1.1.1.3.2" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.3.2.cmml">e</mi><mrow id="S5.SS2.SSS2.3.p2.1.m1.1.1.3.3" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.3.3.cmml"><mo id="S5.SS2.SSS2.3.p2.1.m1.1.1.3.3a" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.3.3.cmml">−</mo><mi id="S5.SS2.SSS2.3.p2.1.m1.1.1.3.3.2" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.3.3.2.cmml">p</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.3.p2.1.m1.1b"><apply id="S5.SS2.SSS2.3.p2.1.m1.1.1.cmml" xref="S5.SS2.SSS2.3.p2.1.m1.1.1"><leq id="S5.SS2.SSS2.3.p2.1.m1.1.1.1.cmml" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.1"></leq><apply id="S5.SS2.SSS2.3.p2.1.m1.1.1.2.cmml" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.2"><minus id="S5.SS2.SSS2.3.p2.1.m1.1.1.2.1.cmml" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.2.1"></minus><cn id="S5.SS2.SSS2.3.p2.1.m1.1.1.2.2.cmml" type="integer" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.2.2">1</cn><ci id="S5.SS2.SSS2.3.p2.1.m1.1.1.2.3.cmml" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.2.3">𝑝</ci></apply><apply id="S5.SS2.SSS2.3.p2.1.m1.1.1.3.cmml" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.SS2.SSS2.3.p2.1.m1.1.1.3.1.cmml" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.3">superscript</csymbol><ci id="S5.SS2.SSS2.3.p2.1.m1.1.1.3.2.cmml" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.3.2">𝑒</ci><apply id="S5.SS2.SSS2.3.p2.1.m1.1.1.3.3.cmml" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.3.3"><minus id="S5.SS2.SSS2.3.p2.1.m1.1.1.3.3.1.cmml" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.3.3"></minus><ci id="S5.SS2.SSS2.3.p2.1.m1.1.1.3.3.2.cmml" xref="S5.SS2.SSS2.3.p2.1.m1.1.1.3.3.2">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.3.p2.1.m1.1c">1-p\leq e^{-p}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.3.p2.1.m1.1d">1 - italic_p ≤ italic_e start_POSTSUPERSCRIPT - italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> when <math alttext="p\in[0,1]" class="ltx_Math" display="inline" id="S5.SS2.SSS2.3.p2.2.m2.2"><semantics id="S5.SS2.SSS2.3.p2.2.m2.2a"><mrow id="S5.SS2.SSS2.3.p2.2.m2.2.3" xref="S5.SS2.SSS2.3.p2.2.m2.2.3.cmml"><mi id="S5.SS2.SSS2.3.p2.2.m2.2.3.2" xref="S5.SS2.SSS2.3.p2.2.m2.2.3.2.cmml">p</mi><mo id="S5.SS2.SSS2.3.p2.2.m2.2.3.1" xref="S5.SS2.SSS2.3.p2.2.m2.2.3.1.cmml">∈</mo><mrow id="S5.SS2.SSS2.3.p2.2.m2.2.3.3.2" xref="S5.SS2.SSS2.3.p2.2.m2.2.3.3.1.cmml"><mo id="S5.SS2.SSS2.3.p2.2.m2.2.3.3.2.1" stretchy="false" xref="S5.SS2.SSS2.3.p2.2.m2.2.3.3.1.cmml">[</mo><mn id="S5.SS2.SSS2.3.p2.2.m2.1.1" xref="S5.SS2.SSS2.3.p2.2.m2.1.1.cmml">0</mn><mo id="S5.SS2.SSS2.3.p2.2.m2.2.3.3.2.2" xref="S5.SS2.SSS2.3.p2.2.m2.2.3.3.1.cmml">,</mo><mn id="S5.SS2.SSS2.3.p2.2.m2.2.2" xref="S5.SS2.SSS2.3.p2.2.m2.2.2.cmml">1</mn><mo id="S5.SS2.SSS2.3.p2.2.m2.2.3.3.2.3" stretchy="false" xref="S5.SS2.SSS2.3.p2.2.m2.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.3.p2.2.m2.2b"><apply id="S5.SS2.SSS2.3.p2.2.m2.2.3.cmml" xref="S5.SS2.SSS2.3.p2.2.m2.2.3"><in id="S5.SS2.SSS2.3.p2.2.m2.2.3.1.cmml" xref="S5.SS2.SSS2.3.p2.2.m2.2.3.1"></in><ci id="S5.SS2.SSS2.3.p2.2.m2.2.3.2.cmml" xref="S5.SS2.SSS2.3.p2.2.m2.2.3.2">𝑝</ci><interval closure="closed" id="S5.SS2.SSS2.3.p2.2.m2.2.3.3.1.cmml" xref="S5.SS2.SSS2.3.p2.2.m2.2.3.3.2"><cn id="S5.SS2.SSS2.3.p2.2.m2.1.1.cmml" type="integer" xref="S5.SS2.SSS2.3.p2.2.m2.1.1">0</cn><cn id="S5.SS2.SSS2.3.p2.2.m2.2.2.cmml" type="integer" xref="S5.SS2.SSS2.3.p2.2.m2.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.3.p2.2.m2.2c">p\in[0,1]</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.3.p2.2.m2.2d">italic_p ∈ [ 0 , 1 ]</annotation></semantics></math> in the last inequality. For a choice of <math alttext="t=\frac{2}{p}\log_{e}n+1" class="ltx_Math" display="inline" id="S5.SS2.SSS2.3.p2.3.m3.1"><semantics id="S5.SS2.SSS2.3.p2.3.m3.1a"><mrow id="S5.SS2.SSS2.3.p2.3.m3.1.1" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.cmml"><mi id="S5.SS2.SSS2.3.p2.3.m3.1.1.2" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.2.cmml">t</mi><mo id="S5.SS2.SSS2.3.p2.3.m3.1.1.1" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.1.cmml">=</mo><mrow id="S5.SS2.SSS2.3.p2.3.m3.1.1.3" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.cmml"><mrow id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.cmml"><mfrac id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.2" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.2.cmml"><mn id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.2.2" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.2.2.cmml">2</mn><mi id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.2.3" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.2.3.cmml">p</mi></mfrac><mo id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.1" lspace="0.167em" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.cmml"><msub id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.1" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.1.cmml"><mi id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.1.2" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.1.2.cmml">log</mi><mi id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.1.3" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.1.3.cmml">e</mi></msub><mo id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3a" lspace="0.167em" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.cmml">⁡</mo><mi id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.2" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.2.cmml">n</mi></mrow></mrow><mo id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.1" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.1.cmml">+</mo><mn id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.3" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.3.p2.3.m3.1b"><apply id="S5.SS2.SSS2.3.p2.3.m3.1.1.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1"><eq id="S5.SS2.SSS2.3.p2.3.m3.1.1.1.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.1"></eq><ci id="S5.SS2.SSS2.3.p2.3.m3.1.1.2.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.2">𝑡</ci><apply id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3"><plus id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.1.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.1"></plus><apply id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2"><times id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.1.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.1"></times><apply id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.2.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.2"><divide id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.2.1.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.2"></divide><cn id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.2.2.cmml" type="integer" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.2.2">2</cn><ci id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.2.3.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.2.3">𝑝</ci></apply><apply id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3"><apply id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.1.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.1.1.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.1">subscript</csymbol><log id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.1.2.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.1.2"></log><ci id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.1.3.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.1.3">𝑒</ci></apply><ci id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.2.cmml" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.2.3.2">𝑛</ci></apply></apply><cn id="S5.SS2.SSS2.3.p2.3.m3.1.1.3.3.cmml" type="integer" xref="S5.SS2.SSS2.3.p2.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.3.p2.3.m3.1c">t=\frac{2}{p}\log_{e}n+1</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.3.p2.3.m3.1d">italic_t = divide start_ARG 2 end_ARG start_ARG italic_p end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n + 1</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_table" id="S5.E5"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S5.E5X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathbb{E}[X^{t}]" class="ltx_Math" display="inline" id="S5.E5X.2.1.1.m1.1"><semantics id="S5.E5X.2.1.1.m1.1a"><mrow id="S5.E5X.2.1.1.m1.1.1" xref="S5.E5X.2.1.1.m1.1.1.cmml"><mi id="S5.E5X.2.1.1.m1.1.1.3" xref="S5.E5X.2.1.1.m1.1.1.3.cmml">𝔼</mi><mo id="S5.E5X.2.1.1.m1.1.1.2" xref="S5.E5X.2.1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S5.E5X.2.1.1.m1.1.1.1.1" xref="S5.E5X.2.1.1.m1.1.1.1.2.cmml"><mo id="S5.E5X.2.1.1.m1.1.1.1.1.2" stretchy="false" xref="S5.E5X.2.1.1.m1.1.1.1.2.1.cmml">[</mo><msup id="S5.E5X.2.1.1.m1.1.1.1.1.1" xref="S5.E5X.2.1.1.m1.1.1.1.1.1.cmml"><mi id="S5.E5X.2.1.1.m1.1.1.1.1.1.2" xref="S5.E5X.2.1.1.m1.1.1.1.1.1.2.cmml">X</mi><mi id="S5.E5X.2.1.1.m1.1.1.1.1.1.3" xref="S5.E5X.2.1.1.m1.1.1.1.1.1.3.cmml">t</mi></msup><mo id="S5.E5X.2.1.1.m1.1.1.1.1.3" stretchy="false" xref="S5.E5X.2.1.1.m1.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.E5X.2.1.1.m1.1b"><apply id="S5.E5X.2.1.1.m1.1.1.cmml" xref="S5.E5X.2.1.1.m1.1.1"><times id="S5.E5X.2.1.1.m1.1.1.2.cmml" xref="S5.E5X.2.1.1.m1.1.1.2"></times><ci id="S5.E5X.2.1.1.m1.1.1.3.cmml" xref="S5.E5X.2.1.1.m1.1.1.3">𝔼</ci><apply id="S5.E5X.2.1.1.m1.1.1.1.2.cmml" xref="S5.E5X.2.1.1.m1.1.1.1.1"><csymbol cd="latexml" id="S5.E5X.2.1.1.m1.1.1.1.2.1.cmml" xref="S5.E5X.2.1.1.m1.1.1.1.1.2">delimited-[]</csymbol><apply id="S5.E5X.2.1.1.m1.1.1.1.1.1.cmml" xref="S5.E5X.2.1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.E5X.2.1.1.m1.1.1.1.1.1.1.cmml" xref="S5.E5X.2.1.1.m1.1.1.1.1.1">superscript</csymbol><ci id="S5.E5X.2.1.1.m1.1.1.1.1.1.2.cmml" xref="S5.E5X.2.1.1.m1.1.1.1.1.1.2">𝑋</ci><ci id="S5.E5X.2.1.1.m1.1.1.1.1.1.3.cmml" xref="S5.E5X.2.1.1.m1.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E5X.2.1.1.m1.1c">\displaystyle\mathbb{E}[X^{t}]</annotation><annotation encoding="application/x-llamapun" id="S5.E5X.2.1.1.m1.1d">blackboard_E [ italic_X start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ]</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\left(\frac{e}{\frac{2}{p}\log_{e}n+1}\right)^{\frac{2}{p}% \log_{e}n+1}." class="ltx_Math" display="inline" id="S5.E5X.3.2.2.m1.2"><semantics id="S5.E5X.3.2.2.m1.2a"><mrow id="S5.E5X.3.2.2.m1.2.2.1" xref="S5.E5X.3.2.2.m1.2.2.1.1.cmml"><mrow id="S5.E5X.3.2.2.m1.2.2.1.1" xref="S5.E5X.3.2.2.m1.2.2.1.1.cmml"><mi id="S5.E5X.3.2.2.m1.2.2.1.1.2" xref="S5.E5X.3.2.2.m1.2.2.1.1.2.cmml"></mi><mo id="S5.E5X.3.2.2.m1.2.2.1.1.1" xref="S5.E5X.3.2.2.m1.2.2.1.1.1.cmml">≤</mo><msup id="S5.E5X.3.2.2.m1.2.2.1.1.3" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.cmml"><mrow id="S5.E5X.3.2.2.m1.2.2.1.1.3.2.2" xref="S5.E5X.3.2.2.m1.1.1.cmml"><mo id="S5.E5X.3.2.2.m1.2.2.1.1.3.2.2.1" xref="S5.E5X.3.2.2.m1.1.1.cmml">(</mo><mstyle displaystyle="true" id="S5.E5X.3.2.2.m1.1.1" xref="S5.E5X.3.2.2.m1.1.1.cmml"><mfrac id="S5.E5X.3.2.2.m1.1.1a" xref="S5.E5X.3.2.2.m1.1.1.cmml"><mi id="S5.E5X.3.2.2.m1.1.1.2" xref="S5.E5X.3.2.2.m1.1.1.2.cmml">e</mi><mrow id="S5.E5X.3.2.2.m1.1.1.3" xref="S5.E5X.3.2.2.m1.1.1.3.cmml"><mrow id="S5.E5X.3.2.2.m1.1.1.3.2" xref="S5.E5X.3.2.2.m1.1.1.3.2.cmml"><mfrac id="S5.E5X.3.2.2.m1.1.1.3.2.2" xref="S5.E5X.3.2.2.m1.1.1.3.2.2.cmml"><mn id="S5.E5X.3.2.2.m1.1.1.3.2.2.2" xref="S5.E5X.3.2.2.m1.1.1.3.2.2.2.cmml">2</mn><mi id="S5.E5X.3.2.2.m1.1.1.3.2.2.3" xref="S5.E5X.3.2.2.m1.1.1.3.2.2.3.cmml">p</mi></mfrac><mo id="S5.E5X.3.2.2.m1.1.1.3.2.1" lspace="0.167em" xref="S5.E5X.3.2.2.m1.1.1.3.2.1.cmml">⁢</mo><mrow id="S5.E5X.3.2.2.m1.1.1.3.2.3" xref="S5.E5X.3.2.2.m1.1.1.3.2.3.cmml"><msub id="S5.E5X.3.2.2.m1.1.1.3.2.3.1" xref="S5.E5X.3.2.2.m1.1.1.3.2.3.1.cmml"><mi id="S5.E5X.3.2.2.m1.1.1.3.2.3.1.2" xref="S5.E5X.3.2.2.m1.1.1.3.2.3.1.2.cmml">log</mi><mi id="S5.E5X.3.2.2.m1.1.1.3.2.3.1.3" xref="S5.E5X.3.2.2.m1.1.1.3.2.3.1.3.cmml">e</mi></msub><mo id="S5.E5X.3.2.2.m1.1.1.3.2.3a" lspace="0.167em" xref="S5.E5X.3.2.2.m1.1.1.3.2.3.cmml">⁡</mo><mi id="S5.E5X.3.2.2.m1.1.1.3.2.3.2" xref="S5.E5X.3.2.2.m1.1.1.3.2.3.2.cmml">n</mi></mrow></mrow><mo id="S5.E5X.3.2.2.m1.1.1.3.1" xref="S5.E5X.3.2.2.m1.1.1.3.1.cmml">+</mo><mn id="S5.E5X.3.2.2.m1.1.1.3.3" xref="S5.E5X.3.2.2.m1.1.1.3.3.cmml">1</mn></mrow></mfrac></mstyle><mo id="S5.E5X.3.2.2.m1.2.2.1.1.3.2.2.2" xref="S5.E5X.3.2.2.m1.1.1.cmml">)</mo></mrow><mrow id="S5.E5X.3.2.2.m1.2.2.1.1.3.3" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.cmml"><mrow id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.cmml"><mfrac id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.2" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.2.cmml"><mn id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.2.2" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.2.2.cmml">2</mn><mi id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.2.3" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.2.3.cmml">p</mi></mfrac><mo id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.1" lspace="0.167em" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.1.cmml">⁢</mo><mrow id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.cmml"><msub id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.1" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.1.cmml"><mi id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.1.2" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.1.2.cmml">log</mi><mi id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.1.3" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.1.3.cmml">e</mi></msub><mo id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3a" lspace="0.167em" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.cmml">⁡</mo><mi id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.2" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.2.cmml">n</mi></mrow></mrow><mo id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.1" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.1.cmml">+</mo><mn id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.3" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><mo id="S5.E5X.3.2.2.m1.2.2.1.2" lspace="0em" xref="S5.E5X.3.2.2.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E5X.3.2.2.m1.2b"><apply id="S5.E5X.3.2.2.m1.2.2.1.1.cmml" xref="S5.E5X.3.2.2.m1.2.2.1"><leq id="S5.E5X.3.2.2.m1.2.2.1.1.1.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.1"></leq><csymbol cd="latexml" id="S5.E5X.3.2.2.m1.2.2.1.1.2.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.2">absent</csymbol><apply id="S5.E5X.3.2.2.m1.2.2.1.1.3.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3"><csymbol cd="ambiguous" id="S5.E5X.3.2.2.m1.2.2.1.1.3.1.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3">superscript</csymbol><apply id="S5.E5X.3.2.2.m1.1.1.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.2.2"><divide id="S5.E5X.3.2.2.m1.1.1.1.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.2.2"></divide><ci id="S5.E5X.3.2.2.m1.1.1.2.cmml" xref="S5.E5X.3.2.2.m1.1.1.2">𝑒</ci><apply id="S5.E5X.3.2.2.m1.1.1.3.cmml" xref="S5.E5X.3.2.2.m1.1.1.3"><plus id="S5.E5X.3.2.2.m1.1.1.3.1.cmml" xref="S5.E5X.3.2.2.m1.1.1.3.1"></plus><apply id="S5.E5X.3.2.2.m1.1.1.3.2.cmml" xref="S5.E5X.3.2.2.m1.1.1.3.2"><times id="S5.E5X.3.2.2.m1.1.1.3.2.1.cmml" xref="S5.E5X.3.2.2.m1.1.1.3.2.1"></times><apply id="S5.E5X.3.2.2.m1.1.1.3.2.2.cmml" xref="S5.E5X.3.2.2.m1.1.1.3.2.2"><divide id="S5.E5X.3.2.2.m1.1.1.3.2.2.1.cmml" xref="S5.E5X.3.2.2.m1.1.1.3.2.2"></divide><cn id="S5.E5X.3.2.2.m1.1.1.3.2.2.2.cmml" type="integer" xref="S5.E5X.3.2.2.m1.1.1.3.2.2.2">2</cn><ci id="S5.E5X.3.2.2.m1.1.1.3.2.2.3.cmml" xref="S5.E5X.3.2.2.m1.1.1.3.2.2.3">𝑝</ci></apply><apply id="S5.E5X.3.2.2.m1.1.1.3.2.3.cmml" xref="S5.E5X.3.2.2.m1.1.1.3.2.3"><apply id="S5.E5X.3.2.2.m1.1.1.3.2.3.1.cmml" xref="S5.E5X.3.2.2.m1.1.1.3.2.3.1"><csymbol cd="ambiguous" id="S5.E5X.3.2.2.m1.1.1.3.2.3.1.1.cmml" xref="S5.E5X.3.2.2.m1.1.1.3.2.3.1">subscript</csymbol><log id="S5.E5X.3.2.2.m1.1.1.3.2.3.1.2.cmml" xref="S5.E5X.3.2.2.m1.1.1.3.2.3.1.2"></log><ci id="S5.E5X.3.2.2.m1.1.1.3.2.3.1.3.cmml" xref="S5.E5X.3.2.2.m1.1.1.3.2.3.1.3">𝑒</ci></apply><ci id="S5.E5X.3.2.2.m1.1.1.3.2.3.2.cmml" xref="S5.E5X.3.2.2.m1.1.1.3.2.3.2">𝑛</ci></apply></apply><cn id="S5.E5X.3.2.2.m1.1.1.3.3.cmml" type="integer" xref="S5.E5X.3.2.2.m1.1.1.3.3">1</cn></apply></apply><apply id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3"><plus id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.1.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.1"></plus><apply id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2"><times id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.1.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.1"></times><apply id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.2.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.2"><divide id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.2.1.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.2"></divide><cn id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.2.2.cmml" type="integer" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.2.2">2</cn><ci id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.2.3.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.2.3">𝑝</ci></apply><apply id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3"><apply id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.1.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.1"><csymbol cd="ambiguous" id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.1.1.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.1">subscript</csymbol><log id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.1.2.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.1.2"></log><ci id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.1.3.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.1.3">𝑒</ci></apply><ci id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.2.cmml" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.2.3.2">𝑛</ci></apply></apply><cn id="S5.E5X.3.2.2.m1.2.2.1.1.3.3.3.cmml" type="integer" xref="S5.E5X.3.2.2.m1.2.2.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E5X.3.2.2.m1.2c">\displaystyle\leq\left(\frac{e}{\frac{2}{p}\log_{e}n+1}\right)^{\frac{2}{p}% \log_{e}n+1}.</annotation><annotation encoding="application/x-llamapun" id="S5.E5X.3.2.2.m1.2d">≤ ( divide start_ARG italic_e end_ARG start_ARG divide start_ARG 2 end_ARG start_ARG italic_p end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n + 1 end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 2 end_ARG start_ARG italic_p end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n + 1 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(5)</span></td> </tr> </tbody> </table> <p class="ltx_p" id="S5.SS2.SSS2.3.p2.12">By Markov’s inequality, the probability of having at least one clique of size <math alttext="t" class="ltx_Math" display="inline" id="S5.SS2.SSS2.3.p2.4.m1.1"><semantics id="S5.SS2.SSS2.3.p2.4.m1.1a"><mi id="S5.SS2.SSS2.3.p2.4.m1.1.1" xref="S5.SS2.SSS2.3.p2.4.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.3.p2.4.m1.1b"><ci id="S5.SS2.SSS2.3.p2.4.m1.1.1.cmml" xref="S5.SS2.SSS2.3.p2.4.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.3.p2.4.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.3.p2.4.m1.1d">italic_t</annotation></semantics></math> in <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.3.p2.5.m2.1"><semantics id="S5.SS2.SSS2.3.p2.5.m2.1a"><msup id="S5.SS2.SSS2.3.p2.5.m2.1.1" xref="S5.SS2.SSS2.3.p2.5.m2.1.1.cmml"><mi id="S5.SS2.SSS2.3.p2.5.m2.1.1.2" xref="S5.SS2.SSS2.3.p2.5.m2.1.1.2.cmml">G</mi><mo id="S5.SS2.SSS2.3.p2.5.m2.1.1.3" xref="S5.SS2.SSS2.3.p2.5.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.3.p2.5.m2.1b"><apply id="S5.SS2.SSS2.3.p2.5.m2.1.1.cmml" xref="S5.SS2.SSS2.3.p2.5.m2.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.3.p2.5.m2.1.1.1.cmml" xref="S5.SS2.SSS2.3.p2.5.m2.1.1">superscript</csymbol><ci id="S5.SS2.SSS2.3.p2.5.m2.1.1.2.cmml" xref="S5.SS2.SSS2.3.p2.5.m2.1.1.2">𝐺</ci><ci id="S5.SS2.SSS2.3.p2.5.m2.1.1.3.cmml" xref="S5.SS2.SSS2.3.p2.5.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.3.p2.5.m2.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.3.p2.5.m2.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is at most <math alttext="\left(\frac{ce}{2\log_{e}n}\right)^{\frac{2}{c}\log_{e}n}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.3.p2.6.m3.1"><semantics id="S5.SS2.SSS2.3.p2.6.m3.1a"><msup id="S5.SS2.SSS2.3.p2.6.m3.1.2" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.cmml"><mrow id="S5.SS2.SSS2.3.p2.6.m3.1.2.2.2" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.cmml"><mo id="S5.SS2.SSS2.3.p2.6.m3.1.2.2.2.1" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS2.3.p2.6.m3.1.1" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.cmml"><mrow id="S5.SS2.SSS2.3.p2.6.m3.1.1.2" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.2.cmml"><mi id="S5.SS2.SSS2.3.p2.6.m3.1.1.2.2" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.2.2.cmml">c</mi><mo id="S5.SS2.SSS2.3.p2.6.m3.1.1.2.1" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.2.1.cmml">⁢</mo><mi id="S5.SS2.SSS2.3.p2.6.m3.1.1.2.3" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.2.3.cmml">e</mi></mrow><mrow id="S5.SS2.SSS2.3.p2.6.m3.1.1.3" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.cmml"><mn id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.2" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.2.cmml">2</mn><mo id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.1" lspace="0.167em" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.cmml"><msub id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.1" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.1.cmml"><mi id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.1.2" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.1.2.cmml">log</mi><mi id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.1.3" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.1.3.cmml">e</mi></msub><mo id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3a" lspace="0.167em" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.cmml">⁡</mo><mi id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.2" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.2.cmml">n</mi></mrow></mrow></mfrac><mo id="S5.SS2.SSS2.3.p2.6.m3.1.2.2.2.2" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.cmml">)</mo></mrow><mrow id="S5.SS2.SSS2.3.p2.6.m3.1.2.3" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.cmml"><mfrac id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.2" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.2.cmml"><mn id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.2.2" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.2.2.cmml">2</mn><mi id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.2.3" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.2.3.cmml">c</mi></mfrac><mo id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.1" lspace="0.167em" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.cmml"><msub id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.1" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.1.cmml"><mi id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.1.2" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.1.2.cmml">log</mi><mi id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.1.3" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.1.3.cmml">e</mi></msub><mo id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3a" lspace="0.167em" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.cmml">⁡</mo><mi id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.2" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.2.cmml">n</mi></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.3.p2.6.m3.1b"><apply id="S5.SS2.SSS2.3.p2.6.m3.1.2.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.2"><csymbol cd="ambiguous" id="S5.SS2.SSS2.3.p2.6.m3.1.2.1.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.2">superscript</csymbol><apply id="S5.SS2.SSS2.3.p2.6.m3.1.1.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.2.2"><divide id="S5.SS2.SSS2.3.p2.6.m3.1.1.1.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.2.2"></divide><apply id="S5.SS2.SSS2.3.p2.6.m3.1.1.2.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.2"><times id="S5.SS2.SSS2.3.p2.6.m3.1.1.2.1.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.2.1"></times><ci id="S5.SS2.SSS2.3.p2.6.m3.1.1.2.2.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.2.2">𝑐</ci><ci id="S5.SS2.SSS2.3.p2.6.m3.1.1.2.3.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.2.3">𝑒</ci></apply><apply id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3"><times id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.1.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.1"></times><cn id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.2.cmml" type="integer" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.2">2</cn><apply id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3"><apply id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.1.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.1.1.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.1">subscript</csymbol><log id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.1.2.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.1.2"></log><ci id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.1.3.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.1.3">𝑒</ci></apply><ci id="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.2.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.1.3.3.2">𝑛</ci></apply></apply></apply><apply id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3"><times id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.1.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.1"></times><apply id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.2.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.2"><divide id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.2.1.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.2"></divide><cn id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.2.2.cmml" type="integer" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.2.2">2</cn><ci id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.2.3.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.2.3">𝑐</ci></apply><apply id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3"><apply id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.1.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.1.1.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.1">subscript</csymbol><log id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.1.2.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.1.2"></log><ci id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.1.3.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.1.3">𝑒</ci></apply><ci id="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.2.cmml" xref="S5.SS2.SSS2.3.p2.6.m3.1.2.3.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.3.p2.6.m3.1c">\left(\frac{ce}{2\log_{e}n}\right)^{\frac{2}{c}\log_{e}n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.3.p2.6.m3.1d">( divide start_ARG italic_c italic_e end_ARG start_ARG 2 roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 2 end_ARG start_ARG italic_c end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="p=c" class="ltx_Math" display="inline" id="S5.SS2.SSS2.3.p2.7.m4.1"><semantics id="S5.SS2.SSS2.3.p2.7.m4.1a"><mrow id="S5.SS2.SSS2.3.p2.7.m4.1.1" xref="S5.SS2.SSS2.3.p2.7.m4.1.1.cmml"><mi id="S5.SS2.SSS2.3.p2.7.m4.1.1.2" xref="S5.SS2.SSS2.3.p2.7.m4.1.1.2.cmml">p</mi><mo id="S5.SS2.SSS2.3.p2.7.m4.1.1.1" xref="S5.SS2.SSS2.3.p2.7.m4.1.1.1.cmml">=</mo><mi id="S5.SS2.SSS2.3.p2.7.m4.1.1.3" xref="S5.SS2.SSS2.3.p2.7.m4.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.3.p2.7.m4.1b"><apply id="S5.SS2.SSS2.3.p2.7.m4.1.1.cmml" xref="S5.SS2.SSS2.3.p2.7.m4.1.1"><eq id="S5.SS2.SSS2.3.p2.7.m4.1.1.1.cmml" xref="S5.SS2.SSS2.3.p2.7.m4.1.1.1"></eq><ci id="S5.SS2.SSS2.3.p2.7.m4.1.1.2.cmml" xref="S5.SS2.SSS2.3.p2.7.m4.1.1.2">𝑝</ci><ci id="S5.SS2.SSS2.3.p2.7.m4.1.1.3.cmml" xref="S5.SS2.SSS2.3.p2.7.m4.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.3.p2.7.m4.1c">p=c</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.3.p2.7.m4.1d">italic_p = italic_c</annotation></semantics></math>. This implies that, with high probability, <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.3.p2.8.m5.1"><semantics id="S5.SS2.SSS2.3.p2.8.m5.1a"><msup id="S5.SS2.SSS2.3.p2.8.m5.1.1" xref="S5.SS2.SSS2.3.p2.8.m5.1.1.cmml"><mi id="S5.SS2.SSS2.3.p2.8.m5.1.1.2" xref="S5.SS2.SSS2.3.p2.8.m5.1.1.2.cmml">G</mi><mo id="S5.SS2.SSS2.3.p2.8.m5.1.1.3" xref="S5.SS2.SSS2.3.p2.8.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.3.p2.8.m5.1b"><apply id="S5.SS2.SSS2.3.p2.8.m5.1.1.cmml" xref="S5.SS2.SSS2.3.p2.8.m5.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.3.p2.8.m5.1.1.1.cmml" xref="S5.SS2.SSS2.3.p2.8.m5.1.1">superscript</csymbol><ci id="S5.SS2.SSS2.3.p2.8.m5.1.1.2.cmml" xref="S5.SS2.SSS2.3.p2.8.m5.1.1.2">𝐺</ci><ci id="S5.SS2.SSS2.3.p2.8.m5.1.1.3.cmml" xref="S5.SS2.SSS2.3.p2.8.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.3.p2.8.m5.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.3.p2.8.m5.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> contains no clique of size <math alttext="\frac{2}{c}\log_{e}n+1" class="ltx_Math" display="inline" id="S5.SS2.SSS2.3.p2.9.m6.1"><semantics id="S5.SS2.SSS2.3.p2.9.m6.1a"><mrow id="S5.SS2.SSS2.3.p2.9.m6.1.1" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.cmml"><mrow id="S5.SS2.SSS2.3.p2.9.m6.1.1.2" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.cmml"><mfrac id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.2" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.2.cmml"><mn id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.2.2" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.2.2.cmml">2</mn><mi id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.2.3" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.2.3.cmml">c</mi></mfrac><mo id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.1" lspace="0.167em" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.cmml"><msub id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.1" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.1.cmml"><mi id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.1.2" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.1.2.cmml">log</mi><mi id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.1.3" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.1.3.cmml">e</mi></msub><mo id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3a" lspace="0.167em" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.cmml">⁡</mo><mi id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.2" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.2.cmml">n</mi></mrow></mrow><mo id="S5.SS2.SSS2.3.p2.9.m6.1.1.1" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.1.cmml">+</mo><mn id="S5.SS2.SSS2.3.p2.9.m6.1.1.3" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.3.p2.9.m6.1b"><apply id="S5.SS2.SSS2.3.p2.9.m6.1.1.cmml" xref="S5.SS2.SSS2.3.p2.9.m6.1.1"><plus id="S5.SS2.SSS2.3.p2.9.m6.1.1.1.cmml" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.1"></plus><apply id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.cmml" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2"><times id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.1.cmml" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.1"></times><apply id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.2.cmml" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.2"><divide id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.2.1.cmml" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.2"></divide><cn id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.2.2.cmml" type="integer" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.2.2">2</cn><ci id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.2.3.cmml" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.2.3">𝑐</ci></apply><apply id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.cmml" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3"><apply id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.1.cmml" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.1.1.cmml" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.1">subscript</csymbol><log id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.1.2.cmml" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.1.2"></log><ci id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.1.3.cmml" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.1.3">𝑒</ci></apply><ci id="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.2.cmml" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.2.3.2">𝑛</ci></apply></apply><cn id="S5.SS2.SSS2.3.p2.9.m6.1.1.3.cmml" type="integer" xref="S5.SS2.SSS2.3.p2.9.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.3.p2.9.m6.1c">\frac{2}{c}\log_{e}n+1</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.3.p2.9.m6.1d">divide start_ARG 2 end_ARG start_ARG italic_c end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n + 1</annotation></semantics></math>. Therefore, the largest clique in <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.3.p2.10.m7.1"><semantics id="S5.SS2.SSS2.3.p2.10.m7.1a"><msup id="S5.SS2.SSS2.3.p2.10.m7.1.1" xref="S5.SS2.SSS2.3.p2.10.m7.1.1.cmml"><mi id="S5.SS2.SSS2.3.p2.10.m7.1.1.2" xref="S5.SS2.SSS2.3.p2.10.m7.1.1.2.cmml">G</mi><mo id="S5.SS2.SSS2.3.p2.10.m7.1.1.3" xref="S5.SS2.SSS2.3.p2.10.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.3.p2.10.m7.1b"><apply id="S5.SS2.SSS2.3.p2.10.m7.1.1.cmml" xref="S5.SS2.SSS2.3.p2.10.m7.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.3.p2.10.m7.1.1.1.cmml" xref="S5.SS2.SSS2.3.p2.10.m7.1.1">superscript</csymbol><ci id="S5.SS2.SSS2.3.p2.10.m7.1.1.2.cmml" xref="S5.SS2.SSS2.3.p2.10.m7.1.1.2">𝐺</ci><ci id="S5.SS2.SSS2.3.p2.10.m7.1.1.3.cmml" xref="S5.SS2.SSS2.3.p2.10.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.3.p2.10.m7.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.3.p2.10.m7.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is smaller than <math alttext="\frac{2}{c}\log_{e}n+1" class="ltx_Math" display="inline" id="S5.SS2.SSS2.3.p2.11.m8.1"><semantics id="S5.SS2.SSS2.3.p2.11.m8.1a"><mrow id="S5.SS2.SSS2.3.p2.11.m8.1.1" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.cmml"><mrow id="S5.SS2.SSS2.3.p2.11.m8.1.1.2" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.cmml"><mfrac id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.2" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.2.cmml"><mn id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.2.2" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.2.2.cmml">2</mn><mi id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.2.3" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.2.3.cmml">c</mi></mfrac><mo id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.1" lspace="0.167em" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.cmml"><msub id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.1" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.1.cmml"><mi id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.1.2" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.1.2.cmml">log</mi><mi id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.1.3" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.1.3.cmml">e</mi></msub><mo id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3a" lspace="0.167em" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.cmml">⁡</mo><mi id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.2" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.2.cmml">n</mi></mrow></mrow><mo id="S5.SS2.SSS2.3.p2.11.m8.1.1.1" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.1.cmml">+</mo><mn id="S5.SS2.SSS2.3.p2.11.m8.1.1.3" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.3.p2.11.m8.1b"><apply id="S5.SS2.SSS2.3.p2.11.m8.1.1.cmml" xref="S5.SS2.SSS2.3.p2.11.m8.1.1"><plus id="S5.SS2.SSS2.3.p2.11.m8.1.1.1.cmml" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.1"></plus><apply id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.cmml" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2"><times id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.1.cmml" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.1"></times><apply id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.2.cmml" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.2"><divide id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.2.1.cmml" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.2"></divide><cn id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.2.2.cmml" type="integer" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.2.2">2</cn><ci id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.2.3.cmml" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.2.3">𝑐</ci></apply><apply id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.cmml" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3"><apply id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.1.cmml" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.1.1.cmml" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.1">subscript</csymbol><log id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.1.2.cmml" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.1.2"></log><ci id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.1.3.cmml" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.1.3">𝑒</ci></apply><ci id="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.2.cmml" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.2.3.2">𝑛</ci></apply></apply><cn id="S5.SS2.SSS2.3.p2.11.m8.1.1.3.cmml" type="integer" xref="S5.SS2.SSS2.3.p2.11.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.3.p2.11.m8.1c">\frac{2}{c}\log_{e}n+1</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.3.p2.11.m8.1d">divide start_ARG 2 end_ARG start_ARG italic_c end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n + 1</annotation></semantics></math> with probability <math alttext="1-\left(\frac{ce}{2\log_{e}n}\right)^{\frac{2}{c}\log_{e}n}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.3.p2.12.m9.1"><semantics id="S5.SS2.SSS2.3.p2.12.m9.1a"><mrow id="S5.SS2.SSS2.3.p2.12.m9.1.2" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.cmml"><mn id="S5.SS2.SSS2.3.p2.12.m9.1.2.2" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.2.cmml">1</mn><mo id="S5.SS2.SSS2.3.p2.12.m9.1.2.1" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.1.cmml">−</mo><msup id="S5.SS2.SSS2.3.p2.12.m9.1.2.3" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.cmml"><mrow id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.2.2" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.cmml"><mo id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.2.2.1" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS2.3.p2.12.m9.1.1" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.cmml"><mrow id="S5.SS2.SSS2.3.p2.12.m9.1.1.2" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.2.cmml"><mi id="S5.SS2.SSS2.3.p2.12.m9.1.1.2.2" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.2.2.cmml">c</mi><mo id="S5.SS2.SSS2.3.p2.12.m9.1.1.2.1" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.2.1.cmml">⁢</mo><mi id="S5.SS2.SSS2.3.p2.12.m9.1.1.2.3" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.2.3.cmml">e</mi></mrow><mrow id="S5.SS2.SSS2.3.p2.12.m9.1.1.3" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.cmml"><mn id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.2" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.2.cmml">2</mn><mo id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.1" lspace="0.167em" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.cmml"><msub id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.1" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.1.cmml"><mi id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.1.2" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.1.2.cmml">log</mi><mi id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.1.3" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.1.3.cmml">e</mi></msub><mo id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3a" lspace="0.167em" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.cmml">⁡</mo><mi id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.2" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.2.cmml">n</mi></mrow></mrow></mfrac><mo id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.2.2.2" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.cmml">)</mo></mrow><mrow id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.cmml"><mfrac id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.2" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.2.cmml"><mn id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.2.2" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.2.2.cmml">2</mn><mi id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.2.3" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.2.3.cmml">c</mi></mfrac><mo id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.1" lspace="0.167em" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.cmml"><msub id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.1" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.1.cmml"><mi id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.1.2" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.1.2.cmml">log</mi><mi id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.1.3" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.1.3.cmml">e</mi></msub><mo id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3a" lspace="0.167em" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.cmml">⁡</mo><mi id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.2" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.2.cmml">n</mi></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.3.p2.12.m9.1b"><apply id="S5.SS2.SSS2.3.p2.12.m9.1.2.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2"><minus id="S5.SS2.SSS2.3.p2.12.m9.1.2.1.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.1"></minus><cn id="S5.SS2.SSS2.3.p2.12.m9.1.2.2.cmml" type="integer" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.2">1</cn><apply id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3"><csymbol cd="ambiguous" id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.1.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3">superscript</csymbol><apply id="S5.SS2.SSS2.3.p2.12.m9.1.1.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.2.2"><divide id="S5.SS2.SSS2.3.p2.12.m9.1.1.1.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.2.2"></divide><apply id="S5.SS2.SSS2.3.p2.12.m9.1.1.2.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.2"><times id="S5.SS2.SSS2.3.p2.12.m9.1.1.2.1.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.2.1"></times><ci id="S5.SS2.SSS2.3.p2.12.m9.1.1.2.2.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.2.2">𝑐</ci><ci id="S5.SS2.SSS2.3.p2.12.m9.1.1.2.3.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.2.3">𝑒</ci></apply><apply id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3"><times id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.1.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.1"></times><cn id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.2.cmml" type="integer" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.2">2</cn><apply id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3"><apply id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.1.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.1.1.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.1">subscript</csymbol><log id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.1.2.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.1.2"></log><ci id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.1.3.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.1.3">𝑒</ci></apply><ci id="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.2.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.1.3.3.2">𝑛</ci></apply></apply></apply><apply id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3"><times id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.1.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.1"></times><apply id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.2.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.2"><divide id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.2.1.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.2"></divide><cn id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.2.2.cmml" type="integer" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.2.2">2</cn><ci id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.2.3.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.2.3">𝑐</ci></apply><apply id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3"><apply id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.1.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.1.1.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.1">subscript</csymbol><log id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.1.2.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.1.2"></log><ci id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.1.3.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.1.3">𝑒</ci></apply><ci id="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.2.cmml" xref="S5.SS2.SSS2.3.p2.12.m9.1.2.3.3.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.3.p2.12.m9.1c">1-\left(\frac{ce}{2\log_{e}n}\right)^{\frac{2}{c}\log_{e}n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.3.p2.12.m9.1d">1 - ( divide start_ARG italic_c italic_e end_ARG start_ARG 2 roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 2 end_ARG start_ARG italic_c end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. According to <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem1" title="Lemma 5.1. ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5.1</span></a>, the maximum welfare is upper-bounded as follows:</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex34"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{SW}(\pi^{*})\leq n(t-1)=\frac{2n}{c}\log_{e}n=\mathcal{O}(n\log_{e}n)." class="ltx_Math" display="block" id="S5.Ex34.m1.1"><semantics id="S5.Ex34.m1.1a"><mrow id="S5.Ex34.m1.1.1.1" xref="S5.Ex34.m1.1.1.1.1.cmml"><mrow id="S5.Ex34.m1.1.1.1.1" xref="S5.Ex34.m1.1.1.1.1.cmml"><mrow id="S5.Ex34.m1.1.1.1.1.1" xref="S5.Ex34.m1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex34.m1.1.1.1.1.1.3" xref="S5.Ex34.m1.1.1.1.1.1.3.cmml">𝒮</mi><mo id="S5.Ex34.m1.1.1.1.1.1.2" xref="S5.Ex34.m1.1.1.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex34.m1.1.1.1.1.1.4" xref="S5.Ex34.m1.1.1.1.1.1.4.cmml">𝒲</mi><mo id="S5.Ex34.m1.1.1.1.1.1.2a" xref="S5.Ex34.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex34.m1.1.1.1.1.1.1.1" xref="S5.Ex34.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex34.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex34.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S5.Ex34.m1.1.1.1.1.1.1.1.1" xref="S5.Ex34.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex34.m1.1.1.1.1.1.1.1.1.2" xref="S5.Ex34.m1.1.1.1.1.1.1.1.1.2.cmml">π</mi><mo id="S5.Ex34.m1.1.1.1.1.1.1.1.1.3" xref="S5.Ex34.m1.1.1.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.Ex34.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex34.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex34.m1.1.1.1.1.5" xref="S5.Ex34.m1.1.1.1.1.5.cmml">≤</mo><mrow id="S5.Ex34.m1.1.1.1.1.2" xref="S5.Ex34.m1.1.1.1.1.2.cmml"><mi id="S5.Ex34.m1.1.1.1.1.2.3" xref="S5.Ex34.m1.1.1.1.1.2.3.cmml">n</mi><mo id="S5.Ex34.m1.1.1.1.1.2.2" xref="S5.Ex34.m1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="S5.Ex34.m1.1.1.1.1.2.1.1" xref="S5.Ex34.m1.1.1.1.1.2.1.1.1.cmml"><mo id="S5.Ex34.m1.1.1.1.1.2.1.1.2" stretchy="false" xref="S5.Ex34.m1.1.1.1.1.2.1.1.1.cmml">(</mo><mrow id="S5.Ex34.m1.1.1.1.1.2.1.1.1" xref="S5.Ex34.m1.1.1.1.1.2.1.1.1.cmml"><mi id="S5.Ex34.m1.1.1.1.1.2.1.1.1.2" xref="S5.Ex34.m1.1.1.1.1.2.1.1.1.2.cmml">t</mi><mo id="S5.Ex34.m1.1.1.1.1.2.1.1.1.1" xref="S5.Ex34.m1.1.1.1.1.2.1.1.1.1.cmml">−</mo><mn id="S5.Ex34.m1.1.1.1.1.2.1.1.1.3" xref="S5.Ex34.m1.1.1.1.1.2.1.1.1.3.cmml">1</mn></mrow><mo id="S5.Ex34.m1.1.1.1.1.2.1.1.3" stretchy="false" xref="S5.Ex34.m1.1.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex34.m1.1.1.1.1.6" xref="S5.Ex34.m1.1.1.1.1.6.cmml">=</mo><mrow id="S5.Ex34.m1.1.1.1.1.7" xref="S5.Ex34.m1.1.1.1.1.7.cmml"><mfrac id="S5.Ex34.m1.1.1.1.1.7.2" xref="S5.Ex34.m1.1.1.1.1.7.2.cmml"><mrow id="S5.Ex34.m1.1.1.1.1.7.2.2" xref="S5.Ex34.m1.1.1.1.1.7.2.2.cmml"><mn id="S5.Ex34.m1.1.1.1.1.7.2.2.2" xref="S5.Ex34.m1.1.1.1.1.7.2.2.2.cmml">2</mn><mo id="S5.Ex34.m1.1.1.1.1.7.2.2.1" xref="S5.Ex34.m1.1.1.1.1.7.2.2.1.cmml">⁢</mo><mi id="S5.Ex34.m1.1.1.1.1.7.2.2.3" xref="S5.Ex34.m1.1.1.1.1.7.2.2.3.cmml">n</mi></mrow><mi id="S5.Ex34.m1.1.1.1.1.7.2.3" xref="S5.Ex34.m1.1.1.1.1.7.2.3.cmml">c</mi></mfrac><mo id="S5.Ex34.m1.1.1.1.1.7.1" lspace="0.167em" xref="S5.Ex34.m1.1.1.1.1.7.1.cmml">⁢</mo><mrow id="S5.Ex34.m1.1.1.1.1.7.3" xref="S5.Ex34.m1.1.1.1.1.7.3.cmml"><msub id="S5.Ex34.m1.1.1.1.1.7.3.1" xref="S5.Ex34.m1.1.1.1.1.7.3.1.cmml"><mi id="S5.Ex34.m1.1.1.1.1.7.3.1.2" xref="S5.Ex34.m1.1.1.1.1.7.3.1.2.cmml">log</mi><mi id="S5.Ex34.m1.1.1.1.1.7.3.1.3" xref="S5.Ex34.m1.1.1.1.1.7.3.1.3.cmml">e</mi></msub><mo id="S5.Ex34.m1.1.1.1.1.7.3a" lspace="0.167em" xref="S5.Ex34.m1.1.1.1.1.7.3.cmml">⁡</mo><mi id="S5.Ex34.m1.1.1.1.1.7.3.2" xref="S5.Ex34.m1.1.1.1.1.7.3.2.cmml">n</mi></mrow></mrow><mo id="S5.Ex34.m1.1.1.1.1.8" xref="S5.Ex34.m1.1.1.1.1.8.cmml">=</mo><mrow id="S5.Ex34.m1.1.1.1.1.3" xref="S5.Ex34.m1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex34.m1.1.1.1.1.3.3" xref="S5.Ex34.m1.1.1.1.1.3.3.cmml">𝒪</mi><mo id="S5.Ex34.m1.1.1.1.1.3.2" xref="S5.Ex34.m1.1.1.1.1.3.2.cmml">⁢</mo><mrow id="S5.Ex34.m1.1.1.1.1.3.1.1" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.cmml"><mo id="S5.Ex34.m1.1.1.1.1.3.1.1.2" stretchy="false" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.cmml">(</mo><mrow id="S5.Ex34.m1.1.1.1.1.3.1.1.1" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.cmml"><mi id="S5.Ex34.m1.1.1.1.1.3.1.1.1.2" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.2.cmml">n</mi><mo id="S5.Ex34.m1.1.1.1.1.3.1.1.1.1" lspace="0.167em" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.1.cmml">⁢</mo><mrow id="S5.Ex34.m1.1.1.1.1.3.1.1.1.3" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.cmml"><msub id="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.1" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.1.cmml"><mi id="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.1.2" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.1.2.cmml">log</mi><mi id="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.1.3" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.1.3.cmml">e</mi></msub><mo id="S5.Ex34.m1.1.1.1.1.3.1.1.1.3a" lspace="0.167em" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.cmml">⁡</mo><mi id="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.2" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S5.Ex34.m1.1.1.1.1.3.1.1.3" stretchy="false" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Ex34.m1.1.1.1.2" lspace="0em" xref="S5.Ex34.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex34.m1.1b"><apply id="S5.Ex34.m1.1.1.1.1.cmml" xref="S5.Ex34.m1.1.1.1"><and id="S5.Ex34.m1.1.1.1.1a.cmml" xref="S5.Ex34.m1.1.1.1"></and><apply id="S5.Ex34.m1.1.1.1.1b.cmml" xref="S5.Ex34.m1.1.1.1"><leq id="S5.Ex34.m1.1.1.1.1.5.cmml" xref="S5.Ex34.m1.1.1.1.1.5"></leq><apply id="S5.Ex34.m1.1.1.1.1.1.cmml" xref="S5.Ex34.m1.1.1.1.1.1"><times id="S5.Ex34.m1.1.1.1.1.1.2.cmml" xref="S5.Ex34.m1.1.1.1.1.1.2"></times><ci id="S5.Ex34.m1.1.1.1.1.1.3.cmml" xref="S5.Ex34.m1.1.1.1.1.1.3">𝒮</ci><ci id="S5.Ex34.m1.1.1.1.1.1.4.cmml" xref="S5.Ex34.m1.1.1.1.1.1.4">𝒲</ci><apply id="S5.Ex34.m1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex34.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Ex34.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex34.m1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S5.Ex34.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.Ex34.m1.1.1.1.1.1.1.1.1.2">𝜋</ci><times id="S5.Ex34.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex34.m1.1.1.1.1.1.1.1.1.3"></times></apply></apply><apply id="S5.Ex34.m1.1.1.1.1.2.cmml" xref="S5.Ex34.m1.1.1.1.1.2"><times id="S5.Ex34.m1.1.1.1.1.2.2.cmml" xref="S5.Ex34.m1.1.1.1.1.2.2"></times><ci id="S5.Ex34.m1.1.1.1.1.2.3.cmml" xref="S5.Ex34.m1.1.1.1.1.2.3">𝑛</ci><apply id="S5.Ex34.m1.1.1.1.1.2.1.1.1.cmml" xref="S5.Ex34.m1.1.1.1.1.2.1.1"><minus id="S5.Ex34.m1.1.1.1.1.2.1.1.1.1.cmml" xref="S5.Ex34.m1.1.1.1.1.2.1.1.1.1"></minus><ci id="S5.Ex34.m1.1.1.1.1.2.1.1.1.2.cmml" xref="S5.Ex34.m1.1.1.1.1.2.1.1.1.2">𝑡</ci><cn id="S5.Ex34.m1.1.1.1.1.2.1.1.1.3.cmml" type="integer" xref="S5.Ex34.m1.1.1.1.1.2.1.1.1.3">1</cn></apply></apply></apply><apply id="S5.Ex34.m1.1.1.1.1c.cmml" xref="S5.Ex34.m1.1.1.1"><eq id="S5.Ex34.m1.1.1.1.1.6.cmml" xref="S5.Ex34.m1.1.1.1.1.6"></eq><share href="https://arxiv.org/html/2503.06017v1#S5.Ex34.m1.1.1.1.1.2.cmml" id="S5.Ex34.m1.1.1.1.1d.cmml" xref="S5.Ex34.m1.1.1.1"></share><apply id="S5.Ex34.m1.1.1.1.1.7.cmml" xref="S5.Ex34.m1.1.1.1.1.7"><times id="S5.Ex34.m1.1.1.1.1.7.1.cmml" xref="S5.Ex34.m1.1.1.1.1.7.1"></times><apply id="S5.Ex34.m1.1.1.1.1.7.2.cmml" xref="S5.Ex34.m1.1.1.1.1.7.2"><divide id="S5.Ex34.m1.1.1.1.1.7.2.1.cmml" xref="S5.Ex34.m1.1.1.1.1.7.2"></divide><apply id="S5.Ex34.m1.1.1.1.1.7.2.2.cmml" xref="S5.Ex34.m1.1.1.1.1.7.2.2"><times id="S5.Ex34.m1.1.1.1.1.7.2.2.1.cmml" xref="S5.Ex34.m1.1.1.1.1.7.2.2.1"></times><cn id="S5.Ex34.m1.1.1.1.1.7.2.2.2.cmml" type="integer" xref="S5.Ex34.m1.1.1.1.1.7.2.2.2">2</cn><ci id="S5.Ex34.m1.1.1.1.1.7.2.2.3.cmml" xref="S5.Ex34.m1.1.1.1.1.7.2.2.3">𝑛</ci></apply><ci id="S5.Ex34.m1.1.1.1.1.7.2.3.cmml" xref="S5.Ex34.m1.1.1.1.1.7.2.3">𝑐</ci></apply><apply id="S5.Ex34.m1.1.1.1.1.7.3.cmml" xref="S5.Ex34.m1.1.1.1.1.7.3"><apply id="S5.Ex34.m1.1.1.1.1.7.3.1.cmml" xref="S5.Ex34.m1.1.1.1.1.7.3.1"><csymbol cd="ambiguous" id="S5.Ex34.m1.1.1.1.1.7.3.1.1.cmml" xref="S5.Ex34.m1.1.1.1.1.7.3.1">subscript</csymbol><log id="S5.Ex34.m1.1.1.1.1.7.3.1.2.cmml" xref="S5.Ex34.m1.1.1.1.1.7.3.1.2"></log><ci id="S5.Ex34.m1.1.1.1.1.7.3.1.3.cmml" xref="S5.Ex34.m1.1.1.1.1.7.3.1.3">𝑒</ci></apply><ci id="S5.Ex34.m1.1.1.1.1.7.3.2.cmml" xref="S5.Ex34.m1.1.1.1.1.7.3.2">𝑛</ci></apply></apply></apply><apply id="S5.Ex34.m1.1.1.1.1e.cmml" xref="S5.Ex34.m1.1.1.1"><eq id="S5.Ex34.m1.1.1.1.1.8.cmml" xref="S5.Ex34.m1.1.1.1.1.8"></eq><share href="https://arxiv.org/html/2503.06017v1#S5.Ex34.m1.1.1.1.1.7.cmml" id="S5.Ex34.m1.1.1.1.1f.cmml" xref="S5.Ex34.m1.1.1.1"></share><apply id="S5.Ex34.m1.1.1.1.1.3.cmml" xref="S5.Ex34.m1.1.1.1.1.3"><times id="S5.Ex34.m1.1.1.1.1.3.2.cmml" xref="S5.Ex34.m1.1.1.1.1.3.2"></times><ci id="S5.Ex34.m1.1.1.1.1.3.3.cmml" xref="S5.Ex34.m1.1.1.1.1.3.3">𝒪</ci><apply id="S5.Ex34.m1.1.1.1.1.3.1.1.1.cmml" xref="S5.Ex34.m1.1.1.1.1.3.1.1"><times id="S5.Ex34.m1.1.1.1.1.3.1.1.1.1.cmml" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.1"></times><ci id="S5.Ex34.m1.1.1.1.1.3.1.1.1.2.cmml" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.2">𝑛</ci><apply id="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.cmml" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.3"><apply id="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.1.cmml" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.1"><csymbol cd="ambiguous" id="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.1.1.cmml" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.1">subscript</csymbol><log id="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.1.2.cmml" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.1.2"></log><ci id="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.1.3.cmml" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.1.3">𝑒</ci></apply><ci id="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.2.cmml" xref="S5.Ex34.m1.1.1.1.1.3.1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex34.m1.1c">\mathcal{SW}(\pi^{*})\leq n(t-1)=\frac{2n}{c}\log_{e}n=\mathcal{O}(n\log_{e}n).</annotation><annotation encoding="application/x-llamapun" id="S5.Ex34.m1.1d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ≤ italic_n ( italic_t - 1 ) = divide start_ARG 2 italic_n end_ARG start_ARG italic_c end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n = caligraphic_O ( italic_n roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS2.SSS2.3.p2.13">This completes the proof. ∎</p> </div> </div> <div class="ltx_para" id="S5.SS2.SSS2.p3"> <p class="ltx_p" id="S5.SS2.SSS2.p3.1">We now prove our main result for the high perturbation regime. Again, the theorem extends to random balanced <math alttext="k" class="ltx_Math" display="inline" id="S5.SS2.SSS2.p3.1.m1.1"><semantics id="S5.SS2.SSS2.p3.1.m1.1a"><mi id="S5.SS2.SSS2.p3.1.m1.1.1" xref="S5.SS2.SSS2.p3.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.p3.1.m1.1b"><ci id="S5.SS2.SSS2.p3.1.m1.1.1.cmml" xref="S5.SS2.SSS2.p3.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.p3.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.p3.1.m1.1d">italic_k</annotation></semantics></math>-partite graphs.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S5.Thmtheorem11"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem11.1.1.1">Theorem 5.11</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem11.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem11.p1"> <p class="ltx_p" id="S5.Thmtheorem11.p1.5"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem11.p1.5.5">Consider aversion-to-enemies games given by random Turán graphs <math alttext="G=(n,k,p)" class="ltx_Math" display="inline" id="S5.Thmtheorem11.p1.1.1.m1.3"><semantics id="S5.Thmtheorem11.p1.1.1.m1.3a"><mrow id="S5.Thmtheorem11.p1.1.1.m1.3.4" xref="S5.Thmtheorem11.p1.1.1.m1.3.4.cmml"><mi id="S5.Thmtheorem11.p1.1.1.m1.3.4.2" xref="S5.Thmtheorem11.p1.1.1.m1.3.4.2.cmml">G</mi><mo id="S5.Thmtheorem11.p1.1.1.m1.3.4.1" xref="S5.Thmtheorem11.p1.1.1.m1.3.4.1.cmml">=</mo><mrow id="S5.Thmtheorem11.p1.1.1.m1.3.4.3.2" xref="S5.Thmtheorem11.p1.1.1.m1.3.4.3.1.cmml"><mo id="S5.Thmtheorem11.p1.1.1.m1.3.4.3.2.1" stretchy="false" xref="S5.Thmtheorem11.p1.1.1.m1.3.4.3.1.cmml">(</mo><mi id="S5.Thmtheorem11.p1.1.1.m1.1.1" xref="S5.Thmtheorem11.p1.1.1.m1.1.1.cmml">n</mi><mo id="S5.Thmtheorem11.p1.1.1.m1.3.4.3.2.2" xref="S5.Thmtheorem11.p1.1.1.m1.3.4.3.1.cmml">,</mo><mi id="S5.Thmtheorem11.p1.1.1.m1.2.2" xref="S5.Thmtheorem11.p1.1.1.m1.2.2.cmml">k</mi><mo id="S5.Thmtheorem11.p1.1.1.m1.3.4.3.2.3" xref="S5.Thmtheorem11.p1.1.1.m1.3.4.3.1.cmml">,</mo><mi id="S5.Thmtheorem11.p1.1.1.m1.3.3" xref="S5.Thmtheorem11.p1.1.1.m1.3.3.cmml">p</mi><mo id="S5.Thmtheorem11.p1.1.1.m1.3.4.3.2.4" stretchy="false" xref="S5.Thmtheorem11.p1.1.1.m1.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem11.p1.1.1.m1.3b"><apply id="S5.Thmtheorem11.p1.1.1.m1.3.4.cmml" xref="S5.Thmtheorem11.p1.1.1.m1.3.4"><eq id="S5.Thmtheorem11.p1.1.1.m1.3.4.1.cmml" xref="S5.Thmtheorem11.p1.1.1.m1.3.4.1"></eq><ci id="S5.Thmtheorem11.p1.1.1.m1.3.4.2.cmml" xref="S5.Thmtheorem11.p1.1.1.m1.3.4.2">𝐺</ci><vector id="S5.Thmtheorem11.p1.1.1.m1.3.4.3.1.cmml" xref="S5.Thmtheorem11.p1.1.1.m1.3.4.3.2"><ci id="S5.Thmtheorem11.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem11.p1.1.1.m1.1.1">𝑛</ci><ci id="S5.Thmtheorem11.p1.1.1.m1.2.2.cmml" xref="S5.Thmtheorem11.p1.1.1.m1.2.2">𝑘</ci><ci id="S5.Thmtheorem11.p1.1.1.m1.3.3.cmml" xref="S5.Thmtheorem11.p1.1.1.m1.3.3">𝑝</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem11.p1.1.1.m1.3c">G=(n,k,p)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem11.p1.1.1.m1.3d">italic_G = ( italic_n , italic_k , italic_p )</annotation></semantics></math>, where <math alttext="p=c" class="ltx_Math" display="inline" id="S5.Thmtheorem11.p1.2.2.m2.1"><semantics id="S5.Thmtheorem11.p1.2.2.m2.1a"><mrow id="S5.Thmtheorem11.p1.2.2.m2.1.1" xref="S5.Thmtheorem11.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem11.p1.2.2.m2.1.1.2" xref="S5.Thmtheorem11.p1.2.2.m2.1.1.2.cmml">p</mi><mo id="S5.Thmtheorem11.p1.2.2.m2.1.1.1" xref="S5.Thmtheorem11.p1.2.2.m2.1.1.1.cmml">=</mo><mi id="S5.Thmtheorem11.p1.2.2.m2.1.1.3" xref="S5.Thmtheorem11.p1.2.2.m2.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem11.p1.2.2.m2.1b"><apply id="S5.Thmtheorem11.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem11.p1.2.2.m2.1.1"><eq id="S5.Thmtheorem11.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem11.p1.2.2.m2.1.1.1"></eq><ci id="S5.Thmtheorem11.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem11.p1.2.2.m2.1.1.2">𝑝</ci><ci id="S5.Thmtheorem11.p1.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem11.p1.2.2.m2.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem11.p1.2.2.m2.1c">p=c</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem11.p1.2.2.m2.1d">italic_p = italic_c</annotation></semantics></math> for some constant <math alttext="c\in(0,1)" class="ltx_Math" display="inline" id="S5.Thmtheorem11.p1.3.3.m3.2"><semantics id="S5.Thmtheorem11.p1.3.3.m3.2a"><mrow id="S5.Thmtheorem11.p1.3.3.m3.2.3" xref="S5.Thmtheorem11.p1.3.3.m3.2.3.cmml"><mi id="S5.Thmtheorem11.p1.3.3.m3.2.3.2" xref="S5.Thmtheorem11.p1.3.3.m3.2.3.2.cmml">c</mi><mo id="S5.Thmtheorem11.p1.3.3.m3.2.3.1" xref="S5.Thmtheorem11.p1.3.3.m3.2.3.1.cmml">∈</mo><mrow id="S5.Thmtheorem11.p1.3.3.m3.2.3.3.2" xref="S5.Thmtheorem11.p1.3.3.m3.2.3.3.1.cmml"><mo id="S5.Thmtheorem11.p1.3.3.m3.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem11.p1.3.3.m3.2.3.3.1.cmml">(</mo><mn id="S5.Thmtheorem11.p1.3.3.m3.1.1" xref="S5.Thmtheorem11.p1.3.3.m3.1.1.cmml">0</mn><mo id="S5.Thmtheorem11.p1.3.3.m3.2.3.3.2.2" xref="S5.Thmtheorem11.p1.3.3.m3.2.3.3.1.cmml">,</mo><mn id="S5.Thmtheorem11.p1.3.3.m3.2.2" xref="S5.Thmtheorem11.p1.3.3.m3.2.2.cmml">1</mn><mo id="S5.Thmtheorem11.p1.3.3.m3.2.3.3.2.3" stretchy="false" xref="S5.Thmtheorem11.p1.3.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem11.p1.3.3.m3.2b"><apply id="S5.Thmtheorem11.p1.3.3.m3.2.3.cmml" xref="S5.Thmtheorem11.p1.3.3.m3.2.3"><in id="S5.Thmtheorem11.p1.3.3.m3.2.3.1.cmml" xref="S5.Thmtheorem11.p1.3.3.m3.2.3.1"></in><ci id="S5.Thmtheorem11.p1.3.3.m3.2.3.2.cmml" xref="S5.Thmtheorem11.p1.3.3.m3.2.3.2">𝑐</ci><interval closure="open" id="S5.Thmtheorem11.p1.3.3.m3.2.3.3.1.cmml" xref="S5.Thmtheorem11.p1.3.3.m3.2.3.3.2"><cn id="S5.Thmtheorem11.p1.3.3.m3.1.1.cmml" type="integer" xref="S5.Thmtheorem11.p1.3.3.m3.1.1">0</cn><cn id="S5.Thmtheorem11.p1.3.3.m3.2.2.cmml" type="integer" xref="S5.Thmtheorem11.p1.3.3.m3.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem11.p1.3.3.m3.2c">c\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem11.p1.3.3.m3.2d">italic_c ∈ ( 0 , 1 )</annotation></semantics></math>. Then there exists a polynomial-time algorithm that returns a partition that provides a <math alttext="\mathcal{O}(\log n)" class="ltx_Math" display="inline" id="S5.Thmtheorem11.p1.4.4.m4.1"><semantics id="S5.Thmtheorem11.p1.4.4.m4.1a"><mrow id="S5.Thmtheorem11.p1.4.4.m4.1.1" xref="S5.Thmtheorem11.p1.4.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem11.p1.4.4.m4.1.1.3" xref="S5.Thmtheorem11.p1.4.4.m4.1.1.3.cmml">𝒪</mi><mo id="S5.Thmtheorem11.p1.4.4.m4.1.1.2" xref="S5.Thmtheorem11.p1.4.4.m4.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1" xref="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1" xref="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1.1" xref="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1.1.cmml">log</mi><mo id="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1a" lspace="0.167em" xref="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1.cmml">⁡</mo><mi id="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1.2" xref="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem11.p1.4.4.m4.1b"><apply id="S5.Thmtheorem11.p1.4.4.m4.1.1.cmml" xref="S5.Thmtheorem11.p1.4.4.m4.1.1"><times id="S5.Thmtheorem11.p1.4.4.m4.1.1.2.cmml" xref="S5.Thmtheorem11.p1.4.4.m4.1.1.2"></times><ci id="S5.Thmtheorem11.p1.4.4.m4.1.1.3.cmml" xref="S5.Thmtheorem11.p1.4.4.m4.1.1.3">𝒪</ci><apply id="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1.cmml" xref="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1"><log id="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1.1"></log><ci id="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem11.p1.4.4.m4.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem11.p1.4.4.m4.1c">\mathcal{O}(\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem11.p1.4.4.m4.1d">caligraphic_O ( roman_log italic_n )</annotation></semantics></math>-approximation of the maximum welfare with probability <math alttext="1-ne^{-\Theta\left(\frac{n}{k}\right)}-\left(\frac{ce}{2\log_{e}n}\right)^{% \frac{2}{c}\log_{e}n+1}" class="ltx_Math" display="inline" id="S5.Thmtheorem11.p1.5.5.m5.2"><semantics id="S5.Thmtheorem11.p1.5.5.m5.2a"><mrow id="S5.Thmtheorem11.p1.5.5.m5.2.3" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.cmml"><mn id="S5.Thmtheorem11.p1.5.5.m5.2.3.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.2.cmml">1</mn><mo id="S5.Thmtheorem11.p1.5.5.m5.2.3.1" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.1.cmml">−</mo><mrow id="S5.Thmtheorem11.p1.5.5.m5.2.3.3" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.3.cmml"><mi id="S5.Thmtheorem11.p1.5.5.m5.2.3.3.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.3.2.cmml">n</mi><mo id="S5.Thmtheorem11.p1.5.5.m5.2.3.3.1" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.3.1.cmml">⁢</mo><msup id="S5.Thmtheorem11.p1.5.5.m5.2.3.3.3" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.3.3.cmml"><mi id="S5.Thmtheorem11.p1.5.5.m5.2.3.3.3.