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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds</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/2411.12976v1/"/></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/2411.12976v1#S1" title="In Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</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/2411.12976v1#S1.SS1" title="In 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Background: Directed cuts, bias, and oblivious algorithms</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.SS2" title="In 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.2 </span>Results</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.SS3" title="In 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.3 </span>Motivations</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.SS3.SSS0.Px1" title="In 1.3 Motivations ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title">Downstream applications.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.SS3.SSS0.Px2" title="In 1.3 Motivations ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title">Oblivious algorithms for other problems.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.SS4" title="In 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.4 </span>Structure of rest of the paper</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.SS5" title="In 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.5 </span>Code</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S2" title="In Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Preliminaries</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/2411.12976v1#S2.SS1" title="In 2 Preliminaries ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Directed graphs</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S2.SS2" title="In 2 Preliminaries ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span><span class="ltx_text ltx_markedasmath ltx_font_smallcaps">Max-DiCut</span> and oblivious algorithms</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S2.SS3" title="In 2 Preliminaries ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Linear program</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S3" title="In Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Improved oblivious algorithms (<span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.5</span>)</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S4" title="In Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Lower bounds</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/2411.12976v1#S4.SS1" title="In 4 Lower bounds ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Methodology for finding hard instances</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S4.SS2" title="In 4 Lower bounds ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Bounds against <math alttext="\mathsf{PLSigmoid}_{1/2}" class="ltx_Math" display="inline"><semantics><msub><mi>𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply><divide></divide><cn type="integer">1</cn><cn type="integer">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex">\mathsf{PLSigmoid}_{1/2}</annotation><annotation encoding="application/x-llamapun">sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math> (<span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.6</span>)</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S4.SS3" title="In 4 Lower bounds ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Lower bound for PL sigmoid functions (<span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.7</span>)</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S5" title="In Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Lower bound for arbitrary selection functions (<span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.9</span>)</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S6" title="In Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Lower bounds for antisymmetric selection functions (<span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.8</span>)</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#A1" title="In Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Recap: The prior lower bound of <span class="ltx_ERROR undefined">\textcite</span>FJ15 (<span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.2</span>)</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <div class="ltx_para" id="p1"> <span class="ltx_ERROR undefined" id="p1.1">\addbibresource</span> <p class="ltx_p" id="p1.2">csps.bib </p> </div> <h1 class="ltx_title ltx_title_document">Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Samuel Hwang </span><span class="ltx_author_notes">Harvard College, Harvard University, Cambridge, MA, USA. Email: <span class="ltx_text ltx_font_typewriter" id="id16.1.id1">samuelhwang@college.harvard.edu</span>.