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.3.3.2.cmml">e</mi><mrow id="S5.Thmtheorem11.p1.5.5.m5.1.1.1" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.cmml"><mo id="S5.Thmtheorem11.p1.5.5.m5.1.1.1a" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.cmml">−</mo><mrow id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3.cmml"><mi id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3.2" mathvariant="normal" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3.2.cmml">Θ</mi><mo id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3.1" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3.3.2" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.1.cmml"><mo id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3.3.2.1" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.1.cmml">(</mo><mfrac id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.1" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.1.cmml"><mi id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.1.2" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.1.2.cmml">n</mi><mi id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.1.3" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.1.3.cmml">k</mi></mfrac><mo id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3.3.2.2" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><mo id="S5.Thmtheorem11.p1.5.5.m5.2.3.1a" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.1.cmml">−</mo><msup id="S5.Thmtheorem11.p1.5.5.m5.2.3.4" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.cmml"><mrow id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.2.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.cmml"><mo id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.2.2.1" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.cmml">(</mo><mfrac id="S5.Thmtheorem11.p1.5.5.m5.2.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.cmml"><mrow id="S5.Thmtheorem11.p1.5.5.m5.2.2.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.2.cmml"><mi id="S5.Thmtheorem11.p1.5.5.m5.2.2.2.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.2.2.cmml">c</mi><mo id="S5.Thmtheorem11.p1.5.5.m5.2.2.2.1" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem11.p1.5.5.m5.2.2.2.3" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.2.3.cmml">e</mi></mrow><mrow id="S5.Thmtheorem11.p1.5.5.m5.2.2.3" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.cmml"><mn id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.2.cmml">2</mn><mo id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.1" lspace="0.167em" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.cmml"><msub id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.1" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.1.cmml"><mi id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.1.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.1.2.cmml">log</mi><mi id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.1.3" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.1.3.cmml">e</mi></msub><mo id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3a" lspace="0.167em" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.cmml">⁡</mo><mi id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.2.cmml">n</mi></mrow></mrow></mfrac><mo id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.2.2.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.cmml">)</mo></mrow><mrow id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.cmml"><mrow id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.cmml"><mfrac id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.2.cmml"><mn id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.2.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.2.2.cmml">2</mn><mi id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.2.3" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.2.3.cmml">c</mi></mfrac><mo id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.1" lspace="0.167em" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.cmml"><msub id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.1" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.1.cmml"><mi id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.1.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.1.2.cmml">log</mi><mi id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.1.3" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.1.3.cmml">e</mi></msub><mo id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3a" lspace="0.167em" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.cmml">⁡</mo><mi id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.2" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.2.cmml">n</mi></mrow></mrow><mo id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.1" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.1.cmml">+</mo><mn id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.3" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem11.p1.5.5.m5.2b"><apply id="S5.Thmtheorem11.p1.5.5.m5.2.3.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3"><minus id="S5.Thmtheorem11.p1.5.5.m5.2.3.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.1"></minus><cn id="S5.Thmtheorem11.p1.5.5.m5.2.3.2.cmml" type="integer" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.2">1</cn><apply id="S5.Thmtheorem11.p1.5.5.m5.2.3.3.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.3"><times id="S5.Thmtheorem11.p1.5.5.m5.2.3.3.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.3.1"></times><ci id="S5.Thmtheorem11.p1.5.5.m5.2.3.3.2.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.3.2">𝑛</ci><apply id="S5.Thmtheorem11.p1.5.5.m5.2.3.3.3.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem11.p1.5.5.m5.2.3.3.3.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.3.3">superscript</csymbol><ci id="S5.Thmtheorem11.p1.5.5.m5.2.3.3.3.2.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.3.3.2">𝑒</ci><apply id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1"><minus id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.2.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1"></minus><apply id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3"><times id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3.1"></times><ci id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3.2.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3.2">Θ</ci><apply id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3.3.2"><divide id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.1.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.3.3.2"></divide><ci id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.1.2.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.1.2">𝑛</ci><ci id="S5.Thmtheorem11.p1.5.5.m5.1.1.1.1.3.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply><apply id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4"><csymbol cd="ambiguous" id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4">superscript</csymbol><apply id="S5.Thmtheorem11.p1.5.5.m5.2.2.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.2.2"><divide id="S5.Thmtheorem11.p1.5.5.m5.2.2.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.2.2"></divide><apply id="S5.Thmtheorem11.p1.5.5.m5.2.2.2.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.2"><times id="S5.Thmtheorem11.p1.5.5.m5.2.2.2.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.2.1"></times><ci id="S5.Thmtheorem11.p1.5.5.m5.2.2.2.2.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.2.2">𝑐</ci><ci id="S5.Thmtheorem11.p1.5.5.m5.2.2.2.3.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.2.3">𝑒</ci></apply><apply id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3"><times id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.1"></times><cn id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.2.cmml" type="integer" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.2">2</cn><apply id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3"><apply id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.1"><csymbol cd="ambiguous" id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.1.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.1">subscript</csymbol><log id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.1.2.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.1.2"></log><ci id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.1.3.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.1.3">𝑒</ci></apply><ci id="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.2.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.2.3.3.2">𝑛</ci></apply></apply></apply><apply id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3"><plus id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.1"></plus><apply id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2"><times id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.1"></times><apply id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.2.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.2"><divide id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.2.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.2"></divide><cn id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.2.2.cmml" type="integer" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.2.2">2</cn><ci id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.2.3.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.2.3">𝑐</ci></apply><apply id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3"><apply id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.1"><csymbol cd="ambiguous" id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.1.1.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.1">subscript</csymbol><log id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.1.2.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.1.2"></log><ci id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.1.3.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.1.3">𝑒</ci></apply><ci id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.2.cmml" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.2.3.2">𝑛</ci></apply></apply><cn id="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.3.cmml" type="integer" xref="S5.Thmtheorem11.p1.5.5.m5.2.3.4.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem11.p1.5.5.m5.2c">1-ne^{-\Theta\left(\frac{n}{k}\right)}-\left(\frac{ce}{2\log_{e}n}\right)^{% \frac{2}{c}\log_{e}n+1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem11.p1.5.5.m5.2d">1 - italic_n italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT - ( divide start_ARG italic_c italic_e end_ARG start_ARG 2 roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 2 end_ARG start_ARG italic_c end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n + 1 end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S5.SS2.SSS2.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.SS2.SSS2.4.p1"> <p class="ltx_p" id="S5.SS2.SSS2.4.p1.9"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem9" title="Lemma 5.9. ‣ 5.2.2 High Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5.9</span></a> implies that <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg2" title="In 5.2.2 High Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">2</span></a> returns a partition <math alttext="\pi" class="ltx_Math" display="inline" id="S5.SS2.SSS2.4.p1.1.m1.1"><semantics id="S5.SS2.SSS2.4.p1.1.m1.1a"><mi id="S5.SS2.SSS2.4.p1.1.m1.1.1" xref="S5.SS2.SSS2.4.p1.1.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.4.p1.1.m1.1b"><ci id="S5.SS2.SSS2.4.p1.1.m1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.1.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.4.p1.1.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.4.p1.1.m1.1d">italic_π</annotation></semantics></math> where <math alttext="\mathcal{SW}(\pi)=\Omega(n)" class="ltx_Math" display="inline" id="S5.SS2.SSS2.4.p1.2.m2.2"><semantics id="S5.SS2.SSS2.4.p1.2.m2.2a"><mrow id="S5.SS2.SSS2.4.p1.2.m2.2.3" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.cmml"><mrow id="S5.SS2.SSS2.4.p1.2.m2.2.3.2" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS2.4.p1.2.m2.2.3.2.2" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.2.2.cmml">𝒮</mi><mo id="S5.SS2.SSS2.4.p1.2.m2.2.3.2.1" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS2.4.p1.2.m2.2.3.2.3" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.2.3.cmml">𝒲</mi><mo id="S5.SS2.SSS2.4.p1.2.m2.2.3.2.1a" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.2.m2.2.3.2.4.2" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.2.cmml"><mo id="S5.SS2.SSS2.4.p1.2.m2.2.3.2.4.2.1" stretchy="false" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.2.cmml">(</mo><mi id="S5.SS2.SSS2.4.p1.2.m2.1.1" xref="S5.SS2.SSS2.4.p1.2.m2.1.1.cmml">π</mi><mo id="S5.SS2.SSS2.4.p1.2.m2.2.3.2.4.2.2" stretchy="false" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.SS2.SSS2.4.p1.2.m2.2.3.1" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.1.cmml">=</mo><mrow id="S5.SS2.SSS2.4.p1.2.m2.2.3.3" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.3.cmml"><mi id="S5.SS2.SSS2.4.p1.2.m2.2.3.3.2" mathvariant="normal" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.3.2.cmml">Ω</mi><mo id="S5.SS2.SSS2.4.p1.2.m2.2.3.3.1" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.2.m2.2.3.3.3.2" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.3.cmml"><mo id="S5.SS2.SSS2.4.p1.2.m2.2.3.3.3.2.1" stretchy="false" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.3.cmml">(</mo><mi id="S5.SS2.SSS2.4.p1.2.m2.2.2" xref="S5.SS2.SSS2.4.p1.2.m2.2.2.cmml">n</mi><mo id="S5.SS2.SSS2.4.p1.2.m2.2.3.3.3.2.2" stretchy="false" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.4.p1.2.m2.2b"><apply id="S5.SS2.SSS2.4.p1.2.m2.2.3.cmml" xref="S5.SS2.SSS2.4.p1.2.m2.2.3"><eq id="S5.SS2.SSS2.4.p1.2.m2.2.3.1.cmml" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.1"></eq><apply id="S5.SS2.SSS2.4.p1.2.m2.2.3.2.cmml" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.2"><times id="S5.SS2.SSS2.4.p1.2.m2.2.3.2.1.cmml" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.2.1"></times><ci id="S5.SS2.SSS2.4.p1.2.m2.2.3.2.2.cmml" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.2.2">𝒮</ci><ci id="S5.SS2.SSS2.4.p1.2.m2.2.3.2.3.cmml" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.2.3">𝒲</ci><ci id="S5.SS2.SSS2.4.p1.2.m2.1.1.cmml" xref="S5.SS2.SSS2.4.p1.2.m2.1.1">𝜋</ci></apply><apply id="S5.SS2.SSS2.4.p1.2.m2.2.3.3.cmml" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.3"><times id="S5.SS2.SSS2.4.p1.2.m2.2.3.3.1.cmml" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.3.1"></times><ci id="S5.SS2.SSS2.4.p1.2.m2.2.3.3.2.cmml" xref="S5.SS2.SSS2.4.p1.2.m2.2.3.3.2">Ω</ci><ci id="S5.SS2.SSS2.4.p1.2.m2.2.2.cmml" xref="S5.SS2.SSS2.4.p1.2.m2.2.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.4.p1.2.m2.2c">\mathcal{SW}(\pi)=\Omega(n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.4.p1.2.m2.2d">caligraphic_S caligraphic_W ( italic_π ) = roman_Ω ( italic_n )</annotation></semantics></math> with probability <math alttext="1-ne^{-\Theta\left(\frac{n}{k}\right)}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.4.p1.3.m3.1"><semantics id="S5.SS2.SSS2.4.p1.3.m3.1a"><mrow id="S5.SS2.SSS2.4.p1.3.m3.1.2" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.cmml"><mn id="S5.SS2.SSS2.4.p1.3.m3.1.2.2" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.2.cmml">1</mn><mo id="S5.SS2.SSS2.4.p1.3.m3.1.2.1" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.1.cmml">−</mo><mrow id="S5.SS2.SSS2.4.p1.3.m3.1.2.3" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.3.cmml"><mi id="S5.SS2.SSS2.4.p1.3.m3.1.2.3.2" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.3.2.cmml">n</mi><mo id="S5.SS2.SSS2.4.p1.3.m3.1.2.3.1" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.3.1.cmml">⁢</mo><msup id="S5.SS2.SSS2.4.p1.3.m3.1.2.3.3" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.3.3.cmml"><mi id="S5.SS2.SSS2.4.p1.3.m3.1.2.3.3.2" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.3.3.2.cmml">e</mi><mrow id="S5.SS2.SSS2.4.p1.3.m3.1.1.1" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.cmml"><mo id="S5.SS2.SSS2.4.p1.3.m3.1.1.1a" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.cmml">−</mo><mrow id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3.cmml"><mi id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3.2" mathvariant="normal" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3.2.cmml">Θ</mi><mo id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3.1" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3.3.2" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.1.cmml"><mo id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3.3.2.1" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.1" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.1.2" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.1.2.cmml">n</mi><mi id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.1.3" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.1.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3.3.2.2" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.4.p1.3.m3.1b"><apply id="S5.SS2.SSS2.4.p1.3.m3.1.2.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.2"><minus id="S5.SS2.SSS2.4.p1.3.m3.1.2.1.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.1"></minus><cn id="S5.SS2.SSS2.4.p1.3.m3.1.2.2.cmml" type="integer" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.2">1</cn><apply id="S5.SS2.SSS2.4.p1.3.m3.1.2.3.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.3"><times id="S5.SS2.SSS2.4.p1.3.m3.1.2.3.1.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.3.1"></times><ci id="S5.SS2.SSS2.4.p1.3.m3.1.2.3.2.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.3.2">𝑛</ci><apply id="S5.SS2.SSS2.4.p1.3.m3.1.2.3.3.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.3.3"><csymbol cd="ambiguous" id="S5.SS2.SSS2.4.p1.3.m3.1.2.3.3.1.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.3.3">superscript</csymbol><ci id="S5.SS2.SSS2.4.p1.3.m3.1.2.3.3.2.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.2.3.3.2">𝑒</ci><apply id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1"><minus id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1"></minus><apply id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3"><times id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3.1.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3.1"></times><ci id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3.2.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3.2">Θ</ci><apply id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3.3.2"><divide id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.3.3.2"></divide><ci id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.1.2">𝑛</ci><ci id="S5.SS2.SSS2.4.p1.3.m3.1.1.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.3.m3.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.4.p1.3.m3.1c">1-ne^{-\Theta\left(\frac{n}{k}\right)}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.4.p1.3.m3.1d">1 - italic_n italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT</annotation></semantics></math>. A simple upper bound on the maximum welfare is provided in <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem6" title="Proposition 5.6. ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">5.6</span></a>, which implies <math alttext="\mathcal{SW}(\pi^{*})\leq n(k-1)" class="ltx_Math" display="inline" id="S5.SS2.SSS2.4.p1.4.m4.2"><semantics id="S5.SS2.SSS2.4.p1.4.m4.2a"><mrow id="S5.SS2.SSS2.4.p1.4.m4.2.2" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.cmml"><mrow id="S5.SS2.SSS2.4.p1.4.m4.1.1.1" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.3" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.3.cmml">𝒮</mi><mo id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.2" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.4" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.4.cmml">𝒲</mi><mo id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.2a" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.1" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.1.2" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.1.2.cmml">π</mi><mo id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.1.3" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS2.SSS2.4.p1.4.m4.2.2.3" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.3.cmml">≤</mo><mrow id="S5.SS2.SSS2.4.p1.4.m4.2.2.2" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.cmml"><mi id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.3" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.3.cmml">n</mi><mo id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.2" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.2.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.cmml"><mo id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.2" stretchy="false" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.cmml"><mi id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.2" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.2.cmml">k</mi><mo id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.1" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.1.cmml">−</mo><mn id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.3" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.3" stretchy="false" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.4.p1.4.m4.2b"><apply id="S5.SS2.SSS2.4.p1.4.m4.2.2.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.2.2"><leq id="S5.SS2.SSS2.4.p1.4.m4.2.2.3.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.3"></leq><apply id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1"><times id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.2"></times><ci id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.3">𝒮</ci><ci id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.4.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.4">𝒲</ci><apply id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.1.2">𝜋</ci><times id="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.1.1.1.1.1.1.3"></times></apply></apply><apply id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2"><times id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.2.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.2"></times><ci id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.3.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.3">𝑛</ci><apply id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1"><minus id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.1"></minus><ci id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.2">𝑘</ci><cn id="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.3.cmml" type="integer" xref="S5.SS2.SSS2.4.p1.4.m4.2.2.2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.4.p1.4.m4.2c">\mathcal{SW}(\pi^{*})\leq n(k-1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.4.p1.4.m4.2d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ≤ italic_n ( italic_k - 1 )</annotation></semantics></math>. If <math alttext="k=\mathcal{O}(\log n)" class="ltx_Math" display="inline" id="S5.SS2.SSS2.4.p1.5.m5.1"><semantics id="S5.SS2.SSS2.4.p1.5.m5.1a"><mrow id="S5.SS2.SSS2.4.p1.5.m5.1.1" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.cmml"><mi id="S5.SS2.SSS2.4.p1.5.m5.1.1.3" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.3.cmml">k</mi><mo id="S5.SS2.SSS2.4.p1.5.m5.1.1.2" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.2.cmml">=</mo><mrow id="S5.SS2.SSS2.4.p1.5.m5.1.1.1" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS2.4.p1.5.m5.1.1.1.3" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1.3.cmml">𝒪</mi><mo id="S5.SS2.SSS2.4.p1.5.m5.1.1.1.2" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1.cmml"><mo id="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1.1" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1.1.cmml">log</mi><mo id="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1a" lspace="0.167em" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1.cmml">⁡</mo><mi id="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1.2" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.4.p1.5.m5.1b"><apply id="S5.SS2.SSS2.4.p1.5.m5.1.1.cmml" xref="S5.SS2.SSS2.4.p1.5.m5.1.1"><eq id="S5.SS2.SSS2.4.p1.5.m5.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.2"></eq><ci id="S5.SS2.SSS2.4.p1.5.m5.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.3">𝑘</ci><apply id="S5.SS2.SSS2.4.p1.5.m5.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1"><times id="S5.SS2.SSS2.4.p1.5.m5.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1.2"></times><ci id="S5.SS2.SSS2.4.p1.5.m5.1.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1.3">𝒪</ci><apply id="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1"><log id="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1.1"></log><ci id="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.5.m5.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.4.p1.5.m5.1c">k=\mathcal{O}(\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.4.p1.5.m5.1d">italic_k = caligraphic_O ( roman_log italic_n )</annotation></semantics></math>, this results in <math alttext="\mathcal{SW}(\pi^{*})=\mathcal{O}(n\log n)" class="ltx_Math" display="inline" id="S5.SS2.SSS2.4.p1.6.m6.2"><semantics id="S5.SS2.SSS2.4.p1.6.m6.2a"><mrow id="S5.SS2.SSS2.4.p1.6.m6.2.2" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.cmml"><mrow id="S5.SS2.SSS2.4.p1.6.m6.1.1.1" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.3" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.3.cmml">𝒮</mi><mo id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.2" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.4" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.4.cmml">𝒲</mi><mo id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.2a" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.1.cmml"><mo id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.1.cmml">(</mo><msup id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.1" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.1.2" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.1.2.cmml">π</mi><mo id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.1.3" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS2.SSS2.4.p1.6.m6.2.2.3" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.3.cmml">=</mo><mrow id="S5.SS2.SSS2.4.p1.6.m6.2.2.2" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.3" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.3.cmml">𝒪</mi><mo id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.2" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.2.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.cmml"><mo id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.2" stretchy="false" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.cmml"><mi id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.2" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.2.cmml">n</mi><mo id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.1" lspace="0.167em" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.3" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.3.cmml"><mi id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.3.1" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.3.1.cmml">log</mi><mo id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.3a" lspace="0.167em" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.3.cmml">⁡</mo><mi id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.3.2" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.3" stretchy="false" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.4.p1.6.m6.2b"><apply id="S5.SS2.SSS2.4.p1.6.m6.2.2.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.2.2"><eq id="S5.SS2.SSS2.4.p1.6.m6.2.2.3.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.3"></eq><apply id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1"><times id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.2"></times><ci id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.3">𝒮</ci><ci id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.4.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.4">𝒲</ci><apply id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1">superscript</csymbol><ci id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.1.2">𝜋</ci><times id="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.1.1.1.1.1.1.3"></times></apply></apply><apply id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2"><times id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.2.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.2"></times><ci id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.3.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.3">𝒪</ci><apply id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1"><times id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.1"></times><ci id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.2">𝑛</ci><apply id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.3"><log id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.3.1.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.3.1"></log><ci id="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.3.2.cmml" xref="S5.SS2.SSS2.4.p1.6.m6.2.2.2.1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.4.p1.6.m6.2c">\mathcal{SW}(\pi^{*})=\mathcal{O}(n\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.4.p1.6.m6.2d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = caligraphic_O ( italic_n roman_log italic_n )</annotation></semantics></math>. However, when <math alttext="k=\Omega(\log n)" class="ltx_Math" display="inline" id="S5.SS2.SSS2.4.p1.7.m7.1"><semantics id="S5.SS2.SSS2.4.p1.7.m7.1a"><mrow id="S5.SS2.SSS2.4.p1.7.m7.1.1" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.cmml"><mi id="S5.SS2.SSS2.4.p1.7.m7.1.1.3" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.3.cmml">k</mi><mo id="S5.SS2.SSS2.4.p1.7.m7.1.1.2" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.2.cmml">=</mo><mrow id="S5.SS2.SSS2.4.p1.7.m7.1.1.1" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1.cmml"><mi id="S5.SS2.SSS2.4.p1.7.m7.1.1.1.3" mathvariant="normal" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1.3.cmml">Ω</mi><mo id="S5.SS2.SSS2.4.p1.7.m7.1.1.1.2" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1.cmml"><mo id="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1.1" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1.1.cmml">log</mi><mo id="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1a" lspace="0.167em" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1.cmml">⁡</mo><mi id="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1.2" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.4.p1.7.m7.1b"><apply id="S5.SS2.SSS2.4.p1.7.m7.1.1.cmml" xref="S5.SS2.SSS2.4.p1.7.m7.1.1"><eq id="S5.SS2.SSS2.4.p1.7.m7.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.2"></eq><ci id="S5.SS2.SSS2.4.p1.7.m7.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.3">𝑘</ci><apply id="S5.SS2.SSS2.4.p1.7.m7.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1"><times id="S5.SS2.SSS2.4.p1.7.m7.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1.2"></times><ci id="S5.SS2.SSS2.4.p1.7.m7.1.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1.3">Ω</ci><apply id="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1"><log id="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1.1"></log><ci id="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.7.m7.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.4.p1.7.m7.1c">k=\Omega(\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.4.p1.7.m7.1d">italic_k = roman_Ω ( roman_log italic_n )</annotation></semantics></math>, it does not provide a useful guarantee. Instead, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem10" title="Lemma 5.10. ‣ 5.2.2 High Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5.10</span></a> shows that even in this case, <math alttext="\mathcal{SW}(\pi^{*})=\mathcal{O}(n\log n)" class="ltx_Math" display="inline" id="S5.SS2.SSS2.4.p1.8.m8.2"><semantics id="S5.SS2.SSS2.4.p1.8.m8.2a"><mrow id="S5.SS2.SSS2.4.p1.8.m8.2.2" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.cmml"><mrow id="S5.SS2.SSS2.4.p1.8.m8.1.1.1" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.3" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.3.cmml">𝒮</mi><mo id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.2" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.4" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.4.cmml">𝒲</mi><mo id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.2a" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.1.cmml"><mo id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.2" stretchy="false" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.1.cmml">(</mo><msup id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.1" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.1.2" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.1.2.cmml">π</mi><mo id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.1.3" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.3" stretchy="false" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS2.SSS2.4.p1.8.m8.2.2.3" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.3.cmml">=</mo><mrow id="S5.SS2.SSS2.4.p1.8.m8.2.2.2" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.3" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.3.cmml">𝒪</mi><mo id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.2" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.2.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.cmml"><mo id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.2" stretchy="false" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.cmml"><mi id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.2" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.2.cmml">n</mi><mo id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.1" lspace="0.167em" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.3" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.3.cmml"><mi id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.3.1" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.3.1.cmml">log</mi><mo id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.3a" lspace="0.167em" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.3.cmml">⁡</mo><mi id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.3.2" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.3" stretchy="false" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.4.p1.8.m8.2b"><apply id="S5.SS2.SSS2.4.p1.8.m8.2.2.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.2.2"><eq id="S5.SS2.SSS2.4.p1.8.m8.2.2.3.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.3"></eq><apply id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1"><times id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.2"></times><ci id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.3">𝒮</ci><ci id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.4.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.4">𝒲</ci><apply id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1">superscript</csymbol><ci id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.1.2">𝜋</ci><times id="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.1.1.1.1.1.1.3"></times></apply></apply><apply id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2"><times id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.2.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.2"></times><ci id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.3.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.3">𝒪</ci><apply id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1"><times id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.1"></times><ci id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.2">𝑛</ci><apply id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.3"><log id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.3.1.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.3.1"></log><ci id="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.3.2.cmml" xref="S5.SS2.SSS2.4.p1.8.m8.2.2.2.1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.4.p1.8.m8.2c">\mathcal{SW}(\pi^{*})=\mathcal{O}(n\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.4.p1.8.m8.2d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = caligraphic_O ( italic_n roman_log italic_n )</annotation></semantics></math> with probability <math alttext="1-\left(\frac{ce}{2\log_{e}n}\right)^{\frac{2}{c}\log_{e}n}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.4.p1.9.m9.1"><semantics id="S5.SS2.SSS2.4.p1.9.m9.1a"><mrow id="S5.SS2.SSS2.4.p1.9.m9.1.2" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.cmml"><mn id="S5.SS2.SSS2.4.p1.9.m9.1.2.2" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.2.cmml">1</mn><mo id="S5.SS2.SSS2.4.p1.9.m9.1.2.1" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.1.cmml">−</mo><msup id="S5.SS2.SSS2.4.p1.9.m9.1.2.3" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.cmml"><mrow id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.2.2" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.cmml"><mo id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.2.2.1" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS2.4.p1.9.m9.1.1" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.cmml"><mrow id="S5.SS2.SSS2.4.p1.9.m9.1.1.2" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.2.cmml"><mi id="S5.SS2.SSS2.4.p1.9.m9.1.1.2.2" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.2.2.cmml">c</mi><mo id="S5.SS2.SSS2.4.p1.9.m9.1.1.2.1" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.2.1.cmml">⁢</mo><mi id="S5.SS2.SSS2.4.p1.9.m9.1.1.2.3" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.2.3.cmml">e</mi></mrow><mrow id="S5.SS2.SSS2.4.p1.9.m9.1.1.3" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.cmml"><mn id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.2" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.2.cmml">2</mn><mo id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.1" lspace="0.167em" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.cmml"><msub id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.1" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.1.cmml"><mi id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.1.2" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.1.2.cmml">log</mi><mi id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.1.3" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.1.3.cmml">e</mi></msub><mo id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3a" lspace="0.167em" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.cmml">⁡</mo><mi id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.2" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.2.cmml">n</mi></mrow></mrow></mfrac><mo id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.2.2.2" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.cmml">)</mo></mrow><mrow id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.cmml"><mfrac id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.2" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.2.cmml"><mn id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.2.2" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.2.2.cmml">2</mn><mi id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.2.3" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.2.3.cmml">c</mi></mfrac><mo id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.1" lspace="0.167em" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.cmml"><msub id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.1" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.1.cmml"><mi id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.1.2" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.1.2.cmml">log</mi><mi id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.1.3" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.1.3.cmml">e</mi></msub><mo id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3a" lspace="0.167em" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.cmml">⁡</mo><mi id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.2" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.2.cmml">n</mi></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.4.p1.9.m9.1b"><apply id="S5.SS2.SSS2.4.p1.9.m9.1.2.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2"><minus id="S5.SS2.SSS2.4.p1.9.m9.1.2.1.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.1"></minus><cn id="S5.SS2.SSS2.4.p1.9.m9.1.2.2.cmml" type="integer" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.2">1</cn><apply id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3"><csymbol cd="ambiguous" id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.1.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3">superscript</csymbol><apply id="S5.SS2.SSS2.4.p1.9.m9.1.1.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.2.2"><divide id="S5.SS2.SSS2.4.p1.9.m9.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.2.2"></divide><apply id="S5.SS2.SSS2.4.p1.9.m9.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.2"><times id="S5.SS2.SSS2.4.p1.9.m9.1.1.2.1.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.2.1"></times><ci id="S5.SS2.SSS2.4.p1.9.m9.1.1.2.2.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.2.2">𝑐</ci><ci id="S5.SS2.SSS2.4.p1.9.m9.1.1.2.3.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.2.3">𝑒</ci></apply><apply id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3"><times id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.1.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.1"></times><cn id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.2.cmml" type="integer" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.2">2</cn><apply id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3"><apply id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.1.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.1.1.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.1">subscript</csymbol><log id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.1.2.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.1.2"></log><ci id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.1.3.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.1.3">𝑒</ci></apply><ci id="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.2.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.1.3.3.2">𝑛</ci></apply></apply></apply><apply id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3"><times id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.1.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.1"></times><apply id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.2.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.2"><divide id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.2.1.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.2"></divide><cn id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.2.2.cmml" type="integer" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.2.2">2</cn><ci id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.2.3.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.2.3">𝑐</ci></apply><apply id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3"><apply id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.1.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.1.1.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.1">subscript</csymbol><log id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.1.2.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.1.2"></log><ci id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.1.3.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.1.3">𝑒</ci></apply><ci id="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.2.cmml" xref="S5.SS2.SSS2.4.p1.9.m9.1.2.3.3.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.4.p1.9.m9.1c">1-\left(\frac{ce}{2\log_{e}n}\right)^{\frac{2}{c}\log_{e}n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.4.p1.9.m9.1d">1 - ( divide start_ARG italic_c italic_e end_ARG start_ARG 2 roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 2 end_ARG start_ARG italic_c end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. By a union bound, we have</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex35"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{SW}(\pi)\geq\Omega\left(\frac{1}{\log_{e}n}\right)\mathcal{SW}(\pi^{*% })." class="ltx_Math" display="block" id="S5.Ex35.m1.3"><semantics id="S5.Ex35.m1.3a"><mrow id="S5.Ex35.m1.3.3.1" xref="S5.Ex35.m1.3.3.1.1.cmml"><mrow id="S5.Ex35.m1.3.3.1.1" xref="S5.Ex35.m1.3.3.1.1.cmml"><mrow id="S5.Ex35.m1.3.3.1.1.3" xref="S5.Ex35.m1.3.3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex35.m1.3.3.1.1.3.2" xref="S5.Ex35.m1.3.3.1.1.3.2.cmml">𝒮</mi><mo id="S5.Ex35.m1.3.3.1.1.3.1" xref="S5.Ex35.m1.3.3.1.1.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex35.m1.3.3.1.1.3.3" xref="S5.Ex35.m1.3.3.1.1.3.3.cmml">𝒲</mi><mo id="S5.Ex35.m1.3.3.1.1.3.1a" xref="S5.Ex35.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S5.Ex35.m1.3.3.1.1.3.4.2" xref="S5.Ex35.m1.3.3.1.1.3.cmml"><mo id="S5.Ex35.m1.3.3.1.1.3.4.2.1" stretchy="false" xref="S5.Ex35.m1.3.3.1.1.3.cmml">(</mo><mi id="S5.Ex35.m1.1.1" xref="S5.Ex35.m1.1.1.cmml">π</mi><mo id="S5.Ex35.m1.3.3.1.1.3.4.2.2" stretchy="false" xref="S5.Ex35.m1.3.3.1.1.3.cmml">)</mo></mrow></mrow><mo id="S5.Ex35.m1.3.3.1.1.2" xref="S5.Ex35.m1.3.3.1.1.2.cmml">≥</mo><mrow id="S5.Ex35.m1.3.3.1.1.1" xref="S5.Ex35.m1.3.3.1.1.1.cmml"><mi id="S5.Ex35.m1.3.3.1.1.1.3" mathvariant="normal" xref="S5.Ex35.m1.3.3.1.1.1.3.cmml">Ω</mi><mo id="S5.Ex35.m1.3.3.1.1.1.2" xref="S5.Ex35.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex35.m1.3.3.1.1.1.4.2" xref="S5.Ex35.m1.2.2.cmml"><mo id="S5.Ex35.m1.3.3.1.1.1.4.2.1" xref="S5.Ex35.m1.2.2.cmml">(</mo><mfrac id="S5.Ex35.m1.2.2" xref="S5.Ex35.m1.2.2.cmml"><mn id="S5.Ex35.m1.2.2.2" xref="S5.Ex35.m1.2.2.2.cmml">1</mn><mrow id="S5.Ex35.m1.2.2.3" xref="S5.Ex35.m1.2.2.3.cmml"><msub id="S5.Ex35.m1.2.2.3.1" xref="S5.Ex35.m1.2.2.3.1.cmml"><mi id="S5.Ex35.m1.2.2.3.1.2" xref="S5.Ex35.m1.2.2.3.1.2.cmml">log</mi><mi id="S5.Ex35.m1.2.2.3.1.3" xref="S5.Ex35.m1.2.2.3.1.3.cmml">e</mi></msub><mo id="S5.Ex35.m1.2.2.3a" lspace="0.167em" xref="S5.Ex35.m1.2.2.3.cmml">⁡</mo><mi id="S5.Ex35.m1.2.2.3.2" xref="S5.Ex35.m1.2.2.3.2.cmml">n</mi></mrow></mfrac><mo id="S5.Ex35.m1.3.3.1.1.1.4.2.2" xref="S5.Ex35.m1.2.2.cmml">)</mo></mrow><mo id="S5.Ex35.m1.3.3.1.1.1.2a" xref="S5.Ex35.m1.3.3.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex35.m1.3.3.1.1.1.5" xref="S5.Ex35.m1.3.3.1.1.1.5.cmml">𝒮</mi><mo id="S5.Ex35.m1.3.3.1.1.1.2b" xref="S5.Ex35.m1.3.3.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex35.m1.3.3.1.1.1.6" xref="S5.Ex35.m1.3.3.1.1.1.6.cmml">𝒲</mi><mo id="S5.Ex35.m1.3.3.1.1.1.2c" xref="S5.Ex35.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex35.m1.3.3.1.1.1.1.1" xref="S5.Ex35.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S5.Ex35.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S5.Ex35.m1.3.3.1.1.1.1.1.1.cmml">(</mo><msup id="S5.Ex35.m1.3.3.1.1.1.1.1.1" xref="S5.Ex35.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S5.Ex35.m1.3.3.1.1.1.1.1.1.2" xref="S5.Ex35.m1.3.3.1.1.1.1.1.1.2.cmml">π</mi><mo id="S5.Ex35.m1.3.3.1.1.1.1.1.1.3" xref="S5.Ex35.m1.3.3.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.Ex35.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S5.Ex35.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Ex35.m1.3.3.1.2" lspace="0em" xref="S5.Ex35.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex35.m1.3b"><apply id="S5.Ex35.m1.3.3.1.1.cmml" xref="S5.Ex35.m1.3.3.1"><geq id="S5.Ex35.m1.3.3.1.1.2.cmml" xref="S5.Ex35.m1.3.3.1.1.2"></geq><apply id="S5.Ex35.m1.3.3.1.1.3.cmml" xref="S5.Ex35.m1.3.3.1.1.3"><times id="S5.Ex35.m1.3.3.1.1.3.1.cmml" xref="S5.Ex35.m1.3.3.1.1.3.1"></times><ci id="S5.Ex35.m1.3.3.1.1.3.2.cmml" xref="S5.Ex35.m1.3.3.1.1.3.2">𝒮</ci><ci id="S5.Ex35.m1.3.3.1.1.3.3.cmml" xref="S5.Ex35.m1.3.3.1.1.3.3">𝒲</ci><ci id="S5.Ex35.m1.1.1.cmml" xref="S5.Ex35.m1.1.1">𝜋</ci></apply><apply id="S5.Ex35.m1.3.3.1.1.1.cmml" xref="S5.Ex35.m1.3.3.1.1.1"><times id="S5.Ex35.m1.3.3.1.1.1.2.cmml" xref="S5.Ex35.m1.3.3.1.1.1.2"></times><ci id="S5.Ex35.m1.3.3.1.1.1.3.cmml" xref="S5.Ex35.m1.3.3.1.1.1.3">Ω</ci><apply id="S5.Ex35.m1.2.2.cmml" xref="S5.Ex35.m1.3.3.1.1.1.4.2"><divide id="S5.Ex35.m1.2.2.1.cmml" xref="S5.Ex35.m1.3.3.1.1.1.4.2"></divide><cn id="S5.Ex35.m1.2.2.2.cmml" type="integer" xref="S5.Ex35.m1.2.2.2">1</cn><apply id="S5.Ex35.m1.2.2.3.cmml" xref="S5.Ex35.m1.2.2.3"><apply id="S5.Ex35.m1.2.2.3.1.cmml" xref="S5.Ex35.m1.2.2.3.1"><csymbol cd="ambiguous" id="S5.Ex35.m1.2.2.3.1.1.cmml" xref="S5.Ex35.m1.2.2.3.1">subscript</csymbol><log id="S5.Ex35.m1.2.2.3.1.2.cmml" xref="S5.Ex35.m1.2.2.3.1.2"></log><ci id="S5.Ex35.m1.2.2.3.1.3.cmml" xref="S5.Ex35.m1.2.2.3.1.3">𝑒</ci></apply><ci id="S5.Ex35.m1.2.2.3.2.cmml" xref="S5.Ex35.m1.2.2.3.2">𝑛</ci></apply></apply><ci id="S5.Ex35.m1.3.3.1.1.1.5.cmml" xref="S5.Ex35.m1.3.3.1.1.1.5">𝒮</ci><ci id="S5.Ex35.m1.3.3.1.1.1.6.cmml" xref="S5.Ex35.m1.3.3.1.1.1.6">𝒲</ci><apply id="S5.Ex35.m1.3.3.1.1.1.1.1.1.cmml" xref="S5.Ex35.m1.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Ex35.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S5.Ex35.m1.3.3.1.1.1.1.1">superscript</csymbol><ci id="S5.Ex35.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S5.Ex35.m1.3.3.1.1.1.1.1.1.2">𝜋</ci><times id="S5.Ex35.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S5.Ex35.m1.3.3.1.1.1.1.1.1.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex35.m1.3c">\mathcal{SW}(\pi)\geq\Omega\left(\frac{1}{\log_{e}n}\right)\mathcal{SW}(\pi^{*% }).</annotation><annotation encoding="application/x-llamapun" id="S5.Ex35.m1.3d">caligraphic_S caligraphic_W ( italic_π ) ≥ roman_Ω ( divide start_ARG 1 end_ARG start_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_ARG ) caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS2.SSS2.4.p1.10">with probability <math alttext="1-ne^{-\Theta\left(\frac{n}{k}\right)}-\left(\frac{ce}{2\log_{e}n}\right)^{% \frac{2}{c}\log_{e}n+1}" class="ltx_Math" display="inline" id="S5.SS2.SSS2.4.p1.10.m1.2"><semantics id="S5.SS2.SSS2.4.p1.10.m1.2a"><mrow id="S5.SS2.SSS2.4.p1.10.m1.2.3" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.cmml"><mn id="S5.SS2.SSS2.4.p1.10.m1.2.3.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.2.cmml">1</mn><mo id="S5.SS2.SSS2.4.p1.10.m1.2.3.1" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.1.cmml">−</mo><mrow id="S5.SS2.SSS2.4.p1.10.m1.2.3.3" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.3.cmml"><mi id="S5.SS2.SSS2.4.p1.10.m1.2.3.3.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.3.2.cmml">n</mi><mo id="S5.SS2.SSS2.4.p1.10.m1.2.3.3.1" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.3.1.cmml">⁢</mo><msup id="S5.SS2.SSS2.4.p1.10.m1.2.3.3.3" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.3.3.cmml"><mi id="S5.SS2.SSS2.4.p1.10.m1.2.3.3.3.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.3.3.2.cmml">e</mi><mrow id="S5.SS2.SSS2.4.p1.10.m1.1.1.1" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.cmml"><mo id="S5.SS2.SSS2.4.p1.10.m1.1.1.1a" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.cmml">−</mo><mrow id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3.cmml"><mi id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3.2" mathvariant="normal" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3.2.cmml">Θ</mi><mo id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3.1" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3.3.2" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.1.cmml"><mo id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3.3.2.1" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.1.cmml">(</mo><mfrac id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.1" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.1.cmml"><mi id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.1.2" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.1.2.cmml">n</mi><mi id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.1.3" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.1.3.cmml">k</mi></mfrac><mo id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3.3.2.2" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><mo id="S5.SS2.SSS2.4.p1.10.m1.2.3.1a" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.1.cmml">−</mo><msup id="S5.SS2.SSS2.4.p1.10.m1.2.3.4" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.cmml"><mrow id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.2.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.cmml"><mo id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.2.2.1" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.cmml">(</mo><mfrac id="S5.SS2.SSS2.4.p1.10.m1.2.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.cmml"><mrow id="S5.SS2.SSS2.4.p1.10.m1.2.2.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.2.cmml"><mi id="S5.SS2.SSS2.4.p1.10.m1.2.2.2.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.2.2.cmml">c</mi><mo id="S5.SS2.SSS2.4.p1.10.m1.2.2.2.1" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.2.1.cmml">⁢</mo><mi id="S5.SS2.SSS2.4.p1.10.m1.2.2.2.3" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.2.3.cmml">e</mi></mrow><mrow id="S5.SS2.SSS2.4.p1.10.m1.2.2.3" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.cmml"><mn id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.2.cmml">2</mn><mo id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.1" lspace="0.167em" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.cmml"><msub id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.1" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.1.cmml"><mi id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.1.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.1.2.cmml">log</mi><mi id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.1.3" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.1.3.cmml">e</mi></msub><mo id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3a" lspace="0.167em" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.cmml">⁡</mo><mi id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.2.cmml">n</mi></mrow></mrow></mfrac><mo id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.2.2.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.cmml">)</mo></mrow><mrow id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.cmml"><mrow id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.cmml"><mfrac id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.2.cmml"><mn id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.2.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.2.2.cmml">2</mn><mi id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.2.3" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.2.3.cmml">c</mi></mfrac><mo id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.1" lspace="0.167em" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.1.cmml">⁢</mo><mrow id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.cmml"><msub id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.1" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.1.cmml"><mi id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.1.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.1.2.cmml">log</mi><mi id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.1.3" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.1.3.cmml">e</mi></msub><mo id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3a" lspace="0.167em" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.cmml">⁡</mo><mi id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.2" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.2.cmml">n</mi></mrow></mrow><mo id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.1" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.1.cmml">+</mo><mn id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.3" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.SSS2.4.p1.10.m1.2b"><apply id="S5.SS2.SSS2.4.p1.10.m1.2.3.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3"><minus id="S5.SS2.SSS2.4.p1.10.m1.2.3.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.1"></minus><cn id="S5.SS2.SSS2.4.p1.10.m1.2.3.2.cmml" type="integer" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.2">1</cn><apply id="S5.SS2.SSS2.4.p1.10.m1.2.3.3.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.3"><times id="S5.SS2.SSS2.4.p1.10.m1.2.3.3.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.3.1"></times><ci id="S5.SS2.SSS2.4.p1.10.m1.2.3.3.2.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.3.2">𝑛</ci><apply id="S5.SS2.SSS2.4.p1.10.m1.2.3.3.3.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.3.3"><csymbol cd="ambiguous" id="S5.SS2.SSS2.4.p1.10.m1.2.3.3.3.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.3.3">superscript</csymbol><ci id="S5.SS2.SSS2.4.p1.10.m1.2.3.3.3.2.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.3.3.2">𝑒</ci><apply id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1"><minus id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1"></minus><apply id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3"><times id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3.1"></times><ci id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3.2.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3.2">Θ</ci><apply id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3.3.2"><divide id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.1.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.3.3.2"></divide><ci id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.1.2.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.1.2">𝑛</ci><ci id="S5.SS2.SSS2.4.p1.10.m1.1.1.1.1.3.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply><apply id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4"><csymbol cd="ambiguous" id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4">superscript</csymbol><apply id="S5.SS2.SSS2.4.p1.10.m1.2.2.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.2.2"><divide id="S5.SS2.SSS2.4.p1.10.m1.2.2.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.2.2"></divide><apply id="S5.SS2.SSS2.4.p1.10.m1.2.2.2.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.2"><times id="S5.SS2.SSS2.4.p1.10.m1.2.2.2.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.2.1"></times><ci id="S5.SS2.SSS2.4.p1.10.m1.2.2.2.2.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.2.2">𝑐</ci><ci id="S5.SS2.SSS2.4.p1.10.m1.2.2.2.3.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.2.3">𝑒</ci></apply><apply id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3"><times id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.1"></times><cn id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.2.cmml" type="integer" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.2">2</cn><apply id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3"><apply id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.1.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.1">subscript</csymbol><log id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.1.2.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.1.2"></log><ci id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.1.3.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.1.3">𝑒</ci></apply><ci id="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.2.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.2.3.3.2">𝑛</ci></apply></apply></apply><apply id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3"><plus id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.1"></plus><apply id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2"><times id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.1"></times><apply id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.2.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.2"><divide id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.2.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.2"></divide><cn id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.2.2.cmml" type="integer" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.2.2">2</cn><ci id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.2.3.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.2.3">𝑐</ci></apply><apply id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3"><apply id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.1"><csymbol cd="ambiguous" id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.1.1.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.1">subscript</csymbol><log id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.1.2.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.1.2"></log><ci id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.1.3.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.1.3">𝑒</ci></apply><ci id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.2.cmml" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.2.3.2">𝑛</ci></apply></apply><cn id="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.3.cmml" type="integer" xref="S5.SS2.SSS2.4.p1.10.m1.2.3.4.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.SSS2.4.p1.10.m1.2c">1-ne^{-\Theta\left(\frac{n}{k}\right)}-\left(\frac{ce}{2\log_{e}n}\right)^{% \frac{2}{c}\log_{e}n+1}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.SSS2.4.p1.10.m1.2d">1 - italic_n italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT - ( divide start_ARG italic_c italic_e end_ARG start_ARG 2 roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 2 end_ARG start_ARG italic_c end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n + 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> </section> </section> <section class="ltx_subsection" id="S5.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.3 </span>Balanced Multipartite Graphs</h3> <div class="ltx_para" id="S5.SS3.p1"> <p class="ltx_p" id="S5.SS3.p1.3">In this section, we show how to extend our results for random Turán graphs to random balanced multipartite graphs. The main idea is a reduction to the case of Turán graphs. Recall that in a balanced <math alttext="k" class="ltx_Math" display="inline" id="S5.SS3.p1.1.m1.1"><semantics id="S5.SS3.p1.1.m1.1a"><mi id="S5.SS3.p1.1.m1.1.1" xref="S5.SS3.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.1.m1.1b"><ci id="S5.SS3.p1.1.m1.1.1.cmml" xref="S5.SS3.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.1.m1.1d">italic_k</annotation></semantics></math>-partite graph, the input has components of comparable sizes, i.e., <math alttext="|V_{k}|\geq q|V_{1}|" class="ltx_Math" display="inline" id="S5.SS3.p1.2.m2.2"><semantics id="S5.SS3.p1.2.m2.2a"><mrow id="S5.SS3.p1.2.m2.2.2" xref="S5.SS3.p1.2.m2.2.2.cmml"><mrow id="S5.SS3.p1.2.m2.1.1.1.1" xref="S5.SS3.p1.2.m2.1.1.1.2.cmml"><mo id="S5.SS3.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S5.SS3.p1.2.m2.1.1.1.2.1.cmml">|</mo><msub id="S5.SS3.p1.2.m2.1.1.1.1.1" xref="S5.SS3.p1.2.m2.1.1.1.1.1.cmml"><mi id="S5.SS3.p1.2.m2.1.1.1.1.1.2" xref="S5.SS3.p1.2.m2.1.1.1.1.1.2.cmml">V</mi><mi id="S5.SS3.p1.2.m2.1.1.1.1.1.3" xref="S5.SS3.p1.2.m2.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S5.SS3.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S5.SS3.p1.2.m2.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.SS3.p1.2.m2.2.2.3" xref="S5.SS3.p1.2.m2.2.2.3.cmml">≥</mo><mrow id="S5.SS3.p1.2.m2.2.2.2" xref="S5.SS3.p1.2.m2.2.2.2.cmml"><mi id="S5.SS3.p1.2.m2.2.2.2.3" xref="S5.SS3.p1.2.m2.2.2.2.3.cmml">q</mi><mo id="S5.SS3.p1.2.m2.2.2.2.2" xref="S5.SS3.p1.2.m2.2.2.2.2.cmml">⁢</mo><mrow id="S5.SS3.p1.2.m2.2.2.2.1.1" xref="S5.SS3.p1.2.m2.2.2.2.1.2.cmml"><mo id="S5.SS3.p1.2.m2.2.2.2.1.1.2" stretchy="false" xref="S5.SS3.p1.2.m2.2.2.2.1.2.1.cmml">|</mo><msub id="S5.SS3.p1.2.m2.2.2.2.1.1.1" xref="S5.SS3.p1.2.m2.2.2.2.1.1.1.cmml"><mi id="S5.SS3.p1.2.m2.2.2.2.1.1.1.2" xref="S5.SS3.p1.2.m2.2.2.2.1.1.1.2.cmml">V</mi><mn id="S5.SS3.p1.2.m2.2.2.2.1.1.1.3" xref="S5.SS3.p1.2.m2.2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS3.p1.2.m2.2.2.2.1.1.3" stretchy="false" xref="S5.SS3.p1.2.m2.2.2.2.1.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.2.m2.2b"><apply id="S5.SS3.p1.2.m2.2.2.cmml" xref="S5.SS3.p1.2.m2.2.2"><geq id="S5.SS3.p1.2.m2.2.2.3.cmml" xref="S5.SS3.p1.2.m2.2.2.3"></geq><apply id="S5.SS3.p1.2.m2.1.1.1.2.cmml" xref="S5.SS3.p1.2.m2.1.1.1.1"><abs id="S5.SS3.p1.2.m2.1.1.1.2.1.cmml" xref="S5.SS3.p1.2.m2.1.1.1.1.2"></abs><apply id="S5.SS3.p1.2.m2.1.1.1.1.1.cmml" xref="S5.SS3.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.2.m2.1.1.1.1.1.1.cmml" xref="S5.SS3.p1.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S5.SS3.p1.2.m2.1.1.1.1.1.2.cmml" xref="S5.SS3.p1.2.m2.1.1.1.1.1.2">𝑉</ci><ci id="S5.SS3.p1.2.m2.1.1.1.1.1.3.cmml" xref="S5.SS3.p1.2.m2.1.1.1.1.1.3">𝑘</ci></apply></apply><apply id="S5.SS3.p1.2.m2.2.2.2.cmml" xref="S5.SS3.p1.2.m2.2.2.2"><times id="S5.SS3.p1.2.m2.2.2.2.2.cmml" xref="S5.SS3.p1.2.m2.2.2.2.2"></times><ci id="S5.SS3.p1.2.m2.2.2.2.3.cmml" xref="S5.SS3.p1.2.m2.2.2.2.3">𝑞</ci><apply id="S5.SS3.p1.2.m2.2.2.2.1.2.cmml" xref="S5.SS3.p1.2.m2.2.2.2.1.1"><abs id="S5.SS3.p1.2.m2.2.2.2.1.2.1.cmml" xref="S5.SS3.p1.2.m2.2.2.2.1.1.2"></abs><apply id="S5.SS3.p1.2.m2.2.2.2.1.1.1.cmml" xref="S5.SS3.p1.2.m2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.p1.2.m2.2.2.2.1.1.1.1.cmml" xref="S5.SS3.p1.2.m2.2.2.2.1.1.1">subscript</csymbol><ci id="S5.SS3.p1.2.m2.2.2.2.1.1.1.2.cmml" xref="S5.SS3.p1.2.m2.2.2.2.1.1.1.2">𝑉</ci><cn id="S5.SS3.p1.2.m2.2.2.2.1.1.1.3.cmml" type="integer" xref="S5.SS3.p1.2.m2.2.2.2.