</span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Noah G. Singer </span><span class="ltx_author_notes">Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA. Supported by an NSF Graduate Research Fellowship (Award DGE2140739). Email: <span class="ltx_text ltx_font_typewriter" id="id17.1.id1">ngsinger@cs.cmu.edu</span>.</span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Santhoshini Velusamy </span><span class="ltx_author_notes">Toyota Technological Institute at Chicago, Chicago, IL, USA. Supported by an NSF CRII award CCF 2348475. Email: <span class="ltx_text ltx_font_typewriter" id="id18.1.id1">santhoshini@ttic.edu</span>.</span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id10.10">In the <em class="ltx_emph ltx_font_italic" id="id10.10.1">maximum directed cut</em> (<span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="id10.10.2">Max-DiCut</span>) problem, the input is a directed graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="id2.2.m2.2"><semantics id="id2.2.m2.2a"><mrow id="id2.2.m2.2.3" xref="id2.2.m2.2.3.cmml"><mi id="id2.2.m2.2.3.2" xref="id2.2.m2.2.3.2.cmml">G</mi><mo id="id2.2.m2.2.3.1" xref="id2.2.m2.2.3.1.cmml">=</mo><mrow id="id2.2.m2.2.3.3.2" xref="id2.2.m2.2.3.3.1.cmml"><mo id="id2.2.m2.2.3.3.2.1" stretchy="false" xref="id2.2.m2.2.3.3.1.cmml">(</mo><mi id="id2.2.m2.1.1" xref="id2.2.m2.1.1.cmml">V</mi><mo id="id2.2.m2.2.3.3.2.2" xref="id2.2.m2.2.3.3.1.cmml">,</mo><mi id="id2.2.m2.2.2" xref="id2.2.m2.2.2.cmml">E</mi><mo id="id2.2.m2.2.3.3.2.3" stretchy="false" xref="id2.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id2.2.m2.2b"><apply id="id2.2.m2.2.3.cmml" xref="id2.2.m2.2.3"><eq id="id2.2.m2.2.3.1.cmml" xref="id2.2.m2.2.3.1"></eq><ci id="id2.2.m2.2.3.2.cmml" xref="id2.2.m2.2.3.2">𝐺</ci><interval closure="open" id="id2.2.m2.2.3.3.1.cmml" xref="id2.2.m2.2.3.3.2"><ci id="id2.2.m2.1.1.cmml" xref="id2.2.m2.1.1">𝑉</ci><ci id="id2.2.m2.2.2.cmml" xref="id2.2.m2.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="id2.2.m2.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="id2.2.m2.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>, and the goal is to pick a partition <math alttext="V=S\cup(V\setminus S)" 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 id="id3.3.m3.1.1.3" xref="id3.3.m3.1.1.3.cmml">V</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" xref="id3.3.m3.1.1.1.cmml"><mi id="id3.3.m3.1.1.1.3" xref="id3.3.m3.1.1.1.3.cmml">S</mi><mo id="id3.3.m3.1.1.1.2" xref="id3.3.m3.1.1.1.2.cmml">∪</mo><mrow id="id3.3.m3.1.1.1.1.1" xref="id3.3.m3.1.1.1.1.1.1.cmml"><mo id="id3.3.m3.1.1.1.1.1.2" stretchy="false" xref="id3.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="id3.3.m3.1.1.1.1.1.1" xref="id3.3.m3.1.1.1.1.1.1.cmml"><mi id="id3.3.m3.1.1.1.1.1.1.2" xref="id3.3.m3.1.1.1.1.1.1.2.cmml">V</mi><mo id="id3.3.m3.1.1.1.1.1.1.1" xref="id3.3.m3.1.1.1.1.1.1.1.cmml">∖</mo><mi id="id3.3.m3.1.1.1.1.1.1.3" xref="id3.3.m3.1.1.1.1.1.1.3.cmml">S</mi></mrow><mo id="id3.3.m3.1.1.1.1.1.3" stretchy="false" xref="id3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></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"><eq id="id3.3.m3.1.1.2.cmml" xref="id3.3.m3.1.1.2"></eq><ci id="id3.3.m3.1.1.3.cmml" xref="id3.3.m3.1.1.3">𝑉</ci><apply id="id3.3.m3.1.1.1.cmml" xref="id3.3.m3.1.1.1"><union id="id3.3.m3.1.1.1.2.cmml" xref="id3.3.m3.1.1.1.2"></union><ci id="id3.3.m3.1.1.1.3.cmml" xref="id3.3.m3.1.1.1.3">𝑆</ci><apply id="id3.3.m3.1.1.1.1.1.1.cmml" xref="id3.3.m3.1.1.1.1.1"><setdiff id="id3.3.m3.1.1.1.1.1.1.1.cmml" xref="id3.3.m3.1.1.1.1.1.1.1"></setdiff><ci id="id3.3.m3.1.1.1.1.1.1.2.cmml" xref="id3.3.m3.1.1.1.1.1.1.2">𝑉</ci><ci id="id3.3.m3.1.1.1.1.1.1.3.cmml" xref="id3.3.m3.1.1.1.1.1.1.3">𝑆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id3.3.m3.1c">V=S\cup(V\setminus S)</annotation><annotation encoding="application/x-llamapun" id="id3.3.m3.1d">italic_V = italic_S ∪ ( italic_V ∖ italic_S )</annotation></semantics></math> of the vertices such that as many edges as possible go <em class="ltx_emph ltx_font_italic" id="id10.10.3">from</em> <math alttext="S" class="ltx_Math" display="inline" id="id4.4.m4.1"><semantics id="id4.4.m4.1a"><mi id="id4.4.m4.1.1" xref="id4.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="id4.4.m4.1b"><ci id="id4.4.m4.1.1.cmml" xref="id4.4.m4.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="id4.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="id4.4.m4.1d">italic_S</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="id10.10.4">to</em> <math alttext="V\setminus S" class="ltx_Math" display="inline" id="id5.5.m5.1"><semantics id="id5.5.m5.1a"><mrow id="id5.5.m5.1.1" xref="id5.5.m5.1.1.cmml"><mi id="id5.5.m5.1.1.2" xref="id5.5.m5.1.1.2.cmml">V</mi><mo id="id5.5.m5.1.1.1" xref="id5.5.m5.1.1.1.cmml">∖</mo><mi id="id5.5.m5.1.1.3" xref="id5.5.m5.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="id5.5.m5.1b"><apply id="id5.5.m5.1.1.cmml" xref="id5.5.m5.1.1"><setdiff id="id5.5.m5.1.1.1.cmml" xref="id5.5.m5.1.1.1"></setdiff><ci id="id5.5.m5.1.1.2.cmml" xref="id5.5.m5.1.1.2">𝑉</ci><ci id="id5.5.m5.1.1.3.cmml" xref="id5.5.m5.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id5.5.m5.1c">V\setminus S</annotation><annotation encoding="application/x-llamapun" id="id5.5.m5.1d">italic_V ∖ italic_S</annotation></semantics></math>. <em class="ltx_emph ltx_font_italic" id="id10.10.5">Oblivious algorithms</em>, introduced by <span class="ltx_ERROR undefined" id="id10.10.6">\textcite</span>FJ15, are a simple class of algorithms for this problem. These algorithms independently and randomly assign each vertex <math alttext="v" class="ltx_Math" display="inline" id="id6.6.m6.1"><semantics id="id6.6.m6.1a"><mi id="id6.6.m6.1.1" xref="id6.6.m6.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="id6.6.m6.1b"><ci id="id6.6.m6.1.1.cmml" xref="id6.6.m6.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="id6.6.m6.1c">v</annotation><annotation encoding="application/x-llamapun" id="id6.6.m6.1d">italic_v</annotation></semantics></math> to either <math alttext="S" class="ltx_Math" display="inline" id="id7.7.m7.1"><semantics id="id7.7.m7.1a"><mi id="id7.7.m7.1.1" xref="id7.7.m7.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="id7.7.m7.1b"><ci id="id7.7.m7.1.1.cmml" xref="id7.7.m7.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="id7.7.m7.1c">S</annotation><annotation encoding="application/x-llamapun" id="id7.7.m7.1d">italic_S</annotation></semantics></math> or <math alttext="V\setminus S" class="ltx_Math" display="inline" id="id8.8.m8.1"><semantics id="id8.8.m8.1a"><mrow id="id8.8.m8.1.1" xref="id8.8.m8.1.1.cmml"><mi id="id8.8.m8.1.1.2" xref="id8.8.m8.1.1.2.cmml">V</mi><mo id="id8.8.m8.1.1.1" xref="id8.8.m8.1.1.1.cmml">∖</mo><mi id="id8.8.m8.1.1.3" xref="id8.8.m8.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="id8.8.m8.1b"><apply id="id8.8.m8.1.1.cmml" xref="id8.8.m8.1.1"><setdiff id="id8.8.m8.1.1.1.cmml" xref="id8.8.m8.1.1.1"></setdiff><ci id="id8.8.m8.1.1.2.cmml" xref="id8.8.m8.1.1.2">𝑉</ci><ci id="id8.8.m8.1.1.3.cmml" xref="id8.8.m8.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id8.8.m8.1c">V\setminus S</annotation><annotation encoding="application/x-llamapun" id="id8.8.m8.1d">italic_V ∖ italic_S</annotation></semantics></math>, and the distribution of <math alttext="v" class="ltx_Math" display="inline" id="id9.9.m9.1"><semantics id="id9.9.m9.1a"><mi id="id9.9.m9.1.1" xref="id9.9.m9.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="id9.9.m9.1b"><ci id="id9.9.m9.1.1.cmml" xref="id9.9.m9.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="id9.9.m9.1c">v</annotation><annotation encoding="application/x-llamapun" id="id9.9.m9.1d">italic_v</annotation></semantics></math>’s assignment is determined using only extremely local information about <math alttext="v" class="ltx_Math" display="inline" id="id10.10.m10.1"><semantics id="id10.10.m10.1a"><mi id="id10.10.m10.1.1" xref="id10.10.m10.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="id10.10.m10.1b"><ci id="id10.10.m10.1.1.cmml" xref="id10.10.m10.