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.2.m2.2c">|V_{k}|\geq q|V_{1}|</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.2.m2.2d">| italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≥ italic_q | italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT |</annotation></semantics></math> holds for some constant <math alttext="q\in(0,1)" class="ltx_Math" display="inline" id="S5.SS3.p1.3.m3.2"><semantics id="S5.SS3.p1.3.m3.2a"><mrow id="S5.SS3.p1.3.m3.2.3" xref="S5.SS3.p1.3.m3.2.3.cmml"><mi id="S5.SS3.p1.3.m3.2.3.2" xref="S5.SS3.p1.3.m3.2.3.2.cmml">q</mi><mo id="S5.SS3.p1.3.m3.2.3.1" xref="S5.SS3.p1.3.m3.2.3.1.cmml">∈</mo><mrow id="S5.SS3.p1.3.m3.2.3.3.2" xref="S5.SS3.p1.3.m3.2.3.3.1.cmml"><mo id="S5.SS3.p1.3.m3.2.3.3.2.1" stretchy="false" xref="S5.SS3.p1.3.m3.2.3.3.1.cmml">(</mo><mn id="S5.SS3.p1.3.m3.1.1" xref="S5.SS3.p1.3.m3.1.1.cmml">0</mn><mo id="S5.SS3.p1.3.m3.2.3.3.2.2" xref="S5.SS3.p1.3.m3.2.3.3.1.cmml">,</mo><mn id="S5.SS3.p1.3.m3.2.2" xref="S5.SS3.p1.3.m3.2.2.cmml">1</mn><mo id="S5.SS3.p1.3.m3.2.3.3.2.3" stretchy="false" xref="S5.SS3.p1.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p1.3.m3.2b"><apply id="S5.SS3.p1.3.m3.2.3.cmml" xref="S5.SS3.p1.3.m3.2.3"><in id="S5.SS3.p1.3.m3.2.3.1.cmml" xref="S5.SS3.p1.3.m3.2.3.1"></in><ci id="S5.SS3.p1.3.m3.2.3.2.cmml" xref="S5.SS3.p1.3.m3.2.3.2">𝑞</ci><interval closure="open" id="S5.SS3.p1.3.m3.2.3.3.1.cmml" xref="S5.SS3.p1.3.m3.2.3.3.2"><cn id="S5.SS3.p1.3.m3.1.1.cmml" type="integer" xref="S5.SS3.p1.3.m3.1.1">0</cn><cn id="S5.SS3.p1.3.m3.2.2.cmml" type="integer" xref="S5.SS3.p1.3.m3.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p1.3.m3.2c">q\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p1.3.m3.2d">italic_q ∈ ( 0 , 1 )</annotation></semantics></math>. We refine the greedy algorithm by forcing the components’ sizes to become equal. This simple procedure is outlined in <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg3" title="In 5.3 Balanced Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">3</span></a>.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg3"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg3.6.1.1">Algorithm 3</span> </span> Reduction to a random Turán graph</figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_left" id="alg3.3.3"><span class="ltx_text ltx_font_bold" id="alg3.3.3.1">Input:</span> <math alttext="\langle G,\varepsilon\rangle" class="ltx_Math" display="inline" id="alg3.1.1.m1.2"><semantics id="alg3.1.1.m1.2a"><mrow id="alg3.1.1.m1.2.3.2" xref="alg3.1.1.m1.2.3.1.cmml"><mo id="alg3.1.1.m1.2.3.2.1" stretchy="false" xref="alg3.1.1.m1.2.3.1.cmml">⟨</mo><mi id="alg3.1.1.m1.1.1" xref="alg3.1.1.m1.1.1.cmml">G</mi><mo id="alg3.1.1.m1.2.3.2.2" xref="alg3.1.1.m1.2.3.1.cmml">,</mo><mi id="alg3.1.1.m1.2.2" xref="alg3.1.1.m1.2.2.cmml">ε</mi><mo id="alg3.1.1.m1.2.3.2.3" stretchy="false" xref="alg3.1.1.m1.2.3.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="alg3.1.1.m1.2b"><list id="alg3.1.1.m1.2.3.1.cmml" xref="alg3.1.1.m1.2.3.2"><ci id="alg3.1.1.m1.1.1.cmml" xref="alg3.1.1.m1.1.1">𝐺</ci><ci id="alg3.1.1.m1.2.2.cmml" xref="alg3.1.1.m1.2.2">𝜀</ci></list></annotation-xml><annotation encoding="application/x-tex" id="alg3.1.1.m1.2c">\langle G,\varepsilon\rangle</annotation><annotation encoding="application/x-llamapun" id="alg3.1.1.m1.2d">⟨ italic_G , italic_ε ⟩</annotation></semantics></math> where <math alttext="G=(\{V_{1},\cdots,V_{k}\},p)" class="ltx_Math" display="inline" id="alg3.2.2.m2.3"><semantics id="alg3.2.2.m2.3a"><mrow id="alg3.2.2.m2.3.3" xref="alg3.2.2.m2.3.3.cmml"><mi id="alg3.2.2.m2.3.3.3" xref="alg3.2.2.m2.3.3.3.cmml">G</mi><mo id="alg3.2.2.m2.3.3.2" xref="alg3.2.2.m2.3.3.2.cmml">=</mo><mrow id="alg3.2.2.m2.3.3.1.1" xref="alg3.2.2.m2.3.3.1.2.cmml"><mo id="alg3.2.2.m2.3.3.1.1.2" stretchy="false" xref="alg3.2.2.m2.3.3.1.2.cmml">(</mo><mrow id="alg3.2.2.m2.3.3.1.1.1.2" xref="alg3.2.2.m2.3.3.1.1.1.3.cmml"><mo id="alg3.2.2.m2.3.3.1.1.1.2.3" stretchy="false" xref="alg3.2.2.m2.3.3.1.1.1.3.cmml">{</mo><msub id="alg3.2.2.m2.3.3.1.1.1.1.1" xref="alg3.2.2.m2.3.3.1.1.1.1.1.cmml"><mi id="alg3.2.2.m2.3.3.1.1.1.1.1.2" xref="alg3.2.2.m2.3.3.1.1.1.1.1.2.cmml">V</mi><mn id="alg3.2.2.m2.3.3.1.1.1.1.1.3" xref="alg3.2.2.m2.3.3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="alg3.2.2.m2.3.3.1.1.1.2.4" xref="alg3.2.2.m2.3.3.1.1.1.3.cmml">,</mo><mi id="alg3.2.2.m2.1.1" mathvariant="normal" xref="alg3.2.2.m2.1.1.cmml">⋯</mi><mo id="alg3.2.2.m2.3.3.1.1.1.2.5" xref="alg3.2.2.m2.3.3.1.1.1.3.cmml">,</mo><msub id="alg3.2.2.m2.3.3.1.1.1.2.2" xref="alg3.2.2.m2.3.3.1.1.1.2.2.cmml"><mi id="alg3.2.2.m2.3.3.1.1.1.2.2.2" xref="alg3.2.2.m2.3.3.1.1.1.2.2.2.cmml">V</mi><mi id="alg3.2.2.m2.3.3.1.1.1.2.2.3" xref="alg3.2.2.m2.3.3.1.1.1.2.2.3.cmml">k</mi></msub><mo id="alg3.2.2.m2.3.3.1.1.1.2.6" stretchy="false" xref="alg3.2.2.m2.3.3.1.1.1.3.cmml">}</mo></mrow><mo id="alg3.2.2.m2.3.3.1.1.3" xref="alg3.2.2.m2.3.3.1.2.cmml">,</mo><mi id="alg3.2.2.m2.2.2" xref="alg3.2.2.m2.2.2.cmml">p</mi><mo id="alg3.2.2.m2.3.3.1.1.4" stretchy="false" xref="alg3.2.2.m2.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.2.2.m2.3b"><apply id="alg3.2.2.m2.3.3.cmml" xref="alg3.2.2.m2.3.3"><eq id="alg3.2.2.m2.3.3.2.cmml" xref="alg3.2.2.m2.3.3.2"></eq><ci id="alg3.2.2.m2.3.3.3.cmml" xref="alg3.2.2.m2.3.3.3">𝐺</ci><interval closure="open" id="alg3.2.2.m2.3.3.1.2.cmml" xref="alg3.2.2.m2.3.3.1.1"><set id="alg3.2.2.m2.3.3.1.1.1.3.cmml" xref="alg3.2.2.m2.3.3.1.1.1.2"><apply id="alg3.2.2.m2.3.3.1.1.1.1.1.cmml" xref="alg3.2.2.m2.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="alg3.2.2.m2.3.3.1.1.1.1.1.1.cmml" xref="alg3.2.2.m2.3.3.1.1.1.1.1">subscript</csymbol><ci id="alg3.2.2.m2.3.3.1.1.1.1.1.2.cmml" xref="alg3.2.2.m2.3.3.1.1.1.1.1.2">𝑉</ci><cn id="alg3.2.2.m2.3.3.1.1.1.1.1.3.cmml" type="integer" xref="alg3.2.2.m2.3.3.1.1.1.1.1.3">1</cn></apply><ci id="alg3.2.2.m2.1.1.cmml" xref="alg3.2.2.m2.1.1">⋯</ci><apply id="alg3.2.2.m2.3.3.1.1.1.2.2.cmml" xref="alg3.2.2.m2.3.3.1.1.1.2.2"><csymbol cd="ambiguous" id="alg3.2.2.m2.3.3.1.1.1.2.2.1.cmml" xref="alg3.2.2.m2.3.3.1.1.1.2.2">subscript</csymbol><ci id="alg3.2.2.m2.3.3.1.1.1.2.2.2.cmml" xref="alg3.2.2.m2.3.3.1.1.1.2.2.2">𝑉</ci><ci id="alg3.2.2.m2.3.3.1.1.1.2.2.3.cmml" xref="alg3.2.2.m2.3.3.1.1.1.2.2.3">𝑘</ci></apply></set><ci id="alg3.2.2.m2.2.2.cmml" xref="alg3.2.2.m2.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.2.2.m2.3c">G=(\{V_{1},\cdots,V_{k}\},p)</annotation><annotation encoding="application/x-llamapun" id="alg3.2.2.m2.3d">italic_G = ( { italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } , italic_p )</annotation></semantics></math> is a random balanced <math alttext="k" class="ltx_Math" display="inline" id="alg3.3.3.m3.1"><semantics id="alg3.3.3.m3.1a"><mi id="alg3.3.3.m3.1.1" xref="alg3.3.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="alg3.3.3.m3.1b"><ci id="alg3.3.3.m3.1.1.cmml" xref="alg3.3.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="alg3.3.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="alg3.3.3.m3.1d">italic_k</annotation></semantics></math>-partite graph</p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_left" id="alg3.4.4"><span class="ltx_text ltx_font_bold" id="alg3.4.4.1">Output:</span> A random Turán graph <math alttext="G^{T}" class="ltx_Math" display="inline" id="alg3.4.4.m1.1"><semantics id="alg3.4.4.m1.1a"><msup id="alg3.4.4.m1.1.1" xref="alg3.4.4.m1.1.1.cmml"><mi id="alg3.4.4.m1.1.1.2" xref="alg3.4.4.m1.1.1.2.cmml">G</mi><mi id="alg3.4.4.m1.1.1.3" xref="alg3.4.4.m1.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="alg3.4.4.m1.1b"><apply id="alg3.4.4.m1.1.1.cmml" xref="alg3.4.4.m1.1.1"><csymbol cd="ambiguous" id="alg3.4.4.m1.1.1.1.cmml" xref="alg3.4.4.m1.1.1">superscript</csymbol><ci id="alg3.4.4.m1.1.1.2.cmml" xref="alg3.4.4.m1.1.1.2">𝐺</ci><ci id="alg3.4.4.m1.1.1.3.cmml" xref="alg3.4.4.m1.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.4.4.m1.1c">G^{T}</annotation><annotation encoding="application/x-llamapun" id="alg3.4.4.m1.1d">italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math></p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <div class="ltx_listing ltx_figure_panel ltx_listing" id="alg3.7"> <div class="ltx_listingline" id="alg3.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l1.1.1.1" style="font-size:80%;">1:</span></span><span class="ltx_text ltx_font_bold" id="alg3.l1.2">for</span> each color class <math alttext="V_{i}" class="ltx_Math" display="inline" id="alg3.l1.m1.1"><semantics id="alg3.l1.m1.1a"><msub id="alg3.l1.m1.1.1" xref="alg3.l1.m1.1.1.cmml"><mi id="alg3.l1.m1.1.1.2" xref="alg3.l1.m1.1.1.2.cmml">V</mi><mi id="alg3.l1.m1.1.1.3" xref="alg3.l1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="alg3.l1.m1.1b"><apply id="alg3.l1.m1.1.1.cmml" xref="alg3.l1.m1.1.1"><csymbol cd="ambiguous" id="alg3.l1.m1.1.1.1.cmml" xref="alg3.l1.m1.1.1">subscript</csymbol><ci id="alg3.l1.m1.1.1.2.cmml" xref="alg3.l1.m1.1.1.2">𝑉</ci><ci id="alg3.l1.m1.1.1.3.cmml" xref="alg3.l1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l1.m1.1c">V_{i}</annotation><annotation encoding="application/x-llamapun" id="alg3.l1.m1.1d">italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="i\in[k]" class="ltx_Math" display="inline" id="alg3.l1.m2.1"><semantics id="alg3.l1.m2.1a"><mrow id="alg3.l1.m2.1.2" xref="alg3.l1.m2.1.2.cmml"><mi id="alg3.l1.m2.1.2.2" xref="alg3.l1.m2.1.2.2.cmml">i</mi><mo id="alg3.l1.m2.1.2.1" xref="alg3.l1.m2.1.2.1.cmml">∈</mo><mrow id="alg3.l1.m2.1.2.3.2" xref="alg3.l1.m2.1.2.3.1.cmml"><mo id="alg3.l1.m2.1.2.3.2.1" stretchy="false" xref="alg3.l1.m2.1.2.3.1.1.cmml">[</mo><mi id="alg3.l1.m2.1.1" xref="alg3.l1.m2.1.1.cmml">k</mi><mo id="alg3.l1.m2.1.2.3.2.2" stretchy="false" xref="alg3.l1.m2.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l1.m2.1b"><apply id="alg3.l1.m2.1.2.cmml" xref="alg3.l1.m2.1.2"><in id="alg3.l1.m2.1.2.1.cmml" xref="alg3.l1.m2.1.2.1"></in><ci id="alg3.l1.m2.1.2.2.cmml" xref="alg3.l1.m2.1.2.2">𝑖</ci><apply id="alg3.l1.m2.1.2.3.1.cmml" xref="alg3.l1.m2.1.2.3.2"><csymbol cd="latexml" id="alg3.l1.m2.1.2.3.1.1.cmml" xref="alg3.l1.m2.1.2.3.2.1">delimited-[]</csymbol><ci id="alg3.l1.m2.1.1.cmml" xref="alg3.l1.m2.1.1">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l1.m2.1c">i\in[k]</annotation><annotation encoding="application/x-llamapun" id="alg3.l1.m2.1d">italic_i ∈ [ italic_k ]</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg3.l1.3">do</span> Select an arbitrary subset of agents <math alttext="V^{\prime}_{i}\subseteq V_{i}" class="ltx_Math" display="inline" id="alg3.l1.m3.1"><semantics id="alg3.l1.m3.1a"><mrow id="alg3.l1.m3.1.1" xref="alg3.l1.m3.1.1.cmml"><msubsup id="alg3.l1.m3.1.1.2" xref="alg3.l1.m3.1.1.2.cmml"><mi id="alg3.l1.m3.1.1.2.2.2" xref="alg3.l1.m3.1.1.2.2.2.cmml">V</mi><mi id="alg3.l1.m3.1.1.2.3" xref="alg3.l1.m3.1.1.2.3.cmml">i</mi><mo id="alg3.l1.m3.1.1.2.2.3" xref="alg3.l1.m3.1.1.2.2.3.cmml">′</mo></msubsup><mo id="alg3.l1.m3.1.1.1" xref="alg3.l1.m3.1.1.1.cmml">⊆</mo><msub id="alg3.l1.m3.1.1.3" xref="alg3.l1.m3.1.1.3.cmml"><mi id="alg3.l1.m3.1.1.3.2" xref="alg3.l1.m3.1.1.3.2.cmml">V</mi><mi id="alg3.l1.m3.1.1.3.3" xref="alg3.l1.m3.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg3.l1.m3.1b"><apply id="alg3.l1.m3.1.1.cmml" xref="alg3.l1.m3.1.1"><subset id="alg3.l1.m3.1.1.1.cmml" xref="alg3.l1.m3.1.1.1"></subset><apply id="alg3.l1.m3.1.1.2.cmml" xref="alg3.l1.m3.1.1.2"><csymbol cd="ambiguous" id="alg3.l1.m3.1.1.2.1.cmml" xref="alg3.l1.m3.1.1.2">subscript</csymbol><apply id="alg3.l1.m3.1.1.2.2.cmml" xref="alg3.l1.m3.1.1.2"><csymbol cd="ambiguous" id="alg3.l1.m3.1.1.2.2.1.cmml" xref="alg3.l1.m3.1.1.2">superscript</csymbol><ci id="alg3.l1.m3.1.1.2.2.2.cmml" xref="alg3.l1.m3.1.1.2.2.2">𝑉</ci><ci id="alg3.l1.m3.1.1.2.2.3.cmml" xref="alg3.l1.m3.1.1.2.2.3">′</ci></apply><ci id="alg3.l1.m3.1.1.2.3.cmml" xref="alg3.l1.m3.1.1.2.3">𝑖</ci></apply><apply id="alg3.l1.m3.1.1.3.cmml" xref="alg3.l1.m3.1.1.3"><csymbol cd="ambiguous" id="alg3.l1.m3.1.1.3.1.cmml" xref="alg3.l1.m3.1.1.3">subscript</csymbol><ci id="alg3.l1.m3.1.1.3.2.cmml" xref="alg3.l1.m3.1.1.3.2">𝑉</ci><ci id="alg3.l1.m3.1.1.3.3.cmml" xref="alg3.l1.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l1.m3.1c">V^{\prime}_{i}\subseteq V_{i}</annotation><annotation encoding="application/x-llamapun" id="alg3.l1.m3.1d">italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊆ italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="|V^{\prime}_{i}|=|V_{k}|" class="ltx_Math" display="inline" id="alg3.l1.m4.2"><semantics id="alg3.l1.m4.2a"><mrow id="alg3.l1.m4.2.2" xref="alg3.l1.m4.2.2.cmml"><mrow id="alg3.l1.m4.1.1.1.1" xref="alg3.l1.m4.1.1.1.2.cmml"><mo id="alg3.l1.m4.1.1.1.1.2" stretchy="false" xref="alg3.l1.m4.1.1.1.2.1.cmml">|</mo><msubsup id="alg3.l1.m4.1.1.1.1.1" xref="alg3.l1.m4.1.1.1.1.1.cmml"><mi id="alg3.l1.m4.1.1.1.1.1.2.2" xref="alg3.l1.m4.1.1.1.1.1.2.2.cmml">V</mi><mi id="alg3.l1.m4.1.1.1.1.1.3" xref="alg3.l1.m4.1.1.1.1.1.3.cmml">i</mi><mo id="alg3.l1.m4.1.1.1.1.1.2.3" xref="alg3.l1.m4.1.1.1.1.1.2.3.cmml">′</mo></msubsup><mo id="alg3.l1.m4.1.1.1.1.3" stretchy="false" xref="alg3.l1.m4.1.1.1.2.1.cmml">|</mo></mrow><mo id="alg3.l1.m4.2.2.3" xref="alg3.l1.m4.2.2.3.cmml">=</mo><mrow id="alg3.l1.m4.2.2.2.1" xref="alg3.l1.m4.2.2.2.2.cmml"><mo id="alg3.l1.m4.2.2.2.1.2" stretchy="false" xref="alg3.l1.m4.2.2.2.2.1.cmml">|</mo><msub id="alg3.l1.m4.2.2.2.1.1" xref="alg3.l1.m4.2.2.2.1.1.cmml"><mi id="alg3.l1.m4.2.2.2.1.1.2" xref="alg3.l1.m4.2.2.2.1.1.2.cmml">V</mi><mi id="alg3.l1.m4.2.2.2.1.1.3" xref="alg3.l1.m4.2.2.2.1.1.3.cmml">k</mi></msub><mo id="alg3.l1.m4.2.2.2.1.3" stretchy="false" xref="alg3.l1.m4.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l1.m4.2b"><apply id="alg3.l1.m4.2.2.cmml" xref="alg3.l1.m4.2.2"><eq id="alg3.l1.m4.2.2.3.cmml" xref="alg3.l1.m4.2.2.3"></eq><apply id="alg3.l1.m4.1.1.1.2.cmml" xref="alg3.l1.m4.1.1.1.1"><abs id="alg3.l1.m4.1.1.1.2.1.cmml" xref="alg3.l1.m4.1.1.1.1.2"></abs><apply id="alg3.l1.m4.1.1.1.1.1.cmml" xref="alg3.l1.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="alg3.l1.m4.1.1.1.1.1.1.cmml" xref="alg3.l1.m4.1.1.1.1.1">subscript</csymbol><apply id="alg3.l1.m4.1.1.1.1.1.2.cmml" xref="alg3.l1.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="alg3.l1.m4.1.1.1.1.1.2.1.cmml" xref="alg3.l1.m4.1.1.1.1.1">superscript</csymbol><ci id="alg3.l1.m4.1.1.1.1.1.2.2.cmml" xref="alg3.l1.m4.1.1.1.1.1.2.2">𝑉</ci><ci id="alg3.l1.m4.1.1.1.1.1.2.3.cmml" xref="alg3.l1.m4.1.1.1.1.1.2.3">′</ci></apply><ci id="alg3.l1.m4.1.1.1.1.1.3.cmml" xref="alg3.l1.m4.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="alg3.l1.m4.2.2.2.2.cmml" xref="alg3.l1.m4.2.2.2.1"><abs id="alg3.l1.m4.2.2.2.2.1.cmml" xref="alg3.l1.m4.2.2.2.1.2"></abs><apply id="alg3.l1.m4.2.2.2.1.1.cmml" xref="alg3.l1.m4.2.2.2.1.1"><csymbol cd="ambiguous" id="alg3.l1.m4.2.2.2.1.1.1.cmml" xref="alg3.l1.m4.2.2.2.1.1">subscript</csymbol><ci id="alg3.l1.m4.2.2.2.1.1.2.cmml" xref="alg3.l1.m4.2.2.2.1.1.2">𝑉</ci><ci id="alg3.l1.m4.2.2.2.1.1.3.cmml" xref="alg3.l1.m4.2.2.2.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l1.m4.2c">|V^{\prime}_{i}|=|V_{k}|</annotation><annotation encoding="application/x-llamapun" id="alg3.l1.m4.2d">| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | = | italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg3.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l2.1.1.1" style="font-size:80%;">2:</span></span>Let <math alttext="G^{T}=(n^{\prime},k,p)" class="ltx_Math" display="inline" id="alg3.l2.m1.3"><semantics id="alg3.l2.m1.3a"><mrow id="alg3.l2.m1.3.3" xref="alg3.l2.m1.3.3.cmml"><msup id="alg3.l2.m1.3.3.3" xref="alg3.l2.m1.3.3.3.cmml"><mi id="alg3.l2.m1.3.3.3.2" xref="alg3.l2.m1.3.3.3.2.cmml">G</mi><mi id="alg3.l2.m1.3.3.3.3" xref="alg3.l2.m1.3.3.3.3.cmml">T</mi></msup><mo id="alg3.l2.m1.3.3.2" xref="alg3.l2.m1.3.3.2.cmml">=</mo><mrow id="alg3.l2.m1.3.3.1.1" xref="alg3.l2.m1.3.3.1.2.cmml"><mo id="alg3.l2.m1.3.3.1.1.2" stretchy="false" xref="alg3.l2.m1.3.3.1.2.cmml">(</mo><msup id="alg3.l2.m1.3.3.1.1.1" xref="alg3.l2.m1.3.3.1.1.1.cmml"><mi id="alg3.l2.m1.3.3.1.1.1.2" xref="alg3.l2.m1.3.3.1.1.1.2.cmml">n</mi><mo id="alg3.l2.m1.3.3.1.1.1.3" xref="alg3.l2.m1.3.3.1.1.1.3.cmml">′</mo></msup><mo id="alg3.l2.m1.3.3.1.1.3" xref="alg3.l2.m1.3.3.1.2.cmml">,</mo><mi id="alg3.l2.m1.1.1" xref="alg3.l2.m1.1.1.cmml">k</mi><mo id="alg3.l2.m1.3.3.1.1.4" xref="alg3.l2.m1.3.3.1.2.cmml">,</mo><mi id="alg3.l2.m1.2.2" xref="alg3.l2.m1.2.2.cmml">p</mi><mo id="alg3.l2.m1.3.3.1.1.5" stretchy="false" xref="alg3.l2.m1.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l2.m1.3b"><apply id="alg3.l2.m1.3.3.cmml" xref="alg3.l2.m1.3.3"><eq id="alg3.l2.m1.3.3.2.cmml" xref="alg3.l2.m1.3.3.2"></eq><apply id="alg3.l2.m1.3.3.3.cmml" xref="alg3.l2.m1.3.3.3"><csymbol cd="ambiguous" id="alg3.l2.m1.3.3.3.1.cmml" xref="alg3.l2.m1.3.3.3">superscript</csymbol><ci id="alg3.l2.m1.3.3.3.2.cmml" xref="alg3.l2.m1.3.3.3.2">𝐺</ci><ci id="alg3.l2.m1.3.3.3.3.cmml" xref="alg3.l2.m1.3.3.3.3">𝑇</ci></apply><vector id="alg3.l2.m1.3.3.1.2.cmml" xref="alg3.l2.m1.3.3.1.1"><apply id="alg3.l2.m1.3.3.1.1.1.cmml" xref="alg3.l2.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="alg3.l2.m1.3.3.1.1.1.1.cmml" xref="alg3.l2.m1.3.3.1.1.1">superscript</csymbol><ci id="alg3.l2.m1.3.3.1.1.1.2.cmml" xref="alg3.l2.m1.3.3.1.1.1.2">𝑛</ci><ci id="alg3.l2.m1.3.3.1.1.1.3.cmml" xref="alg3.l2.m1.3.3.1.1.1.3">′</ci></apply><ci id="alg3.l2.m1.1.1.cmml" xref="alg3.l2.m1.1.1">𝑘</ci><ci id="alg3.l2.m1.2.2.cmml" xref="alg3.l2.m1.2.2">𝑝</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l2.m1.3c">G^{T}=(n^{\prime},k,p)</annotation><annotation encoding="application/x-llamapun" id="alg3.l2.m1.3d">italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = ( italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_k , italic_p )</annotation></semantics></math> denote the Turán graph induced by the vertices in <math alttext="\bigcup_{i=1}^{k}V^{\prime}_{i}" class="ltx_Math" display="inline" id="alg3.l2.m2.1"><semantics id="alg3.l2.m2.1a"><mrow id="alg3.l2.m2.1.1" xref="alg3.l2.m2.1.1.cmml"><msubsup id="alg3.l2.m2.1.1.1" xref="alg3.l2.m2.1.1.1.cmml"><mo id="alg3.l2.m2.1.1.1.2.2" xref="alg3.l2.m2.1.1.1.2.2.cmml">⋃</mo><mrow id="alg3.l2.m2.1.1.1.2.3" xref="alg3.l2.m2.1.1.1.2.3.cmml"><mi id="alg3.l2.m2.1.1.1.2.3.2" xref="alg3.l2.m2.1.1.1.2.3.2.cmml">i</mi><mo id="alg3.l2.m2.1.1.1.2.3.1" xref="alg3.l2.m2.1.1.1.2.3.1.cmml">=</mo><mn id="alg3.l2.m2.1.1.1.2.3.3" xref="alg3.l2.m2.1.1.1.2.3.3.cmml">1</mn></mrow><mi id="alg3.l2.m2.1.1.1.3" xref="alg3.l2.m2.1.1.1.3.cmml">k</mi></msubsup><msubsup id="alg3.l2.m2.1.1.2" xref="alg3.l2.m2.1.1.2.cmml"><mi id="alg3.l2.m2.1.1.2.2.2" xref="alg3.l2.m2.1.1.2.2.2.cmml">V</mi><mi id="alg3.l2.m2.1.1.2.3" xref="alg3.l2.m2.1.1.2.3.cmml">i</mi><mo id="alg3.l2.m2.1.1.2.2.3" xref="alg3.l2.m2.1.1.2.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="alg3.l2.m2.1b"><apply id="alg3.l2.m2.1.1.cmml" xref="alg3.l2.m2.1.1"><apply id="alg3.l2.m2.1.1.1.cmml" xref="alg3.l2.m2.1.1.1"><csymbol cd="ambiguous" id="alg3.l2.m2.1.1.1.1.cmml" xref="alg3.l2.m2.1.1.1">superscript</csymbol><apply id="alg3.l2.m2.1.1.1.2.cmml" xref="alg3.l2.m2.1.1.1"><csymbol cd="ambiguous" id="alg3.l2.m2.1.1.1.2.1.cmml" xref="alg3.l2.m2.1.1.1">subscript</csymbol><union id="alg3.l2.m2.1.1.1.2.2.cmml" xref="alg3.l2.m2.1.1.1.2.2"></union><apply id="alg3.l2.m2.1.1.1.2.3.cmml" xref="alg3.l2.m2.1.1.1.2.3"><eq id="alg3.l2.m2.1.1.1.2.3.1.cmml" xref="alg3.l2.m2.1.1.1.2.3.1"></eq><ci id="alg3.l2.m2.1.1.1.2.3.2.cmml" xref="alg3.l2.m2.1.1.1.2.3.2">𝑖</ci><cn id="alg3.l2.m2.1.1.1.2.3.3.cmml" type="integer" xref="alg3.l2.m2.1.1.1.2.3.3">1</cn></apply></apply><ci id="alg3.l2.m2.1.1.1.3.cmml" xref="alg3.l2.m2.1.1.1.3">𝑘</ci></apply><apply id="alg3.l2.m2.1.1.2.cmml" xref="alg3.l2.m2.1.1.2"><csymbol cd="ambiguous" id="alg3.l2.m2.1.1.2.1.cmml" xref="alg3.l2.m2.1.1.2">subscript</csymbol><apply id="alg3.l2.m2.1.1.2.2.cmml" xref="alg3.l2.m2.1.1.2"><csymbol cd="ambiguous" id="alg3.l2.m2.1.1.2.2.1.cmml" xref="alg3.l2.m2.1.1.2">superscript</csymbol><ci id="alg3.l2.m2.1.1.2.2.2.cmml" xref="alg3.l2.m2.1.1.2.2.2">𝑉</ci><ci id="alg3.l2.m2.1.1.2.2.3.cmml" xref="alg3.l2.m2.1.1.2.2.3">′</ci></apply><ci id="alg3.l2.m2.1.1.2.3.cmml" xref="alg3.l2.m2.1.1.2.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l2.m2.1c">\bigcup_{i=1}^{k}V^{\prime}_{i}</annotation><annotation encoding="application/x-llamapun" id="alg3.l2.m2.1d">⋃ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> where <math alttext="n^{\prime}=k|V_{k}|" class="ltx_Math" display="inline" id="alg3.l2.m3.1"><semantics id="alg3.l2.m3.1a"><mrow id="alg3.l2.m3.1.1" xref="alg3.l2.m3.1.1.cmml"><msup id="alg3.l2.m3.1.1.3" xref="alg3.l2.m3.1.1.3.cmml"><mi id="alg3.l2.m3.1.1.3.2" xref="alg3.l2.m3.1.1.3.2.cmml">n</mi><mo id="alg3.l2.m3.1.1.3.3" xref="alg3.l2.m3.1.1.3.3.cmml">′</mo></msup><mo id="alg3.l2.m3.1.1.2" xref="alg3.l2.m3.1.1.2.cmml">=</mo><mrow id="alg3.l2.m3.1.1.1" xref="alg3.l2.m3.1.1.1.cmml"><mi id="alg3.l2.m3.1.1.1.3" xref="alg3.l2.m3.1.1.1.3.cmml">k</mi><mo id="alg3.l2.m3.1.1.1.2" xref="alg3.l2.m3.1.1.1.2.cmml">⁢</mo><mrow id="alg3.l2.m3.1.1.1.1.1" xref="alg3.l2.m3.1.1.1.1.2.cmml"><mo id="alg3.l2.m3.1.1.1.1.1.2" stretchy="false" xref="alg3.l2.m3.1.1.1.1.2.1.cmml">|</mo><msub id="alg3.l2.m3.1.1.1.1.1.1" xref="alg3.l2.m3.1.1.1.1.1.1.cmml"><mi id="alg3.l2.m3.1.1.1.1.1.1.2" xref="alg3.l2.m3.1.1.1.1.1.1.2.cmml">V</mi><mi id="alg3.l2.m3.1.1.1.1.1.1.3" xref="alg3.l2.m3.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="alg3.l2.m3.1.1.1.1.1.3" stretchy="false" xref="alg3.l2.m3.1.1.1.1.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l2.m3.1b"><apply id="alg3.l2.m3.1.1.cmml" xref="alg3.l2.m3.1.1"><eq id="alg3.l2.m3.1.1.2.cmml" xref="alg3.l2.m3.1.1.2"></eq><apply id="alg3.l2.m3.1.1.3.cmml" xref="alg3.l2.m3.1.1.3"><csymbol cd="ambiguous" id="alg3.l2.m3.1.1.3.1.cmml" xref="alg3.l2.m3.1.1.3">superscript</csymbol><ci id="alg3.l2.m3.1.1.3.2.cmml" xref="alg3.l2.m3.1.1.3.2">𝑛</ci><ci id="alg3.l2.m3.1.1.3.3.cmml" xref="alg3.l2.m3.1.1.3.3">′</ci></apply><apply id="alg3.l2.m3.1.1.1.cmml" xref="alg3.l2.m3.1.1.1"><times id="alg3.l2.m3.1.1.1.2.cmml" xref="alg3.l2.m3.1.1.1.2"></times><ci id="alg3.l2.m3.1.1.1.3.cmml" xref="alg3.l2.m3.1.1.1.3">𝑘</ci><apply id="alg3.l2.m3.1.1.1.1.2.cmml" xref="alg3.l2.m3.1.1.1.1.1"><abs id="alg3.l2.m3.1.1.1.1.2.1.cmml" xref="alg3.l2.m3.1.1.1.1.1.2"></abs><apply id="alg3.l2.m3.1.1.1.1.1.1.cmml" xref="alg3.l2.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg3.l2.m3.1.1.1.1.1.1.1.cmml" xref="alg3.l2.m3.1.1.1.1.1.1">subscript</csymbol><ci id="alg3.l2.m3.1.1.1.1.1.1.2.cmml" xref="alg3.l2.m3.1.1.1.1.1.1.2">𝑉</ci><ci id="alg3.l2.m3.1.1.1.1.1.1.3.cmml" xref="alg3.l2.m3.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l2.m3.1c">n^{\prime}=k|V_{k}|</annotation><annotation encoding="application/x-llamapun" id="alg3.l2.m3.1d">italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_k | italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg3.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l3.1.1.1" style="font-size:80%;">3:</span></span><span class="ltx_text ltx_font_bold" id="alg3.l3.2">return</span> <math alttext="G^{T}=(n^{\prime},k,p)" class="ltx_Math" display="inline" id="alg3.l3.m1.3"><semantics id="alg3.l3.m1.3a"><mrow id="alg3.l3.m1.3.3" xref="alg3.l3.m1.3.3.cmml"><msup id="alg3.l3.m1.3.3.3" xref="alg3.l3.m1.3.3.3.cmml"><mi id="alg3.l3.m1.3.3.3.2" xref="alg3.l3.m1.3.3.3.2.cmml">G</mi><mi id="alg3.l3.m1.3.3.3.3" xref="alg3.l3.m1.3.3.3.3.cmml">T</mi></msup><mo id="alg3.l3.m1.3.3.2" xref="alg3.l3.m1.3.3.2.cmml">=</mo><mrow id="alg3.l3.m1.3.3.1.1" xref="alg3.l3.m1.3.3.1.2.cmml"><mo id="alg3.l3.m1.3.3.1.1.2" stretchy="false" xref="alg3.l3.m1.3.3.1.2.cmml">(</mo><msup id="alg3.l3.m1.3.3.1.1.1" xref="alg3.l3.m1.3.3.1.1.1.cmml"><mi id="alg3.l3.m1.3.3.1.1.1.2" xref="alg3.l3.m1.3.3.1.1.1.2.cmml">n</mi><mo id="alg3.l3.m1.3.3.1.1.1.3" xref="alg3.l3.m1.3.3.1.1.1.3.cmml">′</mo></msup><mo id="alg3.l3.m1.3.3.1.1.3" xref="alg3.l3.m1.3.3.1.2.cmml">,</mo><mi id="alg3.l3.m1.1.1" xref="alg3.l3.m1.1.1.cmml">k</mi><mo id="alg3.l3.m1.3.3.1.1.4" xref="alg3.l3.m1.3.3.1.2.cmml">,</mo><mi id="alg3.l3.m1.2.2" xref="alg3.l3.m1.2.2.cmml">p</mi><mo id="alg3.l3.m1.3.3.1.1.5" stretchy="false" xref="alg3.l3.m1.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l3.m1.3b"><apply id="alg3.l3.m1.3.3.cmml" xref="alg3.l3.m1.3.3"><eq id="alg3.l3.m1.3.3.2.cmml" xref="alg3.l3.m1.3.3.2"></eq><apply id="alg3.l3.m1.3.3.3.cmml" xref="alg3.l3.m1.3.3.3"><csymbol cd="ambiguous" id="alg3.l3.m1.3.3.3.1.cmml" xref="alg3.l3.m1.3.3.3">superscript</csymbol><ci id="alg3.l3.m1.3.3.3.2.cmml" xref="alg3.l3.m1.3.3.3.2">𝐺</ci><ci id="alg3.l3.m1.3.3.3.3.cmml" xref="alg3.l3.m1.3.3.3.3">𝑇</ci></apply><vector id="alg3.l3.m1.3.3.1.2.cmml" xref="alg3.l3.m1.3.3.1.1"><apply id="alg3.l3.m1.3.3.1.1.1.cmml" xref="alg3.l3.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="alg3.l3.m1.3.3.1.1.1.1.cmml" xref="alg3.l3.m1.3.3.1.1.1">superscript</csymbol><ci id="alg3.l3.m1.3.3.1.1.1.2.cmml" xref="alg3.l3.m1.3.3.1.1.1.2">𝑛</ci><ci id="alg3.l3.m1.3.3.1.1.1.3.cmml" xref="alg3.l3.m1.3.3.1.1.1.3">′</ci></apply><ci id="alg3.l3.m1.1.1.cmml" xref="alg3.l3.m1.1.1">𝑘</ci><ci id="alg3.l3.m1.2.2.cmml" xref="alg3.l3.m1.2.2">𝑝</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l3.m1.3c">G^{T}=(n^{\prime},k,p)</annotation><annotation encoding="application/x-llamapun" id="alg3.l3.m1.3d">italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = ( italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_k , italic_p )</annotation></semantics></math> </div> </div> </div> </div> </figure> <div class="ltx_para" id="S5.SS3.p2"> <p class="ltx_p" id="S5.SS3.p2.6">The crucial observation is that by considering subsets of agents <math alttext="V^{\prime}_{i}\subseteq V_{i}" class="ltx_Math" display="inline" id="S5.SS3.p2.1.m1.1"><semantics id="S5.SS3.p2.1.m1.1a"><mrow id="S5.SS3.p2.1.m1.1.1" xref="S5.SS3.p2.1.m1.1.1.cmml"><msubsup id="S5.SS3.p2.1.m1.1.1.2" xref="S5.SS3.p2.1.m1.1.1.2.cmml"><mi id="S5.SS3.p2.1.m1.1.1.2.2.2" xref="S5.SS3.p2.1.m1.1.1.2.2.2.cmml">V</mi><mi id="S5.SS3.p2.1.m1.1.1.2.3" xref="S5.SS3.p2.1.m1.1.1.2.3.cmml">i</mi><mo id="S5.SS3.p2.1.m1.1.1.2.2.3" xref="S5.SS3.p2.1.m1.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S5.SS3.p2.1.m1.1.1.1" xref="S5.SS3.p2.1.m1.1.1.1.cmml">⊆</mo><msub id="S5.SS3.p2.1.m1.1.1.3" xref="S5.SS3.p2.1.m1.1.1.3.cmml"><mi id="S5.SS3.p2.1.m1.1.1.3.2" xref="S5.SS3.p2.1.m1.1.1.3.2.cmml">V</mi><mi id="S5.SS3.p2.1.m1.1.1.3.3" xref="S5.SS3.p2.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.1.m1.1b"><apply id="S5.SS3.p2.1.m1.1.1.cmml" xref="S5.SS3.p2.1.m1.1.1"><subset id="S5.SS3.p2.1.m1.1.1.1.cmml" xref="S5.SS3.p2.1.m1.1.1.1"></subset><apply id="S5.SS3.p2.1.m1.1.1.2.cmml" xref="S5.SS3.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.p2.1.m1.1.1.2.1.cmml" xref="S5.SS3.p2.1.m1.1.1.2">subscript</csymbol><apply id="S5.SS3.p2.1.m1.1.1.2.2.cmml" xref="S5.SS3.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.p2.1.m1.1.1.2.2.1.cmml" xref="S5.SS3.p2.1.m1.1.1.2">superscript</csymbol><ci id="S5.SS3.p2.1.m1.1.1.2.2.2.cmml" xref="S5.SS3.p2.1.m1.1.1.2.2.2">𝑉</ci><ci id="S5.SS3.p2.1.m1.1.1.2.2.3.cmml" xref="S5.SS3.p2.1.m1.1.1.2.2.3">′</ci></apply><ci id="S5.SS3.p2.1.m1.1.1.2.3.cmml" xref="S5.SS3.p2.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.SS3.p2.1.m1.1.1.3.cmml" xref="S5.SS3.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.SS3.p2.1.m1.1.1.3.1.cmml" xref="S5.SS3.p2.1.m1.1.1.3">subscript</csymbol><ci id="S5.SS3.p2.1.m1.1.1.3.2.cmml" xref="S5.SS3.p2.1.m1.1.1.3.2">𝑉</ci><ci id="S5.SS3.p2.1.m1.1.1.3.3.cmml" xref="S5.SS3.p2.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.1.m1.1c">V^{\prime}_{i}\subseteq V_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.1.m1.1d">italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊆ italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="|V^{\prime}_{i}|=|V_{k}|" class="ltx_Math" display="inline" id="S5.SS3.p2.2.m2.2"><semantics id="S5.SS3.p2.2.m2.2a"><mrow id="S5.SS3.p2.2.m2.2.2" xref="S5.SS3.p2.2.m2.2.2.cmml"><mrow id="S5.SS3.p2.2.m2.1.1.1.1" xref="S5.SS3.p2.2.m2.1.1.1.2.cmml"><mo id="S5.SS3.p2.2.m2.1.1.1.1.2" stretchy="false" xref="S5.SS3.p2.2.m2.1.1.1.2.1.cmml">|</mo><msubsup id="S5.SS3.p2.2.m2.1.1.1.1.1" xref="S5.SS3.p2.2.m2.1.1.1.1.1.cmml"><mi id="S5.SS3.p2.2.m2.1.1.1.1.1.2.2" xref="S5.SS3.p2.2.m2.1.1.1.1.1.2.2.cmml">V</mi><mi id="S5.SS3.p2.2.m2.1.1.1.1.1.3" xref="S5.SS3.p2.2.m2.1.1.1.1.1.3.cmml">i</mi><mo id="S5.SS3.p2.2.m2.1.1.1.1.1.2.3" xref="S5.SS3.p2.2.m2.1.1.1.1.1.2.3.cmml">′</mo></msubsup><mo id="S5.SS3.p2.2.m2.1.1.1.1.3" stretchy="false" xref="S5.SS3.p2.2.m2.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.SS3.p2.2.m2.2.2.3" xref="S5.SS3.p2.2.m2.2.2.3.cmml">=</mo><mrow id="S5.SS3.p2.2.m2.2.2.2.1" xref="S5.SS3.p2.2.m2.2.2.2.2.cmml"><mo id="S5.SS3.p2.2.m2.2.2.2.1.2" stretchy="false" xref="S5.SS3.p2.2.m2.2.2.2.2.1.cmml">|</mo><msub id="S5.SS3.p2.2.m2.2.2.2.1.1" xref="S5.SS3.p2.2.m2.2.2.2.1.1.cmml"><mi id="S5.SS3.p2.2.m2.2.2.2.1.1.2" xref="S5.SS3.p2.2.m2.2.2.2.1.1.2.cmml">V</mi><mi id="S5.SS3.p2.2.m2.2.2.2.1.1.3" xref="S5.SS3.p2.2.m2.2.2.2.1.1.3.cmml">k</mi></msub><mo id="S5.SS3.p2.2.m2.2.2.2.1.3" stretchy="false" xref="S5.SS3.p2.2.m2.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.2.m2.2b"><apply id="S5.SS3.p2.2.m2.2.2.cmml" xref="S5.SS3.p2.2.m2.2.2"><eq id="S5.SS3.p2.2.m2.2.2.3.cmml" xref="S5.SS3.p2.2.m2.2.2.3"></eq><apply id="S5.SS3.p2.2.m2.1.1.1.2.cmml" xref="S5.SS3.p2.2.m2.1.1.1.1"><abs id="S5.SS3.p2.2.m2.1.1.1.2.1.cmml" xref="S5.SS3.p2.2.m2.1.1.1.1.2"></abs><apply id="S5.SS3.p2.2.m2.1.1.1.1.1.cmml" xref="S5.SS3.p2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.p2.2.m2.1.1.1.1.1.1.cmml" xref="S5.SS3.p2.2.m2.1.1.1.1.1">subscript</csymbol><apply id="S5.SS3.p2.2.m2.1.1.1.1.1.2.cmml" xref="S5.SS3.p2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.p2.2.m2.1.1.1.1.1.2.1.cmml" xref="S5.SS3.p2.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S5.SS3.p2.2.m2.1.1.1.1.1.2.2.cmml" xref="S5.SS3.p2.2.m2.1.1.1.1.1.2.2">𝑉</ci><ci id="S5.SS3.p2.2.m2.1.1.1.1.1.2.3.cmml" xref="S5.SS3.p2.2.m2.1.1.1.1.1.2.3">′</ci></apply><ci id="S5.SS3.p2.2.m2.1.1.1.1.1.3.cmml" xref="S5.SS3.p2.2.m2.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S5.SS3.p2.2.m2.2.2.2.2.cmml" xref="S5.SS3.p2.2.m2.2.2.2.1"><abs id="S5.SS3.p2.2.m2.2.2.2.2.1.cmml" xref="S5.SS3.p2.2.m2.2.2.2.1.2"></abs><apply id="S5.SS3.p2.2.m2.2.2.2.1.1.cmml" xref="S5.SS3.p2.2.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS3.p2.2.m2.2.2.2.1.1.1.cmml" xref="S5.SS3.p2.2.m2.2.2.2.1.1">subscript</csymbol><ci id="S5.SS3.p2.2.m2.2.2.2.1.1.2.cmml" xref="S5.SS3.p2.2.m2.2.2.2.1.1.2">𝑉</ci><ci id="S5.SS3.p2.2.m2.2.2.2.1.1.3.cmml" xref="S5.SS3.p2.2.m2.2.2.2.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.2.m2.2c">|V^{\prime}_{i}|=|V_{k}|</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.2.m2.2d">| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | = | italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |</annotation></semantics></math>, we do not affect the edge distribution, i.e., edges of weight <math alttext="-n" class="ltx_Math" display="inline" id="S5.SS3.p2.3.m3.1"><semantics id="S5.SS3.p2.3.m3.1a"><mrow id="S5.SS3.p2.3.m3.1.1" xref="S5.SS3.p2.3.m3.1.1.cmml"><mo id="S5.SS3.p2.3.m3.1.1a" xref="S5.SS3.p2.3.m3.1.1.cmml">−</mo><mi id="S5.SS3.p2.3.m3.1.1.2" xref="S5.SS3.p2.3.m3.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.3.m3.1b"><apply id="S5.SS3.p2.3.m3.1.1.cmml" xref="S5.SS3.p2.3.m3.1.1"><minus id="S5.SS3.p2.3.m3.1.1.1.cmml" xref="S5.SS3.p2.3.m3.1.1"></minus><ci id="S5.SS3.p2.3.m3.1.1.2.cmml" xref="S5.SS3.p2.3.m3.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.3.m3.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.3.m3.1d">- italic_n</annotation></semantics></math> and <math alttext="1" class="ltx_Math" display="inline" id="S5.SS3.p2.4.m4.1"><semantics id="S5.SS3.p2.4.m4.1a"><mn id="S5.SS3.p2.4.m4.1.1" xref="S5.SS3.p2.4.m4.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.4.m4.1b"><cn id="S5.SS3.p2.4.m4.1.1.cmml" type="integer" xref="S5.SS3.p2.4.m4.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.4.m4.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.4.m4.1d">1</annotation></semantics></math> are still present with probability <math alttext="p" class="ltx_Math" display="inline" id="S5.SS3.p2.5.m5.1"><semantics id="S5.SS3.p2.5.m5.1a"><mi id="S5.SS3.p2.5.m5.1.1" xref="S5.SS3.p2.5.m5.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.5.m5.1b"><ci id="S5.SS3.p2.5.m5.1.1.cmml" xref="S5.SS3.p2.5.m5.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.5.m5.1c">p</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.5.m5.1d">italic_p</annotation></semantics></math> and <math alttext="1-p" class="ltx_Math" display="inline" id="S5.SS3.p2.6.m6.1"><semantics id="S5.SS3.p2.6.m6.1a"><mrow id="S5.SS3.p2.6.m6.1.1" xref="S5.SS3.p2.6.m6.1.1.cmml"><mn id="S5.SS3.p2.6.m6.1.1.2" xref="S5.SS3.p2.6.m6.1.1.2.cmml">1</mn><mo id="S5.SS3.p2.6.m6.1.1.1" xref="S5.SS3.p2.6.m6.1.1.1.cmml">−</mo><mi id="S5.SS3.p2.6.m6.1.1.3" xref="S5.SS3.p2.6.m6.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.6.m6.1b"><apply id="S5.SS3.p2.6.m6.1.1.cmml" xref="S5.SS3.p2.6.m6.1.1"><minus id="S5.SS3.p2.6.m6.1.1.1.cmml" xref="S5.SS3.p2.6.m6.1.1.1"></minus><cn id="S5.SS3.p2.6.m6.1.1.2.cmml" type="integer" xref="S5.SS3.p2.6.m6.1.1.2">1</cn><ci id="S5.SS3.p2.6.m6.1.1.3.cmml" xref="S5.SS3.p2.6.m6.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.6.m6.1c">1-p</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.6.m6.1d">1 - italic_p</annotation></semantics></math>, respectively. Our analysis for both the low and high perturbation regimes proceed by bounding how much the maximum social welfare changes because of the changes in the sizes.</p> </div> <section class="ltx_subsubsection" id="S5.SS3.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">5.3.1 </span>Low Perturbation Regime for Random Balanced Multipartite Graphs</h4> <div class="ltx_para" id="S5.SS3.SSS1.p1"> <p class="ltx_p" id="S5.SS3.SSS1.p1.18"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg4" title="In 5.3.1 Low Perturbation Regime for Random Balanced Multipartite Graphs ‣ 5.3 Balanced Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">4</span></a> takes as input a random balanced <math alttext="k" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.1.m1.1"><semantics id="S5.SS3.SSS1.p1.1.m1.1a"><mi id="S5.SS3.SSS1.p1.1.m1.1.1" xref="S5.SS3.SSS1.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.1.m1.1b"><ci id="S5.SS3.SSS1.p1.1.m1.1.1.cmml" xref="S5.SS3.SSS1.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.1.m1.1d">italic_k</annotation></semantics></math>-partite graph, <math alttext="G=(\{V_{1},\cdots,V_{k}\},p)" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.2.m2.3"><semantics id="S5.SS3.SSS1.p1.2.m2.3a"><mrow id="S5.SS3.SSS1.p1.2.m2.3.3" xref="S5.SS3.SSS1.p1.2.m2.3.3.cmml"><mi id="S5.SS3.SSS1.p1.2.m2.3.3.3" xref="S5.SS3.SSS1.p1.2.m2.3.3.3.cmml">G</mi><mo id="S5.SS3.SSS1.p1.2.m2.3.3.2" xref="S5.SS3.SSS1.p1.2.m2.3.3.2.cmml">=</mo><mrow id="S5.SS3.SSS1.p1.2.m2.3.3.1.1" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.2.cmml"><mo id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.2" stretchy="false" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.2.cmml">(</mo><mrow id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.3.cmml"><mo id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.3" stretchy="false" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.3.cmml">{</mo><msub id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.1.1" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.1.1.2" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.1.1.2.cmml">V</mi><mn id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.1.1.3" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.4" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.3.cmml">,</mo><mi id="S5.SS3.SSS1.p1.2.m2.1.1" mathvariant="normal" xref="S5.SS3.SSS1.p1.2.m2.1.1.cmml">⋯</mi><mo id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.5" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.3.cmml">,</mo><msub id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.2" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.2.cmml"><mi id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.2.2" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.2.2.cmml">V</mi><mi id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.2.3" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.2.3.cmml">k</mi></msub><mo id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.6" stretchy="false" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.3.cmml">}</mo></mrow><mo id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.3" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.2.cmml">,</mo><mi id="S5.SS3.SSS1.p1.2.m2.2.2" xref="S5.SS3.SSS1.p1.2.m2.2.2.cmml">p</mi><mo id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.4" stretchy="false" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.2.m2.3b"><apply id="S5.SS3.SSS1.p1.2.m2.3.3.cmml" xref="S5.SS3.SSS1.p1.2.m2.3.3"><eq id="S5.SS3.SSS1.p1.2.m2.3.3.2.cmml" xref="S5.SS3.SSS1.p1.2.m2.3.3.2"></eq><ci id="S5.SS3.SSS1.p1.2.m2.3.3.3.cmml" xref="S5.SS3.SSS1.p1.2.m2.3.3.3">𝐺</ci><interval closure="open" id="S5.SS3.SSS1.p1.2.m2.3.3.1.2.cmml" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1"><set id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.3.cmml" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2"><apply id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.1.1">subscript</csymbol><ci id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.1.1.2">𝑉</ci><cn id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.1.1.3.cmml" type="integer" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.1.1.3">1</cn></apply><ci id="S5.SS3.SSS1.p1.2.m2.1.1.cmml" xref="S5.SS3.SSS1.p1.2.m2.1.1">⋯</ci><apply id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.2.cmml" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.2"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.2.1.cmml" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.2">subscript</csymbol><ci id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.2.2.cmml" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.2.2">𝑉</ci><ci id="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.2.3.cmml" xref="S5.SS3.SSS1.p1.2.m2.3.3.1.1.1.2.2.3">𝑘</ci></apply></set><ci id="S5.SS3.SSS1.p1.2.m2.2.2.cmml" xref="S5.SS3.SSS1.p1.2.m2.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.2.m2.3c">G=(\{V_{1},\cdots,V_{k}\},p)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.2.m2.3d">italic_G = ( { italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } , italic_p )</annotation></semantics></math> where <math alttext="p=\mathcal{O}(\frac{1}{k})" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.3.m3.1"><semantics id="S5.SS3.SSS1.p1.3.m3.1a"><mrow id="S5.SS3.SSS1.p1.3.m3.1.2" xref="S5.SS3.SSS1.p1.3.m3.1.2.cmml"><mi id="S5.SS3.SSS1.p1.3.m3.1.2.2" xref="S5.SS3.SSS1.p1.3.m3.1.2.2.cmml">p</mi><mo id="S5.SS3.SSS1.p1.3.m3.1.2.1" xref="S5.SS3.SSS1.p1.3.m3.1.2.1.cmml">=</mo><mrow id="S5.SS3.SSS1.p1.3.m3.1.2.3" xref="S5.SS3.SSS1.p1.3.m3.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS1.p1.3.m3.1.2.3.2" xref="S5.SS3.SSS1.p1.3.m3.1.2.3.2.cmml">𝒪</mi><mo id="S5.SS3.SSS1.p1.3.m3.1.2.3.1" xref="S5.SS3.SSS1.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS3.SSS1.p1.3.m3.1.2.3.3.2" xref="S5.SS3.SSS1.p1.3.m3.1.1.cmml"><mo id="S5.SS3.SSS1.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="S5.SS3.SSS1.p1.3.m3.1.1.cmml">(</mo><mfrac id="S5.SS3.SSS1.p1.3.m3.1.1" xref="S5.SS3.SSS1.p1.3.m3.1.1.cmml"><mn id="S5.SS3.SSS1.p1.3.m3.1.1.2" xref="S5.SS3.SSS1.p1.3.m3.1.1.2.cmml">1</mn><mi id="S5.SS3.SSS1.p1.3.m3.1.1.3" xref="S5.SS3.SSS1.p1.3.m3.1.1.3.cmml">k</mi></mfrac><mo id="S5.SS3.SSS1.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="S5.SS3.SSS1.p1.3.m3.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.3.m3.1b"><apply id="S5.SS3.SSS1.p1.3.m3.1.2.cmml" xref="S5.SS3.SSS1.p1.3.m3.1.2"><eq id="S5.SS3.SSS1.p1.3.m3.1.2.1.cmml" xref="S5.SS3.SSS1.p1.3.m3.1.2.1"></eq><ci id="S5.SS3.SSS1.p1.3.m3.1.2.2.cmml" xref="S5.SS3.SSS1.p1.3.m3.1.2.2">𝑝</ci><apply id="S5.SS3.SSS1.p1.3.m3.1.2.3.cmml" xref="S5.SS3.SSS1.p1.3.m3.1.2.3"><times id="S5.SS3.SSS1.p1.3.m3.1.2.3.1.cmml" xref="S5.SS3.SSS1.p1.3.m3.1.2.3.1"></times><ci id="S5.SS3.SSS1.p1.3.m3.1.2.3.2.cmml" xref="S5.SS3.SSS1.p1.3.m3.1.2.3.2">𝒪</ci><apply id="S5.SS3.SSS1.p1.3.m3.1.1.cmml" xref="S5.SS3.SSS1.p1.3.m3.1.2.3.3.2"><divide id="S5.SS3.SSS1.p1.3.m3.1.1.1.cmml" xref="S5.SS3.SSS1.p1.3.m3.1.2.3.3.2"></divide><cn id="S5.SS3.SSS1.p1.3.m3.1.1.2.cmml" type="integer" xref="S5.SS3.SSS1.p1.3.m3.1.1.2">1</cn><ci id="S5.SS3.SSS1.p1.3.m3.1.1.3.cmml" xref="S5.SS3.SSS1.p1.3.m3.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.3.m3.1c">p=\mathcal{O}(\frac{1}{k})</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.3.m3.1d">italic_p = caligraphic_O ( divide start_ARG 1 end_ARG start_ARG italic_k end_ARG )</annotation></semantics></math>. It then applies <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg3" title="In 5.3 Balanced Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">3</span></a> to <math alttext="G" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.4.m4.1"><semantics id="S5.SS3.SSS1.p1.4.m4.1a"><mi id="S5.SS3.SSS1.p1.4.m4.1.1" xref="S5.SS3.SSS1.p1.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.4.m4.1b"><ci id="S5.SS3.SSS1.p1.4.m4.1.1.cmml" xref="S5.SS3.SSS1.p1.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.4.m4.1d">italic_G</annotation></semantics></math>, resulting in a random Turán graph <math alttext="G^{T}" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.5.m5.1"><semantics id="S5.SS3.SSS1.p1.5.m5.1a"><msup id="S5.SS3.SSS1.p1.5.m5.1.1" xref="S5.SS3.SSS1.p1.5.m5.1.1.cmml"><mi id="S5.SS3.SSS1.p1.5.m5.1.1.2" xref="S5.SS3.SSS1.p1.5.m5.1.1.2.cmml">G</mi><mi id="S5.SS3.SSS1.p1.5.m5.1.1.3" xref="S5.SS3.SSS1.p1.5.m5.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.5.m5.1b"><apply id="S5.SS3.SSS1.p1.5.m5.1.1.cmml" xref="S5.SS3.SSS1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.5.m5.1.1.1.cmml" xref="S5.SS3.SSS1.p1.5.m5.1.1">superscript</csymbol><ci id="S5.SS3.SSS1.p1.5.m5.1.1.2.cmml" xref="S5.SS3.SSS1.p1.5.m5.1.1.2">𝐺</ci><ci id="S5.SS3.SSS1.p1.5.m5.1.1.3.cmml" xref="S5.SS3.SSS1.p1.5.m5.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.5.m5.1c">G^{T}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.5.m5.1d">italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math>. In this process, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg3" title="In 5.3 Balanced Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">3</span></a> selects an arbitrary subset of agents from each color class <math alttext="V^{\prime}_{i}\subseteq V_{i}" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.6.m6.1"><semantics id="S5.SS3.SSS1.p1.6.m6.1a"><mrow id="S5.SS3.SSS1.p1.6.m6.1.1" xref="S5.SS3.SSS1.p1.6.m6.1.1.cmml"><msubsup id="S5.SS3.SSS1.p1.6.m6.1.1.2" xref="S5.SS3.SSS1.p1.6.m6.1.1.2.cmml"><mi id="S5.SS3.SSS1.p1.6.m6.1.1.2.2.2" xref="S5.SS3.SSS1.p1.6.m6.1.1.2.2.2.cmml">V</mi><mi id="S5.SS3.SSS1.p1.6.m6.1.1.2.3" xref="S5.SS3.SSS1.p1.6.m6.1.1.2.3.cmml">i</mi><mo id="S5.SS3.SSS1.p1.6.m6.1.1.2.2.3" xref="S5.SS3.SSS1.p1.6.m6.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S5.SS3.SSS1.p1.6.m6.1.1.1" xref="S5.SS3.SSS1.p1.6.m6.1.1.1.cmml">⊆</mo><msub id="S5.SS3.SSS1.p1.6.m6.1.1.3" xref="S5.SS3.SSS1.p1.6.m6.1.1.3.cmml"><mi id="S5.SS3.SSS1.p1.6.m6.1.1.3.2" xref="S5.SS3.SSS1.p1.6.m6.1.1.3.2.cmml">V</mi><mi id="S5.SS3.SSS1.p1.6.m6.1.1.3.3" xref="S5.SS3.SSS1.p1.6.m6.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.6.m6.1b"><apply id="S5.SS3.SSS1.p1.6.m6.1.1.cmml" xref="S5.SS3.SSS1.p1.6.m6.1.1"><subset id="S5.SS3.SSS1.p1.6.m6.1.1.1.cmml" xref="S5.SS3.SSS1.p1.6.m6.1.1.1"></subset><apply id="S5.SS3.SSS1.p1.6.m6.1.1.2.cmml" xref="S5.SS3.SSS1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.6.m6.1.1.2.1.cmml" xref="S5.SS3.SSS1.p1.6.m6.1.1.2">subscript</csymbol><apply id="S5.SS3.SSS1.p1.6.m6.1.1.2.2.cmml" xref="S5.SS3.SSS1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.6.m6.1.1.2.2.1.cmml" xref="S5.SS3.SSS1.p1.6.m6.1.1.2">superscript</csymbol><ci id="S5.SS3.SSS1.p1.6.m6.1.1.2.2.2.cmml" xref="S5.SS3.SSS1.p1.6.m6.1.1.2.2.2">𝑉</ci><ci id="S5.SS3.SSS1.p1.6.m6.1.1.2.2.3.cmml" xref="S5.SS3.SSS1.p1.6.m6.1.1.2.2.3">′</ci></apply><ci id="S5.SS3.SSS1.p1.6.m6.1.1.2.3.cmml" xref="S5.SS3.SSS1.p1.6.m6.1.1.2.3">𝑖</ci></apply><apply id="S5.SS3.SSS1.p1.6.m6.1.1.3.cmml" xref="S5.SS3.SSS1.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.6.m6.1.1.3.1.cmml" xref="S5.SS3.SSS1.p1.6.m6.1.1.3">subscript</csymbol><ci id="S5.SS3.SSS1.p1.6.m6.1.1.3.2.cmml" xref="S5.SS3.SSS1.p1.6.m6.1.1.3.2">𝑉</ci><ci id="S5.SS3.SSS1.p1.6.m6.1.1.3.3.cmml" xref="S5.SS3.SSS1.p1.6.m6.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.6.m6.1c">V^{\prime}_{i}\subseteq V_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.6.m6.1d">italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊆ italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="|V^{\prime}_{i}|=|V_{k}|" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.7.m7.2"><semantics id="S5.SS3.SSS1.p1.7.m7.2a"><mrow id="S5.SS3.SSS1.p1.7.m7.2.2" xref="S5.SS3.SSS1.p1.7.m7.2.2.cmml"><mrow id="S5.SS3.SSS1.p1.7.m7.1.1.1.1" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.2.cmml"><mo id="S5.SS3.SSS1.p1.7.m7.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.2.1.cmml">|</mo><msubsup id="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.2.2" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.2.2.cmml">V</mi><mi id="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.3" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.3.cmml">i</mi><mo id="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.2.3" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.2.3.cmml">′</mo></msubsup><mo id="S5.SS3.SSS1.p1.7.m7.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.SS3.SSS1.p1.7.m7.2.2.3" xref="S5.SS3.SSS1.p1.7.m7.2.2.3.cmml">=</mo><mrow id="S5.SS3.SSS1.p1.7.m7.2.2.2.1" xref="S5.SS3.SSS1.p1.7.m7.2.2.2.2.cmml"><mo id="S5.SS3.SSS1.p1.7.m7.2.2.2.1.2" stretchy="false" xref="S5.SS3.SSS1.p1.7.m7.2.2.2.2.1.cmml">|</mo><msub id="S5.SS3.SSS1.p1.7.m7.2.2.2.1.1" xref="S5.SS3.SSS1.p1.7.m7.2.2.2.1.1.cmml"><mi id="S5.SS3.SSS1.p1.7.m7.2.2.2.1.1.2" xref="S5.SS3.SSS1.p1.7.m7.2.2.2.1.1.2.cmml">V</mi><mi id="S5.SS3.SSS1.p1.7.m7.2.2.2.1.1.3" xref="S5.SS3.SSS1.p1.7.m7.2.2.2.1.1.3.cmml">k</mi></msub><mo id="S5.SS3.SSS1.p1.7.m7.2.2.2.1.3" stretchy="false" xref="S5.SS3.SSS1.p1.7.m7.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.7.m7.2b"><apply id="S5.SS3.SSS1.p1.7.m7.2.2.cmml" xref="S5.SS3.SSS1.p1.7.m7.2.2"><eq id="S5.SS3.SSS1.p1.7.m7.2.2.3.cmml" xref="S5.SS3.SSS1.p1.7.m7.2.2.3"></eq><apply id="S5.SS3.SSS1.p1.7.m7.1.1.1.2.cmml" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.1"><abs id="S5.SS3.SSS1.p1.7.m7.1.1.1.2.1.cmml" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.1.2"></abs><apply id="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1">subscript</csymbol><apply id="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.2.1.cmml" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1">superscript</csymbol><ci id="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.2.2.cmml" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.2.2">𝑉</ci><ci id="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.2.3.cmml" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.2.3">′</ci></apply><ci id="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.3.cmml" xref="S5.SS3.SSS1.p1.7.m7.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S5.SS3.SSS1.p1.7.m7.2.2.2.2.cmml" xref="S5.SS3.SSS1.p1.7.m7.2.2.2.1"><abs id="S5.SS3.SSS1.p1.7.m7.2.2.2.2.1.cmml" xref="S5.SS3.SSS1.p1.7.m7.2.2.2.1.2"></abs><apply id="S5.SS3.SSS1.p1.7.m7.2.2.2.1.1.cmml" xref="S5.SS3.SSS1.p1.7.m7.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.7.m7.2.2.2.1.1.1.cmml" xref="S5.SS3.SSS1.p1.7.m7.2.2.2.1.1">subscript</csymbol><ci id="S5.SS3.SSS1.p1.7.m7.2.2.2.1.1.2.cmml" xref="S5.SS3.SSS1.p1.7.m7.2.2.2.1.1.2">𝑉</ci><ci id="S5.SS3.SSS1.p1.7.m7.2.2.2.1.1.3.cmml" xref="S5.SS3.SSS1.p1.7.m7.2.2.2.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.7.m7.2c">|V^{\prime}_{i}|=|V_{k}|</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.7.m7.2d">| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | = | italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |</annotation></semantics></math> for all <math alttext="i\in[k]" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.8.m8.1"><semantics id="S5.SS3.SSS1.p1.8.m8.1a"><mrow id="S5.SS3.SSS1.p1.8.m8.1.2" xref="S5.SS3.SSS1.p1.8.m8.1.2.cmml"><mi id="S5.SS3.SSS1.p1.8.m8.1.2.2" xref="S5.SS3.SSS1.p1.8.m8.1.2.2.cmml">i</mi><mo id="S5.SS3.SSS1.p1.8.m8.1.2.1" xref="S5.SS3.SSS1.p1.8.m8.1.2.1.cmml">∈</mo><mrow id="S5.SS3.SSS1.p1.8.m8.1.2.3.2" xref="S5.SS3.SSS1.p1.8.m8.1.2.3.1.cmml"><mo id="S5.SS3.SSS1.p1.8.m8.1.2.3.2.1" stretchy="false" xref="S5.SS3.SSS1.p1.8.m8.1.2.3.1.1.cmml">[</mo><mi id="S5.SS3.SSS1.p1.8.m8.1.1" xref="S5.SS3.SSS1.p1.8.m8.1.1.cmml">k</mi><mo id="S5.SS3.SSS1.p1.8.m8.1.2.3.2.2" stretchy="false" xref="S5.SS3.SSS1.p1.8.m8.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.8.m8.1b"><apply id="S5.SS3.SSS1.p1.8.m8.1.2.cmml" xref="S5.SS3.SSS1.p1.8.m8.1.2"><in id="S5.SS3.SSS1.p1.8.m8.1.2.1.cmml" xref="S5.SS3.SSS1.p1.8.m8.1.2.1"></in><ci id="S5.SS3.SSS1.p1.8.m8.1.2.2.cmml" xref="S5.SS3.SSS1.p1.8.m8.1.2.2">𝑖</ci><apply id="S5.SS3.SSS1.p1.8.m8.1.2.3.1.cmml" xref="S5.SS3.SSS1.p1.8.m8.1.2.3.2"><csymbol cd="latexml" id="S5.SS3.SSS1.p1.8.m8.1.2.3.1.1.cmml" xref="S5.SS3.SSS1.p1.8.m8.1.2.3.2.1">delimited-[]</csymbol><ci id="S5.SS3.SSS1.p1.8.m8.1.1.cmml" xref="S5.SS3.SSS1.p1.8.m8.1.1">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.8.m8.1c">i\in[k]</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.8.m8.1d">italic_i ∈ [ italic_k ]</annotation></semantics></math>. Let <math alttext="G^{T}=(n^{\prime},k,p)" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.9.m9.3"><semantics id="S5.SS3.SSS1.p1.9.m9.3a"><mrow id="S5.SS3.SSS1.p1.9.m9.3.3" xref="S5.SS3.SSS1.p1.9.m9.3.3.cmml"><msup id="S5.SS3.SSS1.p1.9.m9.3.3.3" xref="S5.SS3.SSS1.p1.9.m9.3.3.3.cmml"><mi id="S5.SS3.SSS1.p1.9.m9.3.3.3.2" xref="S5.SS3.SSS1.p1.9.m9.3.3.3.2.cmml">G</mi><mi id="S5.SS3.SSS1.p1.9.m9.3.3.3.3" xref="S5.SS3.SSS1.p1.9.m9.3.3.3.3.cmml">T</mi></msup><mo id="S5.SS3.SSS1.p1.9.m9.3.3.2" xref="S5.SS3.SSS1.p1.9.m9.3.3.2.cmml">=</mo><mrow id="S5.SS3.SSS1.p1.9.m9.3.3.1.1" xref="S5.SS3.SSS1.p1.9.m9.3.3.1.2.cmml"><mo id="S5.SS3.SSS1.p1.9.m9.3.3.1.1.2" stretchy="false" xref="S5.SS3.SSS1.p1.9.m9.3.3.1.2.cmml">(</mo><msup id="S5.SS3.SSS1.p1.9.m9.3.3.1.1.1" xref="S5.SS3.SSS1.p1.9.m9.3.3.1.1.1.cmml"><mi id="S5.SS3.SSS1.p1.9.m9.3.3.1.1.1.2" xref="S5.SS3.SSS1.p1.9.m9.3.3.1.1.1.2.cmml">n</mi><mo id="S5.SS3.SSS1.p1.9.m9.3.3.1.1.1.3" xref="S5.SS3.SSS1.p1.9.m9.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S5.SS3.SSS1.p1.9.m9.3.3.1.1.3" xref="S5.SS3.SSS1.p1.9.m9.3.3.1.2.cmml">,</mo><mi id="S5.SS3.SSS1.p1.9.m9.1.1" xref="S5.SS3.SSS1.p1.9.m9.1.1.cmml">k</mi><mo id="S5.SS3.SSS1.p1.9.m9.3.3.1.1.4" xref="S5.SS3.SSS1.p1.9.m9.3.3.1.2.cmml">,</mo><mi id="S5.SS3.SSS1.p1.9.m9.2.2" xref="S5.SS3.SSS1.p1.9.m9.2.2.cmml">p</mi><mo id="S5.SS3.SSS1.p1.9.m9.3.3.1.1.5" stretchy="false" xref="S5.SS3.SSS1.p1.9.m9.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.9.m9.3b"><apply id="S5.SS3.SSS1.p1.9.m9.3.3.cmml" xref="S5.SS3.SSS1.p1.9.m9.3.3"><eq id="S5.SS3.SSS1.p1.9.m9.3.3.2.cmml" xref="S5.SS3.SSS1.p1.9.m9.3.3.2"></eq><apply id="S5.SS3.SSS1.p1.9.m9.3.3.3.cmml" xref="S5.SS3.SSS1.p1.9.m9.3.3.3"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.9.m9.3.3.3.1.cmml" xref="S5.SS3.SSS1.p1.9.m9.3.3.3">superscript</csymbol><ci id="S5.SS3.SSS1.p1.9.m9.3.3.3.2.cmml" xref="S5.SS3.SSS1.p1.9.m9.3.3.3.2">𝐺</ci><ci id="S5.SS3.SSS1.p1.9.m9.3.3.3.3.cmml" xref="S5.SS3.SSS1.p1.9.m9.3.3.3.3">𝑇</ci></apply><vector id="S5.SS3.SSS1.p1.9.m9.3.3.1.2.cmml" xref="S5.SS3.SSS1.p1.9.m9.3.3.1.1"><apply id="S5.SS3.SSS1.p1.9.m9.3.3.1.1.1.cmml" xref="S5.SS3.SSS1.p1.9.m9.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.9.m9.3.3.1.1.1.1.cmml" xref="S5.SS3.SSS1.p1.9.m9.3.3.1.1.1">superscript</csymbol><ci id="S5.SS3.SSS1.p1.9.m9.3.3.1.1.1.2.cmml" xref="S5.SS3.SSS1.p1.9.m9.3.3.1.1.1.2">𝑛</ci><ci id="S5.SS3.SSS1.p1.9.m9.3.3.1.1.1.3.cmml" xref="S5.SS3.SSS1.p1.9.m9.3.3.1.1.1.3">′</ci></apply><ci id="S5.SS3.SSS1.p1.9.m9.1.1.cmml" xref="S5.SS3.SSS1.p1.9.m9.1.1">𝑘</ci><ci id="S5.SS3.SSS1.p1.9.m9.2.2.cmml" xref="S5.SS3.SSS1.p1.9.m9.2.2">𝑝</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.9.m9.3c">G^{T}=(n^{\prime},k,p)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.9.m9.3d">italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = ( italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_k , italic_p )</annotation></semantics></math> denote the Turán graph induced by the vertices in <math alttext="\bigcup_{i=1}^{k}V^{\prime}_{i}" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.10.m10.1"><semantics id="S5.SS3.SSS1.p1.10.m10.1a"><mrow id="S5.SS3.SSS1.p1.10.m10.1.1" xref="S5.SS3.SSS1.p1.10.m10.1.1.cmml"><msubsup id="S5.SS3.SSS1.p1.10.m10.1.1.1" xref="S5.SS3.SSS1.p1.10.m10.1.1.1.cmml"><mo id="S5.SS3.SSS1.p1.10.m10.1.1.1.2.2" xref="S5.SS3.SSS1.p1.10.m10.1.1.1.2.2.cmml">⋃</mo><mrow id="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3" xref="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3.cmml"><mi id="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3.2" xref="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3.2.cmml">i</mi><mo id="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3.1" xref="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3.1.cmml">=</mo><mn id="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3.3" xref="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3.3.cmml">1</mn></mrow><mi id="S5.SS3.SSS1.p1.10.m10.1.1.1.3" xref="S5.SS3.SSS1.p1.10.m10.1.1.1.3.cmml">k</mi></msubsup><msubsup id="S5.SS3.SSS1.p1.10.m10.1.1.2" xref="S5.SS3.SSS1.p1.10.m10.1.1.2.cmml"><mi id="S5.SS3.SSS1.p1.10.m10.1.1.2.2.2" xref="S5.SS3.SSS1.p1.10.m10.1.1.2.2.2.cmml">V</mi><mi id="S5.SS3.SSS1.p1.10.m10.1.1.2.3" xref="S5.SS3.SSS1.p1.10.m10.1.1.2.3.cmml">i</mi><mo id="S5.SS3.SSS1.p1.10.m10.1.1.2.2.3" xref="S5.SS3.SSS1.p1.10.m10.1.1.2.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.10.m10.1b"><apply id="S5.SS3.SSS1.p1.10.m10.1.1.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1"><apply id="S5.SS3.SSS1.p1.10.m10.1.1.1.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.10.m10.1.1.1.1.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.1">superscript</csymbol><apply id="S5.SS3.SSS1.p1.10.m10.1.1.1.2.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.10.m10.1.1.1.2.1.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.1">subscript</csymbol><union id="S5.SS3.SSS1.p1.10.m10.1.1.1.2.2.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.1.2.2"></union><apply id="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3"><eq id="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3.1.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3.1"></eq><ci id="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3.2.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3.2">𝑖</ci><cn id="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3.3.cmml" type="integer" xref="S5.SS3.SSS1.p1.10.m10.1.1.1.2.3.3">1</cn></apply></apply><ci id="S5.SS3.SSS1.p1.10.m10.1.1.1.3.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.1.3">𝑘</ci></apply><apply id="S5.SS3.SSS1.p1.10.m10.1.1.2.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.10.m10.1.1.2.1.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.2">subscript</csymbol><apply id="S5.SS3.SSS1.p1.10.m10.1.1.2.2.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.10.m10.1.1.2.2.1.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.2">superscript</csymbol><ci id="S5.SS3.SSS1.p1.10.m10.1.1.2.2.2.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.2.2.2">𝑉</ci><ci id="S5.SS3.SSS1.p1.10.m10.1.1.2.2.3.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.2.2.3">′</ci></apply><ci id="S5.SS3.SSS1.p1.10.m10.1.1.2.3.cmml" xref="S5.SS3.SSS1.p1.10.m10.1.1.2.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.10.m10.1c">\bigcup_{i=1}^{k}V^{\prime}_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.10.m10.1d">⋃ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="n^{\prime}=k|V_{k}|" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.11.m11.1"><semantics id="S5.SS3.SSS1.p1.11.m11.1a"><mrow id="S5.SS3.SSS1.p1.11.m11.1.1" xref="S5.SS3.SSS1.p1.11.m11.1.1.cmml"><msup id="S5.SS3.SSS1.p1.11.m11.1.1.3" xref="S5.SS3.SSS1.p1.11.m11.1.1.3.cmml"><mi id="S5.SS3.SSS1.p1.11.m11.1.1.3.2" xref="S5.SS3.SSS1.p1.11.m11.1.1.3.2.cmml">n</mi><mo id="S5.SS3.SSS1.p1.11.m11.1.1.3.3" xref="S5.SS3.SSS1.p1.11.m11.1.1.3.3.cmml">′</mo></msup><mo id="S5.SS3.SSS1.p1.11.m11.1.1.2" xref="S5.SS3.SSS1.p1.11.m11.1.1.2.cmml">=</mo><mrow id="S5.SS3.SSS1.p1.11.m11.1.1.1" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.cmml"><mi id="S5.SS3.SSS1.p1.11.m11.1.1.1.3" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.3.cmml">k</mi><mo id="S5.SS3.SSS1.p1.11.m11.1.1.1.2" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.1.2.cmml"><mo id="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.1.2.1.cmml">|</mo><msub id="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.1" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.1.2" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.1.2.cmml">V</mi><mi id="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.1.3" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.1.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.11.m11.1b"><apply id="S5.SS3.SSS1.p1.11.m11.1.1.cmml" xref="S5.SS3.SSS1.p1.11.m11.1.1"><eq id="S5.SS3.SSS1.p1.11.m11.1.1.2.cmml" xref="S5.SS3.SSS1.p1.11.m11.1.1.2"></eq><apply id="S5.SS3.SSS1.p1.11.m11.1.1.3.cmml" xref="S5.SS3.SSS1.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.11.m11.1.1.3.1.cmml" xref="S5.SS3.SSS1.p1.11.m11.1.1.3">superscript</csymbol><ci id="S5.SS3.SSS1.p1.11.m11.1.1.3.2.cmml" xref="S5.SS3.SSS1.p1.11.m11.1.1.3.2">𝑛</ci><ci id="S5.SS3.SSS1.p1.11.m11.1.1.3.3.cmml" xref="S5.SS3.SSS1.p1.11.m11.1.1.3.3">′</ci></apply><apply id="S5.SS3.SSS1.p1.11.m11.1.1.1.cmml" xref="S5.SS3.SSS1.p1.11.m11.1.1.1"><times id="S5.SS3.SSS1.p1.11.m11.1.1.1.2.cmml" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.2"></times><ci id="S5.SS3.SSS1.p1.11.m11.1.1.1.3.cmml" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.3">𝑘</ci><apply id="S5.SS3.SSS1.p1.11.m11.1.1.1.1.2.cmml" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1"><abs id="S5.SS3.SSS1.p1.11.m11.1.1.1.1.2.1.cmml" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.2"></abs><apply id="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.1">subscript</csymbol><ci id="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.1.2">𝑉</ci><ci id="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.1.3.cmml" xref="S5.SS3.SSS1.p1.11.m11.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.11.m11.1c">n^{\prime}=k|V_{k}|</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.11.m11.1d">italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_k | italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |</annotation></semantics></math>. Note that <math alttext="G^{T}" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.12.m12.1"><semantics id="S5.SS3.SSS1.p1.12.m12.1a"><msup id="S5.SS3.SSS1.p1.12.m12.1.1" xref="S5.SS3.SSS1.p1.12.m12.1.1.cmml"><mi id="S5.SS3.SSS1.p1.12.m12.1.1.2" xref="S5.SS3.SSS1.p1.12.m12.1.1.2.cmml">G</mi><mi id="S5.SS3.SSS1.p1.12.m12.1.1.3" xref="S5.SS3.SSS1.p1.12.m12.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.12.m12.1b"><apply id="S5.SS3.SSS1.p1.12.m12.1.1.cmml" xref="S5.SS3.SSS1.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.12.m12.1.1.1.cmml" xref="S5.SS3.SSS1.p1.12.m12.1.1">superscript</csymbol><ci id="S5.SS3.SSS1.p1.12.m12.1.1.2.cmml" xref="S5.SS3.SSS1.p1.12.m12.1.1.2">𝐺</ci><ci id="S5.SS3.SSS1.p1.12.m12.1.1.3.cmml" xref="S5.SS3.SSS1.p1.12.m12.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.12.m12.1c">G^{T}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.12.m12.1d">italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> is a random Turán graph, as the edge weight between any pair of agents within the same color class is <math alttext="-n" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.13.m13.1"><semantics id="S5.SS3.SSS1.p1.13.m13.1a"><mrow id="S5.SS3.SSS1.p1.13.m13.1.1" xref="S5.SS3.SSS1.p1.13.m13.1.1.cmml"><mo id="S5.SS3.SSS1.p1.13.m13.1.1a" xref="S5.SS3.SSS1.p1.13.m13.1.1.cmml">−</mo><mi id="S5.SS3.SSS1.p1.13.m13.1.1.2" xref="S5.SS3.SSS1.p1.13.m13.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.13.m13.1b"><apply id="S5.SS3.SSS1.p1.13.m13.1.1.cmml" xref="S5.SS3.SSS1.p1.13.m13.1.1"><minus id="S5.SS3.SSS1.p1.13.m13.1.1.1.cmml" xref="S5.SS3.SSS1.p1.13.m13.1.1"></minus><ci id="S5.SS3.SSS1.p1.13.m13.1.1.2.cmml" xref="S5.SS3.SSS1.p1.13.m13.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.13.m13.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.13.m13.1d">- italic_n</annotation></semantics></math> and for pairs of agents belonging to different color classes, the edge weights were drawn from a distribution that assigns a weight of <math alttext="-n" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.14.m14.1"><semantics id="S5.SS3.SSS1.p1.14.m14.1a"><mrow id="S5.SS3.SSS1.p1.14.m14.1.1" xref="S5.SS3.SSS1.p1.14.m14.1.1.cmml"><mo id="S5.SS3.SSS1.p1.14.m14.1.1a" xref="S5.SS3.SSS1.p1.14.m14.1.1.cmml">−</mo><mi id="S5.SS3.SSS1.p1.14.m14.1.1.2" xref="S5.SS3.SSS1.p1.14.m14.1.1.2.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.14.m14.1b"><apply id="S5.SS3.SSS1.p1.14.m14.1.1.cmml" xref="S5.SS3.SSS1.p1.14.m14.1.1"><minus id="S5.SS3.SSS1.p1.14.m14.1.1.1.cmml" xref="S5.SS3.SSS1.p1.14.m14.1.1"></minus><ci id="S5.SS3.SSS1.p1.14.m14.1.1.2.cmml" xref="S5.SS3.SSS1.p1.14.m14.1.1.2">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.14.m14.1c">-n</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.14.m14.1d">- italic_n</annotation></semantics></math> with probability <math alttext="p" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.15.m15.1"><semantics id="S5.SS3.SSS1.p1.15.m15.1a"><mi id="S5.SS3.SSS1.p1.15.m15.1.1" xref="S5.SS3.SSS1.p1.15.m15.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.15.m15.1b"><ci id="S5.SS3.SSS1.p1.15.m15.1.1.cmml" xref="S5.SS3.SSS1.p1.15.m15.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.15.m15.1c">p</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.15.m15.1d">italic_p</annotation></semantics></math> and <math alttext="1" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.16.m16.1"><semantics id="S5.SS3.SSS1.p1.16.m16.1a"><mn id="S5.SS3.SSS1.p1.16.m16.1.1" xref="S5.SS3.SSS1.p1.16.m16.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.16.m16.1b"><cn id="S5.SS3.SSS1.p1.16.m16.1.1.cmml" type="integer" xref="S5.SS3.SSS1.p1.16.m16.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.16.m16.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.16.m16.1d">1</annotation></semantics></math> with probability <math alttext="1-p" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.17.m17.1"><semantics id="S5.SS3.SSS1.p1.17.m17.1a"><mrow id="S5.SS3.SSS1.p1.17.m17.1.1" xref="S5.SS3.SSS1.p1.17.m17.1.1.cmml"><mn id="S5.SS3.SSS1.p1.17.m17.1.1.2" xref="S5.SS3.SSS1.p1.17.m17.1.1.2.cmml">1</mn><mo id="S5.SS3.SSS1.p1.17.m17.1.1.1" xref="S5.SS3.SSS1.p1.17.m17.1.1.1.cmml">−</mo><mi id="S5.SS3.SSS1.p1.17.m17.1.1.3" xref="S5.SS3.SSS1.p1.17.m17.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.17.m17.1b"><apply id="S5.SS3.SSS1.p1.17.m17.1.1.cmml" xref="S5.SS3.SSS1.p1.17.m17.1.1"><minus id="S5.SS3.SSS1.p1.17.m17.1.1.1.cmml" xref="S5.SS3.SSS1.p1.17.m17.1.1.1"></minus><cn id="S5.SS3.SSS1.p1.17.m17.1.1.2.cmml" type="integer" xref="S5.SS3.SSS1.p1.17.m17.1.1.2">1</cn><ci id="S5.SS3.SSS1.p1.17.m17.1.1.3.cmml" xref="S5.SS3.SSS1.p1.17.m17.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.17.m17.1c">1-p</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.17.m17.1d">1 - italic_p</annotation></semantics></math>. This distribution is ensured by not looking at any edge weights when selecting the subsets <math alttext="V^{\prime}_{i}" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p1.18.m18.1"><semantics id="S5.SS3.SSS1.p1.18.m18.1a"><msubsup id="S5.SS3.SSS1.p1.18.m18.1.1" xref="S5.SS3.SSS1.p1.18.m18.1.1.cmml"><mi id="S5.SS3.SSS1.p1.18.m18.1.1.2.2" xref="S5.SS3.SSS1.p1.18.m18.1.1.2.2.cmml">V</mi><mi id="S5.SS3.SSS1.p1.18.m18.1.1.3" xref="S5.SS3.SSS1.p1.18.m18.1.1.3.cmml">i</mi><mo id="S5.SS3.SSS1.p1.18.m18.1.1.2.3" xref="S5.SS3.SSS1.p1.18.m18.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p1.18.m18.1b"><apply id="S5.SS3.SSS1.p1.18.m18.1.1.cmml" xref="S5.SS3.SSS1.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.18.m18.1.1.1.cmml" xref="S5.SS3.SSS1.p1.18.m18.1.1">subscript</csymbol><apply id="S5.SS3.SSS1.p1.18.m18.1.1.2.cmml" xref="S5.SS3.SSS1.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p1.18.m18.1.1.2.1.cmml" xref="S5.SS3.SSS1.p1.18.m18.1.1">superscript</csymbol><ci id="S5.SS3.SSS1.p1.18.m18.1.1.2.2.cmml" xref="S5.SS3.SSS1.p1.18.m18.1.1.2.2">𝑉</ci><ci id="S5.SS3.SSS1.p1.18.m18.1.1.2.3.cmml" xref="S5.SS3.SSS1.p1.18.m18.1.1.2.3">′</ci></apply><ci id="S5.SS3.SSS1.p1.18.m18.1.1.3.cmml" xref="S5.SS3.SSS1.p1.18.m18.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p1.18.m18.1c">V^{\prime}_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p1.18.m18.1d">italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS3.SSS1.p2"> <p class="ltx_p" id="S5.SS3.SSS1.p2.7">Since <math alttext="G" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p2.1.m1.1"><semantics id="S5.SS3.SSS1.p2.1.m1.1a"><mi id="S5.SS3.SSS1.p2.1.m1.1.1" xref="S5.SS3.SSS1.p2.