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="id10.10.m10.1c">v</annotation><annotation encoding="application/x-llamapun" id="id10.10.m10.1d">italic_v</annotation></semantics></math>: its <em class="ltx_emph ltx_font_italic" id="id10.10.7">bias</em>, i.e., the relative difference between its out- and in-degrees. These algorithms have natural implementations in certain graph streaming models, where they have important implications <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">SSSV23-dicut</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">SSSV23-random-ordering</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">kallaugher2023exponential</span>]</cite>.</p> <p class="ltx_p" id="id15.15">In this work, we narrow the gap between upper and lower bounds on the best approximation ratio achievable by oblivious algorithms for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="id15.15.1">Max-DiCut</span>. We show that there exists an oblivious algorithm achieving an approximation ratio of at least <math alttext="0.4853" class="ltx_Math" display="inline" id="id12.12.m2.1"><semantics id="id12.12.m2.1a"><mn id="id12.12.m2.1.1" xref="id12.12.m2.1.1.cmml">0.4853</mn><annotation-xml encoding="MathML-Content" id="id12.12.m2.1b"><cn id="id12.12.m2.1.1.cmml" type="float" xref="id12.12.m2.1.1">0.4853</cn></annotation-xml><annotation encoding="application/x-tex" id="id12.12.m2.1c">0.4853</annotation><annotation encoding="application/x-llamapun" id="id12.12.m2.1d">0.4853</annotation></semantics></math>, while every oblivious algorithm obeying a natural symmetry property achieves an approximation ratio of at most <math alttext="0.4889" class="ltx_Math" display="inline" id="id13.13.m3.1"><semantics id="id13.13.m3.1a"><mn id="id13.13.m3.1.1" xref="id13.13.m3.1.1.cmml">0.4889</mn><annotation-xml encoding="MathML-Content" id="id13.13.m3.1b"><cn id="id13.13.m3.1.1.cmml" type="float" xref="id13.13.m3.1.1">0.4889</cn></annotation-xml><annotation encoding="application/x-tex" id="id13.13.m3.1c">0.4889</annotation><annotation encoding="application/x-llamapun" id="id13.13.m3.1d">0.4889</annotation></semantics></math>. The previous known bounds were <math alttext="0.4844" class="ltx_Math" display="inline" id="id14.14.m4.1"><semantics id="id14.14.m4.1a"><mn id="id14.14.m4.1.1" xref="id14.14.m4.1.1.cmml">0.4844</mn><annotation-xml encoding="MathML-Content" id="id14.14.m4.1b"><cn id="id14.14.m4.1.1.cmml" type="float" xref="id14.14.m4.1.1">0.4844</cn></annotation-xml><annotation encoding="application/x-tex" id="id14.14.m4.1c">0.4844</annotation><annotation encoding="application/x-llamapun" id="id14.14.m4.1d">0.4844</annotation></semantics></math> and <math alttext="0.4899" class="ltx_Math" display="inline" id="id15.15.m5.1"><semantics id="id15.15.m5.1a"><mn id="id15.15.m5.1.1" xref="id15.15.m5.1.1.cmml">0.4899</mn><annotation-xml encoding="MathML-Content" id="id15.15.m5.1b"><cn id="id15.15.m5.1.1.cmml" type="float" xref="id15.15.m5.1.1">0.4899</cn></annotation-xml><annotation encoding="application/x-tex" id="id15.15.m5.1c">0.4899</annotation><annotation encoding="application/x-llamapun" id="id15.15.m5.1d">0.4899</annotation></semantics></math>, due to <span class="ltx_ERROR undefined" id="id15.15.2">\textcite</span>Sin23-kand,FJ15, respectively. Our techniques involve designing principled parameterizations of the spaces of algorithms and lower bounds and then executing computer searches through these spaces.</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">In this work, we study a special class of algorithms, called <em class="ltx_emph ltx_font_italic" id="S1.p1.1.1">oblivious algorithms</em>, for a specific constraint satisfaction problem, called <em class="ltx_emph ltx_font_italic" id="S1.p1.1.2">maximum directed cut</em> (<span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.p1.1.3">Max-DiCut</span>). We first informally describe these two notions; see <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S2.SS2" title="2.2 Max-DiCut and oblivious algorithms ‣ 2 Preliminaries ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">2.2</span></a> below for formal definitions.</p> </div> <section class="ltx_subsection" id="S1.