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p2.1.m1.1b"><ci id="S5.SS3.SSS1.p2.1.m1.1.1.cmml" xref="S5.SS3.SSS1.p2.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p2.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p2.1.m1.1d">italic_G</annotation></semantics></math> is balanced, we know that <math alttext="|V_{k}|\geq q|V_{1}|" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p2.2.m2.2"><semantics id="S5.SS3.SSS1.p2.2.m2.2a"><mrow id="S5.SS3.SSS1.p2.2.m2.2.2" xref="S5.SS3.SSS1.p2.2.m2.2.2.cmml"><mrow id="S5.SS3.SSS1.p2.2.m2.1.1.1.1" xref="S5.SS3.SSS1.p2.2.m2.1.1.1.2.cmml"><mo id="S5.SS3.SSS1.p2.2.m2.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS1.p2.2.m2.1.1.1.2.1.cmml">|</mo><msub id="S5.SS3.SSS1.p2.2.m2.1.1.1.1.1" xref="S5.SS3.SSS1.p2.2.m2.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS1.p2.2.m2.1.1.1.1.1.2" xref="S5.SS3.SSS1.p2.2.m2.1.1.1.1.1.2.cmml">V</mi><mi id="S5.SS3.SSS1.p2.2.m2.1.1.1.1.1.3" xref="S5.SS3.SSS1.p2.2.m2.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S5.SS3.SSS1.p2.2.m2.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS1.p2.2.m2.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.SS3.SSS1.p2.2.m2.2.2.3" xref="S5.SS3.SSS1.p2.2.m2.2.2.3.cmml">≥</mo><mrow id="S5.SS3.SSS1.p2.2.m2.2.2.2" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.cmml"><mi id="S5.SS3.SSS1.p2.2.m2.2.2.2.3" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.3.cmml">q</mi><mo id="S5.SS3.SSS1.p2.2.m2.2.2.2.2" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.2.cmml">⁢</mo><mrow id="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.1.2.cmml"><mo id="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.2" stretchy="false" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.1.2.1.cmml">|</mo><msub id="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.1" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.1.cmml"><mi id="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.1.2" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.1.2.cmml">V</mi><mn id="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.1.3" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.3" stretchy="false" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.1.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p2.2.m2.2b"><apply id="S5.SS3.SSS1.p2.2.m2.2.2.cmml" xref="S5.SS3.SSS1.p2.2.m2.2.2"><geq id="S5.SS3.SSS1.p2.2.m2.2.2.3.cmml" xref="S5.SS3.SSS1.p2.2.m2.2.2.3"></geq><apply id="S5.SS3.SSS1.p2.2.m2.1.1.1.2.cmml" xref="S5.SS3.SSS1.p2.2.m2.1.1.1.1"><abs id="S5.SS3.SSS1.p2.2.m2.1.1.1.2.1.cmml" xref="S5.SS3.SSS1.p2.2.m2.1.1.1.1.2"></abs><apply id="S5.SS3.SSS1.p2.2.m2.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p2.2.m2.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p2.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S5.SS3.SSS1.p2.2.m2.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS1.p2.2.m2.1.1.1.1.1.2">𝑉</ci><ci id="S5.SS3.SSS1.p2.2.m2.1.1.1.1.1.3.cmml" xref="S5.SS3.SSS1.p2.2.m2.1.1.1.1.1.3">𝑘</ci></apply></apply><apply id="S5.SS3.SSS1.p2.2.m2.2.2.2.cmml" xref="S5.SS3.SSS1.p2.2.m2.2.2.2"><times id="S5.SS3.SSS1.p2.2.m2.2.2.2.2.cmml" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.2"></times><ci id="S5.SS3.SSS1.p2.2.m2.2.2.2.3.cmml" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.3">𝑞</ci><apply id="S5.SS3.SSS1.p2.2.m2.2.2.2.1.2.cmml" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1"><abs id="S5.SS3.SSS1.p2.2.m2.2.2.2.1.2.1.cmml" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.2"></abs><apply id="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.1.cmml" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.1.1.cmml" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.1">subscript</csymbol><ci id="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.1.2.cmml" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.1.2">𝑉</ci><cn id="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.1.3.cmml" type="integer" xref="S5.SS3.SSS1.p2.2.m2.2.2.2.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p2.2.m2.2c">|V_{k}|\geq q|V_{1}|</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p2.2.m2.2d">| italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≥ italic_q | italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT |</annotation></semantics></math>. Recall that <math alttext="|V_{1}|\geq|V_{i}|" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p2.3.m3.2"><semantics id="S5.SS3.SSS1.p2.3.m3.2a"><mrow id="S5.SS3.SSS1.p2.3.m3.2.2" xref="S5.SS3.SSS1.p2.3.m3.2.2.cmml"><mrow id="S5.SS3.SSS1.p2.3.m3.1.1.1.1" xref="S5.SS3.SSS1.p2.3.m3.1.1.1.2.cmml"><mo id="S5.SS3.SSS1.p2.3.m3.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS1.p2.3.m3.1.1.1.2.1.cmml">|</mo><msub id="S5.SS3.SSS1.p2.3.m3.1.1.1.1.1" xref="S5.SS3.SSS1.p2.3.m3.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS1.p2.3.m3.1.1.1.1.1.2" xref="S5.SS3.SSS1.p2.3.m3.1.1.1.1.1.2.cmml">V</mi><mn id="S5.SS3.SSS1.p2.3.m3.1.1.1.1.1.3" xref="S5.SS3.SSS1.p2.3.m3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS3.SSS1.p2.3.m3.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS1.p2.3.m3.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.SS3.SSS1.p2.3.m3.2.2.3" xref="S5.SS3.SSS1.p2.3.m3.2.2.3.cmml">≥</mo><mrow id="S5.SS3.SSS1.p2.3.m3.2.2.2.1" xref="S5.SS3.SSS1.p2.3.m3.2.2.2.2.cmml"><mo id="S5.SS3.SSS1.p2.3.m3.2.2.2.1.2" stretchy="false" xref="S5.SS3.SSS1.p2.3.m3.2.2.2.2.1.cmml">|</mo><msub id="S5.SS3.SSS1.p2.3.m3.2.2.2.1.1" xref="S5.SS3.SSS1.p2.3.m3.2.2.2.1.1.cmml"><mi id="S5.SS3.SSS1.p2.3.m3.2.2.2.1.1.2" xref="S5.SS3.SSS1.p2.3.m3.2.2.2.1.1.2.cmml">V</mi><mi id="S5.SS3.SSS1.p2.3.m3.2.2.2.1.1.3" xref="S5.SS3.SSS1.p2.3.m3.2.2.2.1.1.3.cmml">i</mi></msub><mo id="S5.SS3.SSS1.p2.3.m3.2.2.2.1.3" stretchy="false" xref="S5.SS3.SSS1.p2.3.m3.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p2.3.m3.2b"><apply id="S5.SS3.SSS1.p2.3.m3.2.2.cmml" xref="S5.SS3.SSS1.p2.3.m3.2.2"><geq id="S5.SS3.SSS1.p2.3.m3.2.2.3.cmml" xref="S5.SS3.SSS1.p2.3.m3.2.2.3"></geq><apply id="S5.SS3.SSS1.p2.3.m3.1.1.1.2.cmml" xref="S5.SS3.SSS1.p2.3.m3.1.1.1.1"><abs id="S5.SS3.SSS1.p2.3.m3.1.1.1.2.1.cmml" xref="S5.SS3.SSS1.p2.3.m3.1.1.1.1.2"></abs><apply id="S5.SS3.SSS1.p2.3.m3.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p2.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p2.3.m3.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p2.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S5.SS3.SSS1.p2.3.m3.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS1.p2.3.m3.1.1.1.1.1.2">𝑉</ci><cn id="S5.SS3.SSS1.p2.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S5.SS3.SSS1.p2.3.m3.1.1.1.1.1.3">1</cn></apply></apply><apply id="S5.SS3.SSS1.p2.3.m3.2.2.2.2.cmml" xref="S5.SS3.SSS1.p2.3.m3.2.2.2.1"><abs id="S5.SS3.SSS1.p2.3.m3.2.2.2.2.1.cmml" xref="S5.SS3.SSS1.p2.3.m3.2.2.2.1.2"></abs><apply id="S5.SS3.SSS1.p2.3.m3.2.2.2.1.1.cmml" xref="S5.SS3.SSS1.p2.3.m3.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p2.3.m3.2.2.2.1.1.1.cmml" xref="S5.SS3.SSS1.p2.3.m3.2.2.2.1.1">subscript</csymbol><ci id="S5.SS3.SSS1.p2.3.m3.2.2.2.1.1.2.cmml" xref="S5.SS3.SSS1.p2.3.m3.2.2.2.1.1.2">𝑉</ci><ci id="S5.SS3.SSS1.p2.3.m3.2.2.2.1.1.3.cmml" xref="S5.SS3.SSS1.p2.3.m3.2.2.2.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p2.3.m3.2c">|V_{1}|\geq|V_{i}|</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p2.3.m3.2d">| italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≥ | italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT |</annotation></semantics></math> for all <math alttext="i\in[k]" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p2.4.m4.1"><semantics id="S5.SS3.SSS1.p2.4.m4.1a"><mrow id="S5.SS3.SSS1.p2.4.m4.1.2" xref="S5.SS3.SSS1.p2.4.m4.1.2.cmml"><mi id="S5.SS3.SSS1.p2.4.m4.1.2.2" xref="S5.SS3.SSS1.p2.4.m4.1.2.2.cmml">i</mi><mo id="S5.SS3.SSS1.p2.4.m4.1.2.1" xref="S5.SS3.SSS1.p2.4.m4.1.2.1.cmml">∈</mo><mrow id="S5.SS3.SSS1.p2.4.m4.1.2.3.2" xref="S5.SS3.SSS1.p2.4.m4.1.2.3.1.cmml"><mo id="S5.SS3.SSS1.p2.4.m4.1.2.3.2.1" stretchy="false" xref="S5.SS3.SSS1.p2.4.m4.1.2.3.1.1.cmml">[</mo><mi id="S5.SS3.SSS1.p2.4.m4.1.1" xref="S5.SS3.SSS1.p2.4.m4.1.1.cmml">k</mi><mo id="S5.SS3.SSS1.p2.4.m4.1.2.3.2.2" stretchy="false" xref="S5.SS3.SSS1.p2.4.m4.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p2.4.m4.1b"><apply id="S5.SS3.SSS1.p2.4.m4.1.2.cmml" xref="S5.SS3.SSS1.p2.4.m4.1.2"><in id="S5.SS3.SSS1.p2.4.m4.1.2.1.cmml" xref="S5.SS3.SSS1.p2.4.m4.1.2.1"></in><ci id="S5.SS3.SSS1.p2.4.m4.1.2.2.cmml" xref="S5.SS3.SSS1.p2.4.m4.1.2.2">𝑖</ci><apply id="S5.SS3.SSS1.p2.4.m4.1.2.3.1.cmml" xref="S5.SS3.SSS1.p2.4.m4.1.2.3.2"><csymbol cd="latexml" id="S5.SS3.SSS1.p2.4.m4.1.2.3.1.1.cmml" xref="S5.SS3.SSS1.p2.4.m4.1.2.3.2.1">delimited-[]</csymbol><ci id="S5.SS3.SSS1.p2.4.m4.1.1.cmml" xref="S5.SS3.SSS1.p2.4.m4.1.1">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p2.4.m4.1c">i\in[k]</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p2.4.m4.1d">italic_i ∈ [ italic_k ]</annotation></semantics></math>, therefore <math alttext="|V_{k}|\geq q|V_{i}|" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p2.5.m5.2"><semantics id="S5.SS3.SSS1.p2.5.m5.2a"><mrow id="S5.SS3.SSS1.p2.5.m5.2.2" xref="S5.SS3.SSS1.p2.5.m5.2.2.cmml"><mrow id="S5.SS3.SSS1.p2.5.m5.1.1.1.1" xref="S5.SS3.SSS1.p2.5.m5.1.1.1.2.cmml"><mo id="S5.SS3.SSS1.p2.5.m5.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS1.p2.5.m5.1.1.1.2.1.cmml">|</mo><msub id="S5.SS3.SSS1.p2.5.m5.1.1.1.1.1" xref="S5.SS3.SSS1.p2.5.m5.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS1.p2.5.m5.1.1.1.1.1.2" xref="S5.SS3.SSS1.p2.5.m5.1.1.1.1.1.2.cmml">V</mi><mi id="S5.SS3.SSS1.p2.5.m5.1.1.1.1.1.3" xref="S5.SS3.SSS1.p2.5.m5.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S5.SS3.SSS1.p2.5.m5.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS1.p2.5.m5.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.SS3.SSS1.p2.5.m5.2.2.3" xref="S5.SS3.SSS1.p2.5.m5.2.2.3.cmml">≥</mo><mrow id="S5.SS3.SSS1.p2.5.m5.2.2.2" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.cmml"><mi id="S5.SS3.SSS1.p2.5.m5.2.2.2.3" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.3.cmml">q</mi><mo id="S5.SS3.SSS1.p2.5.m5.2.2.2.2" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.2.cmml">⁢</mo><mrow id="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.1.2.cmml"><mo id="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.2" stretchy="false" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.1.2.1.cmml">|</mo><msub id="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.1" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.1.cmml"><mi id="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.1.2" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.1.2.cmml">V</mi><mi id="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.1.3" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.1.3.cmml">i</mi></msub><mo id="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.3" stretchy="false" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.1.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p2.5.m5.2b"><apply id="S5.SS3.SSS1.p2.5.m5.2.2.cmml" xref="S5.SS3.SSS1.p2.5.m5.2.2"><geq id="S5.SS3.SSS1.p2.5.m5.2.2.3.cmml" xref="S5.SS3.SSS1.p2.5.m5.2.2.3"></geq><apply id="S5.SS3.SSS1.p2.5.m5.1.1.1.2.cmml" xref="S5.SS3.SSS1.p2.5.m5.1.1.1.1"><abs id="S5.SS3.SSS1.p2.5.m5.1.1.1.2.1.cmml" xref="S5.SS3.SSS1.p2.5.m5.1.1.1.1.2"></abs><apply id="S5.SS3.SSS1.p2.5.m5.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p2.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p2.5.m5.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p2.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S5.SS3.SSS1.p2.5.m5.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS1.p2.5.m5.1.1.1.1.1.2">𝑉</ci><ci id="S5.SS3.SSS1.p2.5.m5.1.1.1.1.1.3.cmml" xref="S5.SS3.SSS1.p2.5.m5.1.1.1.1.1.3">𝑘</ci></apply></apply><apply id="S5.SS3.SSS1.p2.5.m5.2.2.2.cmml" xref="S5.SS3.SSS1.p2.5.m5.2.2.2"><times id="S5.SS3.SSS1.p2.5.m5.2.2.2.2.cmml" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.2"></times><ci id="S5.SS3.SSS1.p2.5.m5.2.2.2.3.cmml" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.3">𝑞</ci><apply id="S5.SS3.SSS1.p2.5.m5.2.2.2.1.2.cmml" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1"><abs id="S5.SS3.SSS1.p2.5.m5.2.2.2.1.2.1.cmml" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.2"></abs><apply id="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.1.cmml" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.1.1.cmml" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.1">subscript</csymbol><ci id="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.1.2.cmml" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.1.2">𝑉</ci><ci id="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.1.3.cmml" xref="S5.SS3.SSS1.p2.5.m5.2.2.2.1.1.1.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p2.5.m5.2c">|V_{k}|\geq q|V_{i}|</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p2.5.m5.2d">| italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≥ italic_q | italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT |</annotation></semantics></math> for all <math alttext="i\in[k]" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p2.6.m6.1"><semantics id="S5.SS3.SSS1.p2.6.m6.1a"><mrow id="S5.SS3.SSS1.p2.6.m6.1.2" xref="S5.SS3.SSS1.p2.6.m6.1.2.cmml"><mi id="S5.SS3.SSS1.p2.6.m6.1.2.2" xref="S5.SS3.SSS1.p2.6.m6.1.2.2.cmml">i</mi><mo id="S5.SS3.SSS1.p2.6.m6.1.2.1" xref="S5.SS3.SSS1.p2.6.m6.1.2.1.cmml">∈</mo><mrow id="S5.SS3.SSS1.p2.6.m6.1.2.3.2" xref="S5.SS3.SSS1.p2.6.m6.1.2.3.1.cmml"><mo id="S5.SS3.SSS1.p2.6.m6.1.2.3.2.1" stretchy="false" xref="S5.SS3.SSS1.p2.6.m6.1.2.3.1.1.cmml">[</mo><mi id="S5.SS3.SSS1.p2.6.m6.1.1" xref="S5.SS3.SSS1.p2.6.m6.1.1.cmml">k</mi><mo id="S5.SS3.SSS1.p2.6.m6.1.2.3.2.2" stretchy="false" xref="S5.SS3.SSS1.p2.6.m6.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p2.6.m6.1b"><apply id="S5.SS3.SSS1.p2.6.m6.1.2.cmml" xref="S5.SS3.SSS1.p2.6.m6.1.2"><in id="S5.SS3.SSS1.p2.6.m6.1.2.1.cmml" xref="S5.SS3.SSS1.p2.6.m6.1.2.1"></in><ci id="S5.SS3.SSS1.p2.6.m6.1.2.2.cmml" xref="S5.SS3.SSS1.p2.6.m6.1.2.2">𝑖</ci><apply id="S5.SS3.SSS1.p2.6.m6.1.2.3.1.cmml" xref="S5.SS3.SSS1.p2.6.m6.1.2.3.2"><csymbol cd="latexml" id="S5.SS3.SSS1.p2.6.m6.1.2.3.1.1.cmml" xref="S5.SS3.SSS1.p2.6.m6.1.2.3.2.1">delimited-[]</csymbol><ci id="S5.SS3.SSS1.p2.6.m6.1.1.cmml" xref="S5.SS3.SSS1.p2.6.m6.1.1">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p2.6.m6.1c">i\in[k]</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p2.6.m6.1d">italic_i ∈ [ italic_k ]</annotation></semantics></math>. Summing this inequality over all <math alttext="i" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p2.7.m7.1"><semantics id="S5.SS3.SSS1.p2.7.m7.1a"><mi id="S5.SS3.SSS1.p2.7.m7.1.1" xref="S5.SS3.SSS1.p2.7.m7.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p2.7.m7.1b"><ci id="S5.SS3.SSS1.p2.7.m7.1.1.cmml" xref="S5.SS3.SSS1.p2.7.m7.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p2.7.m7.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p2.7.m7.1d">italic_i</annotation></semantics></math>, we have</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex36"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="k|V_{k}|\geq q\sum_{i\in[k]}|V_{i}|." class="ltx_Math" display="block" id="S5.Ex36.m1.2"><semantics id="S5.Ex36.m1.2a"><mrow id="S5.Ex36.m1.2.2.1" xref="S5.Ex36.m1.2.2.1.1.cmml"><mrow id="S5.Ex36.m1.2.2.1.1" xref="S5.Ex36.m1.2.2.1.1.cmml"><mrow id="S5.Ex36.m1.2.2.1.1.1" xref="S5.Ex36.m1.2.2.1.1.1.cmml"><mi id="S5.Ex36.m1.2.2.1.1.1.3" xref="S5.Ex36.m1.2.2.1.1.1.3.cmml">k</mi><mo id="S5.Ex36.m1.2.2.1.1.1.2" xref="S5.Ex36.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex36.m1.2.2.1.1.1.1.1" xref="S5.Ex36.m1.2.2.1.1.1.1.2.cmml"><mo id="S5.Ex36.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S5.Ex36.m1.2.2.1.1.1.1.2.1.cmml">|</mo><msub id="S5.Ex36.m1.2.2.1.1.1.1.1.1" xref="S5.Ex36.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S5.Ex36.m1.2.2.1.1.1.1.1.1.2" xref="S5.Ex36.m1.2.2.1.1.1.1.1.1.2.cmml">V</mi><mi id="S5.Ex36.m1.2.2.1.1.1.1.1.1.3" xref="S5.Ex36.m1.2.2.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S5.Ex36.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S5.Ex36.m1.2.2.1.1.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="S5.Ex36.m1.2.2.1.1.3" xref="S5.Ex36.m1.2.2.1.1.3.cmml">≥</mo><mrow id="S5.Ex36.m1.2.2.1.1.2" xref="S5.Ex36.m1.2.2.1.1.2.cmml"><mi id="S5.Ex36.m1.2.2.1.1.2.3" xref="S5.Ex36.m1.2.2.1.1.2.3.cmml">q</mi><mo id="S5.Ex36.m1.2.2.1.1.2.2" xref="S5.Ex36.m1.2.2.1.1.2.2.cmml">⁢</mo><mrow id="S5.Ex36.m1.2.2.1.1.2.1" xref="S5.Ex36.m1.2.2.1.1.2.1.cmml"><munder id="S5.Ex36.m1.2.2.1.1.2.1.2" xref="S5.Ex36.m1.2.2.1.1.2.1.2.cmml"><mo id="S5.Ex36.m1.2.2.1.1.2.1.2.2" movablelimits="false" rspace="0em" xref="S5.Ex36.m1.2.2.1.1.2.1.2.2.cmml">∑</mo><mrow id="S5.Ex36.m1.1.1.1" xref="S5.Ex36.m1.1.1.1.cmml"><mi id="S5.Ex36.m1.1.1.1.3" xref="S5.Ex36.m1.1.1.1.3.cmml">i</mi><mo id="S5.Ex36.m1.1.1.1.2" xref="S5.Ex36.m1.1.1.1.2.cmml">∈</mo><mrow id="S5.Ex36.m1.1.1.1.4.2" xref="S5.Ex36.m1.1.1.1.4.1.cmml"><mo id="S5.Ex36.m1.1.1.1.4.2.1" stretchy="false" xref="S5.Ex36.m1.1.1.1.4.1.1.cmml">[</mo><mi id="S5.Ex36.m1.1.1.1.1" xref="S5.Ex36.m1.1.1.1.1.cmml">k</mi><mo id="S5.Ex36.m1.1.1.1.4.2.2" stretchy="false" xref="S5.Ex36.m1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></munder><mrow id="S5.Ex36.m1.2.2.1.1.2.1.1.1" xref="S5.Ex36.m1.2.2.1.1.2.1.1.2.cmml"><mo id="S5.Ex36.m1.2.2.1.1.2.1.1.1.2" stretchy="false" xref="S5.Ex36.m1.2.2.1.1.2.1.1.2.1.cmml">|</mo><msub id="S5.Ex36.m1.2.2.1.1.2.1.1.1.1" xref="S5.Ex36.m1.2.2.1.1.2.1.1.1.1.cmml"><mi id="S5.Ex36.m1.2.2.1.1.2.1.1.1.1.2" xref="S5.Ex36.m1.2.2.1.1.2.1.1.1.1.2.cmml">V</mi><mi id="S5.Ex36.m1.2.2.1.1.2.1.1.1.1.3" xref="S5.Ex36.m1.2.2.1.1.2.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.Ex36.m1.2.2.1.1.2.1.1.1.3" stretchy="false" xref="S5.Ex36.m1.2.2.1.1.2.1.1.2.1.cmml">|</mo></mrow></mrow></mrow></mrow><mo id="S5.Ex36.m1.2.2.1.2" lspace="0em" xref="S5.Ex36.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex36.m1.2b"><apply id="S5.Ex36.m1.2.2.1.1.cmml" xref="S5.Ex36.m1.2.2.1"><geq id="S5.Ex36.m1.2.2.1.1.3.cmml" xref="S5.Ex36.m1.2.2.1.1.3"></geq><apply id="S5.Ex36.m1.2.2.1.1.1.cmml" xref="S5.Ex36.m1.2.2.1.1.1"><times id="S5.Ex36.m1.2.2.1.1.1.2.cmml" xref="S5.Ex36.m1.2.2.1.1.1.2"></times><ci id="S5.Ex36.m1.2.2.1.1.1.3.cmml" xref="S5.Ex36.m1.2.2.1.1.1.3">𝑘</ci><apply id="S5.Ex36.m1.2.2.1.1.1.1.2.cmml" xref="S5.Ex36.m1.2.2.1.1.1.1.1"><abs id="S5.Ex36.m1.2.2.1.1.1.1.2.1.cmml" xref="S5.Ex36.m1.2.2.1.1.1.1.1.2"></abs><apply id="S5.Ex36.m1.2.2.1.1.1.1.1.1.cmml" xref="S5.Ex36.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Ex36.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S5.Ex36.m1.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S5.Ex36.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S5.Ex36.m1.2.2.1.1.1.1.1.1.2">𝑉</ci><ci id="S5.Ex36.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S5.Ex36.m1.2.2.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply><apply id="S5.Ex36.m1.2.2.1.1.2.cmml" xref="S5.Ex36.m1.2.2.1.1.2"><times id="S5.Ex36.m1.2.2.1.1.2.2.cmml" xref="S5.Ex36.m1.2.2.1.1.2.2"></times><ci id="S5.Ex36.m1.2.2.1.1.2.3.cmml" xref="S5.Ex36.m1.2.2.1.1.2.3">𝑞</ci><apply id="S5.Ex36.m1.2.2.1.1.2.1.cmml" xref="S5.Ex36.m1.2.2.1.1.2.1"><apply id="S5.Ex36.m1.2.2.1.1.2.1.2.cmml" xref="S5.Ex36.m1.2.2.1.1.2.1.2"><csymbol cd="ambiguous" id="S5.Ex36.m1.2.2.1.1.2.1.2.1.cmml" xref="S5.Ex36.m1.2.2.1.1.2.1.2">subscript</csymbol><sum id="S5.Ex36.m1.2.2.1.1.2.1.2.2.cmml" xref="S5.Ex36.m1.2.2.1.1.2.1.2.2"></sum><apply id="S5.Ex36.m1.1.1.1.cmml" xref="S5.Ex36.m1.1.1.1"><in id="S5.Ex36.m1.1.1.1.2.cmml" xref="S5.Ex36.m1.1.1.1.2"></in><ci id="S5.Ex36.m1.1.1.1.3.cmml" xref="S5.Ex36.m1.1.1.1.3">𝑖</ci><apply id="S5.Ex36.m1.1.1.1.4.1.cmml" xref="S5.Ex36.m1.1.1.1.4.2"><csymbol cd="latexml" id="S5.Ex36.m1.1.1.1.4.1.1.cmml" xref="S5.Ex36.m1.1.1.1.4.2.1">delimited-[]</csymbol><ci id="S5.Ex36.m1.1.1.1.1.cmml" xref="S5.Ex36.m1.1.1.1.1">𝑘</ci></apply></apply></apply><apply id="S5.Ex36.m1.2.2.1.1.2.1.1.2.cmml" xref="S5.Ex36.m1.2.2.1.1.2.1.1.1"><abs id="S5.Ex36.m1.2.2.1.1.2.1.1.2.1.cmml" xref="S5.Ex36.m1.2.2.1.1.2.1.1.1.2"></abs><apply id="S5.Ex36.m1.2.2.1.1.2.1.1.1.1.cmml" xref="S5.Ex36.m1.2.2.1.1.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.Ex36.m1.2.2.1.1.2.1.1.1.1.1.cmml" xref="S5.Ex36.m1.2.2.1.1.2.1.1.1.1">subscript</csymbol><ci id="S5.Ex36.m1.2.2.1.1.2.1.1.1.1.2.cmml" xref="S5.Ex36.m1.2.2.1.1.2.1.1.1.1.2">𝑉</ci><ci id="S5.Ex36.m1.2.2.1.1.2.1.1.1.1.3.cmml" xref="S5.Ex36.m1.2.2.1.1.2.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex36.m1.2c">k|V_{k}|\geq q\sum_{i\in[k]}|V_{i}|.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex36.m1.2d">italic_k | italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ≥ italic_q ∑ start_POSTSUBSCRIPT italic_i ∈ [ italic_k ] end_POSTSUBSCRIPT | italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS3.SSS1.p2.14">Given that <math alttext="\sum_{i\in[k]}|V_{i}|=n" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p2.8.m1.2"><semantics id="S5.SS3.SSS1.p2.8.m1.2a"><mrow id="S5.SS3.SSS1.p2.8.m1.2.2" xref="S5.SS3.SSS1.p2.8.m1.2.2.cmml"><mrow id="S5.SS3.SSS1.p2.8.m1.2.2.1" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.cmml"><msub id="S5.SS3.SSS1.p2.8.m1.2.2.1.2" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.2.cmml"><mo id="S5.SS3.SSS1.p2.8.m1.2.2.1.2.2" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.2.2.cmml">∑</mo><mrow id="S5.SS3.SSS1.p2.8.m1.1.1.1" xref="S5.SS3.SSS1.p2.8.m1.1.1.1.cmml"><mi id="S5.SS3.SSS1.p2.8.m1.1.1.1.3" xref="S5.SS3.SSS1.p2.8.m1.1.1.1.3.cmml">i</mi><mo id="S5.SS3.SSS1.p2.8.m1.1.1.1.2" xref="S5.SS3.SSS1.p2.8.m1.1.1.1.2.cmml">∈</mo><mrow id="S5.SS3.SSS1.p2.8.m1.1.1.1.4.2" xref="S5.SS3.SSS1.p2.8.m1.1.1.1.4.1.cmml"><mo id="S5.SS3.SSS1.p2.8.m1.1.1.1.4.2.1" stretchy="false" xref="S5.SS3.SSS1.p2.8.m1.1.1.1.4.1.1.cmml">[</mo><mi id="S5.SS3.SSS1.p2.8.m1.1.1.1.1" xref="S5.SS3.SSS1.p2.8.m1.1.1.1.1.cmml">k</mi><mo id="S5.SS3.SSS1.p2.8.m1.1.1.1.4.2.2" stretchy="false" xref="S5.SS3.SSS1.p2.8.m1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><mrow id="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.1.2.cmml"><mo id="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.2" lspace="0em" stretchy="false" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.1.2.1.cmml">|</mo><msub id="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.1" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.1.cmml"><mi id="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.1.2" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.1.2.cmml">V</mi><mi id="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.1.3" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.3" stretchy="false" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="S5.SS3.SSS1.p2.8.m1.2.2.2" xref="S5.SS3.SSS1.p2.8.m1.2.2.2.cmml">=</mo><mi id="S5.SS3.SSS1.p2.8.m1.2.2.3" xref="S5.SS3.SSS1.p2.8.m1.2.2.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p2.8.m1.2b"><apply id="S5.SS3.SSS1.p2.8.m1.2.2.cmml" xref="S5.SS3.SSS1.p2.8.m1.2.2"><eq id="S5.SS3.SSS1.p2.8.m1.2.2.2.cmml" xref="S5.SS3.SSS1.p2.8.m1.2.2.2"></eq><apply id="S5.SS3.SSS1.p2.8.m1.2.2.1.cmml" xref="S5.SS3.SSS1.p2.8.m1.2.2.1"><apply id="S5.SS3.SSS1.p2.8.m1.2.2.1.2.cmml" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.2"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p2.8.m1.2.2.1.2.1.cmml" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.2">subscript</csymbol><sum id="S5.SS3.SSS1.p2.8.m1.2.2.1.2.2.cmml" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.2.2"></sum><apply id="S5.SS3.SSS1.p2.8.m1.1.1.1.cmml" xref="S5.SS3.SSS1.p2.8.m1.1.1.1"><in id="S5.SS3.SSS1.p2.8.m1.1.1.1.2.cmml" xref="S5.SS3.SSS1.p2.8.m1.1.1.1.2"></in><ci id="S5.SS3.SSS1.p2.8.m1.1.1.1.3.cmml" xref="S5.SS3.SSS1.p2.8.m1.1.1.1.3">𝑖</ci><apply id="S5.SS3.SSS1.p2.8.m1.1.1.1.4.1.cmml" xref="S5.SS3.SSS1.p2.8.m1.1.1.1.4.2"><csymbol cd="latexml" id="S5.SS3.SSS1.p2.8.m1.1.1.1.4.1.1.cmml" xref="S5.SS3.SSS1.p2.8.m1.1.1.1.4.2.1">delimited-[]</csymbol><ci id="S5.SS3.SSS1.p2.8.m1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p2.8.m1.1.1.1.1">𝑘</ci></apply></apply></apply><apply id="S5.SS3.SSS1.p2.8.m1.2.2.1.1.2.cmml" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1"><abs id="S5.SS3.SSS1.p2.8.m1.2.2.1.1.2.1.cmml" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.2"></abs><apply id="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.1.cmml" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.1">subscript</csymbol><ci id="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.1.2.cmml" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.1.2">𝑉</ci><ci id="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.1.3.cmml" xref="S5.SS3.SSS1.p2.8.m1.2.2.1.1.1.1.3">𝑖</ci></apply></apply></apply><ci id="S5.SS3.SSS1.p2.8.m1.2.2.3.cmml" xref="S5.SS3.SSS1.p2.8.m1.2.2.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p2.8.m1.2c">\sum_{i\in[k]}|V_{i}|=n</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p2.8.m1.2d">∑ start_POSTSUBSCRIPT italic_i ∈ [ italic_k ] end_POSTSUBSCRIPT | italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | = italic_n</annotation></semantics></math> and <math alttext="k|V_{k}|=n^{\prime}" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p2.9.m2.1"><semantics id="S5.SS3.SSS1.p2.9.m2.1a"><mrow id="S5.SS3.SSS1.p2.9.m2.1.1" xref="S5.SS3.SSS1.p2.9.m2.1.1.cmml"><mrow id="S5.SS3.SSS1.p2.9.m2.1.1.1" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.cmml"><mi id="S5.SS3.SSS1.p2.9.m2.1.1.1.3" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.3.cmml">k</mi><mo id="S5.SS3.SSS1.p2.9.m2.1.1.1.2" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.1.2.cmml"><mo id="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.1.2.1.cmml">|</mo><msub id="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.1" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.1.2" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.1.2.cmml">V</mi><mi id="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.1.3" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="S5.SS3.SSS1.p2.9.m2.1.1.2" xref="S5.SS3.SSS1.p2.9.m2.1.1.2.cmml">=</mo><msup id="S5.SS3.SSS1.p2.9.m2.1.1.3" xref="S5.SS3.SSS1.p2.9.m2.1.1.3.cmml"><mi id="S5.SS3.SSS1.p2.9.m2.1.1.3.2" xref="S5.SS3.SSS1.p2.9.m2.1.1.3.2.cmml">n</mi><mo id="S5.SS3.SSS1.p2.9.m2.1.1.3.3" xref="S5.SS3.SSS1.p2.9.m2.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p2.9.m2.1b"><apply id="S5.SS3.SSS1.p2.9.m2.1.1.cmml" xref="S5.SS3.SSS1.p2.9.m2.1.1"><eq id="S5.SS3.SSS1.p2.9.m2.1.1.2.cmml" xref="S5.SS3.SSS1.p2.9.m2.1.1.2"></eq><apply id="S5.SS3.SSS1.p2.9.m2.1.1.1.cmml" xref="S5.SS3.SSS1.p2.9.m2.1.1.1"><times id="S5.SS3.SSS1.p2.9.m2.1.1.1.2.cmml" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.2"></times><ci id="S5.SS3.SSS1.p2.9.m2.1.1.1.3.cmml" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.3">𝑘</ci><apply id="S5.SS3.SSS1.p2.9.m2.1.1.1.1.2.cmml" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1"><abs id="S5.SS3.SSS1.p2.9.m2.1.1.1.1.2.1.cmml" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.2"></abs><apply id="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.1">subscript</csymbol><ci id="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.1.2">𝑉</ci><ci id="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.1.3.cmml" xref="S5.SS3.SSS1.p2.9.m2.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply><apply id="S5.SS3.SSS1.p2.9.m2.1.1.3.cmml" xref="S5.SS3.SSS1.p2.9.m2.1.1.3"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p2.9.m2.1.1.3.1.cmml" xref="S5.SS3.SSS1.p2.9.m2.1.1.3">superscript</csymbol><ci id="S5.SS3.SSS1.p2.9.m2.1.1.3.2.cmml" xref="S5.SS3.SSS1.p2.9.m2.1.1.3.2">𝑛</ci><ci id="S5.SS3.SSS1.p2.9.m2.1.1.3.3.cmml" xref="S5.SS3.SSS1.p2.9.m2.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p2.9.m2.1c">k|V_{k}|=n^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p2.9.m2.1d">italic_k | italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | = italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, it follows that <math alttext="n^{\prime}\geq nq" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p2.10.m3.1"><semantics id="S5.SS3.SSS1.p2.10.m3.1a"><mrow id="S5.SS3.SSS1.p2.10.m3.1.1" xref="S5.SS3.SSS1.p2.10.m3.1.1.cmml"><msup id="S5.SS3.SSS1.p2.10.m3.1.1.2" xref="S5.SS3.SSS1.p2.10.m3.1.1.2.cmml"><mi id="S5.SS3.SSS1.p2.10.m3.1.1.2.2" xref="S5.SS3.SSS1.p2.10.m3.1.1.2.2.cmml">n</mi><mo id="S5.SS3.SSS1.p2.10.m3.1.1.2.3" xref="S5.SS3.SSS1.p2.10.m3.1.1.2.3.cmml">′</mo></msup><mo id="S5.SS3.SSS1.p2.10.m3.1.1.1" xref="S5.SS3.SSS1.p2.10.m3.1.1.1.cmml">≥</mo><mrow id="S5.SS3.SSS1.p2.10.m3.1.1.3" xref="S5.SS3.SSS1.p2.10.m3.1.1.3.cmml"><mi id="S5.SS3.SSS1.p2.10.m3.1.1.3.2" xref="S5.SS3.SSS1.p2.10.m3.1.1.3.2.cmml">n</mi><mo id="S5.SS3.SSS1.p2.10.m3.1.1.3.1" xref="S5.SS3.SSS1.p2.10.m3.1.1.3.1.cmml">⁢</mo><mi id="S5.SS3.SSS1.p2.10.m3.1.1.3.3" xref="S5.SS3.SSS1.p2.10.m3.1.1.3.3.cmml">q</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p2.10.m3.1b"><apply id="S5.SS3.SSS1.p2.10.m3.1.1.cmml" xref="S5.SS3.SSS1.p2.10.m3.1.1"><geq id="S5.SS3.SSS1.p2.10.m3.1.1.1.cmml" xref="S5.SS3.SSS1.p2.10.m3.1.1.1"></geq><apply id="S5.SS3.SSS1.p2.10.m3.1.1.2.cmml" xref="S5.SS3.SSS1.p2.10.m3.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p2.10.m3.1.1.2.1.cmml" xref="S5.SS3.SSS1.p2.10.m3.1.1.2">superscript</csymbol><ci id="S5.SS3.SSS1.p2.10.m3.1.1.2.2.cmml" xref="S5.SS3.SSS1.p2.10.m3.1.1.2.2">𝑛</ci><ci id="S5.SS3.SSS1.p2.10.m3.1.1.2.3.cmml" xref="S5.SS3.SSS1.p2.10.m3.1.1.2.3">′</ci></apply><apply id="S5.SS3.SSS1.p2.10.m3.1.1.3.cmml" xref="S5.SS3.SSS1.p2.10.m3.1.1.3"><times id="S5.SS3.SSS1.p2.10.m3.1.1.3.1.cmml" xref="S5.SS3.SSS1.p2.10.m3.1.1.3.1"></times><ci id="S5.SS3.SSS1.p2.10.m3.1.1.3.2.cmml" xref="S5.SS3.SSS1.p2.10.m3.1.1.3.2">𝑛</ci><ci id="S5.SS3.SSS1.p2.10.m3.1.1.3.3.cmml" xref="S5.SS3.SSS1.p2.10.m3.1.1.3.3">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p2.10.m3.1c">n^{\prime}\geq nq</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p2.10.m3.1d">italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≥ italic_n italic_q</annotation></semantics></math>. Since <math alttext="G^{T}" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p2.11.m4.1"><semantics id="S5.SS3.SSS1.p2.11.m4.1a"><msup id="S5.SS3.SSS1.p2.11.m4.1.1" xref="S5.SS3.SSS1.p2.11.m4.1.1.cmml"><mi id="S5.SS3.SSS1.p2.11.m4.1.1.2" xref="S5.SS3.SSS1.p2.11.m4.1.1.2.cmml">G</mi><mi id="S5.SS3.SSS1.p2.11.m4.1.1.3" xref="S5.SS3.SSS1.p2.11.m4.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p2.11.m4.1b"><apply id="S5.SS3.SSS1.p2.11.m4.1.1.cmml" xref="S5.SS3.SSS1.p2.11.m4.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p2.11.m4.1.1.1.cmml" xref="S5.SS3.SSS1.p2.11.m4.1.1">superscript</csymbol><ci id="S5.SS3.SSS1.p2.11.m4.1.1.2.cmml" xref="S5.SS3.SSS1.p2.11.m4.1.1.2">𝐺</ci><ci id="S5.SS3.SSS1.p2.11.m4.1.1.3.cmml" xref="S5.SS3.SSS1.p2.11.m4.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p2.11.m4.1c">G^{T}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p2.11.m4.1d">italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> is a subgraph of <math alttext="G" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p2.12.m5.1"><semantics id="S5.SS3.SSS1.p2.12.m5.1a"><mi id="S5.SS3.SSS1.p2.12.m5.1.1" xref="S5.SS3.SSS1.p2.12.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p2.12.m5.1b"><ci id="S5.SS3.SSS1.p2.12.m5.1.1.cmml" xref="S5.SS3.SSS1.p2.12.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p2.12.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p2.12.m5.1d">italic_G</annotation></semantics></math>, we also have <math alttext="n^{\prime}\leq n" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p2.13.m6.1"><semantics id="S5.SS3.SSS1.p2.13.m6.1a"><mrow id="S5.SS3.SSS1.p2.13.m6.1.1" xref="S5.SS3.SSS1.p2.13.m6.1.1.cmml"><msup id="S5.SS3.SSS1.p2.13.m6.1.1.2" xref="S5.SS3.SSS1.p2.13.m6.1.1.2.cmml"><mi id="S5.SS3.SSS1.p2.13.m6.1.1.2.2" xref="S5.SS3.SSS1.p2.13.m6.1.1.2.2.cmml">n</mi><mo id="S5.SS3.SSS1.p2.13.m6.1.1.2.3" xref="S5.SS3.SSS1.p2.13.m6.1.1.2.3.cmml">′</mo></msup><mo id="S5.SS3.SSS1.p2.13.m6.1.1.1" xref="S5.SS3.SSS1.p2.13.m6.1.1.1.cmml">≤</mo><mi id="S5.SS3.SSS1.p2.13.m6.1.1.3" xref="S5.SS3.SSS1.p2.13.m6.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p2.13.m6.1b"><apply id="S5.SS3.SSS1.p2.13.m6.1.1.cmml" xref="S5.SS3.SSS1.p2.13.m6.1.1"><leq id="S5.SS3.SSS1.p2.13.m6.1.1.1.cmml" xref="S5.SS3.SSS1.p2.13.m6.1.1.1"></leq><apply id="S5.SS3.SSS1.p2.13.m6.1.1.2.cmml" xref="S5.SS3.SSS1.p2.13.m6.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p2.13.m6.1.1.2.1.cmml" xref="S5.SS3.SSS1.p2.13.m6.1.1.2">superscript</csymbol><ci id="S5.SS3.SSS1.p2.13.m6.1.1.2.2.cmml" xref="S5.SS3.SSS1.p2.13.m6.1.1.2.2">𝑛</ci><ci id="S5.SS3.SSS1.p2.13.m6.1.1.2.3.cmml" xref="S5.SS3.SSS1.p2.13.m6.1.1.2.3">′</ci></apply><ci id="S5.SS3.SSS1.p2.13.m6.1.1.3.cmml" xref="S5.SS3.SSS1.p2.13.m6.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p2.13.m6.1c">n^{\prime}\leq n</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p2.13.m6.1d">italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_n</annotation></semantics></math>, which implies <math alttext="n^{\prime}=\Theta(n)" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p2.14.m7.1"><semantics id="S5.SS3.SSS1.p2.14.m7.1a"><mrow id="S5.SS3.SSS1.p2.14.m7.1.2" xref="S5.SS3.SSS1.p2.14.m7.1.2.cmml"><msup id="S5.SS3.SSS1.p2.14.m7.1.2.2" xref="S5.SS3.SSS1.p2.14.m7.1.2.2.cmml"><mi id="S5.SS3.SSS1.p2.14.m7.1.2.2.2" xref="S5.SS3.SSS1.p2.14.m7.1.2.2.2.cmml">n</mi><mo id="S5.SS3.SSS1.p2.14.m7.1.2.2.3" xref="S5.SS3.SSS1.p2.14.m7.1.2.2.3.cmml">′</mo></msup><mo id="S5.SS3.SSS1.p2.14.m7.1.2.1" xref="S5.SS3.SSS1.p2.14.m7.1.2.1.cmml">=</mo><mrow id="S5.SS3.SSS1.p2.14.m7.1.2.3" xref="S5.SS3.SSS1.p2.14.m7.1.2.3.cmml"><mi id="S5.SS3.SSS1.p2.14.m7.1.2.3.2" mathvariant="normal" xref="S5.SS3.SSS1.p2.14.m7.1.2.3.2.cmml">Θ</mi><mo id="S5.SS3.SSS1.p2.14.m7.1.2.3.1" xref="S5.SS3.SSS1.p2.14.m7.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS3.SSS1.p2.14.m7.1.2.3.3.2" xref="S5.SS3.SSS1.p2.14.m7.1.2.3.cmml"><mo id="S5.SS3.SSS1.p2.14.m7.1.2.3.3.2.1" stretchy="false" xref="S5.SS3.SSS1.p2.14.m7.1.2.3.cmml">(</mo><mi id="S5.SS3.SSS1.p2.14.m7.1.1" xref="S5.SS3.SSS1.p2.14.m7.1.1.cmml">n</mi><mo id="S5.SS3.SSS1.p2.14.m7.1.2.3.3.2.2" stretchy="false" xref="S5.SS3.SSS1.p2.14.m7.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p2.14.m7.1b"><apply id="S5.SS3.SSS1.p2.14.m7.1.2.cmml" xref="S5.SS3.SSS1.p2.14.m7.1.2"><eq id="S5.SS3.SSS1.p2.14.m7.1.2.1.cmml" xref="S5.SS3.SSS1.p2.14.m7.1.2.1"></eq><apply id="S5.SS3.SSS1.p2.14.m7.1.2.2.cmml" xref="S5.SS3.SSS1.p2.14.m7.1.2.2"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p2.14.m7.1.2.2.1.cmml" xref="S5.SS3.SSS1.p2.14.m7.1.2.2">superscript</csymbol><ci id="S5.SS3.SSS1.p2.14.m7.1.2.2.2.cmml" xref="S5.SS3.SSS1.p2.14.m7.1.2.2.2">𝑛</ci><ci id="S5.SS3.SSS1.p2.14.m7.1.2.2.3.cmml" xref="S5.SS3.SSS1.p2.14.m7.1.2.2.3">′</ci></apply><apply id="S5.SS3.SSS1.p2.14.m7.1.2.3.cmml" xref="S5.SS3.SSS1.p2.14.m7.1.2.3"><times id="S5.SS3.SSS1.p2.14.m7.1.2.3.1.cmml" xref="S5.SS3.SSS1.p2.14.m7.1.2.3.1"></times><ci id="S5.SS3.SSS1.p2.14.m7.1.2.3.2.cmml" xref="S5.SS3.SSS1.p2.14.m7.1.2.3.2">Θ</ci><ci id="S5.SS3.SSS1.p2.14.m7.1.1.cmml" xref="S5.SS3.SSS1.p2.14.m7.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p2.14.m7.1c">n^{\prime}=\Theta(n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p2.14.m7.1d">italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = roman_Θ ( italic_n )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS3.SSS1.p3"> <p class="ltx_p" id="S5.SS3.SSS1.p3.2">After completing the first phase, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg4" title="In 5.3.1 Low Perturbation Regime for Random Balanced Multipartite Graphs ‣ 5.3 Balanced Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">4</span></a> proceeds by applying <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg1" title="In 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a> on <math alttext="G^{T}" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p3.1.m1.1"><semantics id="S5.SS3.SSS1.p3.1.m1.1a"><msup id="S5.SS3.SSS1.p3.1.m1.1.1" xref="S5.SS3.SSS1.p3.1.m1.1.1.cmml"><mi id="S5.SS3.SSS1.p3.1.m1.1.1.2" xref="S5.SS3.SSS1.p3.1.m1.1.1.2.cmml">G</mi><mi id="S5.SS3.SSS1.p3.1.m1.1.1.3" xref="S5.SS3.SSS1.p3.1.m1.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p3.1.m1.1b"><apply id="S5.SS3.SSS1.p3.1.m1.1.1.cmml" xref="S5.SS3.SSS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p3.1.m1.1.1.1.cmml" xref="S5.SS3.SSS1.p3.1.m1.1.1">superscript</csymbol><ci id="S5.SS3.SSS1.p3.1.m1.1.1.2.cmml" xref="S5.SS3.SSS1.p3.1.m1.1.1.2">𝐺</ci><ci id="S5.SS3.SSS1.p3.1.m1.1.1.3.cmml" xref="S5.SS3.SSS1.p3.1.m1.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p3.1.m1.1c">G^{T}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p3.1.m1.1d">italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> to partition the agents in <math alttext="\bigcup_{i=1}^{k}V^{\prime}_{i}" class="ltx_Math" display="inline" id="S5.SS3.SSS1.p3.2.m2.1"><semantics id="S5.SS3.SSS1.p3.2.m2.1a"><mrow id="S5.SS3.SSS1.p3.2.m2.1.1" xref="S5.SS3.SSS1.p3.2.m2.1.1.cmml"><msubsup id="S5.SS3.SSS1.p3.2.m2.1.1.1" xref="S5.SS3.SSS1.p3.2.m2.1.1.1.cmml"><mo id="S5.SS3.SSS1.p3.2.m2.1.1.1.2.2" xref="S5.SS3.SSS1.p3.2.m2.1.1.1.2.2.cmml">⋃</mo><mrow id="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3" xref="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3.cmml"><mi id="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3.2" xref="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3.2.cmml">i</mi><mo id="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3.1" xref="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3.1.cmml">=</mo><mn id="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3.3" xref="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3.3.cmml">1</mn></mrow><mi id="S5.SS3.SSS1.p3.2.m2.1.1.1.3" xref="S5.SS3.SSS1.p3.2.m2.1.1.1.3.cmml">k</mi></msubsup><msubsup id="S5.SS3.SSS1.p3.2.m2.1.1.2" xref="S5.SS3.SSS1.p3.2.m2.1.1.2.cmml"><mi id="S5.SS3.SSS1.p3.2.m2.1.1.2.2.2" xref="S5.SS3.SSS1.p3.2.m2.1.1.2.2.2.cmml">V</mi><mi id="S5.SS3.SSS1.p3.2.m2.1.1.2.3" xref="S5.SS3.SSS1.p3.2.m2.1.1.2.3.cmml">i</mi><mo id="S5.SS3.SSS1.p3.2.m2.1.1.2.2.3" xref="S5.SS3.SSS1.p3.2.m2.1.1.2.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.p3.2.m2.1b"><apply id="S5.SS3.SSS1.p3.2.m2.1.1.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1"><apply id="S5.SS3.SSS1.p3.2.m2.1.1.1.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p3.2.m2.1.1.1.1.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.1">superscript</csymbol><apply id="S5.SS3.SSS1.p3.2.m2.1.1.1.2.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p3.2.m2.1.1.1.2.1.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.1">subscript</csymbol><union id="S5.SS3.SSS1.p3.2.m2.1.1.1.2.2.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.1.2.2"></union><apply id="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3"><eq id="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3.1.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3.1"></eq><ci id="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3.2.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3.2">𝑖</ci><cn id="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3.3.cmml" type="integer" xref="S5.SS3.SSS1.p3.2.m2.1.1.1.2.3.3">1</cn></apply></apply><ci id="S5.SS3.SSS1.p3.2.m2.1.1.1.3.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.1.3">𝑘</ci></apply><apply id="S5.SS3.SSS1.p3.2.m2.1.1.2.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p3.2.m2.1.1.2.1.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.2">subscript</csymbol><apply id="S5.SS3.SSS1.p3.2.m2.1.1.2.2.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.SSS1.p3.2.m2.1.1.2.2.1.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.2">superscript</csymbol><ci id="S5.SS3.SSS1.p3.2.m2.1.1.2.2.2.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.2.2.2">𝑉</ci><ci id="S5.SS3.SSS1.p3.2.m2.1.1.2.2.3.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.2.2.3">′</ci></apply><ci id="S5.SS3.SSS1.p3.2.m2.1.1.2.3.cmml" xref="S5.SS3.SSS1.p3.2.m2.1.1.2.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.p3.2.m2.1c">\bigcup_{i=1}^{k}V^{\prime}_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.p3.2.m2.1d">⋃ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, and then assigns any remaining agents to singleton coalitions.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg4"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg4.9.1.1">Algorithm 4</span> </span> Constant-factor approximation algorithm for random balanced <math alttext="k" class="ltx_Math" display="inline" id="alg4.2.m1.1"><semantics id="alg4.2.m1.1b"><mi id="alg4.2.m1.1.1" xref="alg4.2.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="alg4.2.m1.1c"><ci id="alg4.2.m1.1.1.cmml" xref="alg4.2.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="alg4.2.m1.1d">k</annotation><annotation encoding="application/x-llamapun" id="alg4.2.m1.1e">italic_k</annotation></semantics></math>-partite graph</figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_left" id="alg4.6.4"><span class="ltx_text ltx_font_bold" id="alg4.6.4.1">Input:</span> <math alttext="\langle G,\varepsilon\rangle" class="ltx_Math" display="inline" id="alg4.3.1.m1.2"><semantics id="alg4.3.1.m1.2a"><mrow id="alg4.3.1.m1.2.3.2" xref="alg4.3.1.m1.2.3.1.cmml"><mo id="alg4.3.1.m1.2.3.2.1" stretchy="false" xref="alg4.3.1.m1.2.3.1.cmml">⟨</mo><mi id="alg4.3.1.m1.1.1" xref="alg4.3.1.m1.1.1.cmml">G</mi><mo id="alg4.3.1.m1.2.3.2.2" xref="alg4.3.1.m1.2.3.1.cmml">,</mo><mi id="alg4.3.1.m1.2.2" xref="alg4.3.1.m1.2.2.cmml">ε</mi><mo id="alg4.3.1.m1.2.3.2.3" stretchy="false" xref="alg4.3.1.m1.2.3.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="alg4.3.1.m1.2b"><list id="alg4.3.1.m1.2.3.1.cmml" xref="alg4.3.1.m1.2.3.2"><ci id="alg4.3.1.m1.1.1.cmml" xref="alg4.3.1.m1.1.1">𝐺</ci><ci id="alg4.3.1.m1.2.2.cmml" xref="alg4.3.1.m1.2.2">𝜀</ci></list></annotation-xml><annotation encoding="application/x-tex" id="alg4.3.1.m1.2c">\langle G,\varepsilon\rangle</annotation><annotation encoding="application/x-llamapun" id="alg4.3.1.m1.2d">⟨ italic_G , italic_ε ⟩</annotation></semantics></math> where <math alttext="G=(\{V_{1},\cdots,V_{k}\},p)" class="ltx_Math" display="inline" id="alg4.4.2.m2.3"><semantics id="alg4.4.2.m2.3a"><mrow id="alg4.4.2.m2.3.3" xref="alg4.4.2.m2.3.3.cmml"><mi id="alg4.4.2.m2.3.3.3" xref="alg4.4.2.m2.3.3.3.cmml">G</mi><mo id="alg4.4.2.m2.3.3.2" xref="alg4.4.2.m2.3.3.2.cmml">=</mo><mrow id="alg4.4.2.m2.3.3.1.1" xref="alg4.4.2.m2.3.3.1.2.cmml"><mo id="alg4.4.2.m2.3.3.1.1.2" stretchy="false" xref="alg4.4.2.m2.3.3.1.2.cmml">(</mo><mrow id="alg4.4.2.m2.3.3.1.1.1.2" xref="alg4.4.2.m2.3.3.1.1.1.3.cmml"><mo id="alg4.4.2.m2.3.3.1.1.1.2.3" stretchy="false" xref="alg4.4.2.m2.3.3.1.1.1.3.cmml">{</mo><msub id="alg4.4.2.m2.3.3.1.1.1.1.1" xref="alg4.4.2.m2.3.3.1.1.1.1.1.cmml"><mi id="alg4.4.2.m2.3.3.1.1.1.1.1.2" xref="alg4.4.2.m2.3.3.1.1.1.1.1.2.cmml">V</mi><mn id="alg4.4.2.m2.3.3.1.1.1.1.1.3" xref="alg4.4.2.m2.3.3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="alg4.4.2.m2.3.3.1.1.1.2.4" xref="alg4.4.2.m2.3.3.1.1.1.3.cmml">,</mo><mi id="alg4.4.2.m2.1.1" mathvariant="normal" xref="alg4.4.2.m2.1.1.cmml">⋯</mi><mo id="alg4.4.2.m2.3.3.1.1.1.2.5" xref="alg4.4.2.m2.3.3.1.1.1.3.cmml">,</mo><msub id="alg4.4.2.m2.3.3.1.1.1.2.2" xref="alg4.4.2.m2.3.3.1.1.1.2.2.cmml"><mi id="alg4.4.2.m2.3.3.1.1.1.2.2.2" xref="alg4.4.2.m2.3.3.1.1.1.2.2.2.cmml">V</mi><mi id="alg4.4.2.m2.3.3.1.1.1.2.2.3" xref="alg4.4.2.m2.3.3.1.1.1.2.2.3.cmml">k</mi></msub><mo id="alg4.4.2.m2.3.3.1.1.1.2.6" stretchy="false" xref="alg4.4.2.m2.3.3.1.1.1.3.cmml">}</mo></mrow><mo id="alg4.4.2.m2.3.3.1.1.3" xref="alg4.4.2.m2.3.3.1.2.cmml">,</mo><mi id="alg4.4.2.m2.2.2" xref="alg4.4.2.m2.2.2.cmml">p</mi><mo id="alg4.4.2.m2.3.3.1.1.4" stretchy="false" xref="alg4.4.2.m2.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg4.4.2.m2.3b"><apply id="alg4.4.2.m2.3.3.cmml" xref="alg4.4.2.m2.3.3"><eq id="alg4.4.2.m2.3.3.2.cmml" xref="alg4.4.2.m2.3.3.2"></eq><ci id="alg4.4.2.m2.3.3.3.cmml" xref="alg4.4.2.m2.3.3.3">𝐺</ci><interval closure="open" id="alg4.4.2.m2.3.3.1.2.cmml" xref="alg4.4.2.m2.3.3.1.1"><set id="alg4.4.2.m2.3.3.1.1.1.3.cmml" xref="alg4.4.2.m2.3.3.1.1.1.2"><apply id="alg4.4.2.m2.3.3.1.1.1.1.1.cmml" xref="alg4.4.2.m2.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="alg4.4.2.m2.3.3.1.1.1.1.1.1.cmml" xref="alg4.4.2.m2.3.3.1.1.1.1.1">subscript</csymbol><ci id="alg4.4.2.m2.3.3.1.1.1.1.1.2.cmml" xref="alg4.4.2.m2.3.3.1.1.1.1.1.2">𝑉</ci><cn id="alg4.4.2.m2.3.3.1.1.1.1.1.3.cmml" type="integer" xref="alg4.4.2.m2.3.3.1.1.1.1.1.3">1</cn></apply><ci id="alg4.4.2.m2.1.1.cmml" xref="alg4.4.2.m2.1.1">⋯</ci><apply id="alg4.4.2.m2.3.3.1.1.1.2.2.cmml" xref="alg4.4.2.m2.3.3.1.1.1.2.2"><csymbol cd="ambiguous" id="alg4.4.2.m2.3.3.1.1.1.2.2.1.cmml" xref="alg4.4.2.m2.3.3.1.1.1.2.2">subscript</csymbol><ci id="alg4.4.2.m2.3.3.1.1.1.2.2.2.cmml" xref="alg4.4.2.m2.3.3.1.1.1.2.2.2">𝑉</ci><ci id="alg4.4.2.m2.3.3.1.1.1.2.2.3.cmml" xref="alg4.4.2.m2.3.3.1.1.1.2.2.3">𝑘</ci></apply></set><ci id="alg4.4.2.m2.2.2.cmml" xref="alg4.4.2.m2.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.4.2.m2.3c">G=(\{V_{1},\cdots,V_{k}\},p)</annotation><annotation encoding="application/x-llamapun" id="alg4.4.2.m2.3d">italic_G = ( { italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } , italic_p )</annotation></semantics></math> is a random balanced <math alttext="k" class="ltx_Math" display="inline" id="alg4.5.3.m3.1"><semantics id="alg4.5.3.m3.1a"><mi id="alg4.5.3.m3.1.1" xref="alg4.5.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="alg4.5.3.m3.1b"><ci id="alg4.5.3.m3.1.1.cmml" xref="alg4.5.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="alg4.5.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="alg4.5.3.m3.1d">italic_k</annotation></semantics></math>-partite graph and <math alttext="p=\mathcal{O}(\frac{1}{k})" class="ltx_Math" display="inline" id="alg4.6.4.m4.1"><semantics id="alg4.6.4.m4.1a"><mrow id="alg4.6.4.m4.1.2" xref="alg4.6.4.m4.1.2.cmml"><mi id="alg4.6.4.m4.1.2.2" xref="alg4.6.4.m4.1.2.2.cmml">p</mi><mo id="alg4.6.4.m4.1.2.1" xref="alg4.6.4.m4.1.2.1.cmml">=</mo><mrow id="alg4.6.4.m4.1.2.3" xref="alg4.6.4.m4.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="alg4.6.4.m4.1.2.3.2" xref="alg4.6.4.m4.1.2.3.2.cmml">𝒪</mi><mo id="alg4.6.4.m4.1.2.3.1" xref="alg4.6.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="alg4.6.4.m4.1.2.3.3.2" xref="alg4.6.4.m4.1.1.cmml"><mo id="alg4.6.4.m4.1.2.3.3.2.1" stretchy="false" xref="alg4.6.4.m4.1.1.cmml">(</mo><mfrac id="alg4.6.4.m4.1.1" xref="alg4.6.4.m4.1.1.cmml"><mn id="alg4.6.4.m4.1.1.2" xref="alg4.6.4.m4.1.1.2.cmml">1</mn><mi id="alg4.6.4.m4.1.1.3" xref="alg4.6.4.m4.1.1.3.cmml">k</mi></mfrac><mo id="alg4.6.4.m4.1.2.3.3.2.2" stretchy="false" xref="alg4.6.4.m4.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg4.6.4.m4.1b"><apply id="alg4.6.4.m4.1.2.cmml" xref="alg4.6.4.m4.1.2"><eq id="alg4.6.4.m4.1.2.1.cmml" xref="alg4.6.4.m4.1.2.1"></eq><ci id="alg4.6.4.m4.1.2.2.cmml" xref="alg4.6.4.m4.1.2.2">𝑝</ci><apply id="alg4.6.4.m4.1.2.3.cmml" xref="alg4.6.4.m4.1.2.3"><times id="alg4.6.4.m4.1.2.3.1.cmml" xref="alg4.6.4.m4.1.2.3.1"></times><ci id="alg4.6.4.m4.1.2.3.2.cmml" xref="alg4.6.4.m4.1.2.3.2">𝒪</ci><apply id="alg4.6.4.m4.1.1.cmml" xref="alg4.6.4.m4.1.2.3.3.2"><divide id="alg4.6.4.m4.1.1.1.cmml" xref="alg4.6.4.m4.1.2.3.3.2"></divide><cn id="alg4.6.4.m4.1.1.2.cmml" type="integer" xref="alg4.6.4.m4.1.1.2">1</cn><ci id="alg4.6.4.m4.1.1.3.cmml" xref="alg4.6.4.m4.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.6.4.m4.1c">p=\mathcal{O}(\frac{1}{k})</annotation><annotation encoding="application/x-llamapun" id="alg4.6.4.m4.1d">italic_p = caligraphic_O ( divide start_ARG 1 end_ARG start_ARG italic_k end_ARG )</annotation></semantics></math></p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_left" id="alg4.7.5"><span class="ltx_text ltx_font_bold" id="alg4.7.5.1">Output:</span> Partition <math alttext="\pi" class="ltx_Math" display="inline" id="alg4.7.5.m1.1"><semantics id="alg4.7.5.m1.1a"><mi id="alg4.7.5.m1.1.1" xref="alg4.7.5.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="alg4.7.5.m1.1b"><ci id="alg4.7.5.m1.1.1.cmml" xref="alg4.7.5.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="alg4.7.5.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="alg4.7.5.m1.1d">italic_π</annotation></semantics></math></p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <div class="ltx_listing ltx_figure_panel ltx_listing" id="alg4.10"> <div class="ltx_listingline" id="alg4.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l1.1.1.1" style="font-size:80%;">1:</span></span>Reduce to a random Turán graph <math alttext="G^{T}=(n^{\prime},k,p)" class="ltx_Math" display="inline" id="alg4.l1.m1.3"><semantics id="alg4.l1.m1.3a"><mrow id="alg4.l1.m1.3.3" xref="alg4.l1.m1.3.3.cmml"><msup id="alg4.l1.m1.3.3.3" xref="alg4.l1.m1.3.3.3.cmml"><mi id="alg4.l1.m1.3.3.3.2" xref="alg4.l1.m1.3.3.3.2.cmml">G</mi><mi id="alg4.l1.m1.3.3.3.3" xref="alg4.l1.m1.3.3.3.3.cmml">T</mi></msup><mo id="alg4.l1.m1.3.3.2" xref="alg4.l1.m1.3.3.2.cmml">=</mo><mrow id="alg4.l1.m1.3.3.1.1" xref="alg4.l1.m1.3.3.1.2.cmml"><mo id="alg4.l1.m1.3.3.1.1.2" stretchy="false" xref="alg4.l1.m1.3.3.1.2.cmml">(</mo><msup id="alg4.l1.m1.3.3.1.1.1" xref="alg4.l1.m1.3.3.1.1.1.cmml"><mi id="alg4.l1.m1.3.3.1.1.1.2" xref="alg4.l1.m1.3.3.1.1.1.2.cmml">n</mi><mo id="alg4.l1.m1.3.3.1.1.1.3" xref="alg4.l1.m1.3.3.1.1.1.3.cmml">′</mo></msup><mo id="alg4.l1.m1.3.3.1.1.3" xref="alg4.l1.m1.3.3.1.2.cmml">,</mo><mi id="alg4.l1.m1.1.1" xref="alg4.l1.m1.1.1.cmml">k</mi><mo id="alg4.l1.m1.3.3.1.1.4" xref="alg4.l1.m1.3.3.1.2.cmml">,</mo><mi id="alg4.l1.m1.2.2" xref="alg4.l1.m1.2.2.cmml">p</mi><mo id="alg4.l1.m1.3.3.1.1.5" stretchy="false" xref="alg4.l1.m1.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg4.l1.m1.3b"><apply id="alg4.l1.m1.3.3.cmml" xref="alg4.l1.m1.3.3"><eq id="alg4.l1.m1.3.3.2.cmml" xref="alg4.l1.m1.3.3.2"></eq><apply id="alg4.l1.m1.3.3.3.cmml" xref="alg4.l1.m1.3.3.3"><csymbol cd="ambiguous" id="alg4.l1.m1.3.3.3.1.cmml" xref="alg4.l1.m1.3.3.3">superscript</csymbol><ci id="alg4.l1.m1.3.3.3.2.cmml" xref="alg4.l1.m1.3.3.3.2">𝐺</ci><ci id="alg4.l1.m1.3.3.3.3.cmml" xref="alg4.l1.m1.3.3.3.3">𝑇</ci></apply><vector id="alg4.l1.m1.3.3.1.2.cmml" xref="alg4.l1.m1.3.3.1.1"><apply id="alg4.l1.m1.3.3.1.1.1.cmml" xref="alg4.l1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="alg4.l1.m1.3.3.1.1.1.1.cmml" xref="alg4.l1.m1.3.3.1.1.1">superscript</csymbol><ci id="alg4.l1.m1.3.3.1.1.1.2.cmml" xref="alg4.l1.m1.3.3.1.1.1.2">𝑛</ci><ci id="alg4.l1.m1.3.3.1.1.1.3.cmml" xref="alg4.l1.m1.3.3.1.1.1.3">′</ci></apply><ci id="alg4.l1.m1.1.1.cmml" xref="alg4.l1.m1.1.1">𝑘</ci><ci id="alg4.l1.m1.2.2.cmml" xref="alg4.l1.m1.2.2">𝑝</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l1.m1.3c">G^{T}=(n^{\prime},k,p)</annotation><annotation encoding="application/x-llamapun" id="alg4.l1.m1.3d">italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = ( italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_k , italic_p )</annotation></semantics></math> by running <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg3" title="In 5.3 Balanced Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">3</span></a> on <math alttext="\langle G,\varepsilon\rangle" class="ltx_Math" display="inline" id="alg4.l1.m2.2"><semantics id="alg4.l1.m2.2a"><mrow id="alg4.l1.m2.2.3.2" xref="alg4.l1.m2.2.3.1.cmml"><mo id="alg4.l1.m2.2.3.2.1" stretchy="false" xref="alg4.l1.m2.2.3.1.cmml">⟨</mo><mi id="alg4.l1.m2.1.1" xref="alg4.l1.m2.1.1.cmml">G</mi><mo id="alg4.l1.m2.2.3.2.2" xref="alg4.l1.m2.2.3.1.cmml">,</mo><mi id="alg4.l1.m2.2.2" xref="alg4.l1.m2.2.2.cmml">ε</mi><mo id="alg4.l1.m2.2.3.2.3" stretchy="false" xref="alg4.l1.m2.2.3.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="alg4.l1.m2.2b"><list id="alg4.l1.m2.2.3.1.cmml" xref="alg4.l1.m2.2.3.2"><ci id="alg4.l1.m2.1.1.cmml" xref="alg4.l1.m2.1.1">𝐺</ci><ci id="alg4.l1.m2.2.2.cmml" xref="alg4.l1.m2.2.2">𝜀</ci></list></annotation-xml><annotation encoding="application/x-tex" id="alg4.l1.m2.2c">\langle G,\varepsilon\rangle</annotation><annotation encoding="application/x-llamapun" id="alg4.l1.m2.2d">⟨ italic_G , italic_ε ⟩</annotation></semantics></math>, denote <math alttext="k" class="ltx_Math" display="inline" id="alg4.l1.m3.1"><semantics id="alg4.l1.m3.1a"><mi id="alg4.l1.m3.1.1" xref="alg4.l1.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="alg4.l1.m3.1b"><ci id="alg4.l1.m3.1.1.cmml" xref="alg4.l1.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="alg4.l1.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="alg4.l1.m3.1d">italic_k</annotation></semantics></math> color classes as <math alttext="V^{\prime}_{i}" class="ltx_Math" display="inline" id="alg4.l1.m4.1"><semantics id="alg4.l1.m4.1a"><msubsup id="alg4.l1.m4.1.1" xref="alg4.l1.m4.1.1.cmml"><mi id="alg4.l1.m4.1.1.2.2" xref="alg4.l1.m4.1.1.2.2.cmml">V</mi><mi id="alg4.l1.m4.1.1.3" xref="alg4.l1.m4.1.1.3.cmml">i</mi><mo id="alg4.l1.m4.1.1.2.3" xref="alg4.l1.m4.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="alg4.l1.m4.1b"><apply id="alg4.l1.m4.1.1.cmml" xref="alg4.l1.m4.1.1"><csymbol cd="ambiguous" id="alg4.l1.m4.1.1.1.cmml" xref="alg4.l1.m4.1.1">subscript</csymbol><apply id="alg4.l1.m4.1.1.2.cmml" xref="alg4.l1.m4.1.1"><csymbol cd="ambiguous" id="alg4.l1.m4.1.1.2.1.cmml" xref="alg4.l1.m4.1.1">superscript</csymbol><ci id="alg4.l1.m4.1.1.2.2.cmml" xref="alg4.l1.m4.1.1.2.2">𝑉</ci><ci id="alg4.l1.m4.1.1.2.3.cmml" xref="alg4.l1.m4.1.1.2.3">′</ci></apply><ci id="alg4.l1.m4.1.1.3.cmml" xref="alg4.l1.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l1.m4.1c">V^{\prime}_{i}</annotation><annotation encoding="application/x-llamapun" id="alg4.l1.m4.1d">italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="i\in[k]" class="ltx_Math" display="inline" id="alg4.l1.m5.1"><semantics id="alg4.l1.m5.1a"><mrow id="alg4.l1.m5.1.2" xref="alg4.l1.m5.1.2.cmml"><mi id="alg4.l1.m5.1.2.2" xref="alg4.l1.m5.1.2.2.cmml">i</mi><mo id="alg4.l1.m5.1.2.1" xref="alg4.l1.m5.1.2.1.cmml">∈</mo><mrow id="alg4.l1.m5.1.2.3.2" xref="alg4.l1.m5.1.2.3.1.cmml"><mo id="alg4.l1.m5.1.2.3.2.1" stretchy="false" xref="alg4.l1.m5.1.2.3.1.1.cmml">[</mo><mi id="alg4.l1.m5.1.1" xref="alg4.l1.m5.1.1.cmml">k</mi><mo id="alg4.l1.m5.1.2.3.2.2" stretchy="false" xref="alg4.l1.m5.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg4.l1.m5.1b"><apply id="alg4.l1.m5.1.2.cmml" xref="alg4.l1.m5.1.2"><in id="alg4.l1.m5.1.2.1.cmml" xref="alg4.l1.m5.1.2.1"></in><ci id="alg4.l1.m5.1.2.2.cmml" xref="alg4.l1.m5.1.2.2">𝑖</ci><apply id="alg4.l1.m5.1.2.3.1.cmml" xref="alg4.l1.m5.1.2.3.2"><csymbol cd="latexml" id="alg4.l1.m5.1.2.3.1.1.cmml" xref="alg4.l1.m5.1.2.3.2.1">delimited-[]</csymbol><ci id="alg4.l1.m5.1.1.cmml" xref="alg4.l1.m5.1.1">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l1.m5.1c">i\in[k]</annotation><annotation encoding="application/x-llamapun" id="alg4.l1.m5.1d">italic_i ∈ [ italic_k ]</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg4.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l2.1.1.1" style="font-size:80%;">2:</span></span>Let <math alttext="\pi" class="ltx_Math" display="inline" id="alg4.l2.m1.1"><semantics id="alg4.l2.m1.1a"><mi id="alg4.l2.m1.1.1" xref="alg4.l2.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="alg4.l2.m1.1b"><ci id="alg4.l2.m1.1.1.cmml" xref="alg4.l2.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="alg4.l2.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="alg4.l2.m1.1d">italic_π</annotation></semantics></math> be the partition on vertices of <math alttext="G^{T}" class="ltx_Math" display="inline" id="alg4.l2.m2.1"><semantics id="alg4.l2.m2.1a"><msup id="alg4.l2.m2.1.1" xref="alg4.l2.m2.1.1.cmml"><mi id="alg4.l2.m2.1.1.2" xref="alg4.l2.m2.1.1.2.cmml">G</mi><mi id="alg4.l2.m2.1.1.3" xref="alg4.l2.m2.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="alg4.l2.m2.1b"><apply id="alg4.l2.m2.1.1.cmml" xref="alg4.l2.m2.1.1"><csymbol cd="ambiguous" id="alg4.l2.m2.1.1.1.cmml" xref="alg4.l2.m2.1.1">superscript</csymbol><ci id="alg4.l2.m2.1.1.2.cmml" xref="alg4.l2.m2.1.1.2">𝐺</ci><ci id="alg4.l2.m2.1.1.3.cmml" xref="alg4.l2.m2.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l2.m2.1c">G^{T}</annotation><annotation encoding="application/x-llamapun" id="alg4.l2.m2.1d">italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> after running <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg1" title="In 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a> on <math alttext="(G^{T},S=\{V^{\prime}_{1},\cdots,V^{\prime}_{k}\},\varepsilon)" class="ltx_Math" display="inline" id="alg4.l2.m3.4"><semantics id="alg4.l2.m3.4a"><mrow id="alg4.l2.m3.4.4.1"><mo id="alg4.l2.m3.4.4.1.2" stretchy="false">(</mo><mrow id="alg4.l2.m3.4.4.1.1.1" xref="alg4.l2.m3.4.4.1.1.2.cmml"><mrow id="alg4.l2.m3.4.4.1.1.1.1" xref="alg4.l2.m3.4.4.1.1.1.1.cmml"><mrow id="alg4.l2.m3.4.4.1.1.1.1.1.1" xref="alg4.l2.m3.4.4.1.1.1.1.1.2.cmml"><msup id="alg4.l2.m3.4.4.1.1.1.1.1.1.1" xref="alg4.l2.m3.4.4.1.1.1.1.1.1.1.cmml"><mi id="alg4.l2.m3.4.4.1.1.1.1.1.1.1.2" xref="alg4.l2.m3.4.4.1.1.1.1.1.1.1.2.cmml">G</mi><mi id="alg4.l2.m3.4.4.1.1.1.1.1.1.1.3" xref="alg4.l2.m3.4.4.1.1.1.1.1.1.1.3.cmml">T</mi></msup><mo id="alg4.l2.m3.4.4.1.1.1.1.1.1.2" xref="alg4.l2.m3.4.4.1.1.1.1.1.2.cmml">,</mo><mi id="alg4.l2.m3.2.2" xref="alg4.l2.m3.2.2.cmml">S</mi></mrow><mo id="alg4.l2.m3.4.4.1.1.1.1.4" xref="alg4.l2.m3.4.4.1.1.1.1.4.cmml">=</mo><mrow id="alg4.l2.m3.4.4.1.1.1.1.3.2" xref="alg4.l2.m3.4.4.1.1.1.1.3.3.cmml"><mo id="alg4.l2.m3.4.4.1.1.1.1.3.2.3" stretchy="false" xref="alg4.l2.m3.4.4.1.1.1.1.3.3.cmml">{</mo><msubsup id="alg4.l2.m3.4.4.1.1.1.1.2.1.1" xref="alg4.l2.m3.4.4.1.1.1.1.2.1.1.cmml"><mi id="alg4.l2.m3.4.4.1.1.1.1.2.1.1.2.2" xref="alg4.l2.m3.4.4.1.1.1.1.2.1.1.2.2.cmml">V</mi><mn id="alg4.l2.m3.4.4.1.1.1.1.2.1.1.3" xref="alg4.l2.m3.4.4.1.1.1.1.2.1.1.3.cmml">1</mn><mo id="alg4.l2.m3.4.4.1.1.1.1.2.1.1.2.3" xref="alg4.l2.m3.4.4.1.1.1.1.2.1.1.2.3.cmml">′</mo></msubsup><mo id="alg4.l2.m3.4.4.1.1.1.1.3.2.4" xref="alg4.l2.m3.4.4.1.1.1.1.3.3.cmml">,</mo><mi id="alg4.l2.m3.1.1" mathvariant="normal" xref="alg4.l2.m3.1.1.cmml">⋯</mi><mo id="alg4.l2.m3.4.4.1.1.1.1.3.2.5" xref="alg4.l2.m3.4.4.1.1.1.1.3.3.cmml">,</mo><msubsup id="alg4.l2.m3.4.4.1.1.1.1.3.2.2" xref="alg4.l2.m3.4.4.1.1.1.1.3.2.2.cmml"><mi id="alg4.l2.m3.4.4.1.1.1.1.3.2.2.2.2" xref="alg4.l2.m3.4.4.1.1.1.1.3.2.2.2.2.cmml">V</mi><mi id="alg4.l2.m3.4.4.1.1.1.1.3.2.2.3" xref="alg4.l2.m3.4.4.1.1.1.1.3.2.2.3.cmml">k</mi><mo id="alg4.l2.m3.4.4.1.1.1.1.3.2.2.2.3" xref="alg4.l2.m3.4.4.1.1.1.1.3.2.2.2.3.cmml">′</mo></msubsup><mo id="alg4.l2.m3.4.4.1.1.1.1.3.2.6" stretchy="false" xref="alg4.l2.m3.4.4.1.1.1.1.3.3.cmml">}</mo></mrow></mrow><mo id="alg4.l2.m3.4.4.1.1.1.2" xref="alg4.l2.m3.4.4.1.1.2a.cmml">,</mo><mi id="alg4.l2.m3.3.3" xref="alg4.l2.m3.3.3.cmml">ε</mi></mrow><mo id="alg4.l2.m3.4.4.1.3" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="alg4.l2.m3.4b"><apply id="alg4.l2.m3.4.4.1.1.2.cmml" xref="alg4.l2.m3.4.4.1.1.1"><csymbol cd="ambiguous" id="alg4.l2.m3.4.4.1.1.2a.cmml" xref="alg4.l2.m3.4.4.1.1.1.2">formulae-sequence</csymbol><apply id="alg4.l2.m3.4.4.1.1.1.1.cmml" xref="alg4.l2.m3.4.4.1.1.1.1"><eq id="alg4.l2.m3.4.4.1.1.1.1.4.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.4"></eq><list id="alg4.l2.m3.4.4.1.1.1.1.1.2.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.1.1"><apply id="alg4.l2.m3.4.4.1.1.1.1.1.1.1.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg4.l2.m3.4.4.1.1.1.1.1.1.1.1.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.1.1.1">superscript</csymbol><ci id="alg4.l2.m3.4.4.1.1.1.1.1.1.1.2.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.1.1.1.2">𝐺</ci><ci id="alg4.l2.m3.4.4.1.1.1.1.1.1.1.3.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.1.1.1.3">𝑇</ci></apply><ci id="alg4.l2.m3.2.2.cmml" xref="alg4.l2.m3.2.2">𝑆</ci></list><set id="alg4.l2.m3.4.4.1.1.1.1.3.3.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.3.2"><apply id="alg4.l2.m3.4.4.1.1.1.1.2.1.1.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.2.1.1"><csymbol cd="ambiguous" id="alg4.l2.m3.4.4.1.1.1.1.2.1.1.1.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.2.1.1">subscript</csymbol><apply id="alg4.l2.m3.4.4.1.1.1.1.2.1.1.2.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.2.1.1"><csymbol cd="ambiguous" id="alg4.l2.m3.4.4.1.1.1.1.2.1.1.2.1.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.2.1.1">superscript</csymbol><ci id="alg4.l2.m3.4.4.1.1.1.1.2.1.1.2.2.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.2.1.1.2.2">𝑉</ci><ci id="alg4.l2.m3.4.4.1.1.1.1.2.1.1.2.3.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.2.1.1.2.3">′</ci></apply><cn id="alg4.l2.m3.4.4.1.1.1.1.2.1.1.3.cmml" type="integer" xref="alg4.l2.m3.4.4.1.1.1.1.2.1.1.3">1</cn></apply><ci id="alg4.l2.m3.1.1.cmml" xref="alg4.l2.m3.1.1">⋯</ci><apply id="alg4.l2.m3.4.4.1.1.1.1.3.2.2.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="alg4.l2.m3.4.4.1.1.1.1.3.2.2.1.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.3.2.2">subscript</csymbol><apply id="alg4.l2.m3.4.4.1.1.1.1.3.2.2.2.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="alg4.l2.m3.4.4.1.1.1.1.3.2.2.2.1.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.3.2.2">superscript</csymbol><ci id="alg4.l2.m3.4.4.1.1.1.1.3.2.2.2.2.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.3.2.2.2.2">𝑉</ci><ci id="alg4.l2.m3.4.4.1.1.1.1.3.2.2.2.3.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.3.2.2.2.3">′</ci></apply><ci id="alg4.l2.m3.4.4.1.1.1.1.3.2.2.3.cmml" xref="alg4.l2.m3.4.4.1.1.1.1.3.2.2.3">𝑘</ci></apply></set></apply><ci id="alg4.l2.m3.3.3.cmml" xref="alg4.l2.m3.3.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l2.m3.4c">(G^{T},S=\{V^{\prime}_{1},\cdots,V^{\prime}_{k}\},\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="alg4.l2.m3.4d">( italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT , italic_S = { italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } , italic_ε )</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg4.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l3.1.1.1" style="font-size:80%;">3:</span></span><span class="ltx_text ltx_font_bold" id="alg4.l3.2">for all</span> <math alttext="v\in\bigcup_{i\in[k]}V_{i}\setminus\left\{\bigcup_{i\in[k]}V^{\prime}_{i}\right\}" class="ltx_Math" display="inline" id="alg4.l3.m1.3"><semantics id="alg4.l3.m1.3a"><mrow id="alg4.l3.m1.3.3" xref="alg4.l3.m1.3.3.cmml"><mi id="alg4.l3.m1.3.3.3" xref="alg4.l3.m1.3.3.3.cmml">v</mi><mo id="alg4.l3.m1.3.3.2" rspace="0.111em" xref="alg4.l3.m1.3.3.2.cmml">∈</mo><mrow id="alg4.l3.m1.3.3.1" xref="alg4.l3.m1.3.3.1.cmml"><mrow id="alg4.l3.m1.3.3.1.3" xref="alg4.l3.m1.3.3.1.3.cmml"><msub id="alg4.l3.m1.3.3.1.3.1" xref="alg4.l3.m1.3.3.1.3.1.cmml"><mo id="alg4.l3.m1.3.3.1.3.1.2" xref="alg4.l3.m1.3.3.1.3.1.2.cmml">⋃</mo><mrow id="alg4.l3.m1.1.1.1" xref="alg4.l3.m1.1.1.1.cmml"><mi id="alg4.l3.m1.1.1.1.3" xref="alg4.l3.m1.1.1.1.3.cmml">i</mi><mo id="alg4.l3.m1.1.1.1.2" xref="alg4.l3.m1.1.1.1.2.cmml">∈</mo><mrow id="alg4.l3.m1.1.1.1.4.2" xref="alg4.l3.m1.1.1.1.4.1.cmml"><mo id="alg4.l3.m1.1.1.1.4.2.1" stretchy="false" xref="alg4.l3.m1.1.1.1.4.1.1.cmml">[</mo><mi id="alg4.l3.m1.1.1.1.1" xref="alg4.l3.m1.1.1.1.1.cmml">k</mi><mo id="alg4.l3.m1.1.1.1.4.2.2" stretchy="false" xref="alg4.l3.m1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><msub id="alg4.l3.m1.3.3.1.3.2" xref="alg4.l3.m1.3.3.1.3.2.cmml"><mi id="alg4.l3.m1.3.3.1.3.2.2" xref="alg4.l3.m1.3.3.1.3.2.2.cmml">V</mi><mi id="alg4.l3.m1.3.3.1.3.2.3" xref="alg4.l3.m1.3.3.1.3.2.3.cmml">i</mi></msub></mrow><mo id="alg4.l3.m1.3.3.1.2" xref="alg4.l3.m1.3.3.1.2.cmml">∖</mo><mrow id="alg4.l3.m1.3.3.1.1.1" xref="alg4.l3.m1.3.3.1.1.2.cmml"><mo id="alg4.l3.m1.3.3.1.1.1.2" xref="alg4.l3.m1.3.3.1.1.2.cmml">{</mo><mrow id="alg4.l3.m1.3.3.1.1.1.1" xref="alg4.l3.m1.3.3.1.1.1.1.cmml"><msub id="alg4.l3.m1.3.3.1.1.1.1.1" xref="alg4.l3.m1.3.3.1.1.1.1.1.cmml"><mo id="alg4.l3.m1.3.3.1.1.1.1.1.2" lspace="0em" xref="alg4.l3.m1.3.3.1.1.1.1.1.2.cmml">⋃</mo><mrow id="alg4.l3.m1.2.2.1" xref="alg4.l3.m1.2.2.1.cmml"><mi id="alg4.l3.m1.2.2.1.3" xref="alg4.l3.m1.2.2.1.3.cmml">i</mi><mo id="alg4.l3.m1.2.2.1.2" xref="alg4.l3.m1.2.2.1.2.cmml">∈</mo><mrow id="alg4.l3.m1.2.2.1.4.2" xref="alg4.l3.m1.2.2.1.4.1.cmml"><mo id="alg4.l3.m1.2.2.1.4.2.1" stretchy="false" xref="alg4.l3.m1.2.2.1.4.1.1.cmml">[</mo><mi id="alg4.l3.m1.2.2.1.1" xref="alg4.l3.m1.2.2.1.1.cmml">k</mi><mo id="alg4.l3.m1.2.2.1.4.2.2" stretchy="false" xref="alg4.l3.m1.2.2.1.4.1.1.cmml">]</mo></mrow></mrow></msub><msubsup id="alg4.l3.m1.3.3.1.1.1.1.2" xref="alg4.l3.m1.3.3.1.1.1.1.2.cmml"><mi id="alg4.l3.m1.3.3.1.1.1.1.2.2.2" xref="alg4.l3.m1.3.3.1.1.1.1.2.2.2.cmml">V</mi><mi id="alg4.l3.m1.3.3.1.1.1.1.2.3" xref="alg4.l3.m1.3.3.1.1.1.1.2.3.cmml">i</mi><mo id="alg4.l3.m1.3.3.1.1.1.1.2.2.3" xref="alg4.l3.m1.3.3.1.1.1.1.2.2.3.cmml">′</mo></msubsup></mrow><mo id="alg4.l3.m1.3.3.1.1.1.3" xref="alg4.l3.m1.3.3.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg4.l3.m1.3b"><apply id="alg4.l3.m1.3.3.cmml" xref="alg4.l3.m1.3.3"><in id="alg4.l3.m1.3.3.2.cmml" xref="alg4.l3.m1.3.3.2"></in><ci id="alg4.l3.m1.3.3.3.cmml" xref="alg4.l3.m1.3.3.3">𝑣</ci><apply id="alg4.l3.m1.3.3.1.cmml" xref="alg4.l3.m1.3.3.1"><setdiff id="alg4.l3.m1.3.3.1.2.cmml" xref="alg4.l3.m1.3.3.1.2"></setdiff><apply id="alg4.l3.m1.3.3.1.3.cmml" xref="alg4.l3.m1.3.3.1.3"><apply id="alg4.l3.m1.3.3.1.3.1.cmml" xref="alg4.l3.m1.3.3.1.3.1"><csymbol cd="ambiguous" id="alg4.l3.m1.3.3.1.3.1.1.cmml" xref="alg4.l3.m1.3.3.1.3.1">subscript</csymbol><union id="alg4.l3.m1.3.3.1.3.1.2.cmml" xref="alg4.l3.m1.3.3.1.3.1.2"></union><apply id="alg4.l3.m1.1.1.1.cmml" xref="alg4.l3.m1.1.1.1"><in id="alg4.l3.m1.1.1.1.2.cmml" xref="alg4.l3.m1.1.1.1.2"></in><ci id="alg4.l3.m1.1.1.1.3.cmml" xref="alg4.l3.m1.1.1.1.3">𝑖</ci><apply id="alg4.l3.m1.1.1.1.4.1.cmml" xref="alg4.l3.m1.1.1.1.4.2"><csymbol cd="latexml" id="alg4.l3.m1.1.1.1.4.1.1.cmml" xref="alg4.l3.m1.1.1.1.4.2.1">delimited-[]</csymbol><ci id="alg4.l3.m1.1.1.1.1.cmml" xref="alg4.l3.m1.1.1.1.1">𝑘</ci></apply></apply></apply><apply id="alg4.l3.m1.3.3.1.3.2.cmml" xref="alg4.l3.m1.3.3.1.3.2"><csymbol cd="ambiguous" id="alg4.l3.m1.3.3.1.3.2.1.cmml" xref="alg4.l3.m1.3.3.1.3.2">subscript</csymbol><ci id="alg4.l3.m1.3.3.1.3.2.2.cmml" xref="alg4.l3.m1.3.3.1.3.2.2">𝑉</ci><ci id="alg4.l3.m1.3.3.1.3.2.3.cmml" xref="alg4.l3.m1.3.3.1.3.2.3">𝑖</ci></apply></apply><set id="alg4.l3.m1.3.3.1.1.2.cmml" xref="alg4.l3.m1.3.3.1.1.1"><apply id="alg4.l3.m1.3.3.1.1.1.1.cmml" xref="alg4.l3.m1.3.3.1.1.1.1"><apply id="alg4.l3.m1.3.3.1.1.1.1.1.cmml" xref="alg4.l3.m1.