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.1 </span>Background: Directed cuts, bias, and oblivious algorithms</h3> <div class="ltx_para" id="S1.SS1.p1"> <p class="ltx_p" id="S1.SS1.p1.9">An input instance to <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS1.p1.9.1">Max-DiCut</span> is a <em class="ltx_emph ltx_font_italic" id="S1.SS1.p1.9.2">directed graph</em> <math alttext="G=([n],E)" class="ltx_Math" display="inline" id="S1.SS1.p1.2.m2.3"><semantics id="S1.SS1.p1.2.m2.3a"><mrow id="S1.SS1.p1.2.m2.3.3" xref="S1.SS1.p1.2.m2.3.3.cmml"><mi id="S1.SS1.p1.2.m2.3.3.3" xref="S1.SS1.p1.2.m2.3.3.3.cmml">G</mi><mo id="S1.SS1.p1.2.m2.3.3.2" xref="S1.SS1.p1.2.m2.3.3.2.cmml">=</mo><mrow id="S1.SS1.p1.2.m2.3.3.1.1" xref="S1.SS1.p1.2.m2.3.3.1.2.cmml"><mo id="S1.SS1.p1.2.m2.3.3.1.1.2" stretchy="false" xref="S1.SS1.p1.2.m2.3.3.1.2.cmml">(</mo><mrow id="S1.SS1.p1.2.m2.3.3.1.1.1.2" xref="S1.SS1.p1.2.m2.3.3.1.1.1.1.cmml"><mo id="S1.SS1.p1.2.m2.3.3.1.1.1.2.1" stretchy="false" xref="S1.SS1.p1.2.m2.3.3.1.1.1.1.1.cmml">[</mo><mi id="S1.SS1.p1.2.m2.1.1" xref="S1.SS1.p1.2.m2.1.1.cmml">n</mi><mo id="S1.SS1.p1.2.m2.3.3.1.1.1.2.2" stretchy="false" xref="S1.SS1.p1.2.m2.3.3.1.1.1.1.1.cmml">]</mo></mrow><mo id="S1.SS1.p1.2.m2.3.3.1.1.3" xref="S1.SS1.p1.2.m2.3.3.1.2.cmml">,</mo><mi id="S1.SS1.p1.2.m2.2.2" xref="S1.SS1.p1.2.m2.2.2.cmml">E</mi><mo id="S1.SS1.p1.2.m2.3.3.1.1.4" stretchy="false" xref="S1.SS1.p1.2.m2.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.2.m2.3b"><apply id="S1.SS1.p1.2.m2.3.3.cmml" xref="S1.SS1.p1.2.m2.3.3"><eq id="S1.SS1.p1.2.m2.3.3.2.cmml" xref="S1.SS1.p1.2.m2.3.3.2"></eq><ci id="S1.SS1.p1.2.m2.3.3.3.cmml" xref="S1.SS1.p1.2.m2.3.3.3">𝐺</ci><interval closure="open" id="S1.SS1.p1.2.m2.3.3.1.2.cmml" xref="S1.SS1.p1.2.m2.3.3.1.1"><apply id="S1.SS1.p1.2.m2.3.3.1.1.1.1.cmml" xref="S1.SS1.p1.2.m2.3.3.1.1.1.2"><csymbol cd="latexml" id="S1.SS1.p1.2.m2.3.3.1.1.1.1.1.cmml" xref="S1.SS1.p1.2.m2.3.3.1.1.1.2.1">delimited-[]</csymbol><ci id="S1.SS1.p1.2.m2.1.1.cmml" xref="S1.SS1.p1.2.m2.1.1">𝑛</ci></apply><ci id="S1.SS1.p1.2.m2.2.2.cmml" xref="S1.SS1.p1.2.m2.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.2.m2.3c">G=([n],E)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.2.m2.3d">italic_G = ( [ italic_n ] , italic_E )</annotation></semantics></math> (possibly with edge weights) on vertex set <math alttext="[n]=\{1,\ldots,n\}" class="ltx_Math" display="inline" id="S1.SS1.p1.3.m3.4"><semantics id="S1.SS1.p1.3.m3.4a"><mrow id="S1.SS1.p1.3.m3.4.5" xref="S1.SS1.p1.3.m3.4.5.cmml"><mrow id="S1.SS1.p1.3.m3.4.5.2.2" xref="S1.SS1.p1.3.m3.4.5.2.1.cmml"><mo id="S1.SS1.p1.3.m3.4.5.2.2.1" stretchy="false" xref="S1.SS1.p1.3.m3.4.5.2.1.1.cmml">[</mo><mi id="S1.SS1.p1.3.m3.1.1" xref="S1.SS1.p1.3.m3.1.1.cmml">n</mi><mo id="S1.SS1.p1.3.m3.4.5.2.2.2" stretchy="false" xref="S1.SS1.p1.3.m3.4.5.2.1.1.cmml">]</mo></mrow><mo id="S1.SS1.p1.3.m3.4.5.1" xref="S1.SS1.p1.3.m3.4.5.1.cmml">=</mo><mrow id="S1.SS1.p1.3.m3.4.5.3.2" xref="S1.SS1.p1.3.m3.4.5.3.1.cmml"><mo id="S1.SS1.p1.3.m3.4.5.3.2.1" stretchy="false" xref="S1.SS1.p1.3.m3.4.5.3.1.cmml">{</mo><mn id="S1.SS1.p1.3.m3.2.2" xref="S1.SS1.p1.3.m3.2.2.cmml">1</mn><mo id="S1.SS1.p1.3.m3.4.5.3.2.2" xref="S1.SS1.p1.3.m3.4.5.3.1.cmml">,</mo><mi id="S1.SS1.p1.3.m3.3.3" mathvariant="normal" xref="S1.SS1.p1.3.m3.3.3.cmml">…</mi><mo id="S1.SS1.p1.3.m3.4.5.3.2.3" xref="S1.SS1.p1.3.m3.4.5.3.1.cmml">,</mo><mi id="S1.SS1.p1.3.m3.4.4" xref="S1.SS1.p1.3.m3.4.4.cmml">n</mi><mo id="S1.SS1.p1.3.m3.4.5.3.2.4" stretchy="false" xref="S1.SS1.p1.3.m3.4.5.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.3.m3.4b"><apply id="S1.SS1.p1.3.m3.4.5.cmml" xref="S1.SS1.p1.3.m3.4.5"><eq id="S1.SS1.p1.3.m3.4.5.1.cmml" xref="S1.SS1.p1.3.m3.4.5.1"></eq><apply id="S1.SS1.p1.3.m3.4.5.2.1.cmml" xref="S1.SS1.p1.3.m3.4.5.2.2"><csymbol cd="latexml" id="S1.SS1.p1.3.m3.4.5.2.1.1.cmml" xref="S1.SS1.p1.3.m3.4.5.2.2.1">delimited-[]</csymbol><ci id="S1.SS1.p1.3.m3.1.1.cmml" xref="S1.SS1.p1.3.m3.1.1">𝑛</ci></apply><set id="S1.SS1.p1.3.m3.4.5.3.1.cmml" xref="S1.SS1.p1.3.m3.4.5.3.2"><cn id="S1.SS1.p1.3.m3.2.2.cmml" type="integer" xref="S1.SS1.p1.3.m3.2.2">1</cn><ci id="S1.SS1.p1.3.m3.3.3.cmml" xref="S1.SS1.p1.3.m3.3.3">…</ci><ci id="S1.SS1.p1.3.m3.4.4.cmml" xref="S1.SS1.p1.3.m3.4.4">𝑛</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.3.m3.4c">[n]=\{1,\ldots,n\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.3.m3.4d">[ italic_n ] = { 1 , … , italic_n }</annotation></semantics></math>. A <em class="ltx_emph ltx_font_italic" id="S1.SS1.p1.9.3">cut</em> is a vector <math alttext="\boldsymbol{x}=(x_{1},\ldots,x_{n})\in\{\pm 1\}^{n}" class="ltx_Math" display="inline" id="S1.SS1.p1.4.m4.4"><semantics id="S1.SS1.p1.4.m4.4a"><mrow id="S1.SS1.p1.4.m4.4.4" xref="S1.SS1.p1.4.m4.4.4.cmml"><mi id="S1.SS1.p1.4.m4.4.4.5" xref="S1.SS1.p1.4.m4.4.4.5.cmml">𝒙</mi><mo id="S1.SS1.p1.4.m4.4.4.6" xref="S1.SS1.p1.4.m4.4.4.6.cmml">=</mo><mrow id="S1.SS1.p1.4.m4.3.3.2.2" xref="S1.SS1.p1.4.m4.3.3.2.3.cmml"><mo id="S1.SS1.p1.4.m4.3.3.2.2.3" stretchy="false" xref="S1.SS1.p1.4.m4.3.3.2.3.cmml">(</mo><msub id="S1.SS1.p1.4.m4.2.2.1.1.1" xref="S1.SS1.p1.4.m4.2.2.1.1.1.cmml"><mi id="S1.SS1.p1.4.m4.2.2.1.1.1.2" xref="S1.SS1.p1.4.m4.2.2.1.1.1.2.cmml">x</mi><mn id="S1.SS1.p1.4.m4.2.2.1.1.1.3" xref="S1.SS1.p1.4.m4.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S1.SS1.p1.4.m4.3.3.2.2.4" xref="S1.SS1.p1.4.m4.3.3.2.3.cmml">,</mo><mi id="S1.SS1.p1.4.m4.1.1" mathvariant="normal" xref="S1.SS1.p1.4.m4.1.1.cmml">…</mi><mo id="S1.SS1.p1.4.m4.3.3.2.2.5" xref="S1.SS1.p1.4.m4.3.3.2.3.cmml">,</mo><msub id="S1.SS1.p1.4.m4.3.3.2.2.2" xref="S1.SS1.p1.4.m4.3.3.2.2.2.cmml"><mi id="S1.SS1.p1.4.m4.3.3.2.2.2.2" xref="S1.SS1.p1.4.m4.3.3.2.2.2.2.cmml">x</mi><mi id="S1.SS1.p1.4.m4.3.3.2.2.2.3" xref="S1.SS1.p1.4.m4.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S1.SS1.p1.4.m4.3.3.2.2.6" stretchy="false" xref="S1.SS1.p1.4.m4.3.3.2.3.cmml">)</mo></mrow><mo id="S1.SS1.p1.4.m4.4.4.7" xref="S1.SS1.p1.4.m4.4.4.7.cmml">∈</mo><msup id="S1.SS1.p1.4.m4.4.4.3" xref="S1.SS1.p1.4.m4.4.4.3.cmml"><mrow id="S1.SS1.p1.4.m4.4.4.3.1.1" xref="S1.SS1.p1.4.m4.4.4.3.1.2.cmml"><mo id="S1.SS1.p1.4.m4.4.4.3.1.1.2" stretchy="false" xref="S1.SS1.p1.4.m4.4.4.3.1.2.cmml">{</mo><mrow id="S1.SS1.p1.4.m4.4.4.3.1.1.1" xref="S1.SS1.p1.4.m4.4.4.3.1.1.1.cmml"><mo id="S1.SS1.p1.4.m4.4.4.3.1.1.1a" xref="S1.SS1.p1.4.m4.4.4.3.1.1.1.cmml">±</mo><mn id="S1.SS1.p1.4.m4.4.4.3.1.1.1.2" xref="S1.SS1.p1.4.m4.4.4.3.1.1.1.2.cmml">1</mn></mrow><mo id="S1.SS1.p1.4.m4.4.4.3.1.1.3" stretchy="false" xref="S1.SS1.p1.4.m4.4.4.3.1.2.cmml">}</mo></mrow><mi id="S1.SS1.p1.4.m4.4.4.3.3" xref="S1.SS1.p1.4.m4.4.4.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.4.m4.4b"><apply id="S1.SS1.p1.4.m4.4.4.cmml" xref="S1.SS1.p1.4.m4.4.4"><and id="S1.SS1.p1.4.m4.4.4a.cmml" xref="S1.SS1.p1.4.m4.4.4"></and><apply id="S1.SS1.p1.4.m4.4.4b.cmml" xref="S1.SS1.p1.4.m4.4.4"><eq id="S1.SS1.p1.4.m4.4.4.6.cmml" xref="S1.SS1.p1.4.m4.4.4.6"></eq><ci id="S1.SS1.p1.4.m4.4.4.5.cmml" xref="S1.SS1.p1.4.m4.4.4.5">𝒙</ci><vector id="S1.SS1.p1.4.m4.3.3.2.3.cmml" xref="S1.SS1.p1.4.m4.3.3.2.2"><apply id="S1.SS1.p1.4.m4.2.2.1.1.1.cmml" xref="S1.SS1.p1.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S1.SS1.p1.4.m4.2.2.1.1.1.1.cmml" xref="S1.SS1.p1.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S1.SS1.p1.4.m4.2.2.1.1.1.2.cmml" xref="S1.SS1.p1.4.m4.2.2.1.1.1.2">𝑥</ci><cn id="S1.SS1.p1.4.m4.2.2.1.1.1.3.cmml" type="integer" xref="S1.SS1.p1.4.m4.2.2.1.1.1.3">1</cn></apply><ci id="S1.SS1.p1.4.m4.1.1.cmml" xref="S1.SS1.p1.4.m4.1.1">…</ci><apply id="S1.SS1.p1.4.m4.3.3.2.2.2.cmml" xref="S1.SS1.p1.4.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="S1.SS1.p1.4.m4.3.3.2.2.2.1.cmml" xref="S1.SS1.p1.4.m4.3.3.2.2.2">subscript</csymbol><ci id="S1.SS1.p1.4.m4.3.3.2.2.2.2.cmml" xref="S1.SS1.p1.4.m4.3.3.2.2.2.2">𝑥</ci><ci id="S1.SS1.p1.4.m4.3.3.2.2.2.3.cmml" xref="S1.SS1.p1.4.m4.3.3.2.2.2.3">𝑛</ci></apply></vector></apply><apply id="S1.SS1.p1.4.m4.4.4c.cmml" xref="S1.SS1.p1.4.m4.4.4"><in id="S1.SS1.p1.4.m4.4.4.7.cmml" xref="S1.SS1.p1.4.m4.4.4.7"></in><share href="https://arxiv.org/html/2411.12976v1#S1.SS1.p1.4.m4.3.3.2.cmml" id="S1.SS1.p1.4.m4.4.4d.cmml" xref="S1.SS1.p1.4.m4.4.4"></share><apply id="S1.SS1.p1.4.m4.4.4.3.cmml" xref="S1.SS1.p1.4.m4.4.4.3"><csymbol cd="ambiguous" id="S1.SS1.p1.4.m4.4.4.3.2.cmml" xref="S1.SS1.p1.4.m4.4.4.3">superscript</csymbol><set id="S1.SS1.p1.4.m4.4.4.3.1.2.cmml" xref="S1.SS1.p1.4.m4.4.4.3.1.1"><apply id="S1.SS1.p1.4.m4.4.4.3.1.1.1.cmml" xref="S1.SS1.p1.4.m4.4.4.3.1.1.1"><csymbol cd="latexml" id="S1.SS1.p1.4.m4.4.4.3.1.1.1.1.cmml" xref="S1.SS1.p1.4.m4.4.4.3.1.1.1">plus-or-minus</csymbol><cn id="S1.SS1.p1.4.m4.4.4.3.1.1.1.2.cmml" type="integer" xref="S1.SS1.p1.4.m4.4.4.3.1.1.1.2">1</cn></apply></set><ci id="S1.SS1.p1.4.m4.4.4.3.3.cmml" xref="S1.SS1.p1.4.m4.4.4.