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="alg4.l3.m1.3.3.1.1.1.1.1.1.cmml" xref="alg4.l3.m1.3.3.1.1.1.1.1">subscript</csymbol><union id="alg4.l3.m1.3.3.1.1.1.1.1.2.cmml" xref="alg4.l3.m1.3.3.1.1.1.1.1.2"></union><apply id="alg4.l3.m1.2.2.1.cmml" xref="alg4.l3.m1.2.2.1"><in id="alg4.l3.m1.2.2.1.2.cmml" xref="alg4.l3.m1.2.2.1.2"></in><ci id="alg4.l3.m1.2.2.1.3.cmml" xref="alg4.l3.m1.2.2.1.3">𝑖</ci><apply id="alg4.l3.m1.2.2.1.4.1.cmml" xref="alg4.l3.m1.2.2.1.4.2"><csymbol cd="latexml" id="alg4.l3.m1.2.2.1.4.1.1.cmml" xref="alg4.l3.m1.2.2.1.4.2.1">delimited-[]</csymbol><ci id="alg4.l3.m1.2.2.1.1.cmml" xref="alg4.l3.m1.2.2.1.1">𝑘</ci></apply></apply></apply><apply id="alg4.l3.m1.3.3.1.1.1.1.2.cmml" xref="alg4.l3.m1.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="alg4.l3.m1.3.3.1.1.1.1.2.1.cmml" xref="alg4.l3.m1.3.3.1.1.1.1.2">subscript</csymbol><apply id="alg4.l3.m1.3.3.1.1.1.1.2.2.cmml" xref="alg4.l3.m1.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="alg4.l3.m1.3.3.1.1.1.1.2.2.1.cmml" xref="alg4.l3.m1.3.3.1.1.1.1.2">superscript</csymbol><ci id="alg4.l3.m1.3.3.1.1.1.1.2.2.2.cmml" xref="alg4.l3.m1.3.3.1.1.1.1.2.2.2">𝑉</ci><ci id="alg4.l3.m1.3.3.1.1.1.1.2.2.3.cmml" xref="alg4.l3.m1.3.3.1.1.1.1.2.2.3">′</ci></apply><ci id="alg4.l3.m1.3.3.1.1.1.1.2.3.cmml" xref="alg4.l3.m1.3.3.1.1.1.1.2.3">𝑖</ci></apply></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l3.m1.3c">v\in\bigcup_{i\in[k]}V_{i}\setminus\left\{\bigcup_{i\in[k]}V^{\prime}_{i}\right\}</annotation><annotation encoding="application/x-llamapun" id="alg4.l3.m1.3d">italic_v ∈ ⋃ start_POSTSUBSCRIPT italic_i ∈ [ italic_k ] end_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∖ { ⋃ start_POSTSUBSCRIPT italic_i ∈ [ italic_k ] end_POSTSUBSCRIPT italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT }</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg4.l3.3">do</span> </div> <div class="ltx_listingline" id="alg4.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l4.1.1.1" style="font-size:80%;">4:</span></span> <math alttext="\pi\leftarrow\pi\cup\{v\}" class="ltx_Math" display="inline" id="alg4.l4.m1.1"><semantics id="alg4.l4.m1.1a"><mrow id="alg4.l4.m1.1.2" xref="alg4.l4.m1.1.2.cmml"><mi id="alg4.l4.m1.1.2.2" xref="alg4.l4.m1.1.2.2.cmml">π</mi><mo id="alg4.l4.m1.1.2.1" stretchy="false" xref="alg4.l4.m1.1.2.1.cmml">←</mo><mrow id="alg4.l4.m1.1.2.3" xref="alg4.l4.m1.1.2.3.cmml"><mi id="alg4.l4.m1.1.2.3.2" xref="alg4.l4.m1.1.2.3.2.cmml">π</mi><mo id="alg4.l4.m1.1.2.3.1" xref="alg4.l4.m1.1.2.3.1.cmml">∪</mo><mrow id="alg4.l4.m1.1.2.3.3.2" xref="alg4.l4.m1.1.2.3.3.1.cmml"><mo id="alg4.l4.m1.1.2.3.3.2.1" stretchy="false" xref="alg4.l4.m1.1.2.3.3.1.cmml">{</mo><mi id="alg4.l4.m1.1.1" xref="alg4.l4.m1.1.1.cmml">v</mi><mo id="alg4.l4.m1.1.2.3.3.2.2" stretchy="false" xref="alg4.l4.m1.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg4.l4.m1.1b"><apply id="alg4.l4.m1.1.2.cmml" xref="alg4.l4.m1.1.2"><ci id="alg4.l4.m1.1.2.1.cmml" xref="alg4.l4.m1.1.2.1">←</ci><ci id="alg4.l4.m1.1.2.2.cmml" xref="alg4.l4.m1.1.2.2">𝜋</ci><apply id="alg4.l4.m1.1.2.3.cmml" xref="alg4.l4.m1.1.2.3"><union id="alg4.l4.m1.1.2.3.1.cmml" xref="alg4.l4.m1.1.2.3.1"></union><ci id="alg4.l4.m1.1.2.3.2.cmml" xref="alg4.l4.m1.1.2.3.2">𝜋</ci><set id="alg4.l4.m1.1.2.3.3.1.cmml" xref="alg4.l4.m1.1.2.3.3.2"><ci id="alg4.l4.m1.1.1.cmml" xref="alg4.l4.m1.1.1">𝑣</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l4.m1.1c">\pi\leftarrow\pi\cup\{v\}</annotation><annotation encoding="application/x-llamapun" id="alg4.l4.m1.1d">italic_π ← italic_π ∪ { italic_v }</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg4.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l5.1.1.1" style="font-size:80%;">5:</span></span><span class="ltx_text ltx_font_bold" id="alg4.l5.2">return</span> <math alttext="\pi" class="ltx_Math" display="inline" id="alg4.l5.m1.1"><semantics id="alg4.l5.m1.1a"><mi id="alg4.l5.m1.1.1" xref="alg4.l5.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="alg4.l5.m1.1b"><ci id="alg4.l5.m1.1.1.cmml" xref="alg4.l5.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="alg4.l5.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="alg4.l5.m1.1d">italic_π</annotation></semantics></math> </div> </div> </div> </div> </figure> <div class="ltx_theorem ltx_theorem_theorem" id="S5.Thmtheorem12"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem12.1.1.1">Theorem 5.12</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem12.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem12.p1"> <p class="ltx_p" id="S5.Thmtheorem12.p1.4"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem12.p1.4.4">Consider aversion-to-enemies games given by random balanced <math alttext="k" class="ltx_Math" display="inline" id="S5.Thmtheorem12.p1.1.1.m1.1"><semantics id="S5.Thmtheorem12.p1.1.1.m1.1a"><mi id="S5.Thmtheorem12.p1.1.1.m1.1.1" xref="S5.Thmtheorem12.p1.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem12.p1.1.1.m1.1b"><ci id="S5.Thmtheorem12.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem12.p1.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem12.p1.1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem12.p1.1.1.m1.1d">italic_k</annotation></semantics></math>-partite graphs <math alttext="G=(\{V_{1},\cdots,V_{k}\},p)" class="ltx_Math" display="inline" id="S5.Thmtheorem12.p1.2.2.m2.3"><semantics id="S5.Thmtheorem12.p1.2.2.m2.3a"><mrow id="S5.Thmtheorem12.p1.2.2.m2.3.3" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.cmml"><mi id="S5.Thmtheorem12.p1.2.2.m2.3.3.3" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.3.cmml">G</mi><mo id="S5.Thmtheorem12.p1.2.2.m2.3.3.2" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.2.cmml">=</mo><mrow id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.2.cmml"><mo id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.2" stretchy="false" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.2.cmml">(</mo><mrow id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.3.cmml"><mo id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.3" stretchy="false" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.3.cmml">{</mo><msub id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.1.1" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.1.1.2" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.1.1.2.cmml">V</mi><mn id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.1.1.3" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.4" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.3.cmml">,</mo><mi id="S5.Thmtheorem12.p1.2.2.m2.1.1" mathvariant="normal" xref="S5.Thmtheorem12.p1.2.2.m2.1.1.cmml">⋯</mi><mo id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.5" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.3.cmml">,</mo><msub id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.2" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.2.cmml"><mi id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.2.2" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.2.2.cmml">V</mi><mi id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.2.3" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.2.3.cmml">k</mi></msub><mo id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.6" stretchy="false" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.3.cmml">}</mo></mrow><mo id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.3" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.2.cmml">,</mo><mi id="S5.Thmtheorem12.p1.2.2.m2.2.2" xref="S5.Thmtheorem12.p1.2.2.m2.2.2.cmml">p</mi><mo id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.4" stretchy="false" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem12.p1.2.2.m2.3b"><apply id="S5.Thmtheorem12.p1.2.2.m2.3.3.cmml" xref="S5.Thmtheorem12.p1.2.2.m2.3.3"><eq id="S5.Thmtheorem12.p1.2.2.m2.3.3.2.cmml" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.2"></eq><ci id="S5.Thmtheorem12.p1.2.2.m2.3.3.3.cmml" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.3">𝐺</ci><interval closure="open" id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.2.cmml" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1"><set id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.3.cmml" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2"><apply id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.1.1.cmml" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.1.1.2">𝑉</ci><cn id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.1.1.3">1</cn></apply><ci id="S5.Thmtheorem12.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem12.p1.2.2.m2.1.1">⋯</ci><apply id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.2.cmml" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.2.1.cmml" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.2">subscript</csymbol><ci id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.2.2.cmml" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.2.2">𝑉</ci><ci id="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.2.3.cmml" xref="S5.Thmtheorem12.p1.2.2.m2.3.3.1.1.1.2.2.3">𝑘</ci></apply></set><ci id="S5.Thmtheorem12.p1.2.2.m2.2.2.cmml" xref="S5.Thmtheorem12.p1.2.2.m2.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem12.p1.2.2.m2.3c">G=(\{V_{1},\cdots,V_{k}\},p)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem12.p1.2.2.m2.3d">italic_G = ( { italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } , italic_p )</annotation></semantics></math>, where <math alttext="p=\mathcal{O}(\frac{1}{k})" class="ltx_Math" display="inline" id="S5.Thmtheorem12.p1.3.3.m3.1"><semantics id="S5.Thmtheorem12.p1.3.3.m3.1a"><mrow id="S5.Thmtheorem12.p1.3.3.m3.1.2" xref="S5.Thmtheorem12.p1.3.3.m3.1.2.cmml"><mi id="S5.Thmtheorem12.p1.3.3.m3.1.2.2" xref="S5.Thmtheorem12.p1.3.3.m3.1.2.2.cmml">p</mi><mo id="S5.Thmtheorem12.p1.3.3.m3.1.2.1" xref="S5.Thmtheorem12.p1.3.3.m3.1.2.1.cmml">=</mo><mrow id="S5.Thmtheorem12.p1.3.3.m3.1.2.3" xref="S5.Thmtheorem12.p1.3.3.m3.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem12.p1.3.3.m3.1.2.3.2" xref="S5.Thmtheorem12.p1.3.3.m3.1.2.3.2.cmml">𝒪</mi><mo id="S5.Thmtheorem12.p1.3.3.m3.1.2.3.1" xref="S5.Thmtheorem12.p1.3.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem12.p1.3.3.m3.1.2.3.3.2" xref="S5.Thmtheorem12.p1.3.3.m3.1.1.cmml"><mo id="S5.Thmtheorem12.p1.3.3.m3.1.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem12.p1.3.3.m3.1.1.cmml">(</mo><mfrac id="S5.Thmtheorem12.p1.3.3.m3.1.1" xref="S5.Thmtheorem12.p1.3.3.m3.1.1.cmml"><mn id="S5.Thmtheorem12.p1.3.3.m3.1.1.2" xref="S5.Thmtheorem12.p1.3.3.m3.1.1.2.cmml">1</mn><mi id="S5.Thmtheorem12.p1.3.3.m3.1.1.3" xref="S5.Thmtheorem12.p1.3.3.m3.1.1.3.cmml">k</mi></mfrac><mo id="S5.Thmtheorem12.p1.3.3.m3.1.2.3.3.2.2" stretchy="false" xref="S5.Thmtheorem12.p1.3.3.m3.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem12.p1.3.3.m3.1b"><apply id="S5.Thmtheorem12.p1.3.3.m3.1.2.cmml" xref="S5.Thmtheorem12.p1.3.3.m3.1.2"><eq id="S5.Thmtheorem12.p1.3.3.m3.1.2.1.cmml" xref="S5.Thmtheorem12.p1.3.3.m3.1.2.1"></eq><ci id="S5.Thmtheorem12.p1.3.3.m3.1.2.2.cmml" xref="S5.Thmtheorem12.p1.3.3.m3.1.2.2">𝑝</ci><apply id="S5.Thmtheorem12.p1.3.3.m3.1.2.3.cmml" xref="S5.Thmtheorem12.p1.3.3.m3.1.2.3"><times id="S5.Thmtheorem12.p1.3.3.m3.1.2.3.1.cmml" xref="S5.Thmtheorem12.p1.3.3.m3.1.2.3.1"></times><ci id="S5.Thmtheorem12.p1.3.3.m3.1.2.3.2.cmml" xref="S5.Thmtheorem12.p1.3.3.m3.1.2.3.2">𝒪</ci><apply id="S5.Thmtheorem12.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem12.p1.3.3.m3.1.2.3.3.2"><divide id="S5.Thmtheorem12.p1.3.3.m3.1.1.1.cmml" xref="S5.Thmtheorem12.p1.3.3.m3.1.2.3.3.2"></divide><cn id="S5.Thmtheorem12.p1.3.3.m3.1.1.2.cmml" type="integer" xref="S5.Thmtheorem12.p1.3.3.m3.1.1.2">1</cn><ci id="S5.Thmtheorem12.p1.3.3.m3.1.1.3.cmml" xref="S5.Thmtheorem12.p1.3.3.m3.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem12.p1.3.3.m3.1c">p=\mathcal{O}(\frac{1}{k})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem12.p1.3.3.m3.1d">italic_p = caligraphic_O ( divide start_ARG 1 end_ARG start_ARG italic_k end_ARG )</annotation></semantics></math>. Then there is a polynomial-time algorithm that returns a constant-factor approximation to maximum welfare with probability <math alttext="1-e^{-\Theta(n)}" class="ltx_Math" display="inline" id="S5.Thmtheorem12.p1.4.4.m4.1"><semantics id="S5.Thmtheorem12.p1.4.4.m4.1a"><mrow id="S5.Thmtheorem12.p1.4.4.m4.1.2" xref="S5.Thmtheorem12.p1.4.4.m4.1.2.cmml"><mn id="S5.Thmtheorem12.p1.4.4.m4.1.2.2" xref="S5.Thmtheorem12.p1.4.4.m4.1.2.2.cmml">1</mn><mo id="S5.Thmtheorem12.p1.4.4.m4.1.2.1" xref="S5.Thmtheorem12.p1.4.4.m4.1.2.1.cmml">−</mo><msup id="S5.Thmtheorem12.p1.4.4.m4.1.2.3" xref="S5.Thmtheorem12.p1.4.4.m4.1.2.3.cmml"><mi id="S5.Thmtheorem12.p1.4.4.m4.1.2.3.2" xref="S5.Thmtheorem12.p1.4.4.m4.1.2.3.2.cmml">e</mi><mrow id="S5.Thmtheorem12.p1.4.4.m4.1.1.1" xref="S5.Thmtheorem12.p1.4.4.m4.1.1.1.cmml"><mo id="S5.Thmtheorem12.p1.4.4.m4.1.1.1a" xref="S5.Thmtheorem12.p1.4.4.m4.1.1.1.cmml">−</mo><mrow id="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3" xref="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.cmml"><mi id="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.2" mathvariant="normal" xref="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.2.cmml">Θ</mi><mo id="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.1" xref="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.3.2" xref="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.cmml"><mo id="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.3.2.1" stretchy="false" xref="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.cmml">(</mo><mi id="S5.Thmtheorem12.p1.4.4.m4.1.1.1.1" xref="S5.Thmtheorem12.p1.4.4.m4.1.1.1.1.cmml">n</mi><mo id="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.3.2.2" stretchy="false" xref="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.cmml">)</mo></mrow></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem12.p1.4.4.m4.1b"><apply id="S5.Thmtheorem12.p1.4.4.m4.1.2.cmml" xref="S5.Thmtheorem12.p1.4.4.m4.1.2"><minus id="S5.Thmtheorem12.p1.4.4.m4.1.2.1.cmml" xref="S5.Thmtheorem12.p1.4.4.m4.1.2.1"></minus><cn id="S5.Thmtheorem12.p1.4.4.m4.1.2.2.cmml" type="integer" xref="S5.Thmtheorem12.p1.4.4.m4.1.2.2">1</cn><apply id="S5.Thmtheorem12.p1.4.4.m4.1.2.3.cmml" xref="S5.Thmtheorem12.p1.4.4.m4.1.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem12.p1.4.4.m4.1.2.3.1.cmml" xref="S5.Thmtheorem12.p1.4.4.m4.1.2.3">superscript</csymbol><ci id="S5.Thmtheorem12.p1.4.4.m4.1.2.3.2.cmml" xref="S5.Thmtheorem12.p1.4.4.m4.1.2.3.2">𝑒</ci><apply id="S5.Thmtheorem12.p1.4.4.m4.1.1.1.cmml" xref="S5.Thmtheorem12.p1.4.4.m4.1.1.1"><minus id="S5.Thmtheorem12.p1.4.4.m4.1.1.1.2.cmml" xref="S5.Thmtheorem12.p1.4.4.m4.1.1.1"></minus><apply id="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.cmml" xref="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3"><times id="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.1.cmml" xref="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.1"></times><ci id="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.2.cmml" xref="S5.Thmtheorem12.p1.4.4.m4.1.1.1.3.2">Θ</ci><ci id="S5.Thmtheorem12.p1.4.4.m4.1.1.1.1.cmml" xref="S5.Thmtheorem12.p1.4.4.m4.1.1.1.1">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem12.p1.4.4.m4.1c">1-e^{-\Theta(n)}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem12.p1.4.4.m4.1d">1 - italic_e start_POSTSUPERSCRIPT - roman_Θ ( italic_n ) end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S5.SS3.SSS1.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.SS3.SSS1.1.p1"> <p class="ltx_p" id="S5.SS3.SSS1.1.p1.7">Since <math alttext="p=\mathcal{O}(\frac{1}{k})" class="ltx_Math" display="inline" id="S5.SS3.SSS1.1.p1.1.m1.1"><semantics id="S5.SS3.SSS1.1.p1.1.m1.1a"><mrow id="S5.SS3.SSS1.1.p1.1.m1.1.2" xref="S5.SS3.SSS1.1.p1.1.m1.1.2.cmml"><mi id="S5.SS3.SSS1.1.p1.1.m1.1.2.2" xref="S5.SS3.SSS1.1.p1.1.m1.1.2.2.cmml">p</mi><mo id="S5.SS3.SSS1.1.p1.1.m1.1.2.1" xref="S5.SS3.SSS1.1.p1.1.m1.1.2.1.cmml">=</mo><mrow id="S5.SS3.SSS1.1.p1.1.m1.1.2.3" xref="S5.SS3.SSS1.1.p1.1.m1.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS1.1.p1.1.m1.1.2.3.2" xref="S5.SS3.SSS1.1.p1.1.m1.1.2.3.2.cmml">𝒪</mi><mo id="S5.SS3.SSS1.1.p1.1.m1.1.2.3.1" xref="S5.SS3.SSS1.1.p1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS3.SSS1.1.p1.1.m1.1.2.3.3.2" xref="S5.SS3.SSS1.1.p1.1.m1.1.1.cmml"><mo id="S5.SS3.SSS1.1.p1.1.m1.1.2.3.3.2.1" stretchy="false" xref="S5.SS3.SSS1.1.p1.1.m1.1.1.cmml">(</mo><mfrac id="S5.SS3.SSS1.1.p1.1.m1.1.1" xref="S5.SS3.SSS1.1.p1.1.m1.1.1.cmml"><mn id="S5.SS3.SSS1.1.p1.1.m1.1.1.2" xref="S5.SS3.SSS1.1.p1.1.m1.1.1.2.cmml">1</mn><mi id="S5.SS3.SSS1.1.p1.1.m1.1.1.3" xref="S5.SS3.SSS1.1.p1.1.m1.1.1.3.cmml">k</mi></mfrac><mo id="S5.SS3.SSS1.1.p1.1.m1.1.2.3.3.2.2" stretchy="false" xref="S5.SS3.SSS1.1.p1.1.m1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.1.p1.1.m1.1b"><apply id="S5.SS3.SSS1.1.p1.1.m1.1.2.cmml" xref="S5.SS3.SSS1.1.p1.1.m1.1.2"><eq id="S5.SS3.SSS1.1.p1.1.m1.1.2.1.cmml" xref="S5.SS3.SSS1.1.p1.1.m1.1.2.1"></eq><ci id="S5.SS3.SSS1.1.p1.1.m1.1.2.2.cmml" xref="S5.SS3.SSS1.1.p1.1.m1.1.2.2">𝑝</ci><apply id="S5.SS3.SSS1.1.p1.1.m1.1.2.3.cmml" xref="S5.SS3.SSS1.1.p1.1.m1.1.2.3"><times id="S5.SS3.SSS1.1.p1.1.m1.1.2.3.1.cmml" xref="S5.SS3.SSS1.1.p1.1.m1.1.2.3.1"></times><ci id="S5.SS3.SSS1.1.p1.1.m1.1.2.3.2.cmml" xref="S5.SS3.SSS1.1.p1.1.m1.1.2.3.2">𝒪</ci><apply id="S5.SS3.SSS1.1.p1.1.m1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.1.m1.1.2.3.3.2"><divide id="S5.SS3.SSS1.1.p1.1.m1.1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.1.m1.1.2.3.3.2"></divide><cn id="S5.SS3.SSS1.1.p1.1.m1.1.1.2.cmml" type="integer" xref="S5.SS3.SSS1.1.p1.1.m1.1.1.2">1</cn><ci id="S5.SS3.SSS1.1.p1.1.m1.1.1.3.cmml" xref="S5.SS3.SSS1.1.p1.1.m1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.1.p1.1.m1.1c">p=\mathcal{O}(\frac{1}{k})</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.1.p1.1.m1.1d">italic_p = caligraphic_O ( divide start_ARG 1 end_ARG start_ARG italic_k end_ARG )</annotation></semantics></math>, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem7" title="Lemma 5.7. ‣ 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5.7</span></a> implies that <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg1" title="In 5.2.1 Low Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a> returns <math alttext="\alpha\frac{n^{\prime}}{k}" class="ltx_Math" display="inline" id="S5.SS3.SSS1.1.p1.2.m2.1"><semantics id="S5.SS3.SSS1.1.p1.2.m2.1a"><mrow id="S5.SS3.SSS1.1.p1.2.m2.1.1" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.cmml"><mi id="S5.SS3.SSS1.1.p1.2.m2.1.1.2" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.2.cmml">α</mi><mo id="S5.SS3.SSS1.1.p1.2.m2.1.1.1" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.1.cmml">⁢</mo><mfrac id="S5.SS3.SSS1.1.p1.2.m2.1.1.3" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.3.cmml"><msup id="S5.SS3.SSS1.1.p1.2.m2.1.1.3.2" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.3.2.cmml"><mi id="S5.SS3.SSS1.1.p1.2.m2.1.1.3.2.2" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.3.2.2.cmml">n</mi><mo id="S5.SS3.SSS1.1.p1.2.m2.1.1.3.2.3" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.3.2.3.cmml">′</mo></msup><mi id="S5.SS3.SSS1.1.p1.2.m2.1.1.3.3" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.3.3.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.1.p1.2.m2.1b"><apply id="S5.SS3.SSS1.1.p1.2.m2.1.1.cmml" xref="S5.SS3.SSS1.1.p1.2.m2.1.1"><times id="S5.SS3.SSS1.1.p1.2.m2.1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.1"></times><ci id="S5.SS3.SSS1.1.p1.2.m2.1.1.2.cmml" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.2">𝛼</ci><apply id="S5.SS3.SSS1.1.p1.2.m2.1.1.3.cmml" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.3"><divide id="S5.SS3.SSS1.1.p1.2.m2.1.1.3.1.cmml" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.3"></divide><apply id="S5.SS3.SSS1.1.p1.2.m2.1.1.3.2.cmml" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.SS3.SSS1.1.p1.2.m2.1.1.3.2.1.cmml" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.3.2">superscript</csymbol><ci id="S5.SS3.SSS1.1.p1.2.m2.1.1.3.2.2.cmml" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.3.2.2">𝑛</ci><ci id="S5.SS3.SSS1.1.p1.2.m2.1.1.3.2.3.cmml" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.3.2.3">′</ci></apply><ci id="S5.SS3.SSS1.1.p1.2.m2.1.1.3.3.cmml" xref="S5.SS3.SSS1.1.p1.2.m2.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.1.p1.2.m2.1c">\alpha\frac{n^{\prime}}{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.1.p1.2.m2.1d">italic_α divide start_ARG italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_k end_ARG</annotation></semantics></math> cliques, each containing at least <math alttext="k\sqrt{1-\varepsilon}" class="ltx_Math" display="inline" id="S5.SS3.SSS1.1.p1.3.m3.1"><semantics id="S5.SS3.SSS1.1.p1.3.m3.1a"><mrow id="S5.SS3.SSS1.1.p1.3.m3.1.1" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.cmml"><mi id="S5.SS3.SSS1.1.p1.3.m3.1.1.2" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.2.cmml">k</mi><mo id="S5.SS3.SSS1.1.p1.3.m3.1.1.1" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.1.cmml">⁢</mo><msqrt id="S5.SS3.SSS1.1.p1.3.m3.1.1.3" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.3.cmml"><mrow id="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2.cmml"><mn id="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2.2" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2.2.cmml">1</mn><mo id="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2.1" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2.1.cmml">−</mo><mi id="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2.3" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.1.p1.3.m3.1b"><apply id="S5.SS3.SSS1.1.p1.3.m3.1.1.cmml" xref="S5.SS3.SSS1.1.p1.3.m3.1.1"><times id="S5.SS3.SSS1.1.p1.3.m3.1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.1"></times><ci id="S5.SS3.SSS1.1.p1.3.m3.1.1.2.cmml" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.2">𝑘</ci><apply id="S5.SS3.SSS1.1.p1.3.m3.1.1.3.cmml" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.3"><root id="S5.SS3.SSS1.1.p1.3.m3.1.1.3a.cmml" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.3"></root><apply id="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2.cmml" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2"><minus id="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2.1.cmml" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2.1"></minus><cn id="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2.2.cmml" type="integer" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2.2">1</cn><ci id="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2.3.cmml" xref="S5.SS3.SSS1.1.p1.3.m3.1.1.3.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.1.p1.3.m3.1c">k\sqrt{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.1.p1.3.m3.1d">italic_k square-root start_ARG 1 - italic_ε end_ARG</annotation></semantics></math> agents, with probability <math alttext="1-e^{-\Theta(n^{\prime})}" class="ltx_Math" display="inline" id="S5.SS3.SSS1.1.p1.4.m4.1"><semantics id="S5.SS3.SSS1.1.p1.4.m4.1a"><mrow id="S5.SS3.SSS1.1.p1.4.m4.1.2" xref="S5.SS3.SSS1.1.p1.4.m4.1.2.cmml"><mn id="S5.SS3.SSS1.1.p1.4.m4.1.2.2" xref="S5.SS3.SSS1.1.p1.4.m4.1.2.2.cmml">1</mn><mo id="S5.SS3.SSS1.1.p1.4.m4.1.2.1" xref="S5.SS3.SSS1.1.p1.4.m4.1.2.1.cmml">−</mo><msup id="S5.SS3.SSS1.1.p1.4.m4.1.2.3" xref="S5.SS3.SSS1.1.p1.4.m4.1.2.3.cmml"><mi id="S5.SS3.SSS1.1.p1.4.m4.1.2.3.2" xref="S5.SS3.SSS1.1.p1.4.m4.1.2.3.2.cmml">e</mi><mrow id="S5.SS3.SSS1.1.p1.4.m4.1.1.1" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.cmml"><mo id="S5.SS3.SSS1.1.p1.4.m4.1.1.1a" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.cmml">−</mo><mrow id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.cmml"><mi id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.3" mathvariant="normal" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.3.cmml">Θ</mi><mo id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.2" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.1.cmml"><mo id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.1.cmml">(</mo><msup id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.1" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.1.2" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.1.2.cmml">n</mi><mo id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.1.3" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.1.p1.4.m4.1b"><apply id="S5.SS3.SSS1.1.p1.4.m4.1.2.cmml" xref="S5.SS3.SSS1.1.p1.4.m4.1.2"><minus id="S5.SS3.SSS1.1.p1.4.m4.1.2.1.cmml" xref="S5.SS3.SSS1.1.p1.4.m4.1.2.1"></minus><cn id="S5.SS3.SSS1.1.p1.4.m4.1.2.2.cmml" type="integer" xref="S5.SS3.SSS1.1.p1.4.m4.1.2.2">1</cn><apply id="S5.SS3.SSS1.1.p1.4.m4.1.2.3.cmml" xref="S5.SS3.SSS1.1.p1.4.m4.1.2.3"><csymbol cd="ambiguous" id="S5.SS3.SSS1.1.p1.4.m4.1.2.3.1.cmml" xref="S5.SS3.SSS1.1.p1.4.m4.1.2.3">superscript</csymbol><ci id="S5.SS3.SSS1.1.p1.4.m4.1.2.3.2.cmml" xref="S5.SS3.SSS1.1.p1.4.m4.1.2.3.2">𝑒</ci><apply id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1"><minus id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.2.cmml" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1"></minus><apply id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1"><times id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.2.cmml" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.2"></times><ci id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.3.cmml" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.3">Θ</ci><apply id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1">superscript</csymbol><ci id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.1.2">𝑛</ci><ci id="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.1.3.cmml" xref="S5.SS3.SSS1.1.p1.4.m4.1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.1.p1.4.m4.1c">1-e^{-\Theta(n^{\prime})}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.1.p1.4.m4.1d">1 - italic_e start_POSTSUPERSCRIPT - roman_Θ ( italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT</annotation></semantics></math>, for any constant <math alttext="\alpha\in(0,1)" class="ltx_Math" display="inline" id="S5.SS3.SSS1.1.p1.5.m5.2"><semantics id="S5.SS3.SSS1.1.p1.5.m5.2a"><mrow id="S5.SS3.SSS1.1.p1.5.m5.2.3" xref="S5.SS3.SSS1.1.p1.5.m5.2.3.cmml"><mi id="S5.SS3.SSS1.1.p1.5.m5.2.3.2" xref="S5.SS3.SSS1.1.p1.5.m5.2.3.2.cmml">α</mi><mo id="S5.SS3.SSS1.1.p1.5.m5.2.3.1" xref="S5.SS3.SSS1.1.p1.5.m5.2.3.1.cmml">∈</mo><mrow id="S5.SS3.SSS1.1.p1.5.m5.2.3.3.2" xref="S5.SS3.SSS1.1.p1.5.m5.2.3.3.1.cmml"><mo id="S5.SS3.SSS1.1.p1.5.m5.2.3.3.2.1" stretchy="false" xref="S5.SS3.SSS1.1.p1.5.m5.2.3.3.1.cmml">(</mo><mn id="S5.SS3.SSS1.1.p1.5.m5.1.1" xref="S5.SS3.SSS1.1.p1.5.m5.1.1.cmml">0</mn><mo id="S5.SS3.SSS1.1.p1.5.m5.2.3.3.2.2" xref="S5.SS3.SSS1.1.p1.5.m5.2.3.3.1.cmml">,</mo><mn id="S5.SS3.SSS1.1.p1.5.m5.2.2" xref="S5.SS3.SSS1.1.p1.5.m5.2.2.cmml">1</mn><mo id="S5.SS3.SSS1.1.p1.5.m5.2.3.3.2.3" stretchy="false" xref="S5.SS3.SSS1.1.p1.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.1.p1.5.m5.2b"><apply id="S5.SS3.SSS1.1.p1.5.m5.2.3.cmml" xref="S5.SS3.SSS1.1.p1.5.m5.2.3"><in id="S5.SS3.SSS1.1.p1.5.m5.2.3.1.cmml" xref="S5.SS3.SSS1.1.p1.5.m5.2.3.1"></in><ci id="S5.SS3.SSS1.1.p1.5.m5.2.3.2.cmml" xref="S5.SS3.SSS1.1.p1.5.m5.2.3.2">𝛼</ci><interval closure="open" id="S5.SS3.SSS1.1.p1.5.m5.2.3.3.1.cmml" xref="S5.SS3.SSS1.1.p1.5.m5.2.3.3.2"><cn id="S5.SS3.SSS1.1.p1.5.m5.1.1.cmml" type="integer" xref="S5.SS3.SSS1.1.p1.5.m5.1.1">0</cn><cn id="S5.SS3.SSS1.1.p1.5.m5.2.2.cmml" type="integer" xref="S5.SS3.SSS1.1.p1.5.m5.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.1.p1.5.m5.2c">\alpha\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.1.p1.5.m5.2d">italic_α ∈ ( 0 , 1 )</annotation></semantics></math>. Each clique contains at least <math alttext="k\sqrt{1-\varepsilon}" class="ltx_Math" display="inline" id="S5.SS3.SSS1.1.p1.6.m6.1"><semantics id="S5.SS3.SSS1.1.p1.6.m6.1a"><mrow id="S5.SS3.SSS1.1.p1.6.m6.1.1" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.cmml"><mi id="S5.SS3.SSS1.1.p1.6.m6.1.1.2" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.2.cmml">k</mi><mo id="S5.SS3.SSS1.1.p1.6.m6.1.1.1" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.1.cmml">⁢</mo><msqrt id="S5.SS3.SSS1.1.p1.6.m6.1.1.3" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.3.cmml"><mrow id="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2.cmml"><mn id="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2.2" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2.2.cmml">1</mn><mo id="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2.1" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2.1.cmml">−</mo><mi id="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2.3" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.1.p1.6.m6.1b"><apply id="S5.SS3.SSS1.1.p1.6.m6.1.1.cmml" xref="S5.SS3.SSS1.1.p1.6.m6.1.1"><times id="S5.SS3.SSS1.1.p1.6.m6.1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.1"></times><ci id="S5.SS3.SSS1.1.p1.6.m6.1.1.2.cmml" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.2">𝑘</ci><apply id="S5.SS3.SSS1.1.p1.6.m6.1.1.3.cmml" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.3"><root id="S5.SS3.SSS1.1.p1.6.m6.1.1.3a.cmml" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.3"></root><apply id="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2.cmml" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2"><minus id="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2.1.cmml" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2.1"></minus><cn id="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2.2.cmml" type="integer" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2.2">1</cn><ci id="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2.3.cmml" xref="S5.SS3.SSS1.1.p1.6.m6.1.1.3.2.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.1.p1.6.m6.1c">k\sqrt{1-\varepsilon}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.1.p1.6.m6.1d">italic_k square-root start_ARG 1 - italic_ε end_ARG</annotation></semantics></math> agents, and the utility of each agent in such a clique is at least <math alttext="k\sqrt{1-\varepsilon}-1" class="ltx_Math" display="inline" id="S5.SS3.SSS1.1.p1.7.m7.1"><semantics id="S5.SS3.SSS1.1.p1.7.m7.1a"><mrow id="S5.SS3.SSS1.1.p1.7.m7.1.1" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.cmml"><mrow id="S5.SS3.SSS1.1.p1.7.m7.1.1.2" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.cmml"><mi id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.2" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.2.cmml">k</mi><mo id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.1" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.1.cmml">⁢</mo><msqrt id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.cmml"><mrow id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2.cmml"><mn id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2.2" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2.2.cmml">1</mn><mo id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2.1" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2.1.cmml">−</mo><mi id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2.3" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><mo id="S5.SS3.SSS1.1.p1.7.m7.1.1.1" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.1.cmml">−</mo><mn id="S5.SS3.SSS1.1.p1.7.m7.1.1.3" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.1.p1.7.m7.1b"><apply id="S5.SS3.SSS1.1.p1.7.m7.1.1.cmml" xref="S5.SS3.SSS1.1.p1.7.m7.1.1"><minus id="S5.SS3.SSS1.1.p1.7.m7.1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.1"></minus><apply id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.cmml" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2"><times id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.1.cmml" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.1"></times><ci id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.2.cmml" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.2">𝑘</ci><apply id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.cmml" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3"><root id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3a.cmml" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3"></root><apply id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2.cmml" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2"><minus id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2.1.cmml" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2.1"></minus><cn id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2.2.cmml" type="integer" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2.2">1</cn><ci id="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2.3.cmml" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.2.3.2.3">𝜀</ci></apply></apply></apply><cn id="S5.SS3.SSS1.1.p1.7.m7.1.1.3.cmml" type="integer" xref="S5.SS3.SSS1.1.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.1.p1.7.m7.1c">k\sqrt{1-\varepsilon}-1</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.1.p1.7.m7.1d">italic_k square-root start_ARG 1 - italic_ε end_ARG - 1</annotation></semantics></math>. Therefore, the social welfare of the partition returned by the algorithm is bounded from below as follows:</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex37"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{SW}(\pi)\geq\alpha\frac{n^{\prime}}{k}k\sqrt{1-\varepsilon}(k\sqrt{1-% \varepsilon}-1)=\alpha n^{\prime}k(1-\varepsilon)-\alpha n^{\prime}\sqrt{1-% \varepsilon}." class="ltx_Math" display="block" id="S5.Ex37.m1.2"><semantics id="S5.Ex37.m1.2a"><mrow id="S5.Ex37.m1.2.2.1" xref="S5.Ex37.m1.2.2.1.1.cmml"><mrow id="S5.Ex37.m1.2.2.1.1" xref="S5.Ex37.m1.2.2.1.1.cmml"><mrow id="S5.Ex37.m1.2.2.1.1.4" xref="S5.Ex37.m1.2.2.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex37.m1.2.2.1.1.4.2" xref="S5.Ex37.m1.2.2.1.1.4.2.cmml">𝒮</mi><mo id="S5.Ex37.m1.2.2.1.1.4.1" xref="S5.Ex37.m1.2.2.1.1.4.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex37.m1.2.2.1.1.4.3" xref="S5.Ex37.m1.2.2.1.1.4.3.cmml">𝒲</mi><mo id="S5.Ex37.m1.2.2.1.1.4.1a" xref="S5.Ex37.m1.2.2.1.1.4.1.cmml">⁢</mo><mrow id="S5.Ex37.m1.2.2.1.1.4.4.2" xref="S5.Ex37.m1.2.2.1.1.4.cmml"><mo id="S5.Ex37.m1.2.2.1.1.4.4.2.1" stretchy="false" xref="S5.Ex37.m1.2.2.1.1.4.cmml">(</mo><mi id="S5.Ex37.m1.1.1" xref="S5.Ex37.m1.1.1.cmml">π</mi><mo id="S5.Ex37.m1.2.2.1.1.4.4.2.2" stretchy="false" xref="S5.Ex37.m1.2.2.1.1.4.cmml">)</mo></mrow></mrow><mo id="S5.Ex37.m1.2.2.1.1.5" xref="S5.Ex37.m1.2.2.1.1.5.cmml">≥</mo><mrow id="S5.Ex37.m1.2.2.1.1.1" xref="S5.Ex37.m1.2.2.1.1.1.cmml"><mi id="S5.Ex37.m1.2.2.1.1.1.3" xref="S5.Ex37.m1.2.2.1.1.1.3.cmml">α</mi><mo id="S5.Ex37.m1.2.2.1.1.1.2" xref="S5.Ex37.m1.2.2.1.1.1.2.cmml">⁢</mo><mfrac id="S5.Ex37.m1.2.2.1.1.1.4" xref="S5.Ex37.m1.2.2.1.1.1.4.cmml"><msup id="S5.Ex37.m1.2.2.1.1.1.4.2" xref="S5.Ex37.m1.2.2.1.1.1.4.2.cmml"><mi id="S5.Ex37.m1.2.2.1.1.1.4.2.2" xref="S5.Ex37.m1.2.2.1.1.1.4.2.2.cmml">n</mi><mo id="S5.Ex37.m1.2.2.1.1.1.4.2.3" xref="S5.Ex37.m1.2.2.1.1.1.4.2.3.cmml">′</mo></msup><mi id="S5.Ex37.m1.2.2.1.1.1.4.3" xref="S5.Ex37.m1.2.2.1.1.1.4.3.cmml">k</mi></mfrac><mo id="S5.Ex37.m1.2.2.1.1.1.2a" xref="S5.Ex37.m1.2.2.1.1.1.2.cmml">⁢</mo><mi id="S5.Ex37.m1.2.2.1.1.1.5" xref="S5.Ex37.m1.2.2.1.1.1.5.cmml">k</mi><mo id="S5.Ex37.m1.2.2.1.1.1.2b" xref="S5.Ex37.m1.2.2.1.1.1.2.cmml">⁢</mo><msqrt id="S5.Ex37.m1.2.2.1.1.1.6" xref="S5.Ex37.m1.2.2.1.1.1.6.cmml"><mrow id="S5.Ex37.m1.2.2.1.1.1.6.2" xref="S5.Ex37.m1.2.2.1.1.1.6.2.cmml"><mn id="S5.Ex37.m1.2.2.1.1.1.6.2.2" xref="S5.Ex37.m1.2.2.1.1.1.6.2.2.cmml">1</mn><mo id="S5.Ex37.m1.2.2.1.1.1.6.2.1" xref="S5.Ex37.m1.2.2.1.1.1.6.2.1.cmml">−</mo><mi id="S5.Ex37.m1.2.2.1.1.1.6.2.3" xref="S5.Ex37.m1.2.2.1.1.1.6.2.3.cmml">ε</mi></mrow></msqrt><mo id="S5.Ex37.m1.2.2.1.1.1.2c" xref="S5.Ex37.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex37.m1.2.2.1.1.1.1.1" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S5.Ex37.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex37.m1.2.2.1.1.1.1.1.1" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.cmml"><mrow id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.cmml"><mi id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.2" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.2.cmml">k</mi><mo id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.1" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.1.cmml">⁢</mo><msqrt id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.cmml"><mrow id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2.cmml"><mn id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2.2" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2.2.cmml">1</mn><mo id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2.1" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2.1.cmml">−</mo><mi id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2.3" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><mo id="S5.Ex37.m1.2.2.1.1.1.1.1.1.1" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.1.cmml">−</mo><mn id="S5.Ex37.m1.2.2.1.1.1.1.1.1.3" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S5.Ex37.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex37.m1.2.2.1.1.6" xref="S5.Ex37.m1.2.2.1.1.6.cmml">=</mo><mrow id="S5.Ex37.m1.2.2.1.1.2" xref="S5.Ex37.m1.2.2.1.1.2.cmml"><mrow id="S5.Ex37.m1.2.2.1.1.2.1" xref="S5.Ex37.m1.2.2.1.1.2.1.cmml"><mi id="S5.Ex37.m1.2.2.1.1.2.1.3" xref="S5.Ex37.m1.2.2.1.1.2.1.3.cmml">α</mi><mo id="S5.Ex37.m1.2.2.1.1.2.1.2" xref="S5.Ex37.m1.2.2.1.1.2.1.2.cmml">⁢</mo><msup id="S5.Ex37.m1.2.2.1.1.2.1.4" xref="S5.Ex37.m1.2.2.1.1.2.1.4.cmml"><mi id="S5.Ex37.m1.2.2.1.1.2.1.4.2" xref="S5.Ex37.m1.2.2.1.1.2.1.4.2.cmml">n</mi><mo id="S5.Ex37.m1.2.2.1.1.2.1.4.3" xref="S5.Ex37.m1.2.2.1.1.2.1.4.3.cmml">′</mo></msup><mo id="S5.Ex37.m1.2.2.1.1.2.1.2a" xref="S5.Ex37.m1.2.2.1.1.2.1.2.cmml">⁢</mo><mi id="S5.Ex37.m1.2.2.1.1.2.1.5" xref="S5.Ex37.m1.2.2.1.1.2.1.5.cmml">k</mi><mo id="S5.Ex37.m1.2.2.1.1.2.1.2b" xref="S5.Ex37.m1.2.2.1.1.2.1.2.cmml">⁢</mo><mrow id="S5.Ex37.m1.2.2.1.1.2.1.1.1" xref="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.cmml"><mo id="S5.Ex37.m1.2.2.1.1.2.1.1.1.2" stretchy="false" xref="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.cmml">(</mo><mrow id="S5.Ex37.m1.2.2.1.1.2.1.1.1.1" xref="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.cmml"><mn id="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.2" xref="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.2.cmml">1</mn><mo id="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.1" xref="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.1.cmml">−</mo><mi id="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.3" xref="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.Ex37.m1.2.2.1.1.2.1.1.1.3" stretchy="false" xref="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex37.m1.2.2.1.1.2.2" xref="S5.Ex37.m1.2.2.1.1.2.2.cmml">−</mo><mrow id="S5.Ex37.m1.2.2.1.1.2.3" xref="S5.Ex37.m1.2.2.1.1.2.3.cmml"><mi id="S5.Ex37.m1.2.2.1.1.2.3.2" xref="S5.Ex37.m1.2.2.1.1.2.3.2.cmml">α</mi><mo id="S5.Ex37.m1.2.2.1.1.2.3.1" xref="S5.Ex37.m1.2.2.1.1.2.3.1.cmml">⁢</mo><msup id="S5.Ex37.m1.2.2.1.1.2.3.3" xref="S5.Ex37.m1.2.2.1.1.2.3.3.cmml"><mi id="S5.Ex37.m1.2.2.1.1.2.3.3.2" xref="S5.Ex37.m1.2.2.1.1.2.3.3.2.cmml">n</mi><mo id="S5.Ex37.m1.2.2.1.1.2.3.3.3" xref="S5.Ex37.m1.2.2.1.1.2.3.3.3.cmml">′</mo></msup><mo id="S5.Ex37.m1.2.2.1.1.2.3.1a" xref="S5.Ex37.m1.2.2.1.1.2.3.1.cmml">⁢</mo><msqrt id="S5.Ex37.m1.2.2.1.1.2.3.4" xref="S5.Ex37.m1.2.2.1.1.2.3.4.cmml"><mrow id="S5.Ex37.m1.2.2.1.1.2.3.4.2" xref="S5.Ex37.m1.2.2.1.1.2.3.4.2.cmml"><mn id="S5.Ex37.m1.2.2.1.1.2.3.4.2.2" xref="S5.Ex37.m1.2.2.1.1.2.3.4.2.2.cmml">1</mn><mo id="S5.Ex37.m1.2.2.1.1.2.3.4.2.1" xref="S5.Ex37.m1.2.2.1.1.2.3.4.2.1.cmml">−</mo><mi id="S5.Ex37.m1.2.2.1.1.2.3.4.2.3" xref="S5.Ex37.m1.2.2.1.1.2.3.4.2.3.cmml">ε</mi></mrow></msqrt></mrow></mrow></mrow><mo id="S5.Ex37.m1.2.2.1.2" lspace="0em" xref="S5.Ex37.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex37.m1.2b"><apply id="S5.Ex37.m1.2.2.1.1.cmml" xref="S5.Ex37.m1.2.2.1"><and id="S5.Ex37.m1.2.2.1.1a.cmml" xref="S5.Ex37.m1.2.2.1"></and><apply id="S5.Ex37.m1.2.2.1.1b.cmml" xref="S5.Ex37.m1.2.2.1"><geq id="S5.Ex37.m1.2.2.1.1.5.cmml" xref="S5.Ex37.m1.2.2.1.1.5"></geq><apply id="S5.Ex37.m1.2.2.1.1.4.cmml" xref="S5.Ex37.m1.2.2.1.1.4"><times id="S5.Ex37.m1.2.2.1.1.4.1.cmml" xref="S5.Ex37.m1.2.2.1.1.4.1"></times><ci id="S5.Ex37.m1.2.2.1.1.4.2.cmml" xref="S5.Ex37.m1.2.2.1.1.4.2">𝒮</ci><ci id="S5.Ex37.m1.2.2.1.1.4.3.cmml" xref="S5.Ex37.m1.2.2.1.1.4.3">𝒲</ci><ci id="S5.Ex37.m1.1.1.cmml" xref="S5.Ex37.m1.1.1">𝜋</ci></apply><apply id="S5.Ex37.m1.2.2.1.1.1.cmml" xref="S5.Ex37.m1.2.2.1.1.1"><times id="S5.Ex37.m1.2.2.1.1.1.2.cmml" xref="S5.Ex37.m1.2.2.1.1.1.2"></times><ci id="S5.Ex37.m1.2.2.1.1.1.3.cmml" xref="S5.Ex37.m1.2.2.1.1.1.3">𝛼</ci><apply id="S5.Ex37.m1.2.2.1.1.1.4.cmml" xref="S5.Ex37.m1.2.2.1.1.1.4"><divide id="S5.Ex37.m1.2.2.1.1.1.4.1.cmml" xref="S5.Ex37.m1.2.2.1.1.1.4"></divide><apply id="S5.Ex37.m1.2.2.1.1.1.4.2.cmml" xref="S5.Ex37.m1.2.2.1.1.1.4.2"><csymbol cd="ambiguous" id="S5.Ex37.m1.2.2.1.1.1.4.2.1.cmml" xref="S5.Ex37.m1.2.2.1.1.1.4.2">superscript</csymbol><ci id="S5.Ex37.m1.2.2.1.1.1.4.2.2.cmml" xref="S5.Ex37.m1.2.2.1.1.1.4.2.2">𝑛</ci><ci id="S5.Ex37.m1.2.2.1.1.1.4.2.3.cmml" xref="S5.Ex37.m1.2.2.1.1.1.4.2.3">′</ci></apply><ci id="S5.Ex37.m1.2.2.1.1.1.4.3.cmml" xref="S5.Ex37.m1.2.2.1.1.1.4.3">𝑘</ci></apply><ci id="S5.Ex37.m1.2.2.1.1.1.5.cmml" xref="S5.Ex37.m1.2.2.1.1.1.5">𝑘</ci><apply id="S5.Ex37.m1.2.2.1.1.1.6.cmml" xref="S5.Ex37.m1.2.2.1.1.1.6"><root id="S5.Ex37.m1.2.2.1.1.1.6a.cmml" xref="S5.Ex37.m1.2.2.1.1.1.6"></root><apply id="S5.Ex37.m1.2.2.1.1.1.6.2.cmml" xref="S5.Ex37.m1.2.2.1.1.1.6.2"><minus id="S5.Ex37.m1.2.2.1.1.1.6.2.1.cmml" xref="S5.Ex37.m1.2.2.1.1.1.6.2.1"></minus><cn id="S5.Ex37.m1.2.2.1.1.1.6.2.2.cmml" type="integer" xref="S5.Ex37.m1.2.2.1.1.1.6.2.2">1</cn><ci id="S5.Ex37.m1.2.2.1.1.1.6.2.3.cmml" xref="S5.Ex37.m1.2.2.1.1.1.6.2.3">𝜀</ci></apply></apply><apply id="S5.Ex37.m1.2.2.1.1.1.1.1.1.cmml" xref="S5.Ex37.m1.2.2.1.1.1.1.1"><minus id="S5.Ex37.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.1"></minus><apply id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2"><times id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.1.cmml" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.1"></times><ci id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.2.cmml" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.2">𝑘</ci><apply id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.cmml" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3"><root id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3a.cmml" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3"></root><apply id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2.cmml" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2"><minus id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2.1.cmml" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2.1"></minus><cn id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2.2.cmml" type="integer" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2.2">1</cn><ci id="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2.3.cmml" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.2.3.2.3">𝜀</ci></apply></apply></apply><cn id="S5.Ex37.m1.2.2.1.1.1.1.1.1.3.cmml" type="integer" xref="S5.Ex37.m1.2.2.1.1.1.1.1.1.3">1</cn></apply></apply></apply><apply id="S5.Ex37.m1.2.2.1.1c.cmml" xref="S5.Ex37.m1.2.2.1"><eq id="S5.Ex37.m1.2.2.1.1.6.cmml" xref="S5.Ex37.m1.2.2.1.1.6"></eq><share href="https://arxiv.org/html/2503.06017v1#S5.Ex37.m1.2.2.1.1.1.cmml" id="S5.Ex37.m1.2.2.1.1d.cmml" xref="S5.Ex37.m1.2.2.1"></share><apply id="S5.Ex37.m1.2.2.1.1.2.cmml" xref="S5.Ex37.m1.2.2.1.1.2"><minus id="S5.Ex37.m1.2.2.1.1.2.2.cmml" xref="S5.Ex37.m1.2.2.1.1.2.2"></minus><apply id="S5.Ex37.m1.2.2.1.1.2.1.cmml" xref="S5.Ex37.m1.2.2.1.1.2.1"><times id="S5.Ex37.m1.2.2.1.1.2.1.2.cmml" xref="S5.Ex37.m1.2.2.1.1.2.1.2"></times><ci id="S5.Ex37.m1.2.2.1.1.2.1.3.cmml" xref="S5.Ex37.m1.2.2.1.1.2.1.3">𝛼</ci><apply id="S5.Ex37.m1.2.2.1.1.2.1.4.cmml" xref="S5.Ex37.m1.2.2.1.1.2.1.4"><csymbol cd="ambiguous" id="S5.Ex37.m1.2.2.1.1.2.1.4.1.cmml" xref="S5.Ex37.m1.2.2.1.1.2.1.4">superscript</csymbol><ci id="S5.Ex37.m1.2.2.1.1.2.1.4.2.cmml" xref="S5.Ex37.m1.2.2.1.1.2.1.4.2">𝑛</ci><ci id="S5.Ex37.m1.2.2.1.1.2.1.4.3.cmml" xref="S5.Ex37.m1.2.2.1.1.2.1.4.3">′</ci></apply><ci id="S5.Ex37.m1.2.2.1.1.2.1.5.cmml" xref="S5.Ex37.m1.2.2.1.1.2.1.5">𝑘</ci><apply id="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.cmml" xref="S5.Ex37.m1.2.2.1.1.2.1.1.1"><minus id="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.1.cmml" xref="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.1"></minus><cn id="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.2.cmml" type="integer" xref="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.2">1</cn><ci id="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.3.cmml" xref="S5.Ex37.m1.2.2.1.1.2.1.1.1.1.3">𝜀</ci></apply></apply><apply id="S5.Ex37.m1.2.2.1.1.2.3.cmml" xref="S5.Ex37.m1.2.2.1.1.2.3"><times id="S5.Ex37.m1.2.2.1.1.2.3.1.cmml" xref="S5.Ex37.m1.2.2.1.1.2.3.1"></times><ci id="S5.Ex37.m1.2.2.1.1.2.3.2.cmml" xref="S5.Ex37.m1.2.2.1.1.2.3.2">𝛼</ci><apply id="S5.Ex37.m1.2.2.1.1.2.3.3.cmml" xref="S5.Ex37.m1.2.2.1.1.2.3.3"><csymbol cd="ambiguous" id="S5.Ex37.m1.2.2.1.1.2.3.3.1.cmml" xref="S5.Ex37.m1.2.2.1.1.2.3.3">superscript</csymbol><ci id="S5.Ex37.m1.2.2.1.1.2.3.3.2.cmml" xref="S5.Ex37.m1.2.2.1.1.2.3.3.2">𝑛</ci><ci id="S5.Ex37.m1.2.2.1.1.2.3.3.3.cmml" xref="S5.Ex37.m1.2.2.1.1.2.3.3.3">′</ci></apply><apply id="S5.Ex37.m1.2.2.1.1.2.3.4.cmml" xref="S5.Ex37.m1.2.2.1.1.2.3.4"><root id="S5.Ex37.m1.2.2.1.1.2.3.4a.cmml" xref="S5.Ex37.m1.2.2.1.1.2.3.4"></root><apply id="S5.Ex37.m1.2.2.1.1.2.3.4.2.cmml" xref="S5.Ex37.m1.2.2.1.1.2.3.4.2"><minus id="S5.Ex37.m1.2.2.1.1.2.3.4.2.1.cmml" xref="S5.Ex37.m1.2.2.1.1.2.3.4.2.1"></minus><cn id="S5.Ex37.m1.2.2.1.1.2.3.4.2.2.cmml" type="integer" xref="S5.Ex37.m1.2.2.1.1.2.3.4.2.2">1</cn><ci id="S5.Ex37.m1.2.2.1.1.2.3.4.2.3.cmml" xref="S5.Ex37.m1.2.2.1.1.2.3.4.2.3">𝜀</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex37.m1.2c">\mathcal{SW}(\pi)\geq\alpha\frac{n^{\prime}}{k}k\sqrt{1-\varepsilon}(k\sqrt{1-% \varepsilon}-1)=\alpha n^{\prime}k(1-\varepsilon)-\alpha n^{\prime}\sqrt{1-% \varepsilon}.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex37.m1.2d">caligraphic_S caligraphic_W ( italic_π ) ≥ italic_α divide start_ARG italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_k end_ARG italic_k square-root start_ARG 1 - italic_ε end_ARG ( italic_k square-root start_ARG 1 - italic_ε end_ARG - 1 ) = italic_α italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_k ( 1 - italic_ε ) - italic_α italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_ε end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS3.SSS1.1.p1.8">On the other hand, <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem6" title="Proposition 5.6. ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">5.6</span></a> implies <math alttext="\mathcal{SW}(\pi^{*})\leq n(k-1)<nk" class="ltx_Math" display="inline" id="S5.SS3.SSS1.1.p1.8.m1.2"><semantics id="S5.SS3.SSS1.1.p1.8.m1.2a"><mrow id="S5.SS3.SSS1.1.p1.8.m1.2.2" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.cmml"><mrow id="S5.SS3.SSS1.1.p1.8.m1.1.1.1" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.3" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.3.cmml">𝒮</mi><mo id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.2" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.4" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.4.cmml">𝒲</mi><mo id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.2a" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.1.cmml"><mo id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.1" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.1.2" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.1.2.cmml">π</mi><mo id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.1.3" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS3.SSS1.1.p1.8.m1.2.2.4" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.4.cmml">≤</mo><mrow id="S5.SS3.SSS1.1.p1.8.m1.2.2.2" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.cmml"><mi id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.3" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.3.cmml">n</mi><mo id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.2" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.2.cmml">⁢</mo><mrow id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.cmml"><mo id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.2" stretchy="false" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.cmml"><mi id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.2" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.2.cmml">k</mi><mo id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.1" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.1.cmml">−</mo><mn id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.3" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.3" stretchy="false" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS3.SSS1.1.p1.8.m1.2.2.5" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.5.cmml"><</mo><mrow id="S5.SS3.SSS1.1.p1.8.m1.2.2.6" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.6.cmml"><mi id="S5.SS3.SSS1.1.p1.8.m1.2.2.6.2" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.6.2.cmml">n</mi><mo id="S5.SS3.SSS1.1.p1.8.m1.2.2.6.1" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.6.1.cmml">⁢</mo><mi id="S5.SS3.SSS1.1.p1.8.m1.2.2.6.3" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.6.3.cmml">k</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.1.p1.8.m1.2b"><apply id="S5.SS3.SSS1.1.p1.8.m1.2.2.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2"><and id="S5.SS3.SSS1.1.p1.8.m1.2.2a.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2"></and><apply id="S5.SS3.SSS1.1.p1.8.m1.2.2b.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2"><leq id="S5.SS3.SSS1.1.p1.8.m1.2.2.4.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.4"></leq><apply id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1"><times id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.2.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.2"></times><ci id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.3.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.3">𝒮</ci><ci id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.4.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.4">𝒲</ci><apply id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1">superscript</csymbol><ci id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.1.2">𝜋</ci><times id="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.1.3.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.1.1.1.1.1.1.3"></times></apply></apply><apply id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2"><times id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.2.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.2"></times><ci id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.3.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.3">𝑛</ci><apply id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1"><minus id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.1"></minus><ci id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.2.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.2">𝑘</ci><cn id="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.3.cmml" type="integer" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.2.1.1.1.3">1</cn></apply></apply></apply><apply id="S5.SS3.SSS1.1.p1.8.m1.2.2c.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2"><lt id="S5.SS3.SSS1.1.p1.8.m1.2.2.5.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.5"></lt><share href="https://arxiv.org/html/2503.06017v1#S5.SS3.SSS1.1.p1.8.m1.2.2.2.cmml" id="S5.SS3.SSS1.1.p1.8.m1.2.2d.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2"></share><apply id="S5.SS3.SSS1.1.p1.8.m1.2.2.6.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.6"><times id="S5.SS3.SSS1.1.p1.8.m1.2.2.6.1.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.6.1"></times><ci id="S5.SS3.SSS1.1.p1.8.m1.2.2.6.2.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.6.2">𝑛</ci><ci id="S5.SS3.SSS1.1.p1.8.m1.2.2.6.3.cmml" xref="S5.SS3.SSS1.1.p1.8.m1.2.2.6.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.1.p1.8.m1.2c">\mathcal{SW}(\pi^{*})\leq n(k-1)<nk</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.1.p1.8.m1.2d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ≤ italic_n ( italic_k - 1 ) < italic_n italic_k</annotation></semantics></math>. This gives the following bound on the ratio of social welfare:</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex38"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{\mathcal{SW}(\pi)}{\mathcal{SW}(\pi^{*})}\geq\frac{n^{\prime}}{n}\left[% \alpha(1-\varepsilon)-\frac{\alpha\sqrt{1-\varepsilon}}{k}\right]\geq q\alpha(% 1-\varepsilon)-\frac{q\alpha\sqrt{1-\varepsilon}}{k}," class="ltx_Math" display="block" id="S5.Ex38.m1.3"><semantics id="S5.Ex38.m1.3a"><mrow id="S5.Ex38.m1.3.3.1" xref="S5.Ex38.m1.3.3.1.1.cmml"><mrow id="S5.Ex38.m1.3.3.1.1" xref="S5.Ex38.m1.3.3.1.1.cmml"><mfrac id="S5.Ex38.m1.2.2" xref="S5.Ex38.m1.2.2.cmml"><mrow id="S5.Ex38.m1.1.1.1" xref="S5.Ex38.m1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex38.m1.1.1.1.3" xref="S5.Ex38.m1.1.1.1.3.cmml">𝒮</mi><mo id="S5.Ex38.m1.1.1.1.2" xref="S5.Ex38.m1.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex38.m1.1.1.1.4" xref="S5.Ex38.m1.1.1.1.4.cmml">𝒲</mi><mo id="S5.Ex38.m1.1.1.1.2a" xref="S5.Ex38.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex38.m1.1.1.1.5.2" xref="S5.Ex38.m1.1.1.1.cmml"><mo id="S5.Ex38.m1.1.1.1.5.2.1" stretchy="false" xref="S5.Ex38.m1.1.1.1.cmml">(</mo><mi id="S5.Ex38.m1.1.1.1.1" xref="S5.Ex38.m1.1.1.1.1.cmml">π</mi><mo id="S5.Ex38.m1.1.1.1.5.2.2" stretchy="false" xref="S5.Ex38.m1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S5.Ex38.m1.2.2.2" xref="S5.Ex38.m1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex38.m1.2.2.2.3" xref="S5.Ex38.m1.2.2.2.3.cmml">𝒮</mi><mo id="S5.Ex38.m1.2.2.2.2" xref="S5.Ex38.m1.2.2.2.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Ex38.m1.2.2.2.4" xref="S5.Ex38.m1.2.2.2.4.cmml">𝒲</mi><mo id="S5.Ex38.m1.2.2.2.2a" xref="S5.Ex38.m1.2.2.2.2.cmml">⁢</mo><mrow id="S5.Ex38.m1.2.2.2.1.1" xref="S5.Ex38.m1.2.2.2.1.1.1.cmml"><mo id="S5.Ex38.m1.2.2.2.1.1.2" stretchy="false" xref="S5.Ex38.m1.2.2.2.1.1.1.cmml">(</mo><msup id="S5.Ex38.m1.2.2.2.1.1.1" xref="S5.Ex38.m1.2.2.2.1.1.1.cmml"><mi id="S5.Ex38.m1.2.2.2.1.1.1.2" xref="S5.Ex38.m1.2.2.2.1.1.1.2.cmml">π</mi><mo id="S5.Ex38.m1.2.2.2.1.1.1.3" xref="S5.Ex38.m1.2.2.2.1.1.1.3.cmml">∗</mo></msup><mo id="S5.Ex38.m1.2.2.2.1.1.3" stretchy="false" xref="S5.Ex38.m1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mfrac><mo id="S5.Ex38.m1.3.3.1.1.4" xref="S5.Ex38.m1.3.3.1.1.4.cmml">≥</mo><mrow id="S5.Ex38.m1.3.3.1.1.1" xref="S5.Ex38.m1.3.3.1.1.1.cmml"><mfrac id="S5.Ex38.m1.3.3.1.1.1.3" xref="S5.Ex38.m1.3.3.1.1.1.3.cmml"><msup id="S5.Ex38.m1.3.3.1.1.1.3.2" xref="S5.Ex38.m1.3.3.1.1.1.3.2.cmml"><mi id="S5.Ex38.m1.3.3.1.1.1.3.2.2" xref="S5.Ex38.m1.3.3.1.1.1.3.2.2.cmml">n</mi><mo id="S5.Ex38.m1.3.3.1.1.1.3.2.3" xref="S5.Ex38.m1.3.3.1.1.1.3.2.3.cmml">′</mo></msup><mi id="S5.Ex38.m1.3.3.1.1.1.3.3" xref="S5.Ex38.m1.3.3.1.1.1.3.3.cmml">n</mi></mfrac><mo id="S5.Ex38.m1.3.3.1.1.1.2" xref="S5.Ex38.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex38.m1.3.3.1.1.1.1.1" xref="S5.Ex38.m1.3.3.1.1.1.1.2.cmml"><mo id="S5.Ex38.m1.3.3.1.1.1.1.1.2" xref="S5.Ex38.m1.3.3.1.1.1.1.2.1.cmml">[</mo><mrow id="S5.Ex38.m1.3.3.1.1.1.1.1.1" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.cmml"><mrow id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.3" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.3.cmml">α</mi><mo id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.2" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.cmml"><mn id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.2" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.1" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.3" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex38.m1.3.3.1.1.1.1.1.1.2" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.2.cmml">−</mo><mfrac id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.cmml"><mrow id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.cmml"><mi id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.2" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.2.cmml">α</mi><mo id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.1" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.1.cmml">⁢</mo><msqrt id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.cmml"><mrow id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2.cmml"><mn id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2.2" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2.2.cmml">1</mn><mo id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2.1" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2.1.cmml">−</mo><mi id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2.3" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2.3.cmml">ε</mi></mrow></msqrt></mrow><mi id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.3" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.3.cmml">k</mi></mfrac></mrow><mo id="S5.Ex38.m1.3.3.1.1.1.1.1.3" xref="S5.Ex38.m1.3.3.1.1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S5.Ex38.m1.3.3.1.1.5" xref="S5.Ex38.m1.3.3.1.1.5.cmml">≥</mo><mrow id="S5.Ex38.m1.3.3.1.1.2" xref="S5.Ex38.m1.3.3.1.1.2.cmml"><mrow id="S5.Ex38.m1.3.3.1.1.2.1" xref="S5.Ex38.m1.3.3.1.1.2.1.cmml"><mi id="S5.Ex38.m1.3.3.1.1.2.1.3" xref="S5.Ex38.m1.3.3.1.1.2.1.3.cmml">q</mi><mo id="S5.Ex38.m1.3.3.1.1.2.1.2" xref="S5.Ex38.m1.3.3.1.1.2.1.2.cmml">⁢</mo><mi id="S5.Ex38.m1.3.3.1.1.2.1.4" xref="S5.Ex38.m1.3.3.1.1.2.1.4.cmml">α</mi><mo id="S5.Ex38.m1.3.3.1.1.2.1.2a" xref="S5.Ex38.m1.3.3.1.1.2.1.2.cmml">⁢</mo><mrow id="S5.Ex38.m1.3.3.1.1.2.1.1.1" xref="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.cmml"><mo id="S5.Ex38.m1.3.3.1.1.2.1.1.1.2" stretchy="false" xref="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.cmml">(</mo><mrow id="S5.Ex38.m1.3.3.1.1.2.1.1.1.1" xref="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.cmml"><mn id="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.2" xref="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.2.cmml">1</mn><mo id="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.1" xref="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.1.cmml">−</mo><mi id="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.3" xref="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.Ex38.m1.3.3.1.1.2.1.1.1.3" stretchy="false" xref="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex38.m1.3.3.1.1.2.2" xref="S5.Ex38.m1.3.3.1.1.2.2.cmml">−</mo><mfrac id="S5.Ex38.m1.3.3.1.1.2.3" xref="S5.Ex38.m1.3.3.1.1.2.3.cmml"><mrow id="S5.Ex38.m1.3.3.1.1.2.3.2" xref="S5.Ex38.m1.3.3.1.1.2.3.2.cmml"><mi id="S5.Ex38.m1.3.3.1.1.2.3.2.2" xref="S5.Ex38.m1.3.3.1.1.2.3.2.2.cmml">q</mi><mo id="S5.Ex38.m1.3.3.1.1.2.3.2.1" xref="S5.Ex38.m1.3.3.1.1.2.3.2.1.cmml">⁢</mo><mi id="S5.Ex38.m1.3.3.1.1.2.3.2.3" xref="S5.Ex38.m1.3.3.1.1.2.3.2.3.cmml">α</mi><mo id="S5.Ex38.m1.3.3.1.1.2.3.2.1a" xref="S5.Ex38.m1.3.3.1.1.2.3.2.1.cmml">⁢</mo><msqrt id="S5.Ex38.m1.3.3.1.1.2.3.2.4" xref="S5.Ex38.m1.3.3.1.1.2.3.2.4.cmml"><mrow id="S5.Ex38.m1.3.3.1.1.2.3.2.4.2" xref="S5.Ex38.m1.3.3.1.1.2.3.2.4.2.cmml"><mn id="S5.Ex38.m1.3.3.1.1.2.3.2.4.2.2" xref="S5.Ex38.m1.3.3.1.1.2.3.2.4.2.2.cmml">1</mn><mo id="S5.Ex38.m1.3.3.1.1.2.3.2.4.2.1" xref="S5.Ex38.m1.3.3.1.1.2.3.2.4.2.1.cmml">−</mo><mi id="S5.Ex38.m1.3.3.1.1.2.3.2.4.2.3" xref="S5.Ex38.m1.3.3.1.1.2.3.2.4.2.3.cmml">ε</mi></mrow></msqrt></mrow><mi id="S5.Ex38.m1.3.3.1.1.2.3.3" xref="S5.Ex38.m1.3.3.1.1.2.3.3.cmml">k</mi></mfrac></mrow></mrow><mo id="S5.Ex38.m1.3.3.1.2" xref="S5.Ex38.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex38.m1.3b"><apply id="S5.Ex38.m1.3.3.1.1.cmml" xref="S5.Ex38.m1.3.3.1"><and id="S5.Ex38.m1.3.3.1.1a.cmml" xref="S5.Ex38.m1.3.3.1"></and><apply id="S5.Ex38.m1.3.3.1.1b.cmml" xref="S5.Ex38.m1.3.3.1"><geq id="S5.Ex38.m1.3.3.1.1.4.cmml" xref="S5.Ex38.m1.3.3.1.1.4"></geq><apply id="S5.Ex38.m1.2.2.cmml" xref="S5.Ex38.m1.2.2"><divide id="S5.Ex38.m1.2.2.3.cmml" xref="S5.Ex38.m1.2.2"></divide><apply id="S5.Ex38.m1.1.1.1.cmml" xref="S5.Ex38.m1.1.1.1"><times id="S5.Ex38.m1.1.1.1.2.cmml" xref="S5.Ex38.m1.1.1.1.2"></times><ci id="S5.Ex38.m1.1.1.1.3.cmml" xref="S5.Ex38.m1.1.1.1.3">𝒮</ci><ci id="S5.Ex38.m1.1.1.1.4.cmml" xref="S5.Ex38.m1.1.1.1.4">𝒲</ci><ci id="S5.Ex38.m1.1.1.1.1.cmml" xref="S5.Ex38.m1.1.1.1.1">𝜋</ci></apply><apply id="S5.Ex38.m1.2.2.2.cmml" xref="S5.Ex38.m1.2.2.2"><times id="S5.Ex38.m1.2.2.2.2.cmml" xref="S5.Ex38.m1.2.2.2.2"></times><ci id="S5.Ex38.m1.2.2.2.3.cmml" xref="S5.Ex38.m1.2.2.2.3">𝒮</ci><ci id="S5.Ex38.m1.2.2.2.4.cmml" xref="S5.Ex38.m1.2.2.2.4">𝒲</ci><apply id="S5.Ex38.m1.2.2.2.1.1.1.cmml" xref="S5.Ex38.m1.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.Ex38.m1.2.2.2.1.1.1.1.cmml" xref="S5.Ex38.m1.2.2.2.1.1">superscript</csymbol><ci id="S5.Ex38.m1.2.2.2.1.1.1.2.cmml" xref="S5.Ex38.m1.2.2.2.1.1.1.2">𝜋</ci><times id="S5.Ex38.m1.2.2.2.1.1.1.3.cmml" xref="S5.Ex38.m1.2.2.2.1.1.1.3"></times></apply></apply></apply><apply id="S5.Ex38.m1.3.3.1.1.1.cmml" xref="S5.Ex38.m1.3.3.1.1.1"><times id="S5.Ex38.m1.3.3.1.1.1.2.cmml" xref="S5.Ex38.m1.3.3.1.1.1.2"></times><apply id="S5.Ex38.m1.3.3.1.1.1.3.cmml" xref="S5.Ex38.m1.3.3.1.1.1.3"><divide id="S5.Ex38.m1.3.3.1.1.1.3.1.cmml" xref="S5.Ex38.m1.3.3.1.1.1.3"></divide><apply id="S5.Ex38.m1.3.3.1.1.1.3.2.cmml" xref="S5.Ex38.m1.3.3.1.1.1.3.2"><csymbol cd="ambiguous" id="S5.Ex38.m1.3.3.1.1.1.3.2.1.cmml" xref="S5.Ex38.m1.3.3.1.1.1.3.2">superscript</csymbol><ci id="S5.Ex38.m1.3.3.1.1.1.3.2.2.cmml" xref="S5.Ex38.m1.3.3.1.1.1.3.2.2">𝑛</ci><ci id="S5.Ex38.m1.3.3.1.1.1.3.2.3.cmml" xref="S5.Ex38.m1.3.3.1.1.1.3.2.3">′</ci></apply><ci id="S5.Ex38.m1.3.3.1.1.1.3.3.cmml" xref="S5.Ex38.m1.3.3.1.1.1.3.3">𝑛</ci></apply><apply id="S5.Ex38.m1.3.3.1.1.1.1.2.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1"><csymbol cd="latexml" id="S5.Ex38.m1.3.3.1.1.1.1.2.1.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S5.Ex38.m1.3.3.1.1.1.1.1.1.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1"><minus id="S5.Ex38.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.2"></minus><apply id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1"><times id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.2"></times><ci id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.3">𝛼</ci><apply id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1"><minus id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.1"></minus><cn id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.2">1</cn><ci id="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.1.1.1.1.3">𝜀</ci></apply></apply><apply id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3"><divide id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.1.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3"></divide><apply id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2"><times id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.1.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.1"></times><ci id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.2.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.2">𝛼</ci><apply id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3"><root id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3a.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3"></root><apply id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2"><minus id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2.1.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2.1"></minus><cn id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2.2.cmml" type="integer" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2.2">1</cn><ci id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2.3.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.2.3.2.3">𝜀</ci></apply></apply></apply><ci id="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.3.cmml" xref="S5.Ex38.m1.3.3.