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.4.m4.4c">\boldsymbol{x}=(x_{1},\ldots,x_{n})\in\{\pm 1\}^{n}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.4.m4.4d">bold_italic_x = ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∈ { ± 1 } start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, and can be viewed as an assignment of a bit to each vertex in the graph. An edge <math alttext="(v_{1},v_{2})" class="ltx_Math" display="inline" id="S1.SS1.p1.5.m5.2"><semantics id="S1.SS1.p1.5.m5.2a"><mrow id="S1.SS1.p1.5.m5.2.2.2" xref="S1.SS1.p1.5.m5.2.2.3.cmml"><mo id="S1.SS1.p1.5.m5.2.2.2.3" stretchy="false" xref="S1.SS1.p1.5.m5.2.2.3.cmml">(</mo><msub id="S1.SS1.p1.5.m5.1.1.1.1" xref="S1.SS1.p1.5.m5.1.1.1.1.cmml"><mi id="S1.SS1.p1.5.m5.1.1.1.1.2" xref="S1.SS1.p1.5.m5.1.1.1.1.2.cmml">v</mi><mn id="S1.SS1.p1.5.m5.1.1.1.1.3" xref="S1.SS1.p1.5.m5.1.1.1.1.3.cmml">1</mn></msub><mo id="S1.SS1.p1.5.m5.2.2.2.4" xref="S1.SS1.p1.5.m5.2.2.3.cmml">,</mo><msub id="S1.SS1.p1.5.m5.2.2.2.2" xref="S1.SS1.p1.5.m5.2.2.2.2.cmml"><mi id="S1.SS1.p1.5.m5.2.2.2.2.2" xref="S1.SS1.p1.5.m5.2.2.2.2.2.cmml">v</mi><mn id="S1.SS1.p1.5.m5.2.2.2.2.3" xref="S1.SS1.p1.5.m5.2.2.2.2.3.cmml">2</mn></msub><mo id="S1.SS1.p1.5.m5.2.2.2.5" stretchy="false" xref="S1.SS1.p1.5.m5.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.5.m5.2b"><interval closure="open" id="S1.SS1.p1.5.m5.2.2.3.cmml" xref="S1.SS1.p1.5.m5.2.2.2"><apply id="S1.SS1.p1.5.m5.1.1.1.1.cmml" xref="S1.SS1.p1.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS1.p1.5.m5.1.1.1.1.1.cmml" xref="S1.SS1.p1.5.m5.1.1.1.1">subscript</csymbol><ci id="S1.SS1.p1.5.m5.1.1.1.1.2.cmml" xref="S1.SS1.p1.5.m5.1.1.1.1.2">𝑣</ci><cn id="S1.SS1.p1.5.m5.1.1.1.1.3.cmml" type="integer" xref="S1.SS1.p1.5.m5.1.1.1.1.3">1</cn></apply><apply id="S1.SS1.p1.5.m5.2.2.2.2.cmml" xref="S1.SS1.p1.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="S1.SS1.p1.5.m5.2.2.2.2.1.cmml" xref="S1.SS1.p1.5.m5.2.2.2.2">subscript</csymbol><ci id="S1.SS1.p1.5.m5.2.2.2.2.2.cmml" xref="S1.SS1.p1.5.m5.2.2.2.2.2">𝑣</ci><cn id="S1.SS1.p1.5.m5.2.2.2.2.3.cmml" type="integer" xref="S1.SS1.p1.5.m5.2.2.2.2.3">2</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.5.m5.2c">(v_{1},v_{2})</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.5.m5.2d">( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S1.SS1.p1.9.4">satisfied</em> by the cut <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S1.SS1.p1.6.m6.1"><semantics id="S1.SS1.p1.6.m6.1a"><mi id="S1.SS1.p1.6.m6.1.1" xref="S1.SS1.p1.6.m6.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.6.m6.1b"><ci id="S1.SS1.p1.6.m6.1.1.cmml" xref="S1.SS1.p1.6.m6.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.6.m6.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.6.m6.1d">bold_italic_x</annotation></semantics></math> if <math alttext="x_{v_{1}}=1" class="ltx_Math" display="inline" id="S1.SS1.p1.7.m7.1"><semantics id="S1.SS1.p1.7.m7.1a"><mrow id="S1.SS1.p1.7.m7.1.1" xref="S1.SS1.p1.7.m7.1.1.cmml"><msub id="S1.SS1.p1.7.m7.1.1.2" xref="S1.SS1.p1.7.m7.1.1.2.cmml"><mi id="S1.SS1.p1.7.m7.1.1.2.2" xref="S1.SS1.p1.7.m7.1.1.2.2.cmml">x</mi><msub id="S1.SS1.p1.7.m7.1.1.2.3" xref="S1.SS1.p1.7.m7.1.1.2.3.cmml"><mi id="S1.SS1.p1.7.m7.1.1.2.3.2" xref="S1.SS1.p1.7.m7.1.1.2.3.2.cmml">v</mi><mn id="S1.SS1.p1.7.m7.1.1.2.3.3" xref="S1.SS1.p1.7.m7.1.1.2.3.3.cmml">1</mn></msub></msub><mo id="S1.SS1.p1.7.m7.1.1.1" xref="S1.SS1.p1.7.m7.1.1.1.cmml">=</mo><mn id="S1.SS1.p1.7.m7.1.1.3" xref="S1.SS1.p1.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.7.m7.1b"><apply id="S1.SS1.p1.7.m7.1.1.cmml" xref="S1.SS1.p1.7.m7.1.1"><eq id="S1.SS1.p1.7.m7.1.1.1.cmml" xref="S1.SS1.p1.7.m7.1.1.1"></eq><apply id="S1.SS1.p1.7.m7.1.1.2.cmml" xref="S1.SS1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p1.7.m7.1.1.2.1.cmml" xref="S1.SS1.p1.7.m7.1.1.2">subscript</csymbol><ci id="S1.SS1.p1.7.m7.1.1.2.2.cmml" xref="S1.SS1.p1.7.m7.1.1.2.2">𝑥</ci><apply id="S1.SS1.p1.7.m7.1.1.2.3.cmml" xref="S1.SS1.p1.7.m7.1.1.2.3"><csymbol cd="ambiguous" id="S1.SS1.p1.7.m7.1.1.2.3.1.cmml" xref="S1.SS1.p1.7.m7.1.1.2.3">subscript</csymbol><ci id="S1.SS1.p1.7.m7.1.1.2.3.2.cmml" xref="S1.SS1.p1.7.m7.1.1.2.3.2">𝑣</ci><cn id="S1.SS1.p1.7.m7.1.1.2.3.3.cmml" type="integer" xref="S1.SS1.p1.7.m7.1.1.2.3.3">1</cn></apply></apply><cn id="S1.SS1.p1.7.m7.1.1.3.cmml" type="integer" xref="S1.SS1.