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></apply></apply><apply id="S5.Ex38.m1.3.3.1.1c.cmml" xref="S5.Ex38.m1.3.3.1"><geq id="S5.Ex38.m1.3.3.1.1.5.cmml" xref="S5.Ex38.m1.3.3.1.1.5"></geq><share href="https://arxiv.org/html/2503.06017v1#S5.Ex38.m1.3.3.1.1.1.cmml" id="S5.Ex38.m1.3.3.1.1d.cmml" xref="S5.Ex38.m1.3.3.1"></share><apply id="S5.Ex38.m1.3.3.1.1.2.cmml" xref="S5.Ex38.m1.3.3.1.1.2"><minus id="S5.Ex38.m1.3.3.1.1.2.2.cmml" xref="S5.Ex38.m1.3.3.1.1.2.2"></minus><apply id="S5.Ex38.m1.3.3.1.1.2.1.cmml" xref="S5.Ex38.m1.3.3.1.1.2.1"><times id="S5.Ex38.m1.3.3.1.1.2.1.2.cmml" xref="S5.Ex38.m1.3.3.1.1.2.1.2"></times><ci id="S5.Ex38.m1.3.3.1.1.2.1.3.cmml" xref="S5.Ex38.m1.3.3.1.1.2.1.3">𝑞</ci><ci id="S5.Ex38.m1.3.3.1.1.2.1.4.cmml" xref="S5.Ex38.m1.3.3.1.1.2.1.4">𝛼</ci><apply id="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.cmml" xref="S5.Ex38.m1.3.3.1.1.2.1.1.1"><minus id="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.1.cmml" xref="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.1"></minus><cn id="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.2.cmml" type="integer" xref="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.2">1</cn><ci id="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.3.cmml" xref="S5.Ex38.m1.3.3.1.1.2.1.1.1.1.3">𝜀</ci></apply></apply><apply id="S5.Ex38.m1.3.3.1.1.2.3.cmml" xref="S5.Ex38.m1.3.3.1.1.2.3"><divide id="S5.Ex38.m1.3.3.1.1.2.3.1.cmml" xref="S5.Ex38.m1.3.3.1.1.2.3"></divide><apply id="S5.Ex38.m1.3.3.1.1.2.3.2.cmml" xref="S5.Ex38.m1.3.3.1.1.2.3.2"><times id="S5.Ex38.m1.3.3.1.1.2.3.2.1.cmml" xref="S5.Ex38.m1.3.3.1.1.2.3.2.1"></times><ci id="S5.Ex38.m1.3.3.1.1.2.3.2.2.cmml" xref="S5.Ex38.m1.3.3.1.1.2.3.2.2">𝑞</ci><ci id="S5.Ex38.m1.3.3.1.1.2.3.2.3.cmml" xref="S5.Ex38.m1.3.3.1.1.2.3.2.3">𝛼</ci><apply id="S5.Ex38.m1.3.3.1.1.2.3.2.4.cmml" xref="S5.Ex38.m1.3.3.1.1.2.3.2.4"><root id="S5.Ex38.m1.3.3.1.1.2.3.2.4a.cmml" xref="S5.Ex38.m1.3.3.1.1.2.3.2.4"></root><apply id="S5.Ex38.m1.3.3.1.1.2.3.2.4.2.cmml" xref="S5.Ex38.m1.3.3.1.1.2.3.2.4.2"><minus id="S5.Ex38.m1.3.3.1.1.2.3.2.4.2.1.cmml" xref="S5.Ex38.m1.3.3.1.1.2.3.2.4.2.1"></minus><cn id="S5.Ex38.m1.3.3.1.1.2.3.2.4.2.2.cmml" type="integer" xref="S5.Ex38.m1.3.3.1.1.2.3.2.4.2.2">1</cn><ci id="S5.Ex38.m1.3.3.1.1.2.3.2.4.2.3.cmml" xref="S5.Ex38.m1.3.3.1.1.2.3.2.4.2.3">𝜀</ci></apply></apply></apply><ci id="S5.Ex38.m1.3.3.1.1.2.3.3.cmml" xref="S5.Ex38.m1.3.3.1.1.2.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex38.m1.3c">\frac{\mathcal{SW}(\pi)}{\mathcal{SW}(\pi^{*})}\geq\frac{n^{\prime}}{n}\left[% \alpha(1-\varepsilon)-\frac{\alpha\sqrt{1-\varepsilon}}{k}\right]\geq q\alpha(% 1-\varepsilon)-\frac{q\alpha\sqrt{1-\varepsilon}}{k},</annotation><annotation encoding="application/x-llamapun" id="S5.Ex38.m1.3d">divide start_ARG caligraphic_S caligraphic_W ( italic_π ) end_ARG start_ARG caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) end_ARG ≥ divide start_ARG italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_n end_ARG [ italic_α ( 1 - italic_ε ) - divide start_ARG italic_α square-root start_ARG 1 - italic_ε end_ARG end_ARG start_ARG italic_k end_ARG ] ≥ italic_q italic_α ( 1 - italic_ε ) - divide start_ARG italic_q italic_α square-root start_ARG 1 - italic_ε end_ARG end_ARG start_ARG italic_k end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS3.SSS1.1.p1.10">where the last inequality follows from <math alttext="n^{\prime}\geq nq" class="ltx_Math" display="inline" id="S5.SS3.SSS1.1.p1.9.m1.1"><semantics id="S5.SS3.SSS1.1.p1.9.m1.1a"><mrow id="S5.SS3.SSS1.1.p1.9.m1.1.1" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.cmml"><msup id="S5.SS3.SSS1.1.p1.9.m1.1.1.2" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.2.cmml"><mi id="S5.SS3.SSS1.1.p1.9.m1.1.1.2.2" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.2.2.cmml">n</mi><mo id="S5.SS3.SSS1.1.p1.9.m1.1.1.2.3" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.2.3.cmml">′</mo></msup><mo id="S5.SS3.SSS1.1.p1.9.m1.1.1.1" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.1.cmml">≥</mo><mrow id="S5.SS3.SSS1.1.p1.9.m1.1.1.3" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.3.cmml"><mi id="S5.SS3.SSS1.1.p1.9.m1.1.1.3.2" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.3.2.cmml">n</mi><mo id="S5.SS3.SSS1.1.p1.9.m1.1.1.3.1" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.3.1.cmml">⁢</mo><mi id="S5.SS3.SSS1.1.p1.9.m1.1.1.3.3" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.3.3.cmml">q</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.1.p1.9.m1.1b"><apply id="S5.SS3.SSS1.1.p1.9.m1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.9.m1.1.1"><geq id="S5.SS3.SSS1.1.p1.9.m1.1.1.1.cmml" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.1"></geq><apply id="S5.SS3.SSS1.1.p1.9.m1.1.1.2.cmml" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.SSS1.1.p1.9.m1.1.1.2.1.cmml" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.2">superscript</csymbol><ci id="S5.SS3.SSS1.1.p1.9.m1.1.1.2.2.cmml" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.2.2">𝑛</ci><ci id="S5.SS3.SSS1.1.p1.9.m1.1.1.2.3.cmml" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.2.3">′</ci></apply><apply id="S5.SS3.SSS1.1.p1.9.m1.1.1.3.cmml" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.3"><times id="S5.SS3.SSS1.1.p1.9.m1.1.1.3.1.cmml" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.3.1"></times><ci id="S5.SS3.SSS1.1.p1.9.m1.1.1.3.2.cmml" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.3.2">𝑛</ci><ci id="S5.SS3.SSS1.1.p1.9.m1.1.1.3.3.cmml" xref="S5.SS3.SSS1.1.p1.9.m1.1.1.3.3">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.1.p1.9.m1.1c">n^{\prime}\geq nq</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.1.p1.9.m1.1d">italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≥ italic_n italic_q</annotation></semantics></math>. Note that for both constant and sublinear values of <math alttext="k" class="ltx_Math" display="inline" id="S5.SS3.SSS1.1.p1.10.m2.1"><semantics id="S5.SS3.SSS1.1.p1.10.m2.1a"><mi id="S5.SS3.SSS1.1.p1.10.m2.1.1" xref="S5.SS3.SSS1.1.p1.10.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS1.1.p1.10.m2.1b"><ci id="S5.SS3.SSS1.1.p1.10.m2.1.1.cmml" xref="S5.SS3.SSS1.1.p1.10.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS1.1.p1.10.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS1.1.p1.10.m2.1d">italic_k</annotation></semantics></math>, the approximation factor is a constant. ∎</p> </div> </div> </section> <section class="ltx_subsubsection" id="S5.SS3.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">5.3.2 </span>High Perturbation Regime for Random Balanced Multipartite Graphs</h4> <div class="ltx_para" id="S5.SS3.SSS2.p1"> <p class="ltx_p" id="S5.SS3.SSS2.p1.7"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg5" title="In 5.3.2 High Perturbation Regime for Random Balanced Multipartite Graphs ‣ 5.3 Balanced Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">5</span></a> takes as input a random balanced <math alttext="k" class="ltx_Math" display="inline" id="S5.SS3.SSS2.p1.1.m1.1"><semantics id="S5.SS3.SSS2.p1.1.m1.1a"><mi id="S5.SS3.SSS2.p1.1.m1.1.1" xref="S5.SS3.SSS2.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.p1.1.m1.1b"><ci id="S5.SS3.SSS2.p1.1.m1.1.1.cmml" xref="S5.SS3.SSS2.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.p1.1.m1.1d">italic_k</annotation></semantics></math>-partite graph <math alttext="G=(\{V_{1},\cdots,V_{k}\},p)" class="ltx_Math" display="inline" id="S5.SS3.SSS2.p1.2.m2.3"><semantics id="S5.SS3.SSS2.p1.2.m2.3a"><mrow id="S5.SS3.SSS2.p1.2.m2.3.3" xref="S5.SS3.SSS2.p1.2.m2.3.3.cmml"><mi id="S5.SS3.SSS2.p1.2.m2.3.3.3" xref="S5.SS3.SSS2.p1.2.m2.3.3.3.cmml">G</mi><mo id="S5.SS3.SSS2.p1.2.m2.3.3.2" xref="S5.SS3.SSS2.p1.2.m2.3.3.2.cmml">=</mo><mrow id="S5.SS3.SSS2.p1.2.m2.3.3.1.1" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.2.cmml"><mo id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.2" stretchy="false" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.2.cmml">(</mo><mrow id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.3.cmml"><mo id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.3" stretchy="false" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.3.cmml">{</mo><msub id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.1.1" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.1.1.2" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.1.1.2.cmml">V</mi><mn id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.1.1.3" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.4" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.3.cmml">,</mo><mi id="S5.SS3.SSS2.p1.2.m2.1.1" mathvariant="normal" xref="S5.SS3.SSS2.p1.2.m2.1.1.cmml">⋯</mi><mo id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.5" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.3.cmml">,</mo><msub id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.2" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.2.cmml"><mi id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.2.2" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.2.2.cmml">V</mi><mi id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.2.3" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.2.3.cmml">k</mi></msub><mo id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.6" stretchy="false" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.3.cmml">}</mo></mrow><mo id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.3" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.2.cmml">,</mo><mi id="S5.SS3.SSS2.p1.2.m2.2.2" xref="S5.SS3.SSS2.p1.2.m2.2.2.cmml">p</mi><mo id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.4" stretchy="false" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.p1.2.m2.3b"><apply id="S5.SS3.SSS2.p1.2.m2.3.3.cmml" xref="S5.SS3.SSS2.p1.2.m2.3.3"><eq id="S5.SS3.SSS2.p1.2.m2.3.3.2.cmml" xref="S5.SS3.SSS2.p1.2.m2.3.3.2"></eq><ci id="S5.SS3.SSS2.p1.2.m2.3.3.3.cmml" xref="S5.SS3.SSS2.p1.2.m2.3.3.3">𝐺</ci><interval closure="open" id="S5.SS3.SSS2.p1.2.m2.3.3.1.2.cmml" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1"><set id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.3.cmml" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2"><apply id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.1.1">subscript</csymbol><ci id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.1.1.2">𝑉</ci><cn id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.1.1.3.cmml" type="integer" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.1.1.3">1</cn></apply><ci id="S5.SS3.SSS2.p1.2.m2.1.1.cmml" xref="S5.SS3.SSS2.p1.2.m2.1.1">⋯</ci><apply id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.2.cmml" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.2"><csymbol cd="ambiguous" id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.2.1.cmml" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.2">subscript</csymbol><ci id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.2.2.cmml" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.2.2">𝑉</ci><ci id="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.2.3.cmml" xref="S5.SS3.SSS2.p1.2.m2.3.3.1.1.1.2.2.3">𝑘</ci></apply></set><ci id="S5.SS3.SSS2.p1.2.m2.2.2.cmml" xref="S5.SS3.SSS2.p1.2.m2.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.p1.2.m2.3c">G=(\{V_{1},\cdots,V_{k}\},p)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.p1.2.m2.3d">italic_G = ( { italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } , italic_p )</annotation></semantics></math>, where <math alttext="p=c" class="ltx_Math" display="inline" id="S5.SS3.SSS2.p1.3.m3.1"><semantics id="S5.SS3.SSS2.p1.3.m3.1a"><mrow id="S5.SS3.SSS2.p1.3.m3.1.1" xref="S5.SS3.SSS2.p1.3.m3.1.1.cmml"><mi id="S5.SS3.SSS2.p1.3.m3.1.1.2" xref="S5.SS3.SSS2.p1.3.m3.1.1.2.cmml">p</mi><mo id="S5.SS3.SSS2.p1.3.m3.1.1.1" xref="S5.SS3.SSS2.p1.3.m3.1.1.1.cmml">=</mo><mi id="S5.SS3.SSS2.p1.3.m3.1.1.3" xref="S5.SS3.SSS2.p1.3.m3.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.p1.3.m3.1b"><apply id="S5.SS3.SSS2.p1.3.m3.1.1.cmml" xref="S5.SS3.SSS2.p1.3.m3.1.1"><eq id="S5.SS3.SSS2.p1.3.m3.1.1.1.cmml" xref="S5.SS3.SSS2.p1.3.m3.1.1.1"></eq><ci id="S5.SS3.SSS2.p1.3.m3.1.1.2.cmml" xref="S5.SS3.SSS2.p1.3.m3.1.1.2">𝑝</ci><ci id="S5.SS3.SSS2.p1.3.m3.1.1.3.cmml" xref="S5.SS3.SSS2.p1.3.m3.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.p1.3.m3.1c">p=c</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.p1.3.m3.1d">italic_p = italic_c</annotation></semantics></math> for some constant <math alttext="c\in(0,1)" class="ltx_Math" display="inline" id="S5.SS3.SSS2.p1.4.m4.2"><semantics id="S5.SS3.SSS2.p1.4.m4.2a"><mrow id="S5.SS3.SSS2.p1.4.m4.2.3" xref="S5.SS3.SSS2.p1.4.m4.2.3.cmml"><mi id="S5.SS3.SSS2.p1.4.m4.2.3.2" xref="S5.SS3.SSS2.p1.4.m4.2.3.2.cmml">c</mi><mo id="S5.SS3.SSS2.p1.4.m4.2.3.1" xref="S5.SS3.SSS2.p1.4.m4.2.3.1.cmml">∈</mo><mrow id="S5.SS3.SSS2.p1.4.m4.2.3.3.2" xref="S5.SS3.SSS2.p1.4.m4.2.3.3.1.cmml"><mo id="S5.SS3.SSS2.p1.4.m4.2.3.3.2.1" stretchy="false" xref="S5.SS3.SSS2.p1.4.m4.2.3.3.1.cmml">(</mo><mn id="S5.SS3.SSS2.p1.4.m4.1.1" xref="S5.SS3.SSS2.p1.4.m4.1.1.cmml">0</mn><mo id="S5.SS3.SSS2.p1.4.m4.2.3.3.2.2" xref="S5.SS3.SSS2.p1.4.m4.2.3.3.1.cmml">,</mo><mn id="S5.SS3.SSS2.p1.4.m4.2.2" xref="S5.SS3.SSS2.p1.4.m4.2.2.cmml">1</mn><mo id="S5.SS3.SSS2.p1.4.m4.2.3.3.2.3" stretchy="false" xref="S5.SS3.SSS2.p1.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.p1.4.m4.2b"><apply id="S5.SS3.SSS2.p1.4.m4.2.3.cmml" xref="S5.SS3.SSS2.p1.4.m4.2.3"><in id="S5.SS3.SSS2.p1.4.m4.2.3.1.cmml" xref="S5.SS3.SSS2.p1.4.m4.2.3.1"></in><ci id="S5.SS3.SSS2.p1.4.m4.2.3.2.cmml" xref="S5.SS3.SSS2.p1.4.m4.2.3.2">𝑐</ci><interval closure="open" id="S5.SS3.SSS2.p1.4.m4.2.3.3.1.cmml" xref="S5.SS3.SSS2.p1.4.m4.2.3.3.2"><cn id="S5.SS3.SSS2.p1.4.m4.1.1.cmml" type="integer" xref="S5.SS3.SSS2.p1.4.m4.1.1">0</cn><cn id="S5.SS3.SSS2.p1.4.m4.2.2.cmml" type="integer" xref="S5.SS3.SSS2.p1.4.m4.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.p1.4.m4.2c">c\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.p1.4.m4.2d">italic_c ∈ ( 0 , 1 )</annotation></semantics></math>. It first applies <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg3" title="In 5.3 Balanced Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">3</span></a> to <math alttext="G" class="ltx_Math" display="inline" id="S5.SS3.SSS2.p1.5.m5.1"><semantics id="S5.SS3.SSS2.p1.5.m5.1a"><mi id="S5.SS3.SSS2.p1.5.m5.1.1" xref="S5.SS3.SSS2.p1.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.p1.5.m5.1b"><ci id="S5.SS3.SSS2.p1.5.m5.1.1.cmml" xref="S5.SS3.SSS2.p1.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.p1.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.p1.5.m5.1d">italic_G</annotation></semantics></math>, producing a random Turán graph <math alttext="G^{T}" class="ltx_Math" display="inline" id="S5.SS3.SSS2.p1.6.m6.1"><semantics id="S5.SS3.SSS2.p1.6.m6.1a"><msup id="S5.SS3.SSS2.p1.6.m6.1.1" xref="S5.SS3.SSS2.p1.6.m6.1.1.cmml"><mi id="S5.SS3.SSS2.p1.6.m6.1.1.2" xref="S5.SS3.SSS2.p1.6.m6.1.1.2.cmml">G</mi><mi id="S5.SS3.SSS2.p1.6.m6.1.1.3" xref="S5.SS3.SSS2.p1.6.m6.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.p1.6.m6.1b"><apply id="S5.SS3.SSS2.p1.6.m6.1.1.cmml" xref="S5.SS3.SSS2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.p1.6.m6.1.1.1.cmml" xref="S5.SS3.SSS2.p1.6.m6.1.1">superscript</csymbol><ci id="S5.SS3.SSS2.p1.6.m6.1.1.2.cmml" xref="S5.SS3.SSS2.p1.6.m6.1.1.2">𝐺</ci><ci id="S5.SS3.SSS2.p1.6.m6.1.1.3.cmml" xref="S5.SS3.SSS2.p1.6.m6.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.p1.6.m6.1c">G^{T}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.p1.6.m6.1d">italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math>. Then, it applies <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg2" title="In 5.2.2 High Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">2</span></a> to partition the agents in <math alttext="G^{T}" class="ltx_Math" display="inline" id="S5.SS3.SSS2.p1.7.m7.1"><semantics id="S5.SS3.SSS2.p1.7.m7.1a"><msup id="S5.SS3.SSS2.p1.7.m7.1.1" xref="S5.SS3.SSS2.p1.7.m7.1.1.cmml"><mi id="S5.SS3.SSS2.p1.7.m7.1.1.2" xref="S5.SS3.SSS2.p1.7.m7.1.1.2.cmml">G</mi><mi id="S5.SS3.SSS2.p1.7.m7.1.1.3" xref="S5.SS3.SSS2.p1.7.m7.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.p1.7.m7.1b"><apply id="S5.SS3.SSS2.p1.7.m7.1.1.cmml" xref="S5.SS3.SSS2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.p1.7.m7.1.1.1.cmml" xref="S5.SS3.SSS2.p1.7.m7.1.1">superscript</csymbol><ci id="S5.SS3.SSS2.p1.7.m7.1.1.2.cmml" xref="S5.SS3.SSS2.p1.7.m7.1.1.2">𝐺</ci><ci id="S5.SS3.SSS2.p1.7.m7.1.1.3.cmml" xref="S5.SS3.SSS2.p1.7.m7.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.p1.7.m7.1c">G^{T}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.p1.7.m7.1d">italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math>, and then assigns any remaining agents to singleton coalitions.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg5"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg5.10.1.1">Algorithm 5</span> </span> Logarithmic-factor approximation algorithm for random balanced <math alttext="k" class="ltx_Math" display="inline" id="alg5.2.m1.1"><semantics id="alg5.2.m1.1b"><mi id="alg5.2.m1.1.1" xref="alg5.2.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="alg5.2.m1.1c"><ci id="alg5.2.m1.1.1.cmml" xref="alg5.2.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="alg5.2.m1.1d">k</annotation><annotation encoding="application/x-llamapun" id="alg5.2.m1.1e">italic_k</annotation></semantics></math>-partite graph</figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_left" id="alg5.7.5"><span class="ltx_text ltx_font_bold" id="alg5.7.5.1">Input:</span> <math alttext="\langle G,\varepsilon\rangle" class="ltx_Math" display="inline" id="alg5.3.1.m1.2"><semantics id="alg5.3.1.m1.2a"><mrow id="alg5.3.1.m1.2.3.2" xref="alg5.3.1.m1.2.3.1.cmml"><mo id="alg5.3.1.m1.2.3.2.1" stretchy="false" xref="alg5.3.1.m1.2.3.1.cmml">⟨</mo><mi id="alg5.3.1.m1.1.1" xref="alg5.3.1.m1.1.1.cmml">G</mi><mo id="alg5.3.1.m1.2.3.2.2" xref="alg5.3.1.m1.2.3.1.cmml">,</mo><mi id="alg5.3.1.m1.2.2" xref="alg5.3.1.m1.2.2.cmml">ε</mi><mo id="alg5.3.1.m1.2.3.2.3" stretchy="false" xref="alg5.3.1.m1.2.3.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="alg5.3.1.m1.2b"><list id="alg5.3.1.m1.2.3.1.cmml" xref="alg5.3.1.m1.2.3.2"><ci id="alg5.3.1.m1.1.1.cmml" xref="alg5.3.1.m1.1.1">𝐺</ci><ci id="alg5.3.1.m1.2.2.cmml" xref="alg5.3.1.m1.2.2">𝜀</ci></list></annotation-xml><annotation encoding="application/x-tex" id="alg5.3.1.m1.2c">\langle G,\varepsilon\rangle</annotation><annotation encoding="application/x-llamapun" id="alg5.3.1.m1.2d">⟨ italic_G , italic_ε ⟩</annotation></semantics></math> where <math alttext="G=(\{V_{1},\cdots,V_{k}\},p)" class="ltx_Math" display="inline" id="alg5.4.2.m2.3"><semantics id="alg5.4.2.m2.3a"><mrow id="alg5.4.2.m2.3.3" xref="alg5.4.2.m2.3.3.cmml"><mi id="alg5.4.2.m2.3.3.3" xref="alg5.4.2.m2.3.3.3.cmml">G</mi><mo id="alg5.4.2.m2.3.3.2" xref="alg5.4.2.m2.3.3.2.cmml">=</mo><mrow id="alg5.4.2.m2.3.3.1.1" xref="alg5.4.2.m2.3.3.1.2.cmml"><mo id="alg5.4.2.m2.3.3.1.1.2" stretchy="false" xref="alg5.4.2.m2.3.3.1.2.cmml">(</mo><mrow id="alg5.4.2.m2.3.3.1.1.1.2" xref="alg5.4.2.m2.3.3.1.1.1.3.cmml"><mo id="alg5.4.2.m2.3.3.1.1.1.2.3" stretchy="false" xref="alg5.4.2.m2.3.3.1.1.1.3.cmml">{</mo><msub id="alg5.4.2.m2.3.3.1.1.1.1.1" xref="alg5.4.2.m2.3.3.1.1.1.1.1.cmml"><mi id="alg5.4.2.m2.3.3.1.1.1.1.1.2" xref="alg5.4.2.m2.3.3.1.1.1.1.1.2.cmml">V</mi><mn id="alg5.4.2.m2.3.3.1.1.1.1.1.3" xref="alg5.4.2.m2.3.3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="alg5.4.2.m2.3.3.1.1.1.2.4" xref="alg5.4.2.m2.3.3.1.1.1.3.cmml">,</mo><mi id="alg5.4.2.m2.1.1" mathvariant="normal" xref="alg5.4.2.m2.1.1.cmml">⋯</mi><mo id="alg5.4.2.m2.3.3.1.1.1.2.5" xref="alg5.4.2.m2.3.3.1.1.1.3.cmml">,</mo><msub id="alg5.4.2.m2.3.3.1.1.1.2.2" xref="alg5.4.2.m2.3.3.1.1.1.2.2.cmml"><mi id="alg5.4.2.m2.3.3.1.1.1.2.2.2" xref="alg5.4.2.m2.3.3.1.1.1.2.2.2.cmml">V</mi><mi id="alg5.4.2.m2.3.3.1.1.1.2.2.3" xref="alg5.4.2.m2.3.3.1.1.1.2.2.3.cmml">k</mi></msub><mo id="alg5.4.2.m2.3.3.1.1.1.2.6" stretchy="false" xref="alg5.4.2.m2.3.3.1.1.1.3.cmml">}</mo></mrow><mo id="alg5.4.2.m2.3.3.1.1.3" xref="alg5.4.2.m2.3.3.1.2.cmml">,</mo><mi id="alg5.4.2.m2.2.2" xref="alg5.4.2.m2.2.2.cmml">p</mi><mo id="alg5.4.2.m2.3.3.1.1.4" stretchy="false" xref="alg5.4.2.m2.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg5.4.2.m2.3b"><apply id="alg5.4.2.m2.3.3.cmml" xref="alg5.4.2.m2.3.3"><eq id="alg5.4.2.m2.3.3.2.cmml" xref="alg5.4.2.m2.3.3.2"></eq><ci id="alg5.4.2.m2.3.3.3.cmml" xref="alg5.4.2.m2.3.3.3">𝐺</ci><interval closure="open" id="alg5.4.2.m2.3.3.1.2.cmml" xref="alg5.4.2.m2.3.3.1.1"><set id="alg5.4.2.m2.3.3.1.1.1.3.cmml" xref="alg5.4.2.m2.3.3.1.1.1.2"><apply id="alg5.4.2.m2.3.3.1.1.1.1.1.cmml" xref="alg5.4.2.m2.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="alg5.4.2.m2.3.3.1.1.1.1.1.1.cmml" xref="alg5.4.2.m2.3.3.1.1.1.1.1">subscript</csymbol><ci id="alg5.4.2.m2.3.3.1.1.1.1.1.2.cmml" xref="alg5.4.2.m2.3.3.1.1.1.1.1.2">𝑉</ci><cn id="alg5.4.2.m2.3.3.1.1.1.1.1.3.cmml" type="integer" xref="alg5.4.2.m2.3.3.1.1.1.1.1.3">1</cn></apply><ci id="alg5.4.2.m2.1.1.cmml" xref="alg5.4.2.m2.1.1">⋯</ci><apply id="alg5.4.2.m2.3.3.1.1.1.2.2.cmml" xref="alg5.4.2.m2.3.3.1.1.1.2.2"><csymbol cd="ambiguous" id="alg5.4.2.m2.3.3.1.1.1.2.2.1.cmml" xref="alg5.4.2.m2.3.3.1.1.1.2.2">subscript</csymbol><ci id="alg5.4.2.m2.3.3.1.1.1.2.2.2.cmml" xref="alg5.4.2.m2.3.3.1.1.1.2.2.2">𝑉</ci><ci id="alg5.4.2.m2.3.3.1.1.1.2.2.3.cmml" xref="alg5.4.2.m2.3.3.1.1.1.2.2.3">𝑘</ci></apply></set><ci id="alg5.4.2.m2.2.2.cmml" xref="alg5.4.2.m2.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.4.2.m2.3c">G=(\{V_{1},\cdots,V_{k}\},p)</annotation><annotation encoding="application/x-llamapun" id="alg5.4.2.m2.3d">italic_G = ( { italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } , italic_p )</annotation></semantics></math> is a random balanced <math alttext="k" class="ltx_Math" display="inline" id="alg5.5.3.m3.1"><semantics id="alg5.5.3.m3.1a"><mi id="alg5.5.3.m3.1.1" xref="alg5.5.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="alg5.5.3.m3.1b"><ci id="alg5.5.3.m3.1.1.cmml" xref="alg5.5.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="alg5.5.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="alg5.5.3.m3.1d">italic_k</annotation></semantics></math>-partite graph and <math alttext="p=c" class="ltx_Math" display="inline" id="alg5.6.4.m4.1"><semantics id="alg5.6.4.m4.1a"><mrow id="alg5.6.4.m4.1.1" xref="alg5.6.4.m4.1.1.cmml"><mi id="alg5.6.4.m4.1.1.2" xref="alg5.6.4.m4.1.1.2.cmml">p</mi><mo id="alg5.6.4.m4.1.1.1" xref="alg5.6.4.m4.1.1.1.cmml">=</mo><mi id="alg5.6.4.m4.1.1.3" xref="alg5.6.4.m4.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="alg5.6.4.m4.1b"><apply id="alg5.6.4.m4.1.1.cmml" xref="alg5.6.4.m4.1.1"><eq id="alg5.6.4.m4.1.1.1.cmml" xref="alg5.6.4.m4.1.1.1"></eq><ci id="alg5.6.4.m4.1.1.2.cmml" xref="alg5.6.4.m4.1.1.2">𝑝</ci><ci id="alg5.6.4.m4.1.1.3.cmml" xref="alg5.6.4.m4.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.6.4.m4.1c">p=c</annotation><annotation encoding="application/x-llamapun" id="alg5.6.4.m4.1d">italic_p = italic_c</annotation></semantics></math> for a constant <math alttext="c\in(0,1)" class="ltx_Math" display="inline" id="alg5.7.5.m5.2"><semantics id="alg5.7.5.m5.2a"><mrow id="alg5.7.5.m5.2.3" xref="alg5.7.5.m5.2.3.cmml"><mi id="alg5.7.5.m5.2.3.2" xref="alg5.7.5.m5.2.3.2.cmml">c</mi><mo id="alg5.7.5.m5.2.3.1" xref="alg5.7.5.m5.2.3.1.cmml">∈</mo><mrow id="alg5.7.5.m5.2.3.3.2" xref="alg5.7.5.m5.2.3.3.1.cmml"><mo id="alg5.7.5.m5.2.3.3.2.1" stretchy="false" xref="alg5.7.5.m5.2.3.3.1.cmml">(</mo><mn id="alg5.7.5.m5.1.1" xref="alg5.7.5.m5.1.1.cmml">0</mn><mo id="alg5.7.5.m5.2.3.3.2.2" xref="alg5.7.5.m5.2.3.3.1.cmml">,</mo><mn id="alg5.7.5.m5.2.2" xref="alg5.7.5.m5.2.2.cmml">1</mn><mo id="alg5.7.5.m5.2.3.3.2.3" stretchy="false" xref="alg5.7.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg5.7.5.m5.2b"><apply id="alg5.7.5.m5.2.3.cmml" xref="alg5.7.5.m5.2.3"><in id="alg5.7.5.m5.2.3.1.cmml" xref="alg5.7.5.m5.2.3.1"></in><ci id="alg5.7.5.m5.2.3.2.cmml" xref="alg5.7.5.m5.2.3.2">𝑐</ci><interval closure="open" id="alg5.7.5.m5.2.3.3.1.cmml" xref="alg5.7.5.m5.2.3.3.2"><cn id="alg5.7.5.m5.1.1.cmml" type="integer" xref="alg5.7.5.m5.1.1">0</cn><cn id="alg5.7.5.m5.2.2.cmml" type="integer" xref="alg5.7.5.m5.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.7.5.m5.2c">c\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="alg5.7.5.m5.2d">italic_c ∈ ( 0 , 1 )</annotation></semantics></math></p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_left" id="alg5.8.6"><span class="ltx_text ltx_font_bold" id="alg5.8.6.1">Output:</span> Partition <math alttext="\pi" class="ltx_Math" display="inline" id="alg5.8.6.m1.1"><semantics id="alg5.8.6.m1.1a"><mi id="alg5.8.6.m1.1.1" xref="alg5.8.6.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="alg5.8.6.m1.1b"><ci id="alg5.8.6.m1.1.1.cmml" xref="alg5.8.6.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="alg5.8.6.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="alg5.8.6.m1.1d">italic_π</annotation></semantics></math></p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <div class="ltx_listing ltx_figure_panel ltx_listing" id="alg5.11"> <div class="ltx_listingline" id="alg5.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l1.1.1.1" style="font-size:80%;">1:</span></span>Reduce to a random Turán graph <math alttext="G^{T}=(n^{\prime},k,p)" class="ltx_Math" display="inline" id="alg5.l1.m1.3"><semantics id="alg5.l1.m1.3a"><mrow id="alg5.l1.m1.3.3" xref="alg5.l1.m1.3.3.cmml"><msup id="alg5.l1.m1.3.3.3" xref="alg5.l1.m1.3.3.3.cmml"><mi id="alg5.l1.m1.3.3.3.2" xref="alg5.l1.m1.3.3.3.2.cmml">G</mi><mi id="alg5.l1.m1.3.3.3.3" xref="alg5.l1.m1.3.3.3.3.cmml">T</mi></msup><mo id="alg5.l1.m1.3.3.2" xref="alg5.l1.m1.3.3.2.cmml">=</mo><mrow id="alg5.l1.m1.3.3.1.1" xref="alg5.l1.m1.3.3.1.2.cmml"><mo id="alg5.l1.m1.3.3.1.1.2" stretchy="false" xref="alg5.l1.m1.3.3.1.2.cmml">(</mo><msup id="alg5.l1.m1.3.3.1.1.1" xref="alg5.l1.m1.3.3.1.1.1.cmml"><mi id="alg5.l1.m1.3.3.1.1.1.2" xref="alg5.l1.m1.3.3.1.1.1.2.cmml">n</mi><mo id="alg5.l1.m1.3.3.1.1.1.3" xref="alg5.l1.m1.3.3.1.1.1.3.cmml">′</mo></msup><mo id="alg5.l1.m1.3.3.1.1.3" xref="alg5.l1.m1.3.3.1.2.cmml">,</mo><mi id="alg5.l1.m1.1.1" xref="alg5.l1.m1.1.1.cmml">k</mi><mo id="alg5.l1.m1.3.3.1.1.4" xref="alg5.l1.m1.3.3.1.2.cmml">,</mo><mi id="alg5.l1.m1.2.2" xref="alg5.l1.m1.2.2.cmml">p</mi><mo id="alg5.l1.m1.3.3.1.1.5" stretchy="false" xref="alg5.l1.m1.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg5.l1.m1.3b"><apply id="alg5.l1.m1.3.3.cmml" xref="alg5.l1.m1.3.3"><eq id="alg5.l1.m1.3.3.2.cmml" xref="alg5.l1.m1.3.3.2"></eq><apply id="alg5.l1.m1.3.3.3.cmml" xref="alg5.l1.m1.3.3.3"><csymbol cd="ambiguous" id="alg5.l1.m1.3.3.3.1.cmml" xref="alg5.l1.m1.3.3.3">superscript</csymbol><ci id="alg5.l1.m1.3.3.3.2.cmml" xref="alg5.l1.m1.3.3.3.2">𝐺</ci><ci id="alg5.l1.m1.3.3.3.3.cmml" xref="alg5.l1.m1.3.3.3.3">𝑇</ci></apply><vector id="alg5.l1.m1.3.3.1.2.cmml" xref="alg5.l1.m1.3.3.1.1"><apply id="alg5.l1.m1.3.3.1.1.1.cmml" xref="alg5.l1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="alg5.l1.m1.3.3.1.1.1.1.cmml" xref="alg5.l1.m1.3.3.1.1.1">superscript</csymbol><ci id="alg5.l1.m1.3.3.1.1.1.2.cmml" xref="alg5.l1.m1.3.3.1.1.1.2">𝑛</ci><ci id="alg5.l1.m1.3.3.1.1.1.3.cmml" xref="alg5.l1.m1.3.3.1.1.1.3">′</ci></apply><ci id="alg5.l1.m1.1.1.cmml" xref="alg5.l1.m1.1.1">𝑘</ci><ci id="alg5.l1.m1.2.2.cmml" xref="alg5.l1.m1.2.2">𝑝</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l1.m1.3c">G^{T}=(n^{\prime},k,p)</annotation><annotation encoding="application/x-llamapun" id="alg5.l1.m1.3d">italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = ( italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_k , italic_p )</annotation></semantics></math> by running <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg3" title="In 5.3 Balanced Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">3</span></a> on <math alttext="\langle G,\varepsilon\rangle" class="ltx_Math" display="inline" id="alg5.l1.m2.2"><semantics id="alg5.l1.m2.2a"><mrow id="alg5.l1.m2.2.3.2" xref="alg5.l1.m2.2.3.1.cmml"><mo id="alg5.l1.m2.2.3.2.1" stretchy="false" xref="alg5.l1.m2.2.3.1.cmml">⟨</mo><mi id="alg5.l1.m2.1.1" xref="alg5.l1.m2.1.1.cmml">G</mi><mo id="alg5.l1.m2.2.3.2.2" xref="alg5.l1.m2.2.3.1.cmml">,</mo><mi id="alg5.l1.m2.2.2" xref="alg5.l1.m2.2.2.cmml">ε</mi><mo id="alg5.l1.m2.2.3.2.3" stretchy="false" xref="alg5.l1.m2.2.3.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="alg5.l1.m2.2b"><list id="alg5.l1.m2.2.3.1.cmml" xref="alg5.l1.m2.2.3.2"><ci id="alg5.l1.m2.1.1.cmml" xref="alg5.l1.m2.1.1">𝐺</ci><ci id="alg5.l1.m2.2.2.cmml" xref="alg5.l1.m2.2.2">𝜀</ci></list></annotation-xml><annotation encoding="application/x-tex" id="alg5.l1.m2.2c">\langle G,\varepsilon\rangle</annotation><annotation encoding="application/x-llamapun" id="alg5.l1.m2.2d">⟨ italic_G , italic_ε ⟩</annotation></semantics></math>, denote <math alttext="k" class="ltx_Math" display="inline" id="alg5.l1.m3.1"><semantics id="alg5.l1.m3.1a"><mi id="alg5.l1.m3.1.1" xref="alg5.l1.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="alg5.l1.m3.1b"><ci id="alg5.l1.m3.1.1.cmml" xref="alg5.l1.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="alg5.l1.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="alg5.l1.m3.1d">italic_k</annotation></semantics></math> color classes as <math alttext="V^{\prime}_{i}" class="ltx_Math" display="inline" id="alg5.l1.m4.1"><semantics id="alg5.l1.m4.1a"><msubsup id="alg5.l1.m4.1.1" xref="alg5.l1.m4.1.1.cmml"><mi id="alg5.l1.m4.1.1.2.2" xref="alg5.l1.m4.1.1.2.2.cmml">V</mi><mi id="alg5.l1.m4.1.1.3" xref="alg5.l1.m4.1.1.3.cmml">i</mi><mo id="alg5.l1.m4.1.1.2.3" xref="alg5.l1.m4.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="alg5.l1.m4.1b"><apply id="alg5.l1.m4.1.1.cmml" xref="alg5.l1.m4.1.1"><csymbol cd="ambiguous" id="alg5.l1.m4.1.1.1.cmml" xref="alg5.l1.m4.1.1">subscript</csymbol><apply id="alg5.l1.m4.1.1.2.cmml" xref="alg5.l1.m4.1.1"><csymbol cd="ambiguous" id="alg5.l1.m4.1.1.2.1.cmml" xref="alg5.l1.m4.1.1">superscript</csymbol><ci id="alg5.l1.m4.1.1.2.2.cmml" xref="alg5.l1.m4.1.1.2.2">𝑉</ci><ci id="alg5.l1.m4.1.1.2.3.cmml" xref="alg5.l1.m4.1.1.2.3">′</ci></apply><ci id="alg5.l1.m4.1.1.3.cmml" xref="alg5.l1.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l1.m4.1c">V^{\prime}_{i}</annotation><annotation encoding="application/x-llamapun" id="alg5.l1.m4.1d">italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="i\in[k]" class="ltx_Math" display="inline" id="alg5.l1.m5.1"><semantics id="alg5.l1.m5.1a"><mrow id="alg5.l1.m5.1.2" xref="alg5.l1.m5.1.2.cmml"><mi id="alg5.l1.m5.1.2.2" xref="alg5.l1.m5.1.2.2.cmml">i</mi><mo id="alg5.l1.m5.1.2.1" xref="alg5.l1.m5.1.2.1.cmml">∈</mo><mrow id="alg5.l1.m5.1.2.3.2" xref="alg5.l1.m5.1.2.3.1.cmml"><mo id="alg5.l1.m5.1.2.3.2.1" stretchy="false" xref="alg5.l1.m5.1.2.3.1.1.cmml">[</mo><mi id="alg5.l1.m5.1.1" xref="alg5.l1.m5.1.1.cmml">k</mi><mo id="alg5.l1.m5.1.2.3.2.2" stretchy="false" xref="alg5.l1.m5.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg5.l1.m5.1b"><apply id="alg5.l1.m5.1.2.cmml" xref="alg5.l1.m5.1.2"><in id="alg5.l1.m5.1.2.1.cmml" xref="alg5.l1.m5.1.2.1"></in><ci id="alg5.l1.m5.1.2.2.cmml" xref="alg5.l1.m5.1.2.2">𝑖</ci><apply id="alg5.l1.m5.1.2.3.1.cmml" xref="alg5.l1.m5.1.2.3.2"><csymbol cd="latexml" id="alg5.l1.m5.1.2.3.1.1.cmml" xref="alg5.l1.m5.1.2.3.2.1">delimited-[]</csymbol><ci id="alg5.l1.m5.1.1.cmml" xref="alg5.l1.m5.1.1">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l1.m5.1c">i\in[k]</annotation><annotation encoding="application/x-llamapun" id="alg5.l1.m5.1d">italic_i ∈ [ italic_k ]</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg5.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l2.1.1.1" style="font-size:80%;">2:</span></span>Let <math alttext="\pi" class="ltx_Math" display="inline" id="alg5.l2.m1.1"><semantics id="alg5.l2.m1.1a"><mi id="alg5.l2.m1.1.1" xref="alg5.l2.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="alg5.l2.m1.1b"><ci id="alg5.l2.m1.1.1.cmml" xref="alg5.l2.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="alg5.l2.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="alg5.l2.m1.1d">italic_π</annotation></semantics></math> be the partition on vertices of <math alttext="G^{T}" class="ltx_Math" display="inline" id="alg5.l2.m2.1"><semantics id="alg5.l2.m2.1a"><msup id="alg5.l2.m2.1.1" xref="alg5.l2.m2.1.1.cmml"><mi id="alg5.l2.m2.1.1.2" xref="alg5.l2.m2.1.1.2.cmml">G</mi><mi id="alg5.l2.m2.1.1.3" xref="alg5.l2.m2.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="alg5.l2.m2.1b"><apply id="alg5.l2.m2.1.1.cmml" xref="alg5.l2.m2.1.1"><csymbol cd="ambiguous" id="alg5.l2.m2.1.1.1.cmml" xref="alg5.l2.m2.1.1">superscript</csymbol><ci id="alg5.l2.m2.1.1.2.cmml" xref="alg5.l2.m2.1.1.2">𝐺</ci><ci id="alg5.l2.m2.1.1.3.cmml" xref="alg5.l2.m2.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l2.m2.1c">G^{T}</annotation><annotation encoding="application/x-llamapun" id="alg5.l2.m2.1d">italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> after running <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg2" title="In 5.2.2 High Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">2</span></a> on <math alttext="(G^{T},\varepsilon)" class="ltx_Math" display="inline" id="alg5.l2.m3.2"><semantics id="alg5.l2.m3.2a"><mrow id="alg5.l2.m3.2.2.1" xref="alg5.l2.m3.2.2.2.cmml"><mo id="alg5.l2.m3.2.2.1.2" stretchy="false" xref="alg5.l2.m3.2.2.2.cmml">(</mo><msup id="alg5.l2.m3.2.2.1.1" xref="alg5.l2.m3.2.2.1.1.cmml"><mi id="alg5.l2.m3.2.2.1.1.2" xref="alg5.l2.m3.2.2.1.1.2.cmml">G</mi><mi id="alg5.l2.m3.2.2.1.1.3" xref="alg5.l2.m3.2.2.1.1.3.cmml">T</mi></msup><mo id="alg5.l2.m3.2.2.1.3" xref="alg5.l2.m3.2.2.2.cmml">,</mo><mi id="alg5.l2.m3.1.1" xref="alg5.l2.m3.1.1.cmml">ε</mi><mo id="alg5.l2.m3.2.2.1.4" stretchy="false" xref="alg5.l2.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="alg5.l2.m3.2b"><interval closure="open" id="alg5.l2.m3.2.2.2.cmml" xref="alg5.l2.m3.2.2.1"><apply id="alg5.l2.m3.2.2.1.1.cmml" xref="alg5.l2.m3.2.2.1.1"><csymbol cd="ambiguous" id="alg5.l2.m3.2.2.1.1.1.cmml" xref="alg5.l2.m3.2.2.1.1">superscript</csymbol><ci id="alg5.l2.m3.2.2.1.1.2.cmml" xref="alg5.l2.m3.2.2.1.1.2">𝐺</ci><ci id="alg5.l2.m3.2.2.1.1.3.cmml" xref="alg5.l2.m3.2.2.1.1.3">𝑇</ci></apply><ci id="alg5.l2.m3.1.1.cmml" xref="alg5.l2.m3.1.1">𝜀</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="alg5.l2.m3.2c">(G^{T},\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="alg5.l2.m3.2d">( italic_G start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT , italic_ε )</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg5.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l3.1.1.1" style="font-size:80%;">3:</span></span><span class="ltx_text ltx_font_bold" id="alg5.l3.2">for all</span> <math alttext="v\in\bigcup_{i\in[k]}V_{i}\setminus\left\{\bigcup_{i\in[k]}V^{\prime}_{i}\right\}" class="ltx_Math" display="inline" id="alg5.l3.m1.3"><semantics id="alg5.l3.m1.3a"><mrow id="alg5.l3.m1.3.3" xref="alg5.l3.m1.3.3.cmml"><mi id="alg5.l3.m1.3.3.3" xref="alg5.l3.m1.3.3.3.cmml">v</mi><mo id="alg5.l3.m1.3.3.2" rspace="0.111em" xref="alg5.l3.m1.3.3.2.cmml">∈</mo><mrow id="alg5.l3.m1.3.3.1" xref="alg5.l3.m1.3.3.1.cmml"><mrow id="alg5.l3.m1.3.3.1.3" xref="alg5.l3.m1.3.3.1.3.cmml"><msub id="alg5.l3.m1.3.3.1.3.1" xref="alg5.l3.m1.3.3.1.3.1.cmml"><mo id="alg5.l3.m1.3.3.1.3.1.2" xref="alg5.l3.m1.3.3.1.3.1.2.cmml">⋃</mo><mrow id="alg5.l3.m1.1.1.1" xref="alg5.l3.m1.1.1.1.cmml"><mi id="alg5.l3.m1.1.1.1.3" xref="alg5.l3.m1.1.1.1.3.cmml">i</mi><mo id="alg5.l3.m1.1.1.1.2" xref="alg5.l3.m1.1.1.1.2.cmml">∈</mo><mrow id="alg5.l3.m1.1.1.1.4.2" xref="alg5.l3.m1.1.1.1.4.1.cmml"><mo id="alg5.l3.m1.1.1.1.4.2.1" stretchy="false" xref="alg5.l3.m1.1.1.1.4.1.1.cmml">[</mo><mi id="alg5.l3.m1.1.1.1.1" xref="alg5.l3.m1.1.1.1.1.cmml">k</mi><mo id="alg5.l3.m1.1.1.1.4.2.2" stretchy="false" xref="alg5.l3.m1.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><msub id="alg5.l3.m1.3.3.1.3.2" xref="alg5.l3.m1.3.3.1.3.2.cmml"><mi id="alg5.l3.m1.3.3.1.3.2.2" xref="alg5.l3.m1.3.3.1.3.2.2.cmml">V</mi><mi id="alg5.l3.m1.3.3.1.3.2.3" xref="alg5.l3.m1.3.3.1.3.2.3.cmml">i</mi></msub></mrow><mo id="alg5.l3.m1.3.3.1.2" xref="alg5.l3.m1.3.3.1.2.cmml">∖</mo><mrow id="alg5.l3.m1.3.3.1.1.1" xref="alg5.l3.m1.3.3.1.1.2.cmml"><mo id="alg5.l3.m1.3.3.1.1.1.2" xref="alg5.l3.m1.3.3.1.1.2.cmml">{</mo><mrow id="alg5.l3.m1.3.3.1.1.1.1" xref="alg5.l3.m1.3.3.1.1.1.1.cmml"><msub id="alg5.l3.m1.3.3.1.1.1.1.1" xref="alg5.l3.m1.3.3.1.1.1.1.1.cmml"><mo id="alg5.l3.m1.3.3.1.1.1.1.1.2" lspace="0em" xref="alg5.l3.m1.3.3.1.1.1.1.1.2.cmml">⋃</mo><mrow id="alg5.l3.m1.2.2.1" xref="alg5.l3.m1.2.2.1.cmml"><mi id="alg5.l3.m1.2.2.1.3" xref="alg5.l3.m1.2.2.1.3.cmml">i</mi><mo id="alg5.l3.m1.2.2.1.2" xref="alg5.l3.m1.2.2.1.2.cmml">∈</mo><mrow id="alg5.l3.m1.2.2.1.4.2" xref="alg5.l3.m1.2.2.1.4.1.cmml"><mo id="alg5.l3.m1.2.2.1.4.2.1" stretchy="false" xref="alg5.l3.m1.2.2.1.4.1.1.cmml">[</mo><mi id="alg5.l3.m1.2.2.1.1" xref="alg5.l3.m1.2.2.1.1.cmml">k</mi><mo id="alg5.l3.m1.2.2.1.4.2.2" stretchy="false" xref="alg5.l3.m1.2.2.1.4.1.1.cmml">]</mo></mrow></mrow></msub><msubsup id="alg5.l3.m1.3.3.1.1.1.1.2" xref="alg5.l3.m1.3.3.1.1.1.1.2.cmml"><mi id="alg5.l3.m1.3.3.1.1.1.1.2.2.2" xref="alg5.l3.m1.3.3.1.1.1.1.2.2.2.cmml">V</mi><mi id="alg5.l3.m1.3.3.1.1.1.1.2.3" xref="alg5.l3.m1.3.3.1.1.1.1.2.3.cmml">i</mi><mo id="alg5.l3.m1.3.3.1.1.1.1.2.2.3" xref="alg5.l3.m1.3.3.1.1.1.1.2.2.3.cmml">′</mo></msubsup></mrow><mo id="alg5.l3.m1.3.3.1.1.1.3" xref="alg5.l3.m1.3.3.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg5.l3.m1.3b"><apply id="alg5.l3.m1.3.3.cmml" xref="alg5.l3.m1.3.3"><in id="alg5.l3.m1.3.3.2.cmml" xref="alg5.l3.m1.3.3.2"></in><ci id="alg5.l3.m1.3.3.3.cmml" xref="alg5.l3.m1.3.3.3">𝑣</ci><apply id="alg5.l3.m1.3.3.1.cmml" xref="alg5.l3.m1.3.3.1"><setdiff id="alg5.l3.m1.3.3.1.2.cmml" xref="alg5.l3.m1.3.3.1.2"></setdiff><apply id="alg5.l3.m1.3.3.1.3.cmml" xref="alg5.l3.m1.3.3.1.3"><apply id="alg5.l3.m1.3.3.1.3.1.cmml" xref="alg5.l3.m1.3.3.1.3.1"><csymbol cd="ambiguous" id="alg5.l3.m1.3.3.1.3.1.1.cmml" xref="alg5.l3.m1.3.3.1.3.1">subscript</csymbol><union id="alg5.l3.m1.3.3.1.3.1.2.cmml" xref="alg5.l3.m1.3.3.1.3.1.2"></union><apply id="alg5.l3.m1.1.1.1.cmml" xref="alg5.l3.m1.1.1.1"><in id="alg5.l3.m1.1.1.1.2.cmml" xref="alg5.l3.m1.1.1.1.2"></in><ci id="alg5.l3.m1.1.1.1.3.cmml" xref="alg5.l3.m1.1.1.1.3">𝑖</ci><apply id="alg5.l3.m1.1.1.1.4.1.cmml" xref="alg5.l3.m1.1.1.1.4.2"><csymbol cd="latexml" id="alg5.l3.m1.1.1.1.4.1.1.cmml" xref="alg5.l3.m1.1.1.1.4.2.1">delimited-[]</csymbol><ci id="alg5.l3.m1.1.1.1.1.cmml" xref="alg5.l3.m1.1.1.1.1">𝑘</ci></apply></apply></apply><apply id="alg5.l3.m1.3.3.1.3.2.cmml" xref="alg5.l3.m1.3.3.1.3.2"><csymbol cd="ambiguous" id="alg5.l3.m1.3.3.1.3.2.1.cmml" xref="alg5.l3.m1.3.3.1.3.2">subscript</csymbol><ci id="alg5.l3.m1.3.3.1.3.2.2.cmml" xref="alg5.l3.m1.3.3.1.3.2.2">𝑉</ci><ci id="alg5.l3.m1.3.3.1.3.2.3.cmml" xref="alg5.l3.m1.3.3.1.3.2.3">𝑖</ci></apply></apply><set id="alg5.l3.m1.3.3.1.1.2.cmml" xref="alg5.l3.m1.3.3.1.1.1"><apply id="alg5.l3.m1.3.3.1.1.1.1.cmml" xref="alg5.l3.m1.3.3.1.1.1.1"><apply id="alg5.l3.m1.3.3.1.1.1.1.1.cmml" xref="alg5.l3.m1.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="alg5.l3.m1.3.3.1.1.1.1.1.1.cmml" xref="alg5.l3.m1.3.3.1.1.1.1.1">subscript</csymbol><union id="alg5.l3.m1.3.3.1.1.1.1.1.2.cmml" xref="alg5.l3.m1.3.3.1.1.1.1.1.2"></union><apply id="alg5.l3.m1.2.2.1.cmml" xref="alg5.l3.m1.2.2.1"><in id="alg5.l3.m1.2.2.1.2.cmml" xref="alg5.l3.m1.2.2.1.2"></in><ci id="alg5.l3.m1.2.2.1.3.cmml" xref="alg5.l3.m1.2.2.1.3">𝑖</ci><apply id="alg5.l3.m1.2.2.1.4.1.cmml" xref="alg5.l3.m1.2.2.1.4.2"><csymbol cd="latexml" id="alg5.l3.m1.2.2.1.4.1.1.cmml" xref="alg5.l3.m1.2.2.1.4.2.1">delimited-[]</csymbol><ci id="alg5.l3.m1.2.2.1.1.cmml" xref="alg5.l3.m1.2.2.1.1">𝑘</ci></apply></apply></apply><apply id="alg5.l3.m1.3.3.1.1.1.1.2.cmml" xref="alg5.l3.m1.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="alg5.l3.m1.3.3.1.1.1.1.2.1.cmml" xref="alg5.l3.m1.3.3.1.1.1.1.2">subscript</csymbol><apply id="alg5.l3.m1.3.3.1.1.1.1.2.2.cmml" xref="alg5.l3.m1.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="alg5.l3.m1.3.3.1.1.1.1.2.2.1.cmml" xref="alg5.l3.m1.3.3.1.1.1.1.2">superscript</csymbol><ci id="alg5.l3.m1.3.3.1.1.1.1.2.2.2.cmml" xref="alg5.l3.m1.3.3.1.1.1.1.2.2.2">𝑉</ci><ci id="alg5.l3.m1.3.3.1.1.1.1.2.2.3.cmml" xref="alg5.l3.m1.3.3.1.1.1.1.2.2.3">′</ci></apply><ci id="alg5.l3.m1.3.3.1.1.1.1.2.3.cmml" xref="alg5.l3.m1.3.3.1.1.1.1.2.3">𝑖</ci></apply></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l3.m1.3c">v\in\bigcup_{i\in[k]}V_{i}\setminus\left\{\bigcup_{i\in[k]}V^{\prime}_{i}\right\}</annotation><annotation encoding="application/x-llamapun" id="alg5.l3.m1.3d">italic_v ∈ ⋃ start_POSTSUBSCRIPT italic_i ∈ [ italic_k ] end_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∖ { ⋃ start_POSTSUBSCRIPT italic_i ∈ [ italic_k ] end_POSTSUBSCRIPT italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT }</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg5.l3.3">do</span> </div> <div class="ltx_listingline" id="alg5.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l4.1.1.1" style="font-size:80%;">4:</span></span> <math alttext="\pi\leftarrow\pi\cup\{v\}" class="ltx_Math" display="inline" id="alg5.l4.m1.1"><semantics id="alg5.l4.m1.1a"><mrow id="alg5.l4.m1.1.2" xref="alg5.l4.m1.1.2.cmml"><mi id="alg5.l4.m1.1.2.2" xref="alg5.l4.m1.1.2.2.cmml">π</mi><mo id="alg5.l4.m1.1.2.1" stretchy="false" xref="alg5.l4.m1.1.2.1.cmml">←</mo><mrow id="alg5.l4.m1.1.2.3" xref="alg5.l4.m1.1.2.3.cmml"><mi id="alg5.l4.m1.1.2.3.2" xref="alg5.l4.m1.1.2.3.2.cmml">π</mi><mo id="alg5.l4.m1.1.2.3.1" xref="alg5.l4.m1.1.2.3.1.cmml">∪</mo><mrow id="alg5.l4.m1.1.2.3.3.2" xref="alg5.l4.m1.1.2.3.3.1.cmml"><mo id="alg5.l4.m1.1.2.3.3.2.1" stretchy="false" xref="alg5.l4.m1.1.2.3.3.1.cmml">{</mo><mi id="alg5.l4.m1.1.1" xref="alg5.l4.m1.1.1.cmml">v</mi><mo id="alg5.l4.m1.1.2.3.3.2.2" stretchy="false" xref="alg5.l4.m1.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg5.l4.m1.1b"><apply id="alg5.l4.m1.1.2.cmml" xref="alg5.l4.m1.1.2"><ci id="alg5.l4.m1.1.2.1.cmml" xref="alg5.l4.m1.1.2.1">←</ci><ci id="alg5.l4.m1.1.2.2.cmml" xref="alg5.l4.m1.1.2.2">𝜋</ci><apply id="alg5.l4.m1.1.2.3.cmml" xref="alg5.l4.m1.1.2.3"><union id="alg5.l4.m1.1.2.3.1.cmml" xref="alg5.l4.m1.1.2.3.1"></union><ci id="alg5.l4.m1.1.2.3.2.cmml" xref="alg5.l4.m1.1.2.3.2">𝜋</ci><set id="alg5.l4.m1.1.2.3.3.1.cmml" xref="alg5.l4.m1.1.2.3.3.2"><ci id="alg5.l4.m1.1.1.cmml" xref="alg5.l4.m1.1.1">𝑣</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l4.m1.1c">\pi\leftarrow\pi\cup\{v\}</annotation><annotation encoding="application/x-llamapun" id="alg5.l4.m1.1d">italic_π ← italic_π ∪ { italic_v }</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg5.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l5.1.1.1" style="font-size:80%;">5:</span></span><span class="ltx_text ltx_font_bold" id="alg5.l5.2">return</span> <math alttext="\pi" class="ltx_Math" display="inline" id="alg5.l5.m1.1"><semantics id="alg5.l5.m1.1a"><mi id="alg5.l5.m1.1.1" xref="alg5.l5.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="alg5.l5.m1.1b"><ci id="alg5.l5.m1.1.1.cmml" xref="alg5.l5.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="alg5.l5.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="alg5.l5.m1.1d">italic_π</annotation></semantics></math> </div> </div> </div> </div> </figure> <div class="ltx_theorem ltx_theorem_theorem" id="S5.Thmtheorem13"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem13.1.1.1">Theorem 5.13</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem13.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem13.p1"> <p class="ltx_p" id="S5.Thmtheorem13.p1.6"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem13.p1.6.6">Consider aversion-to-enemies games given by random balanced <math alttext="k" class="ltx_Math" display="inline" id="S5.Thmtheorem13.p1.1.1.m1.1"><semantics id="S5.Thmtheorem13.p1.1.1.m1.1a"><mi id="S5.Thmtheorem13.p1.1.1.m1.1.1" xref="S5.Thmtheorem13.p1.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem13.p1.1.1.m1.1b"><ci id="S5.Thmtheorem13.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem13.p1.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem13.p1.1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem13.p1.1.1.m1.1d">italic_k</annotation></semantics></math>-partite graphs <math alttext="G=(\{V_{1}\cdots,V_{k}\},p)" class="ltx_Math" display="inline" id="S5.Thmtheorem13.p1.2.2.m2.2"><semantics id="S5.Thmtheorem13.p1.2.2.m2.2a"><mrow id="S5.Thmtheorem13.p1.2.2.m2.2.2" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.cmml"><mi id="S5.Thmtheorem13.p1.2.2.m2.2.2.3" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.3.cmml">G</mi><mo id="S5.Thmtheorem13.p1.2.2.m2.2.2.2" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.2.cmml">=</mo><mrow id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.2.cmml"><mo id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.2" stretchy="false" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.2.cmml">(</mo><mrow id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.3.cmml"><mo id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.3" stretchy="false" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.3.cmml">{</mo><mrow id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.cmml"><msub id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.2" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.2.cmml"><mi id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.2.2" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.2.2.cmml">V</mi><mn id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.2.3" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.1" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.3" mathvariant="normal" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.3.cmml">⋯</mi></mrow><mo id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.4" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.3.cmml">,</mo><msub id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.2" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.2.cmml"><mi id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.2.2" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.2.2.cmml">V</mi><mi id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.2.3" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.2.3.cmml">k</mi></msub><mo id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.5" stretchy="false" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.3.cmml">}</mo></mrow><mo id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.3" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.2.cmml">,</mo><mi id="S5.Thmtheorem13.p1.2.2.m2.1.1" xref="S5.Thmtheorem13.p1.2.2.m2.1.1.cmml">p</mi><mo id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.4" stretchy="false" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem13.p1.2.2.m2.2b"><apply id="S5.Thmtheorem13.p1.2.2.m2.2.2.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.2.2"><eq id="S5.Thmtheorem13.p1.2.2.m2.2.2.2.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.2"></eq><ci id="S5.Thmtheorem13.p1.2.2.m2.2.2.3.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.3">𝐺</ci><interval closure="open" id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.2.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1"><set id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.3.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2"><apply id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1"><times id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.1"></times><apply id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.2.1.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.2">subscript</csymbol><ci id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.2.2.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.2.2">𝑉</ci><cn id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.2.3.cmml" type="integer" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.2.3">1</cn></apply><ci id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.1.1.3">⋯</ci></apply><apply id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.2.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.2.1.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.2">subscript</csymbol><ci id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.2.2.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.2.2">𝑉</ci><ci id="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.2.3.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.2.2.1.1.1.2.2.3">𝑘</ci></apply></set><ci id="S5.Thmtheorem13.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem13.p1.2.2.m2.1.1">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem13.p1.2.2.m2.2c">G=(\{V_{1}\cdots,V_{k}\},p)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem13.p1.2.2.m2.2d">italic_G = ( { italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ , italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } , italic_p )</annotation></semantics></math>, where <math alttext="p=c" class="ltx_Math" display="inline" id="S5.Thmtheorem13.p1.3.3.m3.1"><semantics id="S5.Thmtheorem13.p1.3.3.m3.1a"><mrow id="S5.Thmtheorem13.p1.3.3.m3.1.1" xref="S5.Thmtheorem13.p1.3.3.m3.1.1.cmml"><mi id="S5.Thmtheorem13.p1.3.3.m3.1.1.2" xref="S5.Thmtheorem13.p1.3.3.m3.1.1.2.cmml">p</mi><mo id="S5.Thmtheorem13.p1.3.3.m3.1.1.1" xref="S5.Thmtheorem13.p1.3.3.m3.1.1.1.cmml">=</mo><mi id="S5.Thmtheorem13.p1.3.3.m3.1.1.3" xref="S5.Thmtheorem13.p1.3.3.m3.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem13.p1.3.3.m3.1b"><apply id="S5.Thmtheorem13.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem13.p1.3.3.m3.1.1"><eq id="S5.Thmtheorem13.p1.3.3.m3.1.1.1.cmml" xref="S5.Thmtheorem13.p1.3.3.m3.1.1.1"></eq><ci id="S5.Thmtheorem13.p1.3.3.m3.1.1.2.cmml" xref="S5.Thmtheorem13.p1.3.3.m3.1.1.2">𝑝</ci><ci id="S5.Thmtheorem13.p1.3.3.m3.1.1.3.cmml" xref="S5.Thmtheorem13.p1.3.3.m3.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem13.p1.3.3.m3.1c">p=c</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem13.p1.3.3.m3.1d">italic_p = italic_c</annotation></semantics></math> for some constant <math alttext="c\in(0,1)" class="ltx_Math" display="inline" id="S5.Thmtheorem13.p1.4.4.m4.2"><semantics id="S5.Thmtheorem13.p1.4.4.m4.2a"><mrow id="S5.Thmtheorem13.p1.4.4.m4.2.3" xref="S5.Thmtheorem13.p1.4.4.m4.2.3.cmml"><mi id="S5.Thmtheorem13.p1.4.4.m4.2.3.2" xref="S5.Thmtheorem13.p1.4.4.m4.2.3.2.cmml">c</mi><mo id="S5.Thmtheorem13.p1.4.4.m4.2.3.1" xref="S5.Thmtheorem13.p1.4.4.m4.2.3.1.cmml">∈</mo><mrow id="S5.Thmtheorem13.p1.4.4.m4.2.3.3.2" xref="S5.Thmtheorem13.p1.4.4.m4.2.3.3.1.cmml"><mo id="S5.Thmtheorem13.p1.4.4.m4.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem13.p1.4.4.m4.2.3.3.1.cmml">(</mo><mn id="S5.Thmtheorem13.p1.4.4.m4.1.1" xref="S5.Thmtheorem13.p1.4.4.m4.1.1.cmml">0</mn><mo id="S5.Thmtheorem13.p1.4.4.m4.2.3.3.2.2" xref="S5.Thmtheorem13.p1.4.4.m4.2.3.3.1.cmml">,</mo><mn id="S5.Thmtheorem13.p1.4.4.m4.2.2" xref="S5.Thmtheorem13.p1.4.4.m4.2.2.cmml">1</mn><mo id="S5.Thmtheorem13.p1.4.4.m4.2.3.3.2.3" stretchy="false" xref="S5.Thmtheorem13.p1.4.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem13.p1.4.4.m4.2b"><apply id="S5.Thmtheorem13.p1.4.4.m4.2.3.cmml" xref="S5.Thmtheorem13.p1.4.4.m4.2.3"><in id="S5.Thmtheorem13.p1.4.4.m4.2.3.1.cmml" xref="S5.Thmtheorem13.p1.4.4.m4.2.3.1"></in><ci id="S5.Thmtheorem13.p1.4.4.m4.2.3.2.cmml" xref="S5.Thmtheorem13.p1.4.4.m4.2.3.2">𝑐</ci><interval closure="open" id="S5.Thmtheorem13.p1.4.4.m4.2.3.3.1.cmml" xref="S5.Thmtheorem13.p1.4.4.m4.2.3.3.2"><cn id="S5.Thmtheorem13.p1.4.4.m4.1.1.cmml" type="integer" xref="S5.Thmtheorem13.p1.4.4.m4.1.1">0</cn><cn id="S5.Thmtheorem13.p1.4.4.m4.2.2.cmml" type="integer" xref="S5.Thmtheorem13.p1.4.4.m4.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem13.p1.4.4.m4.2c">c\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem13.p1.4.4.m4.2d">italic_c ∈ ( 0 , 1 )</annotation></semantics></math>. Then there exists a polynomial-time algorithm that returns a partition that provides a <math alttext="\mathcal{O}(\log n)" class="ltx_Math" display="inline" id="S5.Thmtheorem13.p1.5.5.m5.1"><semantics id="S5.Thmtheorem13.p1.5.5.m5.1a"><mrow id="S5.Thmtheorem13.p1.5.5.m5.1.1" xref="S5.Thmtheorem13.p1.5.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem13.p1.5.5.m5.1.1.3" xref="S5.Thmtheorem13.p1.5.5.m5.1.1.3.cmml">𝒪</mi><mo id="S5.Thmtheorem13.p1.5.5.m5.1.1.2" xref="S5.Thmtheorem13.p1.5.5.m5.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1" xref="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1" xref="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1.1" xref="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1.1.cmml">log</mi><mo id="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1a" lspace="0.167em" xref="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1.cmml">⁡</mo><mi id="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1.2" xref="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem13.p1.5.5.m5.1b"><apply id="S5.Thmtheorem13.p1.5.5.m5.1.1.cmml" xref="S5.Thmtheorem13.p1.5.5.m5.1.1"><times id="S5.Thmtheorem13.p1.5.5.m5.1.1.2.cmml" xref="S5.Thmtheorem13.p1.5.5.m5.1.1.2"></times><ci id="S5.Thmtheorem13.p1.5.5.m5.1.1.3.cmml" xref="S5.Thmtheorem13.p1.5.5.m5.1.1.3">𝒪</ci><apply id="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1.cmml" xref="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1"><log id="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1.1"></log><ci id="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem13.p1.5.5.m5.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem13.p1.5.5.m5.1c">\mathcal{O}(\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem13.p1.5.5.m5.1d">caligraphic_O ( roman_log italic_n )</annotation></semantics></math>-approximation of the maximum welfare with probability <math alttext="1-ne^{-\Theta(\frac{n}{k})}-(\frac{ce}{2\log_{e}n})^{\frac{2}{c}\log_{e}n}" class="ltx_Math" display="inline" id="S5.Thmtheorem13.p1.6.6.m6.2"><semantics id="S5.Thmtheorem13.p1.6.6.m6.2a"><mrow id="S5.Thmtheorem13.p1.6.6.m6.2.3" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.cmml"><mn id="S5.Thmtheorem13.p1.6.6.m6.2.3.2" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.2.cmml">1</mn><mo id="S5.Thmtheorem13.p1.6.6.m6.2.3.1" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.1.cmml">−</mo><mrow id="S5.Thmtheorem13.p1.6.6.m6.2.3.3" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.3.cmml"><mi id="S5.Thmtheorem13.p1.6.6.m6.2.3.3.2" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.3.2.cmml">n</mi><mo id="S5.Thmtheorem13.p1.6.6.m6.2.3.3.1" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.3.1.cmml">⁢</mo><msup id="S5.Thmtheorem13.p1.6.6.m6.2.3.3.3" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.3.3.cmml"><mi id="S5.Thmtheorem13.p1.6.6.m6.2.3.3.3.2" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.3.3.2.cmml">e</mi><mrow id="S5.Thmtheorem13.p1.6.6.m6.1.1.1" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.cmml"><mo id="S5.Thmtheorem13.p1.6.6.m6.1.1.1a" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.cmml">−</mo><mrow id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3.cmml"><mi id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3.2" mathvariant="normal" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3.2.cmml">Θ</mi><mo id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3.1" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3.3.2" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.1.cmml"><mo id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3.3.2.1" stretchy="false" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.1.cmml">(</mo><mfrac id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.1" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.1.cmml"><mi id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.1.2" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.1.2.cmml">n</mi><mi id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.1.3" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.1.3.cmml">k</mi></mfrac><mo id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3.3.2.2" stretchy="false" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><mo id="S5.Thmtheorem13.p1.6.6.m6.2.3.1a" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.1.cmml">−</mo><msup id="S5.Thmtheorem13.p1.6.6.m6.2.3.4" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.cmml"><mrow id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.2.2" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.cmml"><mo id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.2.2.1" stretchy="false" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.cmml">(</mo><mfrac id="S5.Thmtheorem13.p1.6.6.m6.2.2" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.cmml"><mrow id="S5.Thmtheorem13.p1.6.6.m6.2.2.2" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.2.cmml"><mi id="S5.Thmtheorem13.p1.6.6.m6.2.2.2.2" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.2.2.cmml">c</mi><mo id="S5.Thmtheorem13.p1.6.6.m6.2.2.2.1" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem13.p1.6.6.m6.2.2.2.3" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.2.3.cmml">e</mi></mrow><mrow id="S5.Thmtheorem13.p1.6.6.m6.2.2.3" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.cmml"><mn id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.2" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.2.cmml">2</mn><mo id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.1" lspace="0.167em" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.cmml"><msub id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.1" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.1.cmml"><mi id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.1.2" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.1.2.cmml">log</mi><mi id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.1.3" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.1.3.cmml">e</mi></msub><mo id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3a" lspace="0.167em" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.cmml">⁡</mo><mi id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.2" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.2.cmml">n</mi></mrow></mrow></mfrac><mo id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.2.2.2" stretchy="false" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.cmml">)</mo></mrow><mrow id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.cmml"><mfrac id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.2" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.2.cmml"><mn id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.2.2" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.2.2.cmml">2</mn><mi id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.2.3" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.2.3.cmml">c</mi></mfrac><mo id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.1" lspace="0.167em" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.cmml"><msub id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.1" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.1.cmml"><mi id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.1.2" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.1.2.cmml">log</mi><mi id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.1.3" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.1.3.cmml">e</mi></msub><mo id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3a" lspace="0.167em" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.cmml">⁡</mo><mi id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.2" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.2.cmml">n</mi></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem13.p1.6.6.m6.2b"><apply id="S5.Thmtheorem13.p1.6.6.m6.2.3.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3"><minus id="S5.Thmtheorem13.p1.6.6.m6.2.3.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.1"></minus><cn id="S5.Thmtheorem13.p1.6.6.m6.2.3.2.cmml" type="integer" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.2">1</cn><apply id="S5.Thmtheorem13.p1.6.6.m6.2.3.3.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.3"><times id="S5.Thmtheorem13.p1.6.6.m6.2.3.3.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.3.1"></times><ci id="S5.Thmtheorem13.p1.6.6.m6.2.3.3.2.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.3.2">𝑛</ci><apply id="S5.Thmtheorem13.p1.6.6.m6.2.3.3.3.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem13.p1.6.6.m6.2.3.3.3.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.3.3">superscript</csymbol><ci id="S5.Thmtheorem13.p1.6.6.m6.2.3.3.3.2.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.3.3.2">𝑒</ci><apply id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1"><minus id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.2.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1"></minus><apply id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3"><times id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3.1"></times><ci id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3.2.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3.2">Θ</ci><apply id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3.3.2"><divide id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.1.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.3.3.2"></divide><ci id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.1.2.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.1.2">𝑛</ci><ci id="S5.Thmtheorem13.p1.6.6.m6.1.1.1.1.3.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply><apply id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4"><csymbol cd="ambiguous" id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4">superscript</csymbol><apply id="S5.Thmtheorem13.p1.6.6.m6.2.2.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.2.2"><divide id="S5.Thmtheorem13.p1.6.6.m6.2.2.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.2.2"></divide><apply id="S5.Thmtheorem13.p1.6.6.m6.2.2.2.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.2"><times id="S5.Thmtheorem13.p1.6.6.m6.2.2.2.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.2.1"></times><ci id="S5.Thmtheorem13.p1.6.6.m6.2.2.2.2.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.2.2">𝑐</ci><ci id="S5.Thmtheorem13.p1.6.6.m6.2.2.2.3.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.2.3">𝑒</ci></apply><apply id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3"><times id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.1"></times><cn id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.2.cmml" type="integer" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.2">2</cn><apply id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3"><apply id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.1"><csymbol cd="ambiguous" id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.1.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.1">subscript</csymbol><log id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.1.2.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.1.2"></log><ci id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.1.3.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.1.3">𝑒</ci></apply><ci id="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.2.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.2.3.3.2">𝑛</ci></apply></apply></apply><apply id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3"><times id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.1"></times><apply id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.2.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.2"><divide id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.2.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.2"></divide><cn id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.2.2.cmml" type="integer" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.2.2">2</cn><ci id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.2.3.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.2.3">𝑐</ci></apply><apply id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3"><apply id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.1"><csymbol cd="ambiguous" id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.1.1.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.1">subscript</csymbol><log id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.1.2.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.1.2"></log><ci id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.1.3.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.1.3">𝑒</ci></apply><ci id="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.2.cmml" xref="S5.Thmtheorem13.p1.6.6.m6.2.3.4.3.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem13.p1.6.6.m6.2c">1-ne^{-\Theta(\frac{n}{k})}-(\frac{ce}{2\log_{e}n})^{\frac{2}{c}\log_{e}n}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem13.p1.6.6.m6.2d">1 - italic_n italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT - ( divide start_ARG italic_c italic_e end_ARG start_ARG 2 roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 2 end_ARG start_ARG italic_c end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S5.SS3.SSS2.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.SS3.SSS2.1.p1"> <p class="ltx_p" id="S5.SS3.SSS2.1.p1.5"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem9" title="Lemma 5.9. ‣ 5.2.2 High Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5.9</span></a> implies that <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#alg2" title="In 5.2.2 High Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">2</span></a> returns a partition <math alttext="\pi" class="ltx_Math" display="inline" id="S5.SS3.SSS2.1.p1.1.m1.1"><semantics id="S5.SS3.SSS2.1.p1.1.m1.1a"><mi id="S5.SS3.SSS2.1.p1.1.m1.1.1" xref="S5.SS3.SSS2.1.p1.1.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.1.p1.1.m1.1b"><ci id="S5.SS3.SSS2.1.p1.1.m1.1.1.cmml" xref="S5.SS3.SSS2.1.p1.1.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.1.p1.1.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.1.p1.1.m1.1d">italic_π</annotation></semantics></math> where <math alttext="\mathcal{SW}(\pi)=\Omega(n^{\prime})" class="ltx_Math" display="inline" id="S5.SS3.SSS2.1.p1.2.m2.2"><semantics id="S5.SS3.SSS2.1.p1.2.m2.2a"><mrow id="S5.SS3.SSS2.1.p1.2.m2.2.2" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.cmml"><mrow id="S5.SS3.SSS2.1.p1.2.m2.2.2.3" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.1.p1.2.m2.2.2.3.2" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.3.2.cmml">𝒮</mi><mo id="S5.SS3.SSS2.1.p1.2.m2.2.2.3.1" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.1.p1.2.m2.2.2.3.3" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.3.3.cmml">𝒲</mi><mo id="S5.SS3.SSS2.1.p1.2.m2.2.2.3.1a" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.3.1.cmml">⁢</mo><mrow id="S5.SS3.SSS2.1.p1.2.m2.2.2.3.4.2" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.3.cmml"><mo id="S5.SS3.SSS2.1.p1.2.m2.2.2.3.4.2.1" stretchy="false" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.3.cmml">(</mo><mi id="S5.SS3.SSS2.1.p1.2.m2.1.1" xref="S5.SS3.SSS2.1.p1.2.m2.1.1.cmml">π</mi><mo id="S5.SS3.SSS2.1.p1.2.m2.2.2.3.4.2.2" stretchy="false" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.SS3.SSS2.1.p1.2.m2.2.2.2" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.2.cmml">=</mo><mrow id="S5.SS3.SSS2.1.p1.2.m2.2.2.1" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1.cmml"><mi id="S5.SS3.SSS2.1.p1.2.m2.2.2.1.3" mathvariant="normal" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1.3.cmml">Ω</mi><mo id="S5.SS3.SSS2.1.p1.2.m2.2.2.1.2" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.1.cmml"><mo id="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.2" stretchy="false" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.1.cmml">(</mo><msup id="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.1" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.1.cmml"><mi id="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.1.2" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.1.2.cmml">n</mi><mo id="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.1.3" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.3" stretchy="false" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.1.p1.2.m2.2b"><apply id="S5.SS3.SSS2.1.p1.2.m2.2.2.cmml" xref="S5.SS3.SSS2.1.p1.2.m2.2.2"><eq id="S5.SS3.SSS2.1.p1.2.m2.2.2.2.cmml" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.2"></eq><apply id="S5.SS3.SSS2.1.p1.2.m2.2.2.3.cmml" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.3"><times id="S5.SS3.SSS2.1.p1.2.m2.2.2.3.1.cmml" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.3.1"></times><ci id="S5.SS3.SSS2.1.p1.2.m2.2.2.3.2.cmml" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.3.2">𝒮</ci><ci id="S5.SS3.SSS2.1.p1.2.m2.2.2.3.3.cmml" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.3.3">𝒲</ci><ci id="S5.SS3.SSS2.1.p1.2.m2.1.1.cmml" xref="S5.SS3.SSS2.1.p1.2.m2.1.1">𝜋</ci></apply><apply id="S5.SS3.SSS2.1.p1.2.m2.2.2.1.cmml" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1"><times id="S5.SS3.SSS2.1.p1.2.m2.2.2.1.2.cmml" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1.2"></times><ci id="S5.SS3.SSS2.1.p1.2.m2.2.2.1.3.cmml" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1.3">Ω</ci><apply id="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.1.cmml" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1">superscript</csymbol><ci id="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.1.2">𝑛</ci><ci id="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.1.3.cmml" xref="S5.SS3.SSS2.1.p1.2.m2.2.2.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.1.p1.2.m2.2c">\mathcal{SW}(\pi)=\Omega(n^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.1.p1.2.m2.2d">caligraphic_S caligraphic_W ( italic_π ) = roman_Ω ( italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> with probability <math alttext="1-n^{\prime}e^{-\Theta(\frac{n^{\prime}}{k})}" class="ltx_Math" display="inline" id="S5.SS3.SSS2.1.p1.3.m3.1"><semantics id="S5.SS3.SSS2.1.p1.3.m3.1a"><mrow id="S5.SS3.SSS2.1.p1.3.m3.1.2" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.cmml"><mn id="S5.SS3.SSS2.1.p1.3.m3.1.2.2" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.2.cmml">1</mn><mo id="S5.SS3.SSS2.1.p1.3.m3.1.2.1" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.1.cmml">−</mo><mrow id="S5.SS3.SSS2.1.p1.3.m3.1.2.3" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3.cmml"><msup id="S5.SS3.SSS2.1.p1.3.m3.1.2.3.2" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3.2.cmml"><mi id="S5.SS3.SSS2.1.p1.3.m3.1.2.3.2.2" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3.2.2.cmml">n</mi><mo id="S5.SS3.SSS2.1.p1.3.m3.1.2.3.2.3" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3.2.3.cmml">′</mo></msup><mo id="S5.SS3.SSS2.1.p1.3.m3.1.2.3.1" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3.1.cmml">⁢</mo><msup id="S5.SS3.SSS2.1.p1.3.m3.1.2.3.3" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3.3.cmml"><mi id="S5.SS3.SSS2.1.p1.3.m3.1.2.3.3.2" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3.3.2.cmml">e</mi><mrow id="S5.SS3.SSS2.1.p1.3.m3.1.1.1" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.cmml"><mo id="S5.SS3.SSS2.1.p1.3.m3.1.1.1a" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.cmml">−</mo><mrow id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3.cmml"><mi id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3.2" mathvariant="normal" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3.2.cmml">Θ</mi><mo id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3.1" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3.3.2" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.cmml"><mo id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3.3.2.1" stretchy="false" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.cmml">(</mo><mfrac id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.cmml"><msup id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.2" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.2.cmml"><mi id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.2.2" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.2.2.cmml">n</mi><mo id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.2.3" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.2.3.cmml">′</mo></msup><mi id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.3" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.3.cmml">k</mi></mfrac><mo id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3.3.2.2" stretchy="false" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.1.p1.3.m3.1b"><apply id="S5.SS3.SSS2.1.p1.3.m3.1.2.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.2"><minus id="S5.SS3.SSS2.1.p1.3.m3.1.2.1.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.1"></minus><cn id="S5.SS3.SSS2.1.p1.3.m3.1.2.2.cmml" type="integer" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.2">1</cn><apply id="S5.SS3.SSS2.1.p1.3.m3.1.2.3.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3"><times id="S5.SS3.SSS2.1.p1.3.m3.1.2.3.1.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3.1"></times><apply id="S5.SS3.SSS2.1.p1.3.m3.1.2.3.2.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="S5.SS3.SSS2.1.p1.3.m3.1.2.3.2.1.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3.2">superscript</csymbol><ci id="S5.SS3.SSS2.1.p1.3.m3.1.2.3.2.2.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3.2.2">𝑛</ci><ci id="S5.SS3.SSS2.1.p1.3.m3.1.2.3.2.3.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3.2.3">′</ci></apply><apply id="S5.SS3.SSS2.1.p1.3.m3.1.2.3.3.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3.3"><csymbol cd="ambiguous" id="S5.SS3.SSS2.1.p1.3.m3.1.2.3.3.1.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3.3">superscript</csymbol><ci id="S5.SS3.SSS2.1.p1.3.m3.1.2.3.3.2.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.2.3.3.2">𝑒</ci><apply id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1"><minus id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.2.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1"></minus><apply id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3"><times id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3.1.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3.1"></times><ci id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3.2.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3.2">Θ</ci><apply id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3.3.2"><divide id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.3.3.2"></divide><apply id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.2.1.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.2">superscript</csymbol><ci id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.2.2.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.2.2">𝑛</ci><ci id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.2.3.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.2.3">′</ci></apply><ci id="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.3.cmml" xref="S5.SS3.SSS2.1.p1.3.m3.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.1.p1.3.m3.1c">1-n^{\prime}e^{-\Theta(\frac{n^{\prime}}{k})}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.1.p1.3.m3.1d">1 - italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT</annotation></semantics></math>. Since <math alttext="n^{\prime}=\Theta(n)" class="ltx_Math" display="inline" id="S5.SS3.SSS2.1.p1.4.m4.1"><semantics id="S5.SS3.SSS2.1.p1.4.m4.1a"><mrow id="S5.SS3.SSS2.1.p1.4.m4.1.2" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.cmml"><msup id="S5.SS3.SSS2.1.p1.4.m4.1.2.2" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.2.cmml"><mi id="S5.SS3.SSS2.1.p1.4.m4.1.2.2.2" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.2.2.cmml">n</mi><mo id="S5.SS3.SSS2.1.p1.4.m4.1.2.2.3" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.2.3.cmml">′</mo></msup><mo id="S5.SS3.SSS2.1.p1.4.m4.1.2.1" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.1.cmml">=</mo><mrow id="S5.SS3.SSS2.1.p1.4.m4.1.2.3" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.3.cmml"><mi id="S5.SS3.SSS2.1.p1.4.m4.1.2.3.2" mathvariant="normal" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.3.2.cmml">Θ</mi><mo id="S5.SS3.SSS2.1.p1.4.m4.1.2.3.1" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS3.SSS2.1.p1.4.m4.1.2.3.3.2" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.3.cmml"><mo id="S5.SS3.SSS2.1.p1.4.m4.1.2.3.3.2.1" stretchy="false" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.3.cmml">(</mo><mi id="S5.SS3.SSS2.1.p1.4.m4.1.1" xref="S5.SS3.SSS2.1.p1.4.m4.1.1.cmml">n</mi><mo id="S5.SS3.SSS2.1.p1.4.m4.1.2.3.3.2.2" stretchy="false" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.1.p1.4.m4.1b"><apply id="S5.SS3.SSS2.1.p1.4.m4.1.2.cmml" xref="S5.SS3.SSS2.1.p1.4.m4.1.2"><eq id="S5.SS3.SSS2.1.p1.4.m4.1.2.1.cmml" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.1"></eq><apply id="S5.SS3.SSS2.1.p1.4.m4.1.2.2.cmml" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.2"><csymbol cd="ambiguous" id="S5.SS3.SSS2.1.p1.4.m4.1.2.2.1.cmml" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.2">superscript</csymbol><ci id="S5.SS3.SSS2.1.p1.4.m4.1.2.2.2.cmml" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.2.2">𝑛</ci><ci id="S5.SS3.SSS2.1.p1.4.m4.1.2.2.3.cmml" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.2.3">′</ci></apply><apply id="S5.SS3.SSS2.1.p1.4.m4.1.2.3.cmml" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.3"><times id="S5.SS3.SSS2.1.p1.4.m4.1.2.3.1.cmml" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.3.1"></times><ci id="S5.SS3.SSS2.1.p1.4.m4.1.2.3.2.cmml" xref="S5.SS3.SSS2.1.p1.4.m4.1.2.3.2">Θ</ci><ci id="S5.SS3.SSS2.1.p1.4.m4.1.1.cmml" xref="S5.SS3.SSS2.1.p1.4.m4.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.1.p1.4.m4.1c">n^{\prime}=\Theta(n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.1.p1.4.m4.1d">italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = roman_Θ ( italic_n )</annotation></semantics></math>, we also have <math alttext="\mathcal{SW}(\pi)=\Omega(n)" class="ltx_Math" display="inline" id="S5.SS3.SSS2.1.p1.5.m5.2"><semantics id="S5.SS3.SSS2.1.p1.5.m5.2a"><mrow id="S5.SS3.SSS2.1.p1.5.m5.2.3" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.cmml"><mrow id="S5.SS3.SSS2.1.p1.5.m5.2.3.2" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.1.p1.5.m5.2.3.2.2" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.2.2.cmml">𝒮</mi><mo id="S5.SS3.SSS2.1.p1.5.m5.2.3.2.1" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.2.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.1.p1.5.m5.2.3.2.3" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.2.3.cmml">𝒲</mi><mo id="S5.SS3.SSS2.1.p1.5.m5.2.3.2.1a" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S5.SS3.SSS2.1.p1.5.m5.2.3.2.4.2" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.2.cmml"><mo id="S5.SS3.SSS2.1.p1.5.m5.2.3.2.4.2.1" stretchy="false" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.2.cmml">(</mo><mi id="S5.SS3.SSS2.1.p1.5.m5.1.1" xref="S5.SS3.SSS2.1.p1.5.m5.1.1.cmml">π</mi><mo id="S5.SS3.SSS2.1.p1.5.m5.2.3.2.4.2.2" stretchy="false" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.SS3.SSS2.1.p1.5.m5.2.3.1" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.1.cmml">=</mo><mrow id="S5.SS3.SSS2.1.p1.5.m5.2.3.3" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.3.cmml"><mi id="S5.SS3.SSS2.1.p1.5.m5.2.3.3.2" mathvariant="normal" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.3.2.cmml">Ω</mi><mo id="S5.SS3.SSS2.1.p1.5.m5.2.3.3.1" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S5.SS3.SSS2.1.p1.5.m5.2.3.3.3.2" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.3.cmml"><mo id="S5.SS3.SSS2.1.p1.5.m5.2.3.3.3.2.1" stretchy="false" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.3.cmml">(</mo><mi id="S5.SS3.SSS2.1.p1.5.m5.2.2" xref="S5.SS3.SSS2.1.p1.5.m5.2.2.cmml">n</mi><mo id="S5.SS3.SSS2.1.p1.5.m5.2.3.3.3.2.2" stretchy="false" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.1.p1.5.m5.2b"><apply id="S5.SS3.SSS2.1.p1.5.m5.2.3.cmml" xref="S5.SS3.SSS2.1.p1.5.m5.2.3"><eq id="S5.SS3.SSS2.1.p1.5.m5.2.3.1.cmml" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.1"></eq><apply id="S5.SS3.SSS2.1.p1.5.m5.2.3.2.cmml" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.2"><times id="S5.SS3.SSS2.1.p1.5.m5.2.3.2.1.cmml" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.2.1"></times><ci id="S5.SS3.SSS2.1.p1.5.m5.2.3.2.2.cmml" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.2.2">𝒮</ci><ci id="S5.SS3.SSS2.1.p1.5.m5.2.3.2.3.cmml" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.2.3">𝒲</ci><ci id="S5.SS3.SSS2.1.p1.5.m5.1.1.cmml" xref="S5.SS3.SSS2.1.p1.5.m5.1.1">𝜋</ci></apply><apply id="S5.SS3.SSS2.1.p1.5.m5.2.3.3.cmml" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.3"><times id="S5.SS3.SSS2.1.p1.5.m5.2.3.3.1.cmml" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.3.1"></times><ci id="S5.SS3.SSS2.1.p1.5.m5.2.3.3.2.cmml" xref="S5.SS3.SSS2.1.p1.5.m5.2.3.3.2">Ω</ci><ci id="S5.SS3.SSS2.1.p1.5.m5.2.2.cmml" xref="S5.SS3.SSS2.1.p1.5.m5.2.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.1.p1.5.m5.2c">\mathcal{SW}(\pi)=\Omega(n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.1.p1.5.m5.2d">caligraphic_S caligraphic_W ( italic_π ) = roman_Ω ( italic_n )</annotation></semantics></math> with the same probability.</p> </div> <div class="ltx_para" id="S5.SS3.SSS2.2.p2"> <p class="ltx_p" id="S5.SS3.SSS2.2.p2.25"><a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem6" title="Proposition 5.6. ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">5.6</span></a> implies <math alttext="\mathcal{SW}(\pi^{*})\leq n(k-1)" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.1.m1.2"><semantics id="S5.SS3.SSS2.2.p2.1.m1.2a"><mrow id="S5.SS3.SSS2.2.p2.1.m1.2.2" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.cmml"><mrow id="S5.SS3.SSS2.2.p2.1.m1.1.1.1" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.3" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.3.cmml">𝒮</mi><mo id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.2" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.4" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.4.cmml">𝒲</mi><mo id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.2a" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.1.cmml"><mo id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.1.2" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.1.2.cmml">π</mi><mo id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.1.3" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS3.SSS2.2.p2.1.m1.2.2.3" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.3.cmml">≤</mo><mrow id="S5.SS3.SSS2.2.p2.1.m1.2.2.2" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.cmml"><mi id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.3" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.3.cmml">n</mi><mo id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.2" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.2.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.cmml"><mo id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.2" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.2.cmml">k</mi><mo id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.1" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.1.cmml">−</mo><mn id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.3" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.1.m1.2b"><apply id="S5.SS3.SSS2.2.p2.1.m1.2.2.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.2.2"><leq id="S5.SS3.SSS2.2.p2.1.m1.2.2.3.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.3"></leq><apply id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1"><times id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.2"></times><ci id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.3">𝒮</ci><ci id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.4.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.4">𝒲</ci><apply id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.1.2">𝜋</ci><times id="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.1.1.1.1.1.1.3"></times></apply></apply><apply id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2"><times id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.2.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.2"></times><ci id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.3.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.3">𝑛</ci><apply id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1"><minus id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.1"></minus><ci id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.2">𝑘</ci><cn id="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.3.cmml" type="integer" xref="S5.SS3.SSS2.2.p2.1.m1.2.2.2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.1.m1.2c">\mathcal{SW}(\pi^{*})\leq n(k-1)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.1.m1.2d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ≤ italic_n ( italic_k - 1 )</annotation></semantics></math>. If <math alttext="k=\mathcal{O}(\log n)" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.2.m2.1"><semantics id="S5.SS3.SSS2.2.p2.2.m2.1a"><mrow id="S5.SS3.SSS2.2.p2.2.m2.1.1" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.2.m2.1.1.3" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.3.cmml">k</mi><mo id="S5.SS3.SSS2.2.p2.2.m2.1.1.2" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.2.cmml">=</mo><mrow id="S5.SS3.SSS2.2.p2.2.m2.1.1.1" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.2.p2.2.m2.1.1.1.3" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1.3.cmml">𝒪</mi><mo id="S5.SS3.SSS2.2.p2.2.m2.1.1.1.2" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1.cmml"><mo id="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1.1.cmml">log</mi><mo id="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1a" lspace="0.167em" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1.cmml">⁡</mo><mi id="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1.2" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.2.m2.1b"><apply id="S5.SS3.SSS2.2.p2.2.m2.1.1.cmml" xref="S5.SS3.SSS2.2.p2.2.m2.1.1"><eq id="S5.SS3.SSS2.2.p2.2.m2.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.2"></eq><ci id="S5.SS3.SSS2.2.p2.2.m2.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.3">𝑘</ci><apply id="S5.SS3.SSS2.2.p2.2.m2.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1"><times id="S5.SS3.SSS2.2.p2.2.m2.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1.2"></times><ci id="S5.SS3.SSS2.2.p2.2.m2.1.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1.3">𝒪</ci><apply id="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1"><log id="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1.1"></log><ci id="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.2.m2.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.2.m2.1c">k=\mathcal{O}(\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.2.m2.1d">italic_k = caligraphic_O ( roman_log italic_n )</annotation></semantics></math>, then <math alttext="\mathcal{SW}(\pi^{*})=\mathcal{O}(n\log n)" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.3.m3.2"><semantics id="S5.SS3.SSS2.2.p2.3.m3.2a"><mrow id="S5.SS3.SSS2.2.p2.3.m3.2.2" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.cmml"><mrow id="S5.SS3.SSS2.2.p2.3.m3.1.1.1" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.3" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.3.cmml">𝒮</mi><mo id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.2" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.4" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.4.cmml">𝒲</mi><mo id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.2a" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.1.cmml"><mo id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.1.cmml">(</mo><msup id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.1.2" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.1.2.cmml">π</mi><mo id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.1.3" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS3.SSS2.2.p2.3.m3.2.2.3" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.3.cmml">=</mo><mrow id="S5.SS3.SSS2.2.p2.3.m3.2.2.2" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.3" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.3.cmml">𝒪</mi><mo id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.2" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.2.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.cmml"><mo id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.2" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.2.cmml">n</mi><mo id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.1" lspace="0.167em" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.3" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.3.cmml"><mi id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.3.1" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.3.1.cmml">log</mi><mo id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.3a" lspace="0.167em" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.3.cmml">⁡</mo><mi id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.3.2" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.3.m3.2b"><apply id="S5.SS3.SSS2.2.p2.3.m3.2.2.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.2.2"><eq id="S5.SS3.SSS2.2.p2.3.m3.2.2.3.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.3"></eq><apply id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1"><times id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.2"></times><ci id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.3">𝒮</ci><ci id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.4.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.4">𝒲</ci><apply id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.1.2">𝜋</ci><times id="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.1.1.1.1.1.1.3"></times></apply></apply><apply id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2"><times id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.2.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.2"></times><ci id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.3.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.3">𝒪</ci><apply id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1"><times id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.1"></times><ci id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.2">𝑛</ci><apply id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.3"><log id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.3.1.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.3.1"></log><ci id="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.3.2.cmml" xref="S5.SS3.SSS2.2.p2.3.m3.2.2.2.1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.3.m3.2c">\mathcal{SW}(\pi^{*})=\mathcal{O}(n\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.3.m3.2d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = caligraphic_O ( italic_n roman_log italic_n )</annotation></semantics></math>. When <math alttext="k=\Omega(\log n)" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.4.m4.1"><semantics id="S5.SS3.SSS2.2.p2.4.m4.1a"><mrow id="S5.SS3.SSS2.2.p2.4.m4.1.1" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.4.m4.1.1.3" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.3.cmml">k</mi><mo id="S5.SS3.SSS2.2.p2.4.m4.1.1.2" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.2.cmml">=</mo><mrow id="S5.SS3.SSS2.2.p2.4.m4.1.1.1" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.4.m4.1.1.1.3" mathvariant="normal" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1.3.cmml">Ω</mi><mo id="S5.SS3.SSS2.2.p2.4.m4.1.1.1.2" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1.cmml"><mo id="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1.1.cmml">log</mi><mo id="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1a" lspace="0.167em" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1.cmml">⁡</mo><mi id="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1.2" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.4.m4.1b"><apply id="S5.SS3.SSS2.2.p2.4.m4.1.1.cmml" xref="S5.SS3.SSS2.2.p2.4.m4.1.1"><eq id="S5.SS3.SSS2.2.p2.4.m4.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.2"></eq><ci id="S5.SS3.SSS2.2.p2.4.m4.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.3">𝑘</ci><apply id="S5.SS3.SSS2.2.p2.4.m4.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1"><times id="S5.SS3.SSS2.2.p2.4.m4.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1.2"></times><ci id="S5.SS3.SSS2.2.p2.4.m4.1.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1.3">Ω</ci><apply id="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1"><log id="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1.1"></log><ci id="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.4.m4.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.4.m4.1c">k=\Omega(\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.4.m4.1d">italic_k = roman_Ω ( roman_log italic_n )</annotation></semantics></math>, we add <math alttext="|V_{1}|-|V_{i}|" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.5.m5.2"><semantics id="S5.SS3.SSS2.2.p2.5.m5.2a"><mrow id="S5.SS3.SSS2.2.p2.5.m5.2.2" xref="S5.SS3.SSS2.2.p2.5.m5.2.2.cmml"><mrow id="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1" xref="S5.SS3.SSS2.2.p2.5.m5.1.1.1.2.cmml"><mo id="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.5.m5.1.1.1.2.1.cmml">|</mo><msub id="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.1.2" xref="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.1.2.cmml">V</mi><mn id="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.1.3" xref="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.5.m5.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.SS3.SSS2.2.p2.5.m5.2.2.3" xref="S5.SS3.SSS2.2.p2.5.m5.2.2.3.cmml">−</mo><mrow id="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1" xref="S5.SS3.SSS2.2.p2.5.m5.2.2.2.2.cmml"><mo id="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.5.m5.2.2.2.2.1.cmml">|</mo><msub id="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.1" xref="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.1.2" xref="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.1.2.cmml">V</mi><mi id="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.1.3" xref="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.1.3.cmml">i</mi></msub><mo id="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.5.m5.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.5.m5.2b"><apply id="S5.SS3.SSS2.2.p2.5.m5.2.2.cmml" xref="S5.SS3.SSS2.2.p2.5.m5.2.2"><minus id="S5.SS3.SSS2.2.p2.5.m5.2.2.3.cmml" xref="S5.SS3.SSS2.2.p2.5.m5.2.2.3"></minus><apply id="S5.SS3.SSS2.2.p2.5.m5.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1"><abs id="S5.SS3.SSS2.2.p2.5.m5.1.1.1.2.1.cmml" xref="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.2"></abs><apply id="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.1.2">𝑉</ci><cn id="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.1.3.cmml" type="integer" xref="S5.SS3.SSS2.2.p2.5.m5.1.1.1.1.1.3">1</cn></apply></apply><apply id="S5.SS3.SSS2.2.p2.5.m5.2.2.2.2.cmml" xref="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1"><abs id="S5.SS3.SSS2.2.p2.5.m5.2.2.2.2.1.cmml" xref="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.2"></abs><apply id="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.1.cmml" xref="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.1">subscript</csymbol><ci id="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.1.2">𝑉</ci><ci id="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.5.m5.2.2.2.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.5.m5.2c">|V_{1}|-|V_{i}|</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.5.m5.2d">| italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | - | italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT |</annotation></semantics></math> dummy vertices to each color class <math alttext="V_{i}" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.6.m6.1"><semantics id="S5.SS3.SSS2.2.p2.6.m6.1a"><msub id="S5.SS3.SSS2.2.p2.6.m6.1.1" xref="S5.SS3.SSS2.2.p2.6.m6.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.6.m6.1.1.2" xref="S5.SS3.SSS2.2.p2.6.m6.1.1.2.cmml">V</mi><mi id="S5.SS3.SSS2.2.p2.6.m6.1.1.3" xref="S5.SS3.SSS2.2.p2.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.6.m6.1b"><apply id="S5.SS3.SSS2.2.p2.6.m6.1.1.cmml" xref="S5.SS3.SSS2.2.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.6.m6.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.6.m6.1.1">subscript</csymbol><ci id="S5.SS3.SSS2.2.p2.6.m6.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.6.m6.1.1.2">𝑉</ci><ci id="S5.SS3.SSS2.2.p2.6.m6.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.6.m6.1c">V_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.6.m6.1d">italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (for all <math alttext="i\in[k]" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.7.m7.1"><semantics id="S5.SS3.SSS2.2.p2.7.m7.1a"><mrow id="S5.SS3.SSS2.2.p2.7.m7.1.2" xref="S5.SS3.SSS2.2.p2.7.m7.1.2.cmml"><mi id="S5.SS3.SSS2.2.p2.7.m7.1.2.2" xref="S5.SS3.SSS2.2.p2.7.m7.1.2.2.cmml">i</mi><mo id="S5.SS3.SSS2.2.p2.7.m7.1.2.1" xref="S5.SS3.SSS2.2.p2.7.m7.1.2.1.cmml">∈</mo><mrow id="S5.SS3.SSS2.2.p2.7.m7.1.2.3.2" xref="S5.SS3.SSS2.2.p2.7.m7.1.2.3.1.cmml"><mo id="S5.SS3.SSS2.2.p2.7.m7.1.2.3.2.1" stretchy="false" xref="S5.SS3.SSS2.2.p2.7.m7.1.2.3.1.1.cmml">[</mo><mi id="S5.SS3.SSS2.2.p2.7.m7.1.1" xref="S5.SS3.SSS2.2.p2.7.m7.1.1.cmml">k</mi><mo id="S5.SS3.SSS2.2.p2.7.m7.1.2.3.2.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.7.m7.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.7.m7.1b"><apply id="S5.SS3.SSS2.2.p2.7.m7.1.2.cmml" xref="S5.SS3.SSS2.2.p2.7.m7.1.2"><in id="S5.SS3.SSS2.2.p2.7.m7.1.2.1.cmml" xref="S5.SS3.SSS2.2.p2.7.m7.1.2.1"></in><ci id="S5.SS3.SSS2.2.p2.7.m7.1.2.2.cmml" xref="S5.SS3.SSS2.2.p2.7.m7.1.2.2">𝑖</ci><apply id="S5.SS3.SSS2.2.p2.7.m7.1.2.3.1.cmml" xref="S5.SS3.SSS2.2.p2.7.m7.1.2.3.2"><csymbol cd="latexml" id="S5.SS3.SSS2.2.p2.7.m7.1.2.3.1.1.cmml" xref="S5.SS3.SSS2.2.p2.7.m7.1.2.3.2.1">delimited-[]</csymbol><ci id="S5.SS3.SSS2.2.p2.7.m7.1.1.cmml" xref="S5.SS3.SSS2.2.p2.7.m7.1.1">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.7.m7.1c">i\in[k]</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.7.m7.1d">italic_i ∈ [ italic_k ]</annotation></semantics></math>) to create new color classes <math alttext="V^{n}_{1},\cdots,V^{n}_{k}" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.8.m8.3"><semantics id="S5.SS3.SSS2.2.p2.8.m8.3a"><mrow id="S5.SS3.SSS2.2.p2.8.m8.3.3.2" xref="S5.SS3.SSS2.2.p2.8.m8.3.3.3.cmml"><msubsup id="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1" xref="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.2.2" xref="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.2.2.cmml">V</mi><mn id="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.3" xref="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.3.cmml">1</mn><mi id="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.2.3" xref="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.2.3.cmml">n</mi></msubsup><mo id="S5.SS3.SSS2.2.p2.8.m8.3.3.2.3" xref="S5.SS3.SSS2.2.p2.8.m8.3.3.3.cmml">,</mo><mi id="S5.SS3.SSS2.2.p2.8.m8.1.1" mathvariant="normal" xref="S5.SS3.SSS2.2.p2.8.m8.1.1.cmml">⋯</mi><mo id="S5.SS3.SSS2.2.p2.8.m8.3.3.2.4" xref="S5.SS3.SSS2.2.p2.8.m8.3.3.3.cmml">,</mo><msubsup id="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2" xref="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.cmml"><mi id="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.2.2" xref="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.2.2.cmml">V</mi><mi id="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.3" xref="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.3.cmml">k</mi><mi id="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.2.3" xref="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.2.3.cmml">n</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.8.m8.3b"><list id="S5.SS3.SSS2.2.p2.8.m8.3.3.3.cmml" xref="S5.SS3.SSS2.2.p2.8.m8.3.3.2"><apply id="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.cmml" xref="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1">subscript</csymbol><apply id="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.2.1.cmml" xref="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.2.2.cmml" xref="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.2.2">𝑉</ci><ci id="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.2.3.cmml" xref="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.2.3">𝑛</ci></apply><cn id="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.3.cmml" type="integer" xref="S5.SS3.SSS2.2.p2.8.m8.2.2.1.1.3">1</cn></apply><ci id="S5.SS3.SSS2.2.p2.8.m8.1.1.cmml" xref="S5.SS3.SSS2.2.p2.8.m8.1.1">⋯</ci><apply id="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.cmml" xref="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.1.cmml" xref="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2">subscript</csymbol><apply id="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.2.cmml" xref="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.2.1.cmml" xref="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.2.2.cmml" xref="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.2.2">𝑉</ci><ci id="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.2.3.cmml" xref="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.2.3">𝑛</ci></apply><ci id="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.3.cmml" xref="S5.SS3.SSS2.2.p2.8.m8.3.3.2.2.3">𝑘</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.8.m8.3c">V^{n}_{1},\cdots,V^{n}_{k}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.8.m8.3d">italic_V start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_V start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, ensuring <math alttext="|V^{n}_{i}|=|V_{1}|" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.9.m9.2"><semantics id="S5.SS3.SSS2.2.p2.9.m9.2a"><mrow id="S5.SS3.SSS2.2.p2.9.m9.2.2" xref="S5.SS3.SSS2.2.p2.9.m9.2.2.cmml"><mrow id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.2.cmml"><mo id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.2.1.cmml">|</mo><msubsup id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.2.2" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.2.2.cmml">V</mi><mi id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.3" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.3.cmml">i</mi><mi id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.2.3" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.2.3.cmml">n</mi></msubsup><mo id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.SS3.SSS2.2.p2.9.m9.2.2.3" xref="S5.SS3.SSS2.2.p2.9.m9.2.2.3.cmml">=</mo><mrow id="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1" xref="S5.SS3.SSS2.2.p2.9.m9.2.2.2.2.cmml"><mo id="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.9.m9.2.2.2.2.1.cmml">|</mo><msub id="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.1" xref="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.1.2" xref="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.1.2.cmml">V</mi><mn id="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.1.3" xref="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.1.3.cmml">1</mn></msub><mo id="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.9.m9.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.9.m9.2b"><apply id="S5.SS3.SSS2.2.p2.9.m9.2.2.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.2.2"><eq id="S5.SS3.SSS2.2.p2.9.m9.2.2.3.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.2.2.3"></eq><apply id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1"><abs id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.2.1.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.2"></abs><apply id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1">subscript</csymbol><apply id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.2.1.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.2.2.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.2.2">𝑉</ci><ci id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.2.3.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.2.3">𝑛</ci></apply><ci id="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S5.SS3.SSS2.2.p2.9.m9.2.2.2.2.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1"><abs id="S5.SS3.SSS2.2.p2.9.m9.2.2.2.2.1.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.2"></abs><apply id="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.1.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.1">subscript</csymbol><ci id="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.1.2">𝑉</ci><cn id="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.1.3.cmml" type="integer" xref="S5.SS3.SSS2.2.p2.9.m9.2.2.2.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.9.m9.2c">|V^{n}_{i}|=|V_{1}|</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.9.m9.2d">| italic_V start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | = | italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT |</annotation></semantics></math> for all <math alttext="i\in[k]" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.10.m10.1"><semantics id="S5.SS3.SSS2.2.p2.10.m10.1a"><mrow id="S5.SS3.SSS2.2.p2.10.m10.1.2" xref="S5.SS3.SSS2.2.p2.10.m10.1.2.cmml"><mi id="S5.SS3.SSS2.2.p2.10.m10.1.2.2" xref="S5.SS3.SSS2.2.p2.10.m10.1.2.2.cmml">i</mi><mo id="S5.SS3.SSS2.2.p2.10.m10.1.2.1" xref="S5.SS3.SSS2.2.p2.10.m10.1.2.1.cmml">∈</mo><mrow id="S5.SS3.SSS2.2.p2.10.m10.1.2.3.2" xref="S5.SS3.SSS2.2.p2.10.m10.1.2.3.1.cmml"><mo id="S5.SS3.SSS2.2.p2.10.m10.1.2.3.2.1" stretchy="false" xref="S5.SS3.SSS2.2.p2.10.m10.1.2.3.1.1.cmml">[</mo><mi id="S5.SS3.SSS2.2.p2.10.m10.1.1" xref="S5.SS3.SSS2.2.p2.10.m10.1.1.cmml">k</mi><mo id="S5.SS3.SSS2.2.p2.10.m10.1.2.3.2.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.10.m10.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.10.m10.1b"><apply id="S5.SS3.SSS2.2.p2.10.m10.1.2.cmml" xref="S5.SS3.SSS2.2.p2.10.m10.1.2"><in id="S5.SS3.SSS2.2.p2.10.m10.1.2.1.cmml" xref="S5.SS3.SSS2.2.p2.10.m10.1.2.1"></in><ci id="S5.SS3.SSS2.2.p2.10.m10.1.2.2.cmml" xref="S5.SS3.SSS2.2.p2.10.m10.1.2.2">𝑖</ci><apply id="S5.SS3.SSS2.2.p2.10.m10.1.2.3.1.cmml" xref="S5.SS3.SSS2.2.p2.10.m10.1.2.3.2"><csymbol cd="latexml" id="S5.SS3.SSS2.2.p2.10.m10.1.2.3.1.1.cmml" xref="S5.SS3.SSS2.2.p2.10.m10.1.2.3.2.1">delimited-[]</csymbol><ci id="S5.SS3.SSS2.2.p2.10.m10.1.1.cmml" xref="S5.SS3.SSS2.2.p2.10.m10.1.1">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.10.m10.1c">i\in[k]</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.10.m10.1d">italic_i ∈ [ italic_k ]</annotation></semantics></math>. For each pair of vertices without an edge, add a weighted edge based on the distribution defined in <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem5" title="Definition 5.5. ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Definition</span> <span class="ltx_text ltx_ref_tag">5.5</span></a>. Let <math alttext="G^{n}" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.11.m11.1"><semantics id="S5.SS3.SSS2.2.p2.11.m11.1a"><msup id="S5.SS3.SSS2.2.p2.11.m11.1.1" xref="S5.SS3.SSS2.2.p2.11.m11.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.11.m11.1.1.2" xref="S5.SS3.SSS2.2.p2.11.m11.1.1.2.cmml">G</mi><mi id="S5.SS3.SSS2.2.p2.11.m11.1.1.3" xref="S5.SS3.SSS2.2.p2.11.m11.1.1.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.11.m11.1b"><apply id="S5.SS3.SSS2.2.p2.11.m11.1.1.cmml" xref="S5.SS3.SSS2.2.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.11.m11.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.11.m11.1.1">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.11.m11.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.11.m11.1.1.2">𝐺</ci><ci id="S5.SS3.SSS2.2.p2.11.m11.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.11.m11.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.11.m11.1c">G^{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.11.m11.1d">italic_G start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> be the resulting random Turán graph, and let <math alttext="n^{\prime\prime}" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.12.m12.1"><semantics id="S5.SS3.SSS2.2.p2.12.m12.1a"><msup id="S5.SS3.SSS2.2.p2.12.m12.1.1" xref="S5.SS3.SSS2.2.p2.12.m12.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.12.m12.1.1.2" xref="S5.SS3.SSS2.2.p2.12.m12.1.1.2.cmml">n</mi><mo id="S5.SS3.SSS2.2.p2.12.m12.1.1.3" xref="S5.SS3.SSS2.2.p2.12.m12.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.12.m12.1b"><apply id="S5.SS3.SSS2.2.p2.12.m12.1.1.cmml" xref="S5.SS3.SSS2.2.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.12.m12.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.12.m12.1.1">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.12.m12.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.12.m12.1.1.2">𝑛</ci><ci id="S5.SS3.SSS2.2.p2.12.m12.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.12.m12.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.12.m12.1c">n^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.12.m12.1d">italic_n start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> represent the total number of vertices in <math alttext="G^{n}" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.13.m13.1"><semantics id="S5.SS3.SSS2.2.p2.13.m13.1a"><msup id="S5.SS3.SSS2.2.p2.13.m13.1.1" xref="S5.SS3.SSS2.2.p2.13.m13.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.13.m13.1.1.2" xref="S5.SS3.SSS2.2.p2.13.m13.1.1.2.cmml">G</mi><mi id="S5.SS3.SSS2.2.p2.13.m13.1.1.3" xref="S5.SS3.SSS2.2.p2.13.m13.1.1.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.13.m13.1b"><apply id="S5.SS3.SSS2.2.p2.13.m13.1.1.cmml" xref="S5.SS3.SSS2.2.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.13.m13.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.13.m13.1.1">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.13.m13.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.13.m13.1.1.2">𝐺</ci><ci id="S5.SS3.SSS2.2.p2.13.m13.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.13.m13.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.13.m13.1c">G^{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.13.m13.1d">italic_G start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. Note that <math alttext="n\leq n^{\prime\prime}=k|V_{1}|\leq k\frac{|V_{k}|}{q}=\frac{n^{\prime}}{q}% \leq\frac{n}{q}" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.14.m14.2"><semantics id="S5.SS3.SSS2.2.p2.14.m14.2a"><mrow id="S5.SS3.SSS2.2.p2.14.m14.2.2" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.cmml"><mi id="S5.SS3.SSS2.2.p2.14.m14.2.2.3" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.3.cmml">n</mi><mo id="S5.SS3.SSS2.2.p2.14.m14.2.2.4" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.4.cmml">≤</mo><msup id="S5.SS3.SSS2.2.p2.14.m14.2.2.5" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.5.cmml"><mi id="S5.SS3.SSS2.2.p2.14.m14.2.2.5.2" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.5.2.cmml">n</mi><mo id="S5.SS3.SSS2.2.p2.14.m14.2.2.5.3" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.5.3.cmml">′′</mo></msup><mo id="S5.SS3.SSS2.2.p2.14.m14.2.2.6" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.6.cmml">=</mo><mrow id="S5.SS3.SSS2.2.p2.14.m14.2.2.1" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.cmml"><mi id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.3" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.3.cmml">k</mi><mo id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.2" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.2.cmml"><mo id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.2.1.cmml">|</mo><msub id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.1" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.1.2" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.1.2.cmml">V</mi><mn id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.1.3" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="S5.SS3.SSS2.2.p2.14.m14.2.2.7" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.7.cmml">≤</mo><mrow id="S5.SS3.SSS2.2.p2.14.m14.2.2.8" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.8.cmml"><mi id="S5.SS3.SSS2.2.p2.14.m14.2.2.8.2" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.8.2.cmml">k</mi><mo id="S5.SS3.SSS2.2.p2.14.m14.2.2.8.1" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.8.1.cmml">⁢</mo><mfrac id="S5.SS3.SSS2.2.p2.14.m14.1.1" xref="S5.SS3.SSS2.2.p2.14.m14.1.1.cmml"><mrow id="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1" xref="S5.SS3.SSS2.2.p2.14.m14.1.1.1.2.cmml"><mo id="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.14.m14.1.1.1.2.1.cmml">|</mo><msub id="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.1.2" xref="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.1.2.cmml">V</mi><mi id="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.1.3" xref="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.14.m14.1.1.1.2.1.cmml">|</mo></mrow><mi id="S5.SS3.SSS2.2.p2.14.m14.1.1.3" xref="S5.SS3.SSS2.2.p2.14.m14.1.1.3.cmml">q</mi></mfrac></mrow><mo id="S5.SS3.SSS2.2.p2.14.m14.2.2.9" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.9.cmml">=</mo><mfrac id="S5.SS3.SSS2.2.p2.14.m14.2.2.10" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.10.cmml"><msup id="S5.SS3.SSS2.2.p2.14.m14.2.2.10.2" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.10.2.cmml"><mi id="S5.SS3.SSS2.2.p2.14.m14.2.2.10.2.2" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.10.2.2.cmml">n</mi><mo id="S5.SS3.SSS2.2.p2.14.m14.2.2.10.2.3" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.10.2.3.cmml">′</mo></msup><mi id="S5.SS3.SSS2.2.p2.14.m14.2.2.10.3" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.10.3.cmml">q</mi></mfrac><mo id="S5.SS3.SSS2.2.p2.14.m14.2.2.11" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.11.cmml">≤</mo><mfrac id="S5.SS3.SSS2.2.p2.14.m14.2.2.12" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.12.cmml"><mi id="S5.SS3.SSS2.2.p2.14.m14.2.2.12.2" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.12.2.cmml">n</mi><mi id="S5.SS3.SSS2.2.p2.14.m14.2.2.12.3" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.12.3.cmml">q</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.14.m14.2b"><apply id="S5.SS3.SSS2.2.p2.14.m14.2.2.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2"><and id="S5.SS3.SSS2.2.p2.14.m14.2.2a.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2"></and><apply id="S5.SS3.SSS2.2.p2.14.m14.2.2b.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2"><leq id="S5.SS3.SSS2.2.p2.14.m14.2.2.4.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.4"></leq><ci id="S5.SS3.SSS2.2.p2.14.m14.2.2.3.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.3">𝑛</ci><apply id="S5.SS3.SSS2.2.p2.14.m14.2.2.5.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.5"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.14.m14.2.2.5.1.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.5">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.14.m14.2.2.5.2.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.5.2">𝑛</ci><ci id="S5.SS3.SSS2.2.p2.14.m14.2.2.5.3.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.5.3">′′</ci></apply></apply><apply id="S5.SS3.SSS2.2.p2.14.m14.2.2c.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2"><eq id="S5.SS3.SSS2.2.p2.14.m14.2.2.6.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.6"></eq><share href="https://arxiv.org/html/2503.06017v1#S5.SS3.SSS2.2.p2.14.m14.2.2.5.cmml" id="S5.SS3.SSS2.2.p2.14.m14.2.2d.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2"></share><apply id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1"><times id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.2.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.2"></times><ci id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.3.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.3">𝑘</ci><apply id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1"><abs id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.2.1.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.2"></abs><apply id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.1">subscript</csymbol><ci id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.1.2">𝑉</ci><cn id="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.1.3.cmml" type="integer" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.1.1.1.1.3">1</cn></apply></apply></apply></apply><apply id="S5.SS3.SSS2.2.p2.14.m14.2.2e.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2"><leq id="S5.SS3.SSS2.2.p2.14.m14.2.2.7.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.7"></leq><share href="https://arxiv.org/html/2503.06017v1#S5.SS3.SSS2.2.p2.14.m14.2.2.1.cmml" id="S5.SS3.SSS2.2.p2.14.m14.2.2f.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2"></share><apply id="S5.SS3.SSS2.2.p2.14.m14.2.2.8.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.8"><times id="S5.SS3.SSS2.2.p2.14.m14.2.2.8.1.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.8.1"></times><ci id="S5.SS3.SSS2.2.p2.14.m14.2.2.8.2.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.8.2">𝑘</ci><apply id="S5.SS3.SSS2.2.p2.14.m14.1.1.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.1.1"><divide id="S5.SS3.SSS2.2.p2.14.m14.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.1.1"></divide><apply id="S5.SS3.SSS2.2.p2.14.m14.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1"><abs id="S5.SS3.SSS2.2.p2.14.m14.1.1.1.2.1.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.2"></abs><apply id="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.1">subscript</csymbol><ci id="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.1.2">𝑉</ci><ci id="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.1.1.1.1.1.3">𝑘</ci></apply></apply><ci id="S5.SS3.SSS2.2.p2.14.m14.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.1.1.3">𝑞</ci></apply></apply></apply><apply id="S5.SS3.SSS2.2.p2.14.m14.2.2g.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2"><eq id="S5.SS3.SSS2.2.p2.14.m14.2.2.9.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.9"></eq><share href="https://arxiv.org/html/2503.06017v1#S5.SS3.SSS2.2.p2.14.m14.2.2.8.cmml" id="S5.SS3.SSS2.2.p2.14.m14.2.2h.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2"></share><apply id="S5.SS3.SSS2.2.p2.14.m14.2.2.10.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.10"><divide id="S5.SS3.SSS2.2.p2.14.m14.2.2.10.1.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.10"></divide><apply id="S5.SS3.SSS2.2.p2.14.m14.2.2.10.2.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.10.2"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.14.m14.2.2.10.2.1.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.10.2">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.14.m14.2.2.10.2.2.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.10.2.2">𝑛</ci><ci id="S5.SS3.SSS2.2.p2.14.m14.2.2.10.2.3.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.10.2.3">′</ci></apply><ci id="S5.SS3.SSS2.2.p2.14.m14.2.2.10.3.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.10.3">𝑞</ci></apply></apply><apply id="S5.SS3.SSS2.2.p2.14.m14.2.2i.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2"><leq id="S5.SS3.SSS2.2.p2.14.m14.2.2.11.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.11"></leq><share href="https://arxiv.org/html/2503.06017v1#S5.SS3.SSS2.2.p2.14.m14.2.2.10.cmml" id="S5.SS3.SSS2.2.p2.14.m14.2.2j.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2"></share><apply id="S5.SS3.SSS2.2.p2.14.m14.2.2.12.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.12"><divide id="S5.SS3.SSS2.2.p2.14.m14.2.2.12.1.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.12"></divide><ci id="S5.SS3.SSS2.2.p2.14.m14.2.2.12.2.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.12.2">𝑛</ci><ci id="S5.SS3.SSS2.2.p2.14.m14.2.2.12.3.cmml" xref="S5.SS3.SSS2.2.p2.14.m14.2.2.12.3">𝑞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.14.m14.2c">n\leq n^{\prime\prime}=k|V_{1}|\leq k\frac{|V_{k}|}{q}=\frac{n^{\prime}}{q}% \leq\frac{n}{q}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.14.m14.2d">italic_n ≤ italic_n start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT = italic_k | italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≤ italic_k divide start_ARG | italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | end_ARG start_ARG italic_q end_ARG = divide start_ARG italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_q end_ARG ≤ divide start_ARG italic_n end_ARG start_ARG italic_q end_ARG</annotation></semantics></math>, implying <math alttext="n^{\prime\prime}=\Theta(n)" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.15.m15.1"><semantics id="S5.SS3.SSS2.2.p2.15.m15.1a"><mrow id="S5.SS3.SSS2.2.p2.15.m15.1.2" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.cmml"><msup id="S5.SS3.SSS2.2.p2.15.m15.1.2.2" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.2.cmml"><mi id="S5.SS3.SSS2.2.p2.15.m15.1.2.2.2" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.2.2.cmml">n</mi><mo id="S5.SS3.SSS2.2.p2.15.m15.1.2.2.3" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.2.3.cmml">′′</mo></msup><mo id="S5.SS3.SSS2.2.p2.15.m15.1.2.1" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.1.cmml">=</mo><mrow id="S5.SS3.SSS2.2.p2.15.m15.1.2.3" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.3.cmml"><mi id="S5.SS3.SSS2.2.p2.15.m15.1.2.3.2" mathvariant="normal" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.3.2.cmml">Θ</mi><mo id="S5.SS3.SSS2.2.p2.15.m15.1.2.3.1" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.3.1.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.15.m15.1.2.3.3.2" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.3.cmml"><mo id="S5.SS3.SSS2.2.p2.15.m15.1.2.3.3.2.1" stretchy="false" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.3.cmml">(</mo><mi id="S5.SS3.SSS2.2.p2.15.m15.1.1" xref="S5.SS3.SSS2.2.p2.15.m15.1.1.cmml">n</mi><mo id="S5.SS3.SSS2.2.p2.15.m15.1.2.3.3.2.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.15.m15.1b"><apply id="S5.SS3.SSS2.2.p2.15.m15.1.2.cmml" xref="S5.SS3.SSS2.2.p2.15.m15.1.2"><eq id="S5.SS3.SSS2.2.p2.15.m15.1.2.1.cmml" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.1"></eq><apply id="S5.SS3.SSS2.2.p2.15.m15.1.2.2.cmml" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.2"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.15.m15.1.2.2.1.cmml" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.2">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.15.m15.1.2.2.2.cmml" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.2.2">𝑛</ci><ci id="S5.SS3.SSS2.2.p2.15.m15.1.2.2.3.cmml" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.2.3">′′</ci></apply><apply id="S5.SS3.SSS2.2.p2.15.m15.1.2.3.cmml" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.3"><times id="S5.SS3.SSS2.2.p2.15.m15.1.2.3.1.cmml" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.3.1"></times><ci id="S5.SS3.SSS2.2.p2.15.m15.1.2.3.2.cmml" xref="S5.SS3.SSS2.2.p2.15.m15.1.2.3.2">Θ</ci><ci id="S5.SS3.SSS2.2.p2.15.m15.1.1.cmml" xref="S5.SS3.SSS2.2.p2.15.m15.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.15.m15.1c">n^{\prime\prime}=\Theta(n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.15.m15.1d">italic_n start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT = roman_Θ ( italic_n )</annotation></semantics></math>. The maximum welfare in <math alttext="G^{n}" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.16.m16.1"><semantics id="S5.SS3.SSS2.2.p2.16.m16.1a"><msup id="S5.SS3.SSS2.2.p2.16.m16.1.1" xref="S5.SS3.SSS2.2.p2.16.m16.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.16.m16.1.1.2" xref="S5.SS3.SSS2.2.p2.16.m16.1.1.2.cmml">G</mi><mi id="S5.SS3.SSS2.2.p2.16.m16.1.1.3" xref="S5.SS3.SSS2.2.p2.16.m16.1.1.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.16.m16.1b"><apply id="S5.SS3.SSS2.2.p2.16.m16.1.1.cmml" xref="S5.SS3.SSS2.2.p2.16.m16.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.16.m16.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.16.m16.1.1">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.16.m16.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.16.m16.1.1.2">𝐺</ci><ci id="S5.SS3.SSS2.2.p2.16.m16.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.16.m16.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.16.m16.1c">G^{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.16.m16.1d">italic_G start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> serves as an upper bound for the maximum welfare in <math alttext="G" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.17.m17.1"><semantics id="S5.SS3.SSS2.2.p2.17.m17.1a"><mi id="S5.SS3.SSS2.2.p2.17.m17.1.1" xref="S5.SS3.SSS2.2.p2.17.m17.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.17.m17.1b"><ci id="S5.SS3.SSS2.2.p2.17.m17.1.1.cmml" xref="S5.SS3.SSS2.2.p2.17.m17.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.17.m17.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.17.m17.1d">italic_G</annotation></semantics></math>, since <math alttext="G" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.18.m18.1"><semantics id="S5.SS3.SSS2.2.p2.18.m18.1a"><mi id="S5.SS3.SSS2.2.p2.18.m18.1.1" xref="S5.SS3.SSS2.2.p2.18.m18.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.18.m18.1b"><ci id="S5.SS3.SSS2.2.p2.18.m18.1.1.cmml" xref="S5.SS3.SSS2.2.p2.18.m18.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.18.m18.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.18.m18.1d">italic_G</annotation></semantics></math> is a subgraph of <math alttext="G^{n}" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.19.m19.1"><semantics id="S5.SS3.SSS2.2.p2.19.m19.1a"><msup id="S5.SS3.SSS2.2.p2.19.m19.1.1" xref="S5.SS3.SSS2.2.p2.19.m19.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.19.m19.1.1.2" xref="S5.SS3.SSS2.2.p2.19.m19.1.1.2.cmml">G</mi><mi id="S5.SS3.SSS2.2.p2.19.m19.1.1.3" xref="S5.SS3.SSS2.2.p2.19.m19.1.1.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.19.m19.1b"><apply id="S5.SS3.SSS2.2.p2.19.m19.1.1.cmml" xref="S5.SS3.SSS2.2.p2.19.m19.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.19.m19.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.19.m19.1.1">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.19.m19.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.19.m19.1.1.2">𝐺</ci><ci id="S5.SS3.SSS2.2.p2.19.m19.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.19.m19.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.19.m19.1c">G^{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.19.m19.1d">italic_G start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>. When <math alttext="k=\Omega(\log n)" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.20.m20.1"><semantics id="S5.SS3.SSS2.2.p2.20.m20.1a"><mrow id="S5.SS3.SSS2.2.p2.20.m20.1.1" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.20.m20.1.1.3" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.3.cmml">k</mi><mo id="S5.SS3.SSS2.2.p2.20.m20.1.1.2" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.2.cmml">=</mo><mrow id="S5.SS3.SSS2.2.p2.20.m20.1.1.1" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.20.m20.1.1.1.3" mathvariant="normal" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1.3.cmml">Ω</mi><mo id="S5.SS3.SSS2.2.p2.20.m20.1.1.1.2" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1.cmml"><mo id="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1.1.cmml">log</mi><mo id="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1a" lspace="0.167em" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1.cmml">⁡</mo><mi id="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1.2" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.20.m20.1b"><apply id="S5.SS3.SSS2.2.p2.20.m20.1.1.cmml" xref="S5.SS3.SSS2.2.p2.20.m20.1.1"><eq id="S5.SS3.SSS2.2.p2.20.m20.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.2"></eq><ci id="S5.SS3.SSS2.2.p2.20.m20.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.3">𝑘</ci><apply id="S5.SS3.SSS2.2.p2.20.m20.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1"><times id="S5.SS3.SSS2.2.p2.20.m20.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1.2"></times><ci id="S5.SS3.SSS2.2.p2.20.m20.1.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1.3">Ω</ci><apply id="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1"><log id="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1.1"></log><ci id="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.20.m20.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.20.m20.1c">k=\Omega(\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.20.m20.1d">italic_k = roman_Ω ( roman_log italic_n )</annotation></semantics></math>, this implies <math alttext="k=\Omega(\log n^{\prime\prime})" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.21.m21.1"><semantics id="S5.SS3.SSS2.2.p2.21.m21.1a"><mrow id="S5.SS3.SSS2.2.p2.21.m21.1.1" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.21.m21.1.1.3" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.3.cmml">k</mi><mo id="S5.SS3.SSS2.2.p2.21.m21.1.1.2" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.2.cmml">=</mo><mrow id="S5.SS3.SSS2.2.p2.21.m21.1.1.1" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.3" mathvariant="normal" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.3.cmml">Ω</mi><mo id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.2" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.cmml"><mo id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.1.cmml">log</mi><mo id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1a" lspace="0.167em" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.cmml">⁡</mo><msup id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.2" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.2.cmml"><mi id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.2.2" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.2.2.cmml">n</mi><mo id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.2.3" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.2.3.cmml">′′</mo></msup></mrow><mo id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.21.m21.1b"><apply id="S5.SS3.SSS2.2.p2.21.m21.1.1.cmml" xref="S5.SS3.SSS2.2.p2.21.m21.1.1"><eq id="S5.SS3.SSS2.2.p2.21.m21.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.2"></eq><ci id="S5.SS3.SSS2.2.p2.21.m21.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.3">𝑘</ci><apply id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1"><times id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.2"></times><ci id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.3">Ω</ci><apply id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1"><log id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.1"></log><apply id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.2.1.cmml" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.2">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.2.2.cmml" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.2.2">𝑛</ci><ci id="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.2.3.cmml" xref="S5.SS3.SSS2.2.p2.21.m21.1.1.1.1.1.1.2.3">′′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.21.m21.1c">k=\Omega(\log n^{\prime\prime})</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.21.m21.1d">italic_k = roman_Ω ( roman_log italic_n start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>. Therefore, by <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S5.Thmtheorem10" title="Lemma 5.10. ‣ 5.2.2 High Perturbation Regime for Random Turán Graphs ‣ 5.2 Random Multipartite Graphs ‣ 5 Beyond Worst-Case Analysis ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5.10</span></a>, the maximum welfare in <math alttext="G^{n}" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.22.m22.1"><semantics id="S5.SS3.SSS2.2.p2.22.m22.1a"><msup id="S5.SS3.SSS2.2.p2.22.m22.1.1" xref="S5.SS3.SSS2.2.p2.22.m22.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.22.m22.1.1.2" xref="S5.SS3.SSS2.2.p2.22.m22.1.1.2.cmml">G</mi><mi id="S5.SS3.SSS2.2.p2.22.m22.1.1.3" xref="S5.SS3.SSS2.2.p2.22.m22.1.1.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.22.m22.1b"><apply id="S5.SS3.SSS2.2.p2.22.m22.1.1.cmml" xref="S5.SS3.SSS2.2.p2.22.m22.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.22.m22.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.22.m22.1.1">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.22.m22.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.22.m22.1.1.2">𝐺</ci><ci id="S5.SS3.SSS2.2.p2.22.m22.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.22.m22.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.22.m22.1c">G^{n}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.22.m22.1d">italic_G start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is <math alttext="\mathcal{O}(n^{\prime\prime}\log n^{\prime\prime})" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.23.m23.1"><semantics id="S5.SS3.SSS2.2.p2.23.m23.1a"><mrow id="S5.SS3.SSS2.2.p2.23.m23.1.1" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.2.p2.23.m23.1.1.3" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.3.cmml">𝒪</mi><mo id="S5.SS3.SSS2.2.p2.23.m23.1.1.2" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.cmml"><mo id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.cmml">(</mo><mrow id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.cmml"><msup id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.2" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.2.cmml"><mi id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.2.2" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.2.2.cmml">n</mi><mo id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.2.3" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.2.3.cmml">′′</mo></msup><mo id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.1" lspace="0.167em" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.cmml"><mi id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.1" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.1.cmml">log</mi><mo id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3a" lspace="0.167em" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.cmml">⁡</mo><msup id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.2" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.2.cmml"><mi id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.2.2" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.2.2.cmml">n</mi><mo id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.2.3" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.2.3.cmml">′′</mo></msup></mrow></mrow><mo id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.23.m23.1b"><apply id="S5.SS3.SSS2.2.p2.23.m23.1.1.cmml" xref="S5.SS3.SSS2.2.p2.23.m23.1.1"><times id="S5.SS3.SSS2.2.p2.23.m23.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.2"></times><ci id="S5.SS3.SSS2.2.p2.23.m23.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.3">𝒪</ci><apply id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1"><times id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.1"></times><apply id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.2.1.cmml" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.2">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.2.2.cmml" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.2.2">𝑛</ci><ci id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.2.3.cmml" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.2.3">′′</ci></apply><apply id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3"><log id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.1.cmml" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.1"></log><apply id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.2.cmml" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.2.1.cmml" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.2">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.2.2.cmml" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.2.2">𝑛</ci><ci id="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.2.3.cmml" xref="S5.SS3.SSS2.2.p2.23.m23.1.1.1.1.1.3.2.3">′′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.23.m23.1c">\mathcal{O}(n^{\prime\prime}\log n^{\prime\prime})</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.23.m23.1d">caligraphic_O ( italic_n start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT roman_log italic_n start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> with probability <math alttext="1-\left(\frac{ce}{2\log_{e}n^{\prime\prime}}\right)^{\frac{2}{c}\log_{e}n^{% \prime\prime}}" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.24.m24.1"><semantics id="S5.SS3.SSS2.2.p2.24.m24.1a"><mrow id="S5.SS3.SSS2.2.p2.24.m24.1.2" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.cmml"><mn id="S5.SS3.SSS2.2.p2.24.m24.1.2.2" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.2.cmml">1</mn><mo id="S5.SS3.SSS2.2.p2.24.m24.1.2.1" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.1.cmml">−</mo><msup id="S5.SS3.SSS2.2.p2.24.m24.1.2.3" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.cmml"><mrow id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.2.2" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.cmml"><mo id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.2.2.1" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.cmml">(</mo><mfrac id="S5.SS3.SSS2.2.p2.24.m24.1.1" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.cmml"><mrow id="S5.SS3.SSS2.2.p2.24.m24.1.1.2" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.2.cmml"><mi id="S5.SS3.SSS2.2.p2.24.m24.1.1.2.2" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.2.2.cmml">c</mi><mo id="S5.SS3.SSS2.2.p2.24.m24.1.1.2.1" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.2.1.cmml">⁢</mo><mi id="S5.SS3.SSS2.2.p2.24.m24.1.1.2.3" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.2.3.cmml">e</mi></mrow><mrow id="S5.SS3.SSS2.2.p2.24.m24.1.1.3" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.cmml"><mn id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.2" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.2.cmml">2</mn><mo id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.1" lspace="0.167em" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.1.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.cmml"><msub id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.1" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.1.cmml"><mi id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.1.2" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.1.2.cmml">log</mi><mi id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.1.3" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.1.3.cmml">e</mi></msub><mo id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3a" lspace="0.167em" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.cmml">⁡</mo><msup id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.2" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.2.cmml"><mi id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.2.2" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.2.2.cmml">n</mi><mo id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.2.3" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.2.3.cmml">′′</mo></msup></mrow></mrow></mfrac><mo id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.2.2.2" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.cmml">)</mo></mrow><mrow id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.cmml"><mfrac id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.2" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.2.cmml"><mn id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.2.2" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.2.2.cmml">2</mn><mi id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.2.3" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.2.3.cmml">c</mi></mfrac><mo id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.1" lspace="0.167em" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.1.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.cmml"><msub id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.1" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.1.cmml"><mi id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.1.2" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.1.2.cmml">log</mi><mi id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.1.3" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.1.3.cmml">e</mi></msub><mo id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3a" lspace="0.167em" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.cmml">⁡</mo><msup id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.2" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.2.cmml"><mi id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.2.2" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.2.2.cmml">n</mi><mo id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.2.3" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.2.3.cmml">′′</mo></msup></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.24.m24.1b"><apply id="S5.SS3.SSS2.2.p2.24.m24.1.2.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2"><minus id="S5.SS3.SSS2.2.p2.24.m24.1.2.1.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.1"></minus><cn id="S5.SS3.SSS2.2.p2.24.m24.1.2.2.cmml" type="integer" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.2">1</cn><apply id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.1.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3">superscript</csymbol><apply id="S5.SS3.SSS2.2.p2.24.m24.1.1.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.2.2"><divide id="S5.SS3.SSS2.2.p2.24.m24.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.2.2"></divide><apply id="S5.SS3.SSS2.2.p2.24.m24.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.2"><times id="S5.SS3.SSS2.2.p2.24.m24.1.1.2.1.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.2.1"></times><ci id="S5.SS3.SSS2.2.p2.24.m24.1.1.2.2.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.2.2">𝑐</ci><ci id="S5.SS3.SSS2.2.p2.24.m24.1.1.2.3.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.2.3">𝑒</ci></apply><apply id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3"><times id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.1.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.1"></times><cn id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.2.cmml" type="integer" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.2">2</cn><apply id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3"><apply id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.1.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.1.1.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.1">subscript</csymbol><log id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.1.2.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.1.2"></log><ci id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.1.3.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.1.3">𝑒</ci></apply><apply id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.2.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.2"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.2.1.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.2">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.2.2.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.2.2">𝑛</ci><ci id="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.2.3.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.1.3.3.2.3">′′</ci></apply></apply></apply></apply><apply id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3"><times id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.1.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.1"></times><apply id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.2.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.2"><divide id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.2.1.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.2"></divide><cn id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.2.2.cmml" type="integer" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.2.2">2</cn><ci id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.2.3.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.2.3">𝑐</ci></apply><apply id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3"><apply id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.1.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.1.1.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.1">subscript</csymbol><log id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.1.2.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.1.2"></log><ci id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.1.3.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.1.3">𝑒</ci></apply><apply id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.2.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.2"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.2.1.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.2">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.2.2.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.2.2">𝑛</ci><ci id="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.2.3.cmml" xref="S5.SS3.SSS2.2.p2.24.m24.1.2.3.3.3.2.3">′′</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.24.m24.1c">1-\left(\frac{ce}{2\log_{e}n^{\prime\prime}}\right)^{\frac{2}{c}\log_{e}n^{% \prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.24.m24.1d">1 - ( divide start_ARG italic_c italic_e end_ARG start_ARG 2 roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 2 end_ARG start_ARG italic_c end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. Thus, in any case, <math alttext="\mathcal{SW}(\pi^{*})=\mathcal{O}(n\log n)" class="ltx_Math" display="inline" id="S5.SS3.SSS2.2.p2.25.m25.2"><semantics id="S5.SS3.SSS2.2.p2.25.m25.2a"><mrow id="S5.SS3.SSS2.2.p2.25.m25.2.2" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.cmml"><mrow id="S5.SS3.SSS2.2.p2.25.m25.1.1.1" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.3" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.3.cmml">𝒮</mi><mo id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.2" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.4" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.4.cmml">𝒲</mi><mo id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.2a" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.1.cmml"><mo id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.1.cmml">(</mo><msup id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.1" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.1.2" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.1.2.cmml">π</mi><mo id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.1.3" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.SS3.SSS2.2.p2.25.m25.2.2.3" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.3.cmml">=</mo><mrow id="S5.SS3.SSS2.2.p2.25.m25.2.2.2" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.3" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.3.cmml">𝒪</mi><mo id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.2" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.2.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.cmml"><mo id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.2" stretchy="false" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.cmml"><mi id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.2" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.2.cmml">n</mi><mo id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.1" lspace="0.167em" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.3" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.3.cmml"><mi id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.3.1" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.3.1.cmml">log</mi><mo id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.3a" lspace="0.167em" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.3.cmml">⁡</mo><mi id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.3.2" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.3" stretchy="false" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.2.p2.25.m25.2b"><apply id="S5.SS3.SSS2.2.p2.25.m25.2.2.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.2.2"><eq id="S5.SS3.SSS2.2.p2.25.m25.2.2.3.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.3"></eq><apply id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1"><times id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.2"></times><ci id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.3">𝒮</ci><ci id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.4.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.4">𝒲</ci><apply id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1">superscript</csymbol><ci id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.1.2">𝜋</ci><times id="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.1.1.1.1.1.1.3"></times></apply></apply><apply id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2"><times id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.2.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.2"></times><ci id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.3.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.3">𝒪</ci><apply id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1"><times id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.1.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.1"></times><ci id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.2.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.2">𝑛</ci><apply id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.3.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.3"><log id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.3.1.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.3.1"></log><ci id="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.3.2.cmml" xref="S5.SS3.SSS2.2.p2.25.m25.2.2.2.1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.2.p2.25.m25.2c">\mathcal{SW}(\pi^{*})=\mathcal{O}(n\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.2.p2.25.m25.2d">caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = caligraphic_O ( italic_n roman_log italic_n )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.SS3.SSS2.3.p3"> <p class="ltx_p" id="S5.SS3.SSS2.3.p3.2">By union bound, with probability at least</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex39"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="1-n^{\prime}e^{-\Theta(\frac{n^{\prime}}{k})}-\left(\frac{ce}{2\log_{e}n^{% \prime\prime}}\right)^{\frac{2}{c}\log_{e}n^{\prime\prime}}\geq 1-ne^{-\Theta(% \frac{n}{k})}-\left(\frac{ce}{2\log_{e}n}\right)^{\frac{2}{c}\log_{e}n}" class="ltx_Math" display="block" id="S5.Ex39.m1.4"><semantics id="S5.Ex39.m1.4a"><mrow id="S5.Ex39.m1.4.5" xref="S5.Ex39.m1.4.5.cmml"><mrow id="S5.Ex39.m1.4.5.2" xref="S5.Ex39.m1.4.5.2.cmml"><mn id="S5.Ex39.m1.4.5.2.2" xref="S5.Ex39.m1.4.5.2.2.cmml">1</mn><mo id="S5.Ex39.m1.4.5.2.1" xref="S5.Ex39.m1.4.5.2.1.cmml">−</mo><mrow id="S5.Ex39.m1.4.5.2.3" xref="S5.Ex39.m1.4.5.2.3.cmml"><msup id="S5.Ex39.m1.4.5.2.3.2" xref="S5.Ex39.m1.4.5.2.3.2.cmml"><mi id="S5.Ex39.m1.4.5.2.3.2.2" xref="S5.Ex39.m1.4.5.2.3.2.2.cmml">n</mi><mo id="S5.Ex39.m1.4.5.2.3.2.3" xref="S5.Ex39.m1.4.5.2.3.2.3.cmml">′</mo></msup><mo id="S5.Ex39.m1.4.5.2.3.1" xref="S5.Ex39.m1.4.5.2.3.1.cmml">⁢</mo><msup id="S5.Ex39.m1.4.5.2.3.3" xref="S5.Ex39.m1.4.5.2.3.3.cmml"><mi id="S5.Ex39.m1.4.5.2.3.3.2" xref="S5.Ex39.m1.4.5.2.3.3.2.cmml">e</mi><mrow id="S5.Ex39.m1.1.1.1" xref="S5.Ex39.m1.1.1.1.cmml"><mo id="S5.Ex39.m1.1.1.1a" xref="S5.Ex39.m1.1.1.1.cmml">−</mo><mrow id="S5.Ex39.m1.1.1.1.3" xref="S5.Ex39.m1.1.1.1.3.cmml"><mi id="S5.Ex39.m1.1.1.1.3.2" mathvariant="normal" xref="S5.Ex39.m1.1.1.1.3.2.cmml">Θ</mi><mo id="S5.Ex39.m1.1.1.1.3.1" xref="S5.Ex39.m1.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.Ex39.m1.1.1.1.3.3.2" xref="S5.Ex39.m1.1.1.1.1.cmml"><mo id="S5.Ex39.m1.1.1.1.3.3.2.1" stretchy="false" xref="S5.Ex39.m1.1.1.1.1.cmml">(</mo><mfrac id="S5.Ex39.m1.1.1.1.1" xref="S5.Ex39.m1.1.1.1.1.cmml"><msup id="S5.Ex39.m1.1.1.1.1.2" xref="S5.Ex39.m1.1.1.1.1.2.cmml"><mi id="S5.Ex39.m1.1.1.1.1.2.2" xref="S5.Ex39.m1.1.1.1.1.2.2.cmml">n</mi><mo id="S5.Ex39.m1.1.1.1.1.2.3" xref="S5.Ex39.m1.1.1.1.1.2.3.cmml">′</mo></msup><mi id="S5.Ex39.m1.1.1.1.1.3" xref="S5.Ex39.m1.1.1.1.1.3.cmml">k</mi></mfrac><mo id="S5.Ex39.m1.1.1.1.3.3.2.2" stretchy="false" xref="S5.Ex39.m1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><mo id="S5.Ex39.m1.4.5.2.1a" xref="S5.Ex39.m1.4.5.2.1.cmml">−</mo><msup id="S5.Ex39.m1.4.5.2.4" xref="S5.Ex39.m1.4.5.2.4.cmml"><mrow id="S5.Ex39.m1.4.5.2.4.2.2" xref="S5.Ex39.m1.3.3.cmml"><mo id="S5.Ex39.m1.4.5.2.4.2.2.1" xref="S5.Ex39.m1.3.3.cmml">(</mo><mfrac id="S5.Ex39.m1.3.3" xref="S5.Ex39.m1.3.3.cmml"><mrow id="S5.Ex39.m1.3.3.2" xref="S5.Ex39.m1.3.3.2.cmml"><mi id="S5.Ex39.m1.3.3.2.2" xref="S5.Ex39.m1.3.3.2.2.cmml">c</mi><mo id="S5.Ex39.m1.3.3.2.1" xref="S5.Ex39.m1.3.3.2.1.cmml">⁢</mo><mi id="S5.Ex39.m1.3.3.2.3" xref="S5.Ex39.m1.3.3.2.3.cmml">e</mi></mrow><mrow id="S5.Ex39.m1.3.3.3" xref="S5.Ex39.m1.3.3.3.cmml"><mn id="S5.Ex39.m1.3.3.3.2" xref="S5.Ex39.m1.3.3.3.2.cmml">2</mn><mo id="S5.Ex39.m1.3.3.3.1" lspace="0.167em" xref="S5.Ex39.m1.3.3.3.1.cmml">⁢</mo><mrow id="S5.Ex39.m1.3.3.3.3" xref="S5.Ex39.m1.3.3.3.3.cmml"><msub id="S5.Ex39.m1.3.3.3.3.1" xref="S5.Ex39.m1.3.3.3.3.1.cmml"><mi id="S5.Ex39.m1.3.3.3.3.1.2" xref="S5.Ex39.m1.3.3.3.3.1.2.cmml">log</mi><mi id="S5.Ex39.m1.3.3.3.3.1.3" xref="S5.Ex39.m1.3.3.3.3.1.3.cmml">e</mi></msub><mo id="S5.Ex39.m1.3.3.3.3a" lspace="0.167em" xref="S5.Ex39.m1.3.3.3.3.cmml">⁡</mo><msup id="S5.Ex39.m1.3.3.3.3.2" xref="S5.Ex39.m1.3.3.3.3.2.cmml"><mi id="S5.Ex39.m1.3.3.3.3.2.2" xref="S5.Ex39.m1.3.3.3.3.2.2.cmml">n</mi><mo id="S5.Ex39.m1.3.3.3.3.2.3" xref="S5.Ex39.m1.3.3.3.3.2.3.cmml">′′</mo></msup></mrow></mrow></mfrac><mo id="S5.Ex39.m1.4.5.2.4.2.2.2" xref="S5.Ex39.m1.3.3.cmml">)</mo></mrow><mrow id="S5.Ex39.m1.4.5.2.4.3" xref="S5.Ex39.m1.4.5.2.4.3.cmml"><mfrac id="S5.Ex39.m1.4.5.2.4.3.2" xref="S5.Ex39.m1.4.5.2.4.3.2.cmml"><mn id="S5.Ex39.m1.4.5.2.4.3.2.2" xref="S5.Ex39.m1.4.5.2.4.3.2.2.cmml">2</mn><mi id="S5.Ex39.m1.4.5.2.4.3.2.3" xref="S5.Ex39.m1.4.5.2.4.3.2.3.cmml">c</mi></mfrac><mo id="S5.Ex39.m1.4.5.2.4.3.1" lspace="0.167em" xref="S5.Ex39.m1.4.5.2.4.3.1.cmml">⁢</mo><mrow id="S5.Ex39.m1.4.5.2.4.3.3" xref="S5.Ex39.m1.4.5.2.4.3.3.cmml"><msub id="S5.Ex39.m1.4.5.2.4.3.3.1" xref="S5.Ex39.m1.4.5.2.4.3.3.1.cmml"><mi id="S5.Ex39.m1.4.5.2.4.3.3.1.2" xref="S5.Ex39.m1.4.5.2.4.3.3.1.2.cmml">log</mi><mi id="S5.Ex39.m1.4.5.2.4.3.3.1.3" xref="S5.Ex39.m1.4.5.2.4.3.3.1.3.cmml">e</mi></msub><mo id="S5.Ex39.m1.4.5.2.4.3.3a" lspace="0.167em" xref="S5.Ex39.m1.4.5.2.4.3.3.cmml">⁡</mo><msup id="S5.Ex39.m1.4.5.2.4.3.3.2" xref="S5.Ex39.m1.4.5.2.4.3.3.2.cmml"><mi id="S5.Ex39.m1.4.5.2.4.3.3.2.2" xref="S5.Ex39.m1.4.5.2.4.3.3.2.2.cmml">n</mi><mo id="S5.Ex39.m1.4.5.2.4.3.3.2.3" xref="S5.Ex39.m1.4.5.2.4.3.3.2.3.cmml">′′</mo></msup></mrow></mrow></msup></mrow><mo id="S5.Ex39.m1.4.5.1" xref="S5.Ex39.m1.4.5.1.cmml">≥</mo><mrow id="S5.Ex39.m1.4.5.3" xref="S5.Ex39.m1.4.5.3.cmml"><mn id="S5.Ex39.m1.4.5.3.2" xref="S5.Ex39.m1.4.5.3.2.cmml">1</mn><mo id="S5.Ex39.m1.4.5.3.1" xref="S5.Ex39.m1.4.5.3.1.cmml">−</mo><mrow id="S5.Ex39.m1.4.5.3.3" xref="S5.Ex39.m1.4.5.3.3.cmml"><mi id="S5.Ex39.m1.4.5.3.3.2" xref="S5.Ex39.m1.4.5.3.3.2.cmml">n</mi><mo id="S5.Ex39.m1.4.5.3.3.1" xref="S5.Ex39.m1.4.5.3.3.1.cmml">⁢</mo><msup id="S5.Ex39.m1.4.5.3.3.3" xref="S5.Ex39.m1.4.5.3.3.3.cmml"><mi id="S5.Ex39.m1.4.5.3.3.3.2" xref="S5.Ex39.m1.4.5.3.3.3.2.cmml">e</mi><mrow id="S5.Ex39.m1.2.2.1" xref="S5.Ex39.m1.2.2.1.cmml"><mo id="S5.Ex39.m1.2.2.1a" xref="S5.Ex39.m1.2.2.1.cmml">−</mo><mrow id="S5.Ex39.m1.2.2.1.3" xref="S5.Ex39.m1.2.2.1.3.cmml"><mi id="S5.Ex39.m1.2.2.1.3.2" mathvariant="normal" xref="S5.Ex39.m1.2.2.1.3.2.cmml">Θ</mi><mo id="S5.Ex39.m1.2.2.1.3.1" xref="S5.Ex39.m1.2.2.1.3.1.cmml">⁢</mo><mrow id="S5.Ex39.m1.2.2.1.3.3.2" xref="S5.Ex39.m1.2.2.1.1.cmml"><mo id="S5.Ex39.m1.2.2.1.3.3.2.1" stretchy="false" xref="S5.Ex39.m1.2.2.1.1.cmml">(</mo><mfrac id="S5.Ex39.m1.2.2.1.1" xref="S5.Ex39.m1.2.2.1.1.cmml"><mi id="S5.Ex39.m1.2.2.1.1.2" xref="S5.Ex39.m1.2.2.1.1.2.cmml">n</mi><mi id="S5.Ex39.m1.2.2.1.1.3" xref="S5.Ex39.m1.2.2.1.1.3.cmml">k</mi></mfrac><mo id="S5.Ex39.m1.2.2.1.3.3.2.2" stretchy="false" xref="S5.Ex39.m1.2.2.1.1.cmml">)</mo></mrow></mrow></mrow></msup></mrow><mo id="S5.Ex39.m1.4.5.3.1a" xref="S5.Ex39.m1.4.5.3.1.cmml">−</mo><msup id="S5.Ex39.m1.4.5.3.4" xref="S5.Ex39.m1.4.5.3.4.cmml"><mrow id="S5.Ex39.m1.4.5.3.4.2.2" xref="S5.Ex39.m1.4.4.cmml"><mo id="S5.Ex39.m1.4.5.3.4.2.2.1" xref="S5.Ex39.m1.4.4.cmml">(</mo><mfrac id="S5.Ex39.m1.4.4" xref="S5.Ex39.m1.4.4.cmml"><mrow id="S5.Ex39.m1.4.4.2" xref="S5.Ex39.m1.4.4.2.cmml"><mi id="S5.Ex39.m1.4.4.2.2" xref="S5.Ex39.m1.4.4.2.2.cmml">c</mi><mo id="S5.Ex39.m1.4.4.2.1" xref="S5.Ex39.m1.4.4.2.1.cmml">⁢</mo><mi id="S5.Ex39.m1.4.4.2.3" xref="S5.Ex39.m1.4.4.2.3.cmml">e</mi></mrow><mrow id="S5.Ex39.m1.4.4.3" xref="S5.Ex39.m1.4.4.3.cmml"><mn id="S5.Ex39.m1.4.4.3.2" xref="S5.Ex39.m1.4.4.3.2.cmml">2</mn><mo id="S5.Ex39.m1.4.4.3.1" lspace="0.167em" xref="S5.Ex39.m1.4.4.3.1.cmml">⁢</mo><mrow id="S5.Ex39.m1.4.4.3.3" xref="S5.Ex39.m1.4.4.3.3.cmml"><msub id="S5.Ex39.m1.4.4.3.3.1" xref="S5.Ex39.m1.4.4.3.3.1.cmml"><mi id="S5.Ex39.m1.4.4.3.3.1.2" xref="S5.Ex39.m1.4.4.3.3.1.2.cmml">log</mi><mi id="S5.Ex39.m1.4.4.3.3.1.3" xref="S5.Ex39.m1.4.4.3.3.1.3.cmml">e</mi></msub><mo id="S5.Ex39.m1.4.4.3.3a" lspace="0.167em" xref="S5.Ex39.m1.4.4.3.3.cmml">⁡</mo><mi id="S5.Ex39.m1.4.4.3.3.2" xref="S5.Ex39.m1.4.4.3.3.2.cmml">n</mi></mrow></mrow></mfrac><mo id="S5.Ex39.m1.4.5.3.4.2.2.2" xref="S5.Ex39.m1.4.4.cmml">)</mo></mrow><mrow id="S5.Ex39.m1.4.5.3.4.3" xref="S5.Ex39.m1.4.5.3.4.3.cmml"><mfrac id="S5.Ex39.m1.4.5.3.4.3.2" xref="S5.Ex39.m1.4.5.3.4.3.2.cmml"><mn id="S5.Ex39.m1.4.5.3.4.3.2.2" xref="S5.Ex39.m1.4.5.3.4.3.2.2.cmml">2</mn><mi id="S5.Ex39.m1.4.5.3.4.3.2.3" xref="S5.Ex39.m1.4.5.3.4.3.2.3.cmml">c</mi></mfrac><mo id="S5.Ex39.m1.4.5.3.4.3.1" lspace="0.167em" xref="S5.Ex39.m1.4.5.3.4.3.1.cmml">⁢</mo><mrow id="S5.Ex39.m1.4.5.3.4.3.3" xref="S5.Ex39.m1.4.5.3.4.3.3.cmml"><msub id="S5.Ex39.m1.4.5.3.4.3.3.1" xref="S5.Ex39.m1.4.5.3.4.3.3.1.cmml"><mi id="S5.Ex39.m1.4.5.3.4.3.3.1.2" xref="S5.Ex39.m1.4.5.3.4.3.3.1.2.cmml">log</mi><mi id="S5.Ex39.m1.4.5.3.4.3.3.1.3" xref="S5.Ex39.m1.4.5.3.4.3.3.1.3.cmml">e</mi></msub><mo id="S5.Ex39.m1.4.5.3.4.3.3a" lspace="0.167em" xref="S5.Ex39.m1.4.5.3.4.3.3.cmml">⁡</mo><mi id="S5.Ex39.m1.4.5.3.4.3.3.2" xref="S5.Ex39.m1.4.5.3.4.3.3.2.cmml">n</mi></mrow></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex39.m1.4b"><apply id="S5.Ex39.m1.4.5.cmml" xref="S5.Ex39.m1.4.5"><geq id="S5.Ex39.m1.4.5.1.cmml" xref="S5.Ex39.m1.4.5.1"></geq><apply id="S5.Ex39.m1.4.5.2.cmml" xref="S5.Ex39.m1.4.5.2"><minus id="S5.Ex39.m1.4.5.2.1.cmml" xref="S5.Ex39.m1.4.5.2.1"></minus><cn id="S5.Ex39.m1.4.5.2.2.cmml" type="integer" xref="S5.Ex39.m1.4.5.2.2">1</cn><apply id="S5.Ex39.m1.4.5.2.3.cmml" xref="S5.Ex39.m1.4.5.2.3"><times id="S5.Ex39.m1.4.5.2.3.1.cmml" xref="S5.Ex39.m1.4.5.2.3.1"></times><apply id="S5.Ex39.m1.4.5.2.3.2.cmml" xref="S5.Ex39.m1.4.5.2.3.2"><csymbol cd="ambiguous" id="S5.Ex39.m1.4.5.2.3.2.1.cmml" xref="S5.Ex39.m1.4.5.2.3.2">superscript</csymbol><ci id="S5.Ex39.m1.4.5.2.3.2.2.cmml" xref="S5.Ex39.m1.4.5.2.3.2.2">𝑛</ci><ci id="S5.Ex39.m1.4.5.2.3.2.3.cmml" xref="S5.Ex39.m1.4.5.2.3.2.3">′</ci></apply><apply id="S5.Ex39.m1.4.5.2.3.3.cmml" xref="S5.Ex39.m1.4.5.2.3.3"><csymbol cd="ambiguous" id="S5.Ex39.m1.4.5.2.3.3.1.cmml" xref="S5.Ex39.m1.4.5.2.3.3">superscript</csymbol><ci id="S5.Ex39.m1.4.5.2.3.3.2.cmml" xref="S5.Ex39.m1.4.5.2.3.3.2">𝑒</ci><apply id="S5.Ex39.m1.1.1.1.cmml" xref="S5.Ex39.m1.1.1.1"><minus id="S5.Ex39.m1.1.1.1.2.cmml" xref="S5.Ex39.m1.1.1.1"></minus><apply id="S5.Ex39.m1.1.1.1.3.cmml" xref="S5.Ex39.m1.1.1.1.3"><times id="S5.Ex39.m1.1.1.1.3.1.cmml" xref="S5.Ex39.m1.1.1.1.3.1"></times><ci id="S5.Ex39.m1.1.1.1.3.2.cmml" xref="S5.Ex39.m1.1.1.1.3.2">Θ</ci><apply id="S5.Ex39.m1.1.1.1.1.cmml" xref="S5.Ex39.m1.1.1.1.3.3.2"><divide id="S5.Ex39.m1.1.1.1.1.1.cmml" xref="S5.Ex39.m1.1.1.1.3.3.2"></divide><apply id="S5.Ex39.m1.1.1.1.1.2.cmml" xref="S5.Ex39.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.Ex39.m1.1.1.1.1.2.1.cmml" xref="S5.Ex39.m1.1.1.1.1.2">superscript</csymbol><ci id="S5.Ex39.m1.1.1.1.1.2.2.cmml" xref="S5.Ex39.m1.1.1.1.1.2.2">𝑛</ci><ci id="S5.Ex39.m1.1.1.1.1.2.3.cmml" xref="S5.Ex39.m1.1.1.1.1.2.3">′</ci></apply><ci id="S5.Ex39.m1.1.1.1.1.3.cmml" xref="S5.Ex39.m1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply><apply id="S5.Ex39.m1.4.5.2.4.cmml" xref="S5.Ex39.m1.4.5.2.4"><csymbol cd="ambiguous" id="S5.Ex39.m1.4.5.2.4.1.cmml" xref="S5.Ex39.m1.4.5.2.4">superscript</csymbol><apply id="S5.Ex39.m1.3.3.cmml" xref="S5.Ex39.m1.4.5.2.4.2.2"><divide id="S5.Ex39.m1.3.3.1.cmml" xref="S5.Ex39.m1.4.5.2.4.2.2"></divide><apply id="S5.Ex39.m1.3.3.2.cmml" xref="S5.Ex39.m1.3.3.2"><times id="S5.Ex39.m1.3.3.2.1.cmml" xref="S5.Ex39.m1.3.3.2.1"></times><ci id="S5.Ex39.m1.3.3.2.2.cmml" xref="S5.Ex39.m1.3.3.2.2">𝑐</ci><ci id="S5.Ex39.m1.3.3.2.3.cmml" xref="S5.Ex39.m1.3.3.2.3">𝑒</ci></apply><apply id="S5.Ex39.m1.3.3.3.cmml" xref="S5.Ex39.m1.3.3.3"><times id="S5.Ex39.m1.3.3.3.1.cmml" xref="S5.Ex39.m1.3.3.3.1"></times><cn id="S5.Ex39.m1.3.3.3.2.cmml" type="integer" xref="S5.Ex39.m1.3.3.3.2">2</cn><apply id="S5.Ex39.m1.3.3.3.3.cmml" xref="S5.Ex39.m1.3.3.3.3"><apply id="S5.Ex39.m1.3.3.3.3.1.cmml" xref="S5.Ex39.m1.3.3.3.3.1"><csymbol cd="ambiguous" id="S5.Ex39.m1.3.3.3.3.1.1.cmml" xref="S5.Ex39.m1.3.3.3.3.1">subscript</csymbol><log id="S5.Ex39.m1.3.3.3.3.1.2.cmml" xref="S5.Ex39.m1.3.3.3.3.1.2"></log><ci id="S5.Ex39.m1.3.3.3.3.1.3.cmml" xref="S5.Ex39.m1.3.3.3.3.1.3">𝑒</ci></apply><apply id="S5.Ex39.m1.3.3.3.3.2.cmml" xref="S5.Ex39.m1.3.3.3.3.2"><csymbol cd="ambiguous" id="S5.Ex39.m1.3.3.3.3.2.1.cmml" xref="S5.Ex39.m1.3.3.3.3.2">superscript</csymbol><ci id="S5.Ex39.m1.3.3.3.3.2.2.cmml" xref="S5.Ex39.m1.3.3.3.3.2.2">𝑛</ci><ci id="S5.Ex39.m1.3.3.3.3.2.3.cmml" xref="S5.Ex39.m1.3.3.3.3.2.3">′′</ci></apply></apply></apply></apply><apply id="S5.Ex39.m1.4.5.2.4.3.cmml" xref="S5.Ex39.m1.4.5.2.4.3"><times id="S5.Ex39.m1.4.5.2.4.3.1.cmml" xref="S5.Ex39.m1.4.5.2.4.3.1"></times><apply id="S5.Ex39.m1.4.5.2.4.3.2.cmml" xref="S5.Ex39.m1.4.5.2.4.3.2"><divide id="S5.Ex39.m1.4.5.2.4.3.2.1.cmml" xref="S5.Ex39.m1.4.5.2.4.3.2"></divide><cn id="S5.Ex39.m1.4.5.2.4.3.2.2.cmml" type="integer" xref="S5.Ex39.m1.4.5.2.4.3.2.2">2</cn><ci id="S5.Ex39.m1.4.5.2.4.3.2.3.cmml" xref="S5.Ex39.m1.4.5.2.4.3.2.3">𝑐</ci></apply><apply id="S5.Ex39.m1.4.5.2.4.3.3.cmml" xref="S5.Ex39.m1.4.5.2.4.3.3"><apply id="S5.Ex39.m1.4.5.2.4.3.3.1.cmml" xref="S5.Ex39.m1.4.5.2.4.3.3.1"><csymbol cd="ambiguous" id="S5.Ex39.m1.4.5.2.4.3.3.1.1.cmml" xref="S5.Ex39.m1.4.5.2.4.3.3.1">subscript</csymbol><log id="S5.Ex39.m1.4.5.2.4.3.3.1.2.cmml" xref="S5.Ex39.m1.4.5.2.4.3.3.1.2"></log><ci id="S5.Ex39.m1.4.5.2.4.3.3.1.3.cmml" xref="S5.Ex39.m1.4.5.2.4.3.3.1.3">𝑒</ci></apply><apply id="S5.Ex39.m1.4.5.2.4.3.3.2.cmml" xref="S5.Ex39.m1.4.5.2.4.3.3.2"><csymbol cd="ambiguous" id="S5.Ex39.m1.4.5.2.4.3.3.2.1.cmml" xref="S5.Ex39.m1.4.5.2.4.3.3.2">superscript</csymbol><ci id="S5.Ex39.m1.4.5.2.4.3.3.2.2.cmml" xref="S5.Ex39.m1.4.5.2.4.3.3.2.2">𝑛</ci><ci id="S5.Ex39.m1.4.5.2.4.3.3.2.3.cmml" xref="S5.Ex39.m1.4.5.2.4.3.3.2.3">′′</ci></apply></apply></apply></apply></apply><apply id="S5.Ex39.m1.4.5.3.cmml" xref="S5.Ex39.m1.4.5.3"><minus id="S5.Ex39.m1.4.5.3.1.cmml" xref="S5.Ex39.m1.4.5.3.1"></minus><cn id="S5.Ex39.m1.4.5.3.2.cmml" type="integer" xref="S5.Ex39.m1.4.5.3.2">1</cn><apply id="S5.Ex39.m1.4.5.3.3.cmml" xref="S5.Ex39.m1.4.5.3.3"><times id="S5.Ex39.m1.4.5.3.3.1.cmml" xref="S5.Ex39.m1.4.5.3.3.1"></times><ci id="S5.Ex39.m1.4.5.3.3.2.cmml" xref="S5.Ex39.m1.4.5.3.3.2">𝑛</ci><apply id="S5.Ex39.m1.4.5.3.3.3.cmml" xref="S5.Ex39.m1.4.5.3.3.3"><csymbol cd="ambiguous" id="S5.Ex39.m1.4.5.3.3.3.1.cmml" xref="S5.Ex39.m1.4.5.3.3.3">superscript</csymbol><ci id="S5.Ex39.m1.4.5.3.3.3.2.cmml" xref="S5.Ex39.m1.4.5.3.3.3.2">𝑒</ci><apply id="S5.Ex39.m1.2.2.1.cmml" xref="S5.Ex39.m1.2.2.1"><minus id="S5.Ex39.m1.2.2.1.2.cmml" xref="S5.Ex39.m1.2.2.1"></minus><apply id="S5.Ex39.m1.2.2.1.3.cmml" xref="S5.Ex39.m1.2.2.1.3"><times id="S5.Ex39.m1.2.2.1.3.1.cmml" xref="S5.Ex39.m1.2.2.1.3.1"></times><ci id="S5.Ex39.m1.2.2.1.3.2.cmml" xref="S5.Ex39.m1.2.2.1.3.2">Θ</ci><apply id="S5.Ex39.m1.2.2.1.1.cmml" xref="S5.Ex39.m1.2.2.1.3.3.2"><divide id="S5.Ex39.m1.2.2.1.1.1.cmml" xref="S5.Ex39.m1.2.2.1.3.3.2"></divide><ci id="S5.Ex39.m1.2.2.1.1.2.cmml" xref="S5.Ex39.m1.2.2.1.1.2">𝑛</ci><ci id="S5.Ex39.m1.2.2.1.1.3.cmml" xref="S5.Ex39.m1.2.2.1.1.3">𝑘</ci></apply></apply></apply></apply></apply><apply id="S5.Ex39.m1.4.5.3.4.cmml" xref="S5.Ex39.m1.4.5.3.4"><csymbol cd="ambiguous" id="S5.Ex39.m1.4.5.3.4.1.cmml" xref="S5.Ex39.m1.4.5.3.4">superscript</csymbol><apply id="S5.Ex39.m1.4.4.cmml" xref="S5.Ex39.m1.4.5.3.4.2.2"><divide id="S5.Ex39.m1.4.4.1.cmml" xref="S5.Ex39.m1.4.5.3.4.2.2"></divide><apply id="S5.Ex39.m1.4.4.2.cmml" xref="S5.Ex39.m1.4.4.2"><times id="S5.Ex39.m1.4.4.2.1.cmml" xref="S5.Ex39.m1.4.4.2.1"></times><ci id="S5.Ex39.m1.4.4.2.2.cmml" xref="S5.Ex39.m1.4.4.2.2">𝑐</ci><ci id="S5.Ex39.m1.4.4.2.3.cmml" xref="S5.Ex39.m1.4.4.2.3">𝑒</ci></apply><apply id="S5.Ex39.m1.4.4.3.cmml" xref="S5.Ex39.m1.4.4.3"><times id="S5.Ex39.m1.4.4.3.1.cmml" xref="S5.Ex39.m1.4.4.3.1"></times><cn id="S5.Ex39.m1.4.4.3.2.cmml" type="integer" xref="S5.Ex39.m1.4.4.3.2">2</cn><apply id="S5.Ex39.m1.4.4.3.3.cmml" xref="S5.Ex39.m1.4.4.3.3"><apply id="S5.Ex39.m1.4.4.3.3.1.cmml" xref="S5.Ex39.m1.4.4.3.3.1"><csymbol cd="ambiguous" id="S5.Ex39.m1.4.4.3.3.1.1.cmml" xref="S5.Ex39.m1.4.4.3.3.1">subscript</csymbol><log id="S5.Ex39.m1.4.4.3.3.1.2.cmml" xref="S5.Ex39.m1.4.4.3.3.1.2"></log><ci id="S5.Ex39.m1.4.4.3.3.1.3.cmml" xref="S5.Ex39.m1.4.4.3.3.1.3">𝑒</ci></apply><ci id="S5.Ex39.m1.4.4.3.3.2.cmml" xref="S5.Ex39.m1.4.4.3.3.2">𝑛</ci></apply></apply></apply><apply id="S5.Ex39.m1.4.5.3.4.3.cmml" xref="S5.Ex39.m1.4.5.3.4.3"><times id="S5.Ex39.m1.4.5.3.4.3.1.cmml" xref="S5.Ex39.m1.4.5.3.4.3.1"></times><apply id="S5.Ex39.m1.4.5.3.4.3.2.cmml" xref="S5.Ex39.m1.4.5.3.4.3.2"><divide id="S5.Ex39.m1.4.5.3.4.3.2.1.cmml" xref="S5.Ex39.m1.4.5.3.4.3.2"></divide><cn id="S5.Ex39.m1.4.5.3.4.3.2.2.cmml" type="integer" xref="S5.Ex39.m1.4.5.3.4.3.2.2">2</cn><ci id="S5.Ex39.m1.4.5.3.4.3.2.3.cmml" xref="S5.Ex39.m1.4.5.3.4.3.2.3">𝑐</ci></apply><apply id="S5.Ex39.m1.4.5.3.4.3.3.cmml" xref="S5.Ex39.m1.4.5.3.4.3.3"><apply id="S5.Ex39.m1.4.5.3.4.3.3.1.cmml" xref="S5.Ex39.m1.4.5.3.4.3.3.1"><csymbol cd="ambiguous" id="S5.Ex39.m1.4.5.3.4.3.3.1.1.cmml" xref="S5.Ex39.m1.4.5.3.4.3.3.1">subscript</csymbol><log id="S5.Ex39.m1.4.5.3.4.3.3.1.2.cmml" xref="S5.Ex39.m1.4.5.3.4.3.3.1.2"></log><ci id="S5.Ex39.m1.4.5.3.4.3.3.1.3.cmml" xref="S5.Ex39.m1.4.5.3.4.3.3.1.3">𝑒</ci></apply><ci id="S5.Ex39.m1.4.5.3.4.3.3.2.cmml" xref="S5.Ex39.m1.4.5.3.4.3.3.2">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex39.m1.4c">1-n^{\prime}e^{-\Theta(\frac{n^{\prime}}{k})}-\left(\frac{ce}{2\log_{e}n^{% \prime\prime}}\right)^{\frac{2}{c}\log_{e}n^{\prime\prime}}\geq 1-ne^{-\Theta(% \frac{n}{k})}-\left(\frac{ce}{2\log_{e}n}\right)^{\frac{2}{c}\log_{e}n}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex39.m1.4d">1 - italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT - ( divide start_ARG italic_c italic_e end_ARG start_ARG 2 roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 2 end_ARG start_ARG italic_c end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ≥ 1 - italic_n italic_e start_POSTSUPERSCRIPT - roman_Θ ( divide start_ARG italic_n end_ARG start_ARG italic_k end_ARG ) end_POSTSUPERSCRIPT - ( divide start_ARG italic_c italic_e end_ARG start_ARG 2 roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 2 end_ARG start_ARG italic_c end_ARG roman_log start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.SS3.SSS2.3.p3.1">we obtain <math alttext="\mathcal{SW}(\pi)\geq\Omega\left(\frac{1}{\log n}\right)\mathcal{SW}(\pi^{*})" class="ltx_Math" display="inline" id="S5.SS3.SSS2.3.p3.1.m1.3"><semantics id="S5.SS3.SSS2.3.p3.1.m1.3a"><mrow id="S5.SS3.SSS2.3.p3.1.m1.3.3" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.cmml"><mrow id="S5.SS3.SSS2.3.p3.1.m1.3.3.3" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.3.p3.1.m1.3.3.3.2" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.3.2.cmml">𝒮</mi><mo id="S5.SS3.SSS2.3.p3.1.m1.3.3.3.1" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.3.p3.1.m1.3.3.3.3" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.3.3.cmml">𝒲</mi><mo id="S5.SS3.SSS2.3.p3.1.m1.3.3.3.1a" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.3.1.cmml">⁢</mo><mrow id="S5.SS3.SSS2.3.p3.1.m1.3.3.3.4.2" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.3.cmml"><mo id="S5.SS3.SSS2.3.p3.1.m1.3.3.3.4.2.1" stretchy="false" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.3.cmml">(</mo><mi id="S5.SS3.SSS2.3.p3.1.m1.1.1" xref="S5.SS3.SSS2.3.p3.1.m1.1.1.cmml">π</mi><mo id="S5.SS3.SSS2.3.p3.1.m1.3.3.3.4.2.2" stretchy="false" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.3.cmml">)</mo></mrow></mrow><mo id="S5.SS3.SSS2.3.p3.1.m1.3.3.2" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.2.cmml">≥</mo><mrow id="S5.SS3.SSS2.3.p3.1.m1.3.3.1" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.cmml"><mi id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.3" mathvariant="normal" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.3.cmml">Ω</mi><mo id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.2" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.4.2" xref="S5.SS3.SSS2.3.p3.1.m1.2.2.cmml"><mo id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.4.2.1" xref="S5.SS3.SSS2.3.p3.1.m1.2.2.cmml">(</mo><mfrac id="S5.SS3.SSS2.3.p3.1.m1.2.2" xref="S5.SS3.SSS2.3.p3.1.m1.2.2.cmml"><mn id="S5.SS3.SSS2.3.p3.1.m1.2.2.2" xref="S5.SS3.SSS2.3.p3.1.m1.2.2.2.cmml">1</mn><mrow id="S5.SS3.SSS2.3.p3.1.m1.2.2.3" xref="S5.SS3.SSS2.3.p3.1.m1.2.2.3.cmml"><mi id="S5.SS3.SSS2.3.p3.1.m1.2.2.3.1" xref="S5.SS3.SSS2.3.p3.1.m1.2.2.3.1.cmml">log</mi><mo id="S5.SS3.SSS2.3.p3.1.m1.2.2.3a" lspace="0.167em" xref="S5.SS3.SSS2.3.p3.1.m1.2.2.3.cmml">⁡</mo><mi id="S5.SS3.SSS2.3.p3.1.m1.2.2.3.2" xref="S5.SS3.SSS2.3.p3.1.m1.2.2.3.2.cmml">n</mi></mrow></mfrac><mo id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.4.2.2" xref="S5.SS3.SSS2.3.p3.1.m1.2.2.cmml">)</mo></mrow><mo id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.2a" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.5" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.5.cmml">𝒮</mi><mo id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.2b" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.6" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.6.cmml">𝒲</mi><mo id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.2c" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.1.cmml"><mo id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.2" stretchy="false" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.1.cmml">(</mo><msup id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.1" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.1.cmml"><mi id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.1.2" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.1.2.cmml">π</mi><mo id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.1.3" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.1.3.cmml">∗</mo></msup><mo id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.3" stretchy="false" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS3.SSS2.3.p3.1.m1.3b"><apply id="S5.SS3.SSS2.3.p3.1.m1.3.3.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3"><geq id="S5.SS3.SSS2.3.p3.1.m1.3.3.2.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.2"></geq><apply id="S5.SS3.SSS2.3.p3.1.m1.3.3.3.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.3"><times id="S5.SS3.SSS2.3.p3.1.m1.3.3.3.1.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.3.1"></times><ci id="S5.SS3.SSS2.3.p3.1.m1.3.3.3.2.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.3.2">𝒮</ci><ci id="S5.SS3.SSS2.3.p3.1.m1.3.3.3.3.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.3.3">𝒲</ci><ci id="S5.SS3.SSS2.3.p3.1.m1.1.1.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.1.1">𝜋</ci></apply><apply id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1"><times id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.2.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.2"></times><ci id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.3.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.3">Ω</ci><apply id="S5.SS3.SSS2.3.p3.1.m1.2.2.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.4.2"><divide id="S5.SS3.SSS2.3.p3.1.m1.2.2.1.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.4.2"></divide><cn id="S5.SS3.SSS2.3.p3.1.m1.2.2.2.cmml" type="integer" xref="S5.SS3.SSS2.3.p3.1.m1.2.2.2">1</cn><apply id="S5.SS3.SSS2.3.p3.1.m1.2.2.3.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.2.2.3"><log id="S5.SS3.SSS2.3.p3.1.m1.2.2.3.1.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.2.2.3.1"></log><ci id="S5.SS3.SSS2.3.p3.1.m1.2.2.3.2.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.2.2.3.2">𝑛</ci></apply></apply><ci id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.5.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.5">𝒮</ci><ci id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.6.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.6">𝒲</ci><apply id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.1.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.1.1.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1">superscript</csymbol><ci id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.1.2.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.1.2">𝜋</ci><times id="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.1.3.cmml" xref="S5.SS3.SSS2.3.p3.1.m1.3.3.1.1.1.1.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.SSS2.3.p3.1.m1.3c">\mathcal{SW}(\pi)\geq\Omega\left(\frac{1}{\log n}\right)\mathcal{SW}(\pi^{*})</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.SSS2.3.p3.1.m1.3d">caligraphic_S caligraphic_W ( italic_π ) ≥ roman_Ω ( divide start_ARG 1 end_ARG start_ARG roman_log italic_n end_ARG ) caligraphic_S caligraphic_W ( italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math>. ∎</p> </div> </div> </section> </section> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6 </span>Conclusion</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.2">We have investigated maximizing social welfare in additively separable hedonic games. This is known to be a very hard problem in a worst-case analysis: approximating welfare better than the <math alttext="n" class="ltx_Math" display="inline" id="S6.p1.1.m1.1"><semantics id="S6.p1.1.m1.1a"><mi id="S6.p1.1.m1.1.1" xref="S6.p1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.p1.1.m1.1b"><ci id="S6.p1.1.m1.1.1.cmml" xref="S6.p1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.p1.1.m1.1d">italic_n</annotation></semantics></math>-approximation provided by maximum weight matchings faces computational boundaries. We have strengthened the existing approximation hardness to games with bounded valuations, in particular when restricting them to <math alttext="\{-1,0,1\}" class="ltx_Math" display="inline" id="S6.p1.2.m2.3"><semantics id="S6.p1.2.m2.3a"><mrow id="S6.p1.2.m2.3.3.1" xref="S6.p1.2.m2.3.3.2.cmml"><mo id="S6.p1.2.m2.3.3.1.2" stretchy="false" xref="S6.p1.2.m2.3.3.2.cmml">{</mo><mrow id="S6.p1.2.m2.3.3.1.1" xref="S6.p1.2.m2.3.3.1.1.cmml"><mo id="S6.p1.2.m2.3.3.1.1a" xref="S6.p1.2.m2.3.3.1.1.cmml">−</mo><mn id="S6.p1.2.m2.3.3.1.1.2" xref="S6.p1.2.m2.3.3.1.1.2.cmml">1</mn></mrow><mo id="S6.p1.2.m2.3.3.1.3" xref="S6.p1.2.m2.3.3.2.cmml">,</mo><mn id="S6.p1.2.m2.1.1" xref="S6.p1.2.m2.1.1.cmml">0</mn><mo id="S6.p1.2.m2.3.3.1.4" xref="S6.p1.2.m2.3.3.2.cmml">,</mo><mn id="S6.p1.2.m2.2.2" xref="S6.p1.2.m2.2.2.cmml">1</mn><mo id="S6.p1.2.m2.3.3.1.5" stretchy="false" xref="S6.p1.2.m2.3.3.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.2.m2.3b"><set id="S6.p1.2.m2.3.3.2.cmml" xref="S6.p1.2.m2.3.3.1"><apply id="S6.p1.2.m2.3.3.1.1.cmml" xref="S6.p1.2.m2.3.3.1.1"><minus id="S6.p1.2.m2.3.3.1.1.1.cmml" xref="S6.p1.2.m2.3.3.1.1"></minus><cn id="S6.p1.2.m2.3.3.1.1.2.cmml" type="integer" xref="S6.p1.2.m2.3.3.1.1.2">1</cn></apply><cn id="S6.p1.2.m2.1.1.cmml" type="integer" xref="S6.p1.2.m2.1.1">0</cn><cn id="S6.p1.2.m2.2.2.cmml" type="integer" xref="S6.p1.2.m2.2.2">1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.2.m2.3c">\{-1,0,1\}</annotation><annotation encoding="application/x-llamapun" id="S6.p1.2.m2.3d">{ - 1 , 0 , 1 }</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.p2"> <p class="ltx_p" id="S6.p2.1">By contrast, we have carved out various possibilities to obtain better approximation guarantees. In games with nonnegative total value, a randomized polynomial-time algorithm achieves a <math alttext="\mathcal{O}(\log n)" class="ltx_Math" display="inline" id="S6.p2.1.m1.1"><semantics id="S6.p2.1.m1.1a"><mrow id="S6.p2.1.m1.1.1" xref="S6.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.p2.1.m1.1.1.3" xref="S6.p2.1.m1.1.1.3.cmml">𝒪</mi><mo id="S6.p2.1.m1.1.1.2" xref="S6.p2.1.m1.1.1.2.cmml">⁢</mo><mrow id="S6.p2.1.m1.1.1.1.1" xref="S6.p2.1.m1.1.1.1.1.1.cmml"><mo id="S6.p2.1.m1.1.1.1.1.2" stretchy="false" xref="S6.p2.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S6.p2.1.m1.1.1.1.1.1" xref="S6.p2.1.m1.1.1.1.1.1.cmml"><mi id="S6.p2.1.m1.1.1.1.1.1.1" xref="S6.p2.1.m1.1.1.1.1.1.1.cmml">log</mi><mo id="S6.p2.1.m1.1.1.1.1.1a" lspace="0.167em" xref="S6.p2.1.m1.1.1.1.1.1.cmml">⁡</mo><mi id="S6.p2.1.m1.1.1.1.1.1.2" xref="S6.p2.1.m1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S6.p2.1.m1.1.1.1.1.3" stretchy="false" xref="S6.p2.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.1.m1.1b"><apply id="S6.p2.1.m1.1.1.cmml" xref="S6.p2.1.m1.1.1"><times id="S6.p2.1.m1.1.1.2.cmml" xref="S6.p2.1.m1.1.1.2"></times><ci id="S6.p2.1.m1.1.1.3.cmml" xref="S6.p2.1.m1.1.1.3">𝒪</ci><apply id="S6.p2.1.m1.1.1.1.1.1.cmml" xref="S6.p2.1.m1.1.1.1.1"><log id="S6.p2.1.m1.1.1.1.1.1.1.cmml" xref="S6.p2.1.m1.1.1.1.1.1.1"></log><ci id="S6.p2.1.m1.1.1.1.1.1.2.cmml" xref="S6.p2.1.m1.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.1.m1.1c">\mathcal{O}(\log n)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.1.m1.1d">caligraphic_O ( roman_log italic_n )</annotation></semantics></math>-approximation. Our proof establishes an interesting connection to the correlation clustering literature. Moreover, going beyond worst-case guarantees, we have defined two stochastic models of aversion-to-enemies games, i.e., the games which cause the inapproximability in the first place. In both models, we perform a high probability analysis. The first stochastic model is based on Erdős-Rényi graphs, where we can efficiently compute partitions that approximate social welfare within a constant factor. The second stochastic model is based on balanced multipartite graphs. We distinguish between a low and high perturbation regime, where we can guarantee constant and logarithmic approximations, respectively.</p> </div> <div class="ltx_para" id="S6.p3"> <p class="ltx_p" id="S6.p3.1">Social welfare is a fundamental objective in ASHGs that deserves further attention in future research. A specific open question is to investigate whether efficient approximation algorithms are possible for symmetric ASHGs with valuations of <math alttext="\{-1,1\}" class="ltx_Math" display="inline" id="S6.p3.1.m1.2"><semantics id="S6.p3.1.m1.2a"><mrow id="S6.p3.1.m1.2.2.1" xref="S6.p3.1.m1.2.2.2.cmml"><mo id="S6.p3.1.m1.2.2.1.2" stretchy="false" xref="S6.p3.1.m1.2.2.2.cmml">{</mo><mrow id="S6.p3.1.m1.2.2.1.1" xref="S6.p3.1.m1.2.2.1.1.cmml"><mo id="S6.p3.1.m1.2.2.1.1a" xref="S6.p3.1.m1.2.2.1.1.cmml">−</mo><mn id="S6.p3.1.m1.2.2.1.1.2" xref="S6.p3.1.m1.2.2.1.1.2.cmml">1</mn></mrow><mo id="S6.p3.1.m1.2.2.1.3" xref="S6.p3.1.m1.2.2.2.cmml">,</mo><mn id="S6.p3.1.m1.1.1" xref="S6.p3.1.m1.1.1.cmml">1</mn><mo id="S6.p3.1.m1.2.2.1.4" stretchy="false" xref="S6.p3.1.m1.2.2.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.p3.1.m1.2b"><set id="S6.p3.1.m1.2.2.2.cmml" xref="S6.p3.1.m1.2.2.1"><apply id="S6.p3.1.m1.2.2.1.1.cmml" xref="S6.p3.1.m1.2.2.1.1"><minus id="S6.p3.1.m1.2.2.1.1.1.cmml" xref="S6.p3.1.m1.2.2.1.1"></minus><cn id="S6.p3.1.m1.2.2.1.1.2.cmml" type="integer" xref="S6.p3.1.m1.2.2.1.1.2">1</cn></apply><cn id="S6.p3.1.m1.1.1.cmml" type="integer" xref="S6.p3.1.m1.1.1">1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.1.m1.2c">\{-1,1\}</annotation><annotation encoding="application/x-llamapun" id="S6.p3.1.m1.2d">{ - 1 , 1 }</annotation></semantics></math>, see our discussion after <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1 Welfare Inapproximability for Restricted Valuations ‣ 4 Deterministic Games ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.1</span></a>. Moreover, considering welfare approximation in suitable classes of random hedonic games might lead to intriguing discoveries. One candidate are random ASHGs with uniformly random valuations <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#bib.bibx11" title="">BK24</a>]</cite>. Finally, we restricted attention to symmetric games, which is <em class="ltx_emph ltx_font_italic" id="S6.p3.1.1">not</em> without loss of generality for aversion-to-enemies games (cf. <a class="ltx_ref" href="https://arxiv.org/html/2503.06017v1#footnote2" title="In 3 Preliminaries ‣ Welfare Approximation in Additively Separable Hedonic Games"><span class="ltx_text ltx_ref_tag">Footnote</span> <span class="ltx_text ltx_ref_tag">2</span></a>). Hence, another direction is to consider asymmetric subclasses of ASHGs.</p> </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Acknowledgements</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">Martin Bullinger is supported by the AI Programme of The Alan Turing Institute, Vaggos Chatziafratis is supported by a UCSC startup grant and a Hellman’s fellowship, and Parnian Shahkar is supported by the National Science Foundation (NSF) under grant CCF-2230414.</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bibx1"> <span class="ltx_tag ltx_tag_bibitem">[ABS13]</span> <span class="ltx_bibblock"> Haris Aziz, Felix Brandt, and Hans Georg Seedig. </span> <span class="ltx_bibblock">Computing desirable partitions in additively separable hedonic games. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx1.1.1">Artificial Intelligence</span>, 195:316–334, 2013. </span> </li> <li class="ltx_bibitem" id="bib.bibx2"> <span class="ltx_tag ltx_tag_bibitem">[AS16]</span> <span class="ltx_bibblock"> Haris Aziz and Rahul Savani. </span> <span class="ltx_bibblock">Hedonic games. </span> <span class="ltx_bibblock">In Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel D. Procaccia, editors, <span class="ltx_text ltx_font_italic" id="bib.bibx2.1.1">Handbook of Computational Social Choice</span>, chapter 15. Cambridge University Press, 2016. </span> </li> <li class="ltx_bibitem" id="bib.bibx3"> <span class="ltx_tag ltx_tag_bibitem">[Bal04]</span> <span class="ltx_bibblock"> Coralio Ballester. </span> <span class="ltx_bibblock">NP-completeness in hedonic games. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx3.1.1">Games and Economic Behavior</span>, 49(1):1–30, 2004. </span> </li> <li class="ltx_bibitem" id="bib.bibx4"> <span class="ltx_tag ltx_tag_bibitem">[BB22]</span> <span class="ltx_bibblock"> Felix Brandt and Martin Bullinger. </span> <span class="ltx_bibblock">Finding and recognizing popular coalition structures. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx4.1.1">Journal of Artificial Intelligence Research</span>, 74:569–626, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bibx5"> <span class="ltx_tag ltx_tag_bibitem">[BBC04]</span> <span class="ltx_bibblock"> Nikhil Bansal, Avrim Blum, and Shuchi Chawla. </span> <span class="ltx_bibblock">Correlation clustering. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx5.1.1">Machine Learning</span>, 56:89–113, 2004. </span> </li> <li class="ltx_bibitem" id="bib.bibx6"> <span class="ltx_tag ltx_tag_bibitem">[BBT24]</span> <span class="ltx_bibblock"> Felix Brandt, Martin Bullinger, and Leo Tappe. </span> <span class="ltx_bibblock">Stability based on single-agent deviations in additively separable hedonic games. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx6.1.1">Artificial Intelligence</span>, 334:104160, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bibx7"> <span class="ltx_tag ltx_tag_bibitem">[BBW23]</span> <span class="ltx_bibblock"> Felix Brandt, Martin Bullinger, and Anaëlle Wilczynski. </span> <span class="ltx_bibblock">Reaching individually stable coalition structures. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx7.1.1">ACM Transactions on Economics and Computation</span>, 11(1–2):4:1–65, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bibx8"> <span class="ltx_tag ltx_tag_bibitem">[BCE<sup class="ltx_sup" id="bib.bibx8.4.4.1"><span class="ltx_text ltx_font_italic" id="bib.bibx8.4.4.1.1">+</span></sup>16]</span> <span class="ltx_bibblock"> Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel D. Procaccia, editors. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx8.7.1">Handbook of Computational Social Choice</span>. </span> <span class="ltx_bibblock">Cambridge University Press, 2016. </span> </li> <li class="ltx_bibitem" id="bib.bibx9"> <span class="ltx_tag ltx_tag_bibitem">[BER24]</span> <span class="ltx_bibblock"> Martin Bullinger, Edith Elkind, and Jörg Rothe. </span> <span class="ltx_bibblock">Cooperative game theory. </span> <span class="ltx_bibblock">In Jörg Rothe, editor, <span class="ltx_text ltx_font_italic" id="bib.bibx9.1.1">Economics and Computation: An Introduction to Algorithmic Game Theory, Computational Social Choice, and Fair Division</span>, chapter 3, pages 139–229. Springer, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bibx10"> <span class="ltx_tag ltx_tag_bibitem">[BJ02]</span> <span class="ltx_bibblock"> Anna Bogomolnaia and Matthew O. Jackson. </span> <span class="ltx_bibblock">The stability of hedonic coalition structures. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx10.1.1">Games and Economic Behavior</span>, 38(2):201–230, 2002. </span> </li> <li class="ltx_bibitem" id="bib.bibx11"> <span class="ltx_tag ltx_tag_bibitem">[BK24]</span> <span class="ltx_bibblock"> Martin Bullinger and Sonja Kraiczy. </span> <span class="ltx_bibblock">Stability in random hedonic games. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bibx11.1.1">Proceedings of the 25th ACM Conference on Economics and Computation (ACM-EC)</span>, page 212, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bibx12"> <span class="ltx_tag ltx_tag_bibitem">[BKS01]</span> <span class="ltx_bibblock"> Suryapratim Banerjee, Hideo Konishi, and Tayfun Sönmez. </span> <span class="ltx_bibblock">Core in a simple coalition formation game. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx12.1.1">Social Choice and Welfare</span>, 18:135–153, 2001. </span> </li> <li class="ltx_bibitem" id="bib.bibx13"> <span class="ltx_tag ltx_tag_bibitem">[BR23]</span> <span class="ltx_bibblock"> Martin Bullinger and René Romen. </span> <span class="ltx_bibblock">Online coalition formation under random arrival or coalition dissolution. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bibx13.1.1">Proceedings of the 31st Annual European Symposium on Algorithms (ESA)</span>, pages 27:1–27:18, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bibx14"> <span class="ltx_tag ltx_tag_bibitem">[Bul20]</span> <span class="ltx_bibblock"> Martin Bullinger. </span> <span class="ltx_bibblock">Pareto-optimality in cardinal hedonic games. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bibx14.1.1">Proceedings of the 19th International Conference on Autonomous Agents and Multiagent Systems (AAMAS)</span>, pages 213–221, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bibx15"> <span class="ltx_tag ltx_tag_bibitem">[CGW05]</span> <span class="ltx_bibblock"> Moses Charikar, Venkatesan Guruswami, and Anthony Wirth. </span> <span class="ltx_bibblock">Clustering with qualitative information. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx15.1.1">Journal of Computer and System Sciences</span>, 71(3):360–383, 2005. </span> </li> <li class="ltx_bibitem" id="bib.bibx16"> <span class="ltx_tag ltx_tag_bibitem">[CH02]</span> <span class="ltx_bibblock"> Katarína Cechlárová and Jana Hajduková. </span> <span class="ltx_bibblock">Computational complexity of stable partitions with B-preferences. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx16.1.1">International Journal of Game Theory</span>, 31(3):353–354, 2002. </span> </li> <li class="ltx_bibitem" id="bib.bibx17"> <span class="ltx_tag ltx_tag_bibitem">[CRM01]</span> <span class="ltx_bibblock"> Katarína Cechlárová and Antonio Romero-Medina. </span> <span class="ltx_bibblock">Stability in coalition formation games. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx17.1.1">International Journal of Game Theory</span>, 29:487–494, 2001. </span> </li> <li class="ltx_bibitem" id="bib.bibx18"> <span class="ltx_tag ltx_tag_bibitem">[CW04]</span> <span class="ltx_bibblock"> Moses Charikar and Anthony Wirth. </span> <span class="ltx_bibblock">Maximizing quadratic programs: Extending grothendieck’s inequality. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bibx18.1.1">Proceedings of the 45th Symposium on Foundations of Computer Science (FOCS)</span>, pages 54–60, 2004. </span> </li> <li class="ltx_bibitem" id="bib.bibx19"> <span class="ltx_tag ltx_tag_bibitem">[DBHS06]</span> <span class="ltx_bibblock"> Dinko Dimitrov, Peter Borm, Ruud Hendrickx, and Shao C. Sung. </span> <span class="ltx_bibblock">Simple priorities and core stability in hedonic games. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx19.1.1">Social Choice and Welfare</span>, 26(2):421–433, 2006. </span> </li> <li class="ltx_bibitem" id="bib.bibx20"> <span class="ltx_tag ltx_tag_bibitem">[DEFI06]</span> <span class="ltx_bibblock"> Erik D. Demaine, Dotan Emanuel, Amos Fiat, and Nicole Immorlica. </span> <span class="ltx_bibblock">Correlation clustering in general weighted graphs. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx20.1.1">Theoretical Computer Science</span>, 361(2-3):172–187, 2006. </span> </li> <li class="ltx_bibitem" id="bib.bibx21"> <span class="ltx_tag ltx_tag_bibitem">[DG80]</span> <span class="ltx_bibblock"> Jacques H. Drèze and Joseph Greenberg. </span> <span class="ltx_bibblock">Hedonic coalitions: Optimality and stability. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx21.1.1">Econometrica</span>, 48(4):987–1003, 1980. </span> </li> <li class="ltx_bibitem" id="bib.bibx22"> <span class="ltx_tag ltx_tag_bibitem">[DMC15]</span> <span class="ltx_bibblock"> Nameirakpam Dhanachandra, Khumanthem Manglem, and Yambem Jina Chanu. </span> <span class="ltx_bibblock">Image segmentation using k-means clustering algorithm and subtractive clustering algorithm. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx22.1.1">Procedia Computer Science</span>, 54:764–771, 2015. </span> </li> <li class="ltx_bibitem" id="bib.bibx23"> <span class="ltx_tag ltx_tag_bibitem">[EFF20]</span> <span class="ltx_bibblock"> Edith Elkind, Angelo Fanelli, and Michele Flammini. </span> <span class="ltx_bibblock">Price of pareto optimality in hedonic games. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx23.1.1">Artificial Intelligence</span>, 288:103357, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bibx24"> <span class="ltx_tag ltx_tag_bibitem">[FFKV23]</span> <span class="ltx_bibblock"> Simone Fioravanti, Michele Flammini, Bojana Kodric, and Giovanna Varricchio. </span> <span class="ltx_bibblock"><math alttext="\varepsilon" class="ltx_Math" display="inline" id="bib.bibx24.1.m1.1"><semantics id="bib.bibx24.1.m1.1a"><mi id="bib.bibx24.1.m1.1.1" xref="bib.bibx24.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="bib.bibx24.1.m1.1b"><ci id="bib.bibx24.1.m1.1.1.cmml" xref="bib.bibx24.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="bib.bibx24.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="bib.bibx24.1.m1.1d">italic_ε</annotation></semantics></math>-fractional core stability in hedonic games. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bibx24.2.1">Proceedings of the 37th Annual Conference on Neural Information Processing Systems (NeurIPS)</span>, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bibx25"> <span class="ltx_tag ltx_tag_bibitem">[FKMZ21]</span> <span class="ltx_bibblock"> Michele Flammini, Bojana Kodric, Gianpiero Monaco, and Qiang Zhang. </span> <span class="ltx_bibblock">Strategyproof mechanisms for additively separable and fractional hedonic games. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx25.1.1">Journal of Artificial Intelligence Research</span>, 70:1253–1279, 2021. </span> </li> <li class="ltx_bibitem" id="bib.bibx26"> <span class="ltx_tag ltx_tag_bibitem">[FKV22]</span> <span class="ltx_bibblock"> Michele Flammini, Bojana Kodric, and Giovanna Varricchio. </span> <span class="ltx_bibblock">Strategyproof mechanisms for friends and enemies games. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx26.1.1">Artificial Intelligence</span>, 302:103610, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bibx27"> <span class="ltx_tag ltx_tag_bibitem">[FMM<sup class="ltx_sup" id="bib.bibx27.4.4.1"><span class="ltx_text ltx_font_italic" id="bib.bibx27.4.4.1.1">+</span></sup>21]</span> <span class="ltx_bibblock"> Michele Flammini, Gianpiero Monaco, Luca Moscardelli, Mordechai Shalom, and Shmuel Zaks. </span> <span class="ltx_bibblock">On the online coalition structure generation problem. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx27.7.1">Journal of Artificial Intelligence Research</span>, 72:1215–1250, 2021. </span> </li> <li class="ltx_bibitem" id="bib.bibx28"> <span class="ltx_tag ltx_tag_bibitem">[GM75]</span> <span class="ltx_bibblock"> Geoffrey R. Grimmett and Colin J. H. McDiarmid. </span> <span class="ltx_bibblock">On colouring random graphs. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bibx28.1.1">Mathematical Proceedings of the Cambridge Philosophical Society</span>, volume 77, pages 313–324, 1975. </span> </li> <li class="ltx_bibitem" id="bib.bibx29"> <span class="ltx_tag ltx_tag_bibitem">[GS19]</span> <span class="ltx_bibblock"> Martin Gairing and Rahul Savani. </span> <span class="ltx_bibblock">Computing stable outcomes in symmetric additively separable hedonic games. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx29.1.1">Mathematics of Operations Research</span>, 44(3):1101–1121, 2019. </span> </li> <li class="ltx_bibitem" id="bib.bibx30"> <span class="ltx_tag ltx_tag_bibitem">[MMV15]</span> <span class="ltx_bibblock"> Konstantin Makarychev, Yury Makarychev, and Aravindan Vijayaraghavan. </span> <span class="ltx_bibblock">Correlation clustering with noisy partial information. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bibx30.1.1">Conference on Learning Theory</span>, pages 1321–1342. PMLR, 2015. </span> </li> <li class="ltx_bibitem" id="bib.bibx31"> <span class="ltx_tag ltx_tag_bibitem">[MS10]</span> <span class="ltx_bibblock"> Claire Mathieu and Warren Schudy. </span> <span class="ltx_bibblock">Correlation clustering with noisy input. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bibx31.1.1">Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)</span>, pages 712–728, 2010. </span> </li> <li class="ltx_bibitem" id="bib.bibx32"> <span class="ltx_tag ltx_tag_bibitem">[New04]</span> <span class="ltx_bibblock"> Mark E. J. Newman. </span> <span class="ltx_bibblock">Detecting community structure in networks. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx32.1.1">The European Physical Journal B - Condensed Matter and Complex Systems</span>, 38(2):321–330, 2004. </span> </li> <li class="ltx_bibitem" id="bib.bibx33"> <span class="ltx_tag ltx_tag_bibitem">[SD10]</span> <span class="ltx_bibblock"> Shao C. Sung and Dinko Dimitrov. </span> <span class="ltx_bibblock">Computational complexity in additive hedonic games. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx33.1.1">European Journal of Operational Research</span>, 203(3):635–639, 2010. </span> </li> <li class="ltx_bibitem" id="bib.bibx34"> <span class="ltx_tag ltx_tag_bibitem">[Swa04]</span> <span class="ltx_bibblock"> Chaitanya Swamy. </span> <span class="ltx_bibblock">Correlation clustering: maximizing agreements via semidefinite programming. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bibx34.1.1">Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)</span>, volume 4, pages 526–527, 2004. </span> </li> <li class="ltx_bibitem" id="bib.bibx35"> <span class="ltx_tag ltx_tag_bibitem">[Woe13]</span> <span class="ltx_bibblock"> Gerhard J. Woeginger. </span> <span class="ltx_bibblock">A hardness result for core stability in additive hedonic games. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bibx35.1.1">Mathematical Social Sciences</span>, 65(2):101–104, 2013. </span> </li> <li class="ltx_bibitem" id="bib.bibx36"> <span class="ltx_tag ltx_tag_bibitem">[Zuc06]</span> <span class="ltx_bibblock"> David Zuckerman. </span> <span class="ltx_bibblock">Linear degree extractors and the inapproximability of max clique and chromatic number. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bibx36.1.1">Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC)</span>, pages 681–690, 2006. </span> </li> </ul> </section> <div class="ltx_pagination ltx_role_newpage"></div> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Sat Mar 8 02:35:24 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>