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.7.m7.1c">x_{v_{1}}=1</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.7.m7.1d">italic_x start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="x_{v_{2}}=0" class="ltx_Math" display="inline" id="S1.SS1.p1.8.m8.1"><semantics id="S1.SS1.p1.8.m8.1a"><mrow id="S1.SS1.p1.8.m8.1.1" xref="S1.SS1.p1.8.m8.1.1.cmml"><msub id="S1.SS1.p1.8.m8.1.1.2" xref="S1.SS1.p1.8.m8.1.1.2.cmml"><mi id="S1.SS1.p1.8.m8.1.1.2.2" xref="S1.SS1.p1.8.m8.1.1.2.2.cmml">x</mi><msub 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The <em class="ltx_emph ltx_font_italic" id="S1.SS1.p1.9.5">value</em> of a cut is the fraction of edges it satisfies, and the <em class="ltx_emph ltx_font_italic" id="S1.SS1.p1.9.6">value</em> of a graph, denoted <math alttext="\mathsf{val}_{G}" class="ltx_Math" display="inline" id="S1.SS1.p1.9.m9.1"><semantics id="S1.SS1.p1.9.m9.1a"><msub id="S1.SS1.p1.9.m9.1.1" xref="S1.SS1.p1.9.m9.1.1.cmml"><mi id="S1.SS1.p1.9.m9.1.1.2" xref="S1.SS1.p1.9.m9.1.1.2.cmml">𝗏𝖺𝗅</mi><mi id="S1.SS1.p1.9.m9.1.1.3" xref="S1.SS1.p1.9.m9.1.1.3.cmml">G</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.9.m9.1b"><apply id="S1.SS1.p1.9.m9.1.1.cmml" xref="S1.SS1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S1.SS1.p1.9.m9.1.1.1.cmml" xref="S1.SS1.p1.9.m9.1.1">subscript</csymbol><ci id="S1.SS1.p1.9.m9.1.1.2.cmml" xref="S1.SS1.p1.9.m9.1.1.2">𝗏𝖺𝗅</ci><ci id="S1.SS1.p1.9.m9.1.1.3.cmml" xref="S1.SS1.p1.9.m9.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.9.m9.1c">\mathsf{val}_{G}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.9.m9.1d">sansserif_val start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math>, is the maximum value over all cuts.</p> </div> <div class="ltx_para" id="S1.SS1.p2"> <p class="ltx_p" id="S1.SS1.p2.10">To define oblivious algorithms, we first define a scalar quantity associated to each vertex in a directed graph called <em class="ltx_emph ltx_font_italic" id="S1.SS1.p2.10.1">bias</em>. 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xref="S1.Ex1.m1.4.4.4.5.2.3">𝐺</ci></apply><ci id="S1.Ex1.m1.4.4.4.2.cmml" xref="S1.Ex1.m1.4.4.4.2">𝑣</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex1.m1.6c">\mathsf{bias}_{G}(v)\stackrel{{\scriptstyle\mathrm{\small def}}}{{=}}\frac{% \mathsf{outdeg}_{G}(v)-\mathsf{indeg}_{G}(v)}{\mathsf{outdeg}_{G}(v)+\mathsf{% indeg}_{G}(v)},</annotation><annotation encoding="application/x-llamapun" id="S1.Ex1.m1.6d">sansserif_bias start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP divide start_ARG sansserif_outdeg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) - sansserif_indeg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) end_ARG start_ARG sansserif_outdeg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) + sansserif_indeg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.SS1.p2.9">where <math alttext="\mathsf{outdeg}_{G}(v)" class="ltx_Math" display="inline" id="S1.SS1.p2.1.m1.1"><semantics id="S1.SS1.p2.1.m1.1a"><mrow id="S1.SS1.p2.1.m1.1.2" xref="S1.SS1.p2.1.m1.1.2.cmml"><msub id="S1.SS1.p2.1.m1.1.2.2" xref="S1.SS1.p2.1.m1.1.2.2.cmml"><mi id="S1.SS1.p2.1.m1.1.2.2.2" xref="S1.SS1.p2.1.m1.1.2.2.2.cmml">𝗈𝗎𝗍𝖽𝖾𝗀</mi><mi id="S1.SS1.p2.1.m1.1.2.2.3" xref="S1.SS1.p2.1.m1.1.2.2.3.cmml">G</mi></msub><mo id="S1.SS1.p2.1.m1.1.2.1" xref="S1.SS1.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.p2.1.m1.1.2.3.2" xref="S1.SS1.p2.1.m1.1.2.cmml"><mo id="S1.SS1.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S1.SS1.p2.1.m1.1.2.cmml">(</mo><mi id="S1.SS1.p2.1.m1.1.1" xref="S1.SS1.p2.1.m1.1.1.cmml">v</mi><mo id="S1.SS1.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S1.SS1.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.1.m1.1b"><apply id="S1.SS1.p2.1.m1.1.2.cmml" xref="S1.SS1.p2.1.m1.1.2"><times id="S1.SS1.p2.1.m1.1.2.1.cmml" xref="S1.SS1.p2.1.m1.1.2.1"></times><apply id="S1.SS1.p2.1.m1.1.2.2.cmml" xref="S1.SS1.p2.1.m1.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.p2.1.m1.1.2.2.1.cmml" xref="S1.SS1.p2.1.m1.1.2.2">subscript</csymbol><ci id="S1.SS1.p2.1.m1.1.2.2.2.cmml" xref="S1.SS1.p2.1.m1.1.2.2.2">𝗈𝗎𝗍𝖽𝖾𝗀</ci><ci id="S1.SS1.p2.1.m1.1.2.2.3.cmml" xref="S1.SS1.p2.1.m1.1.2.2.3">𝐺</ci></apply><ci id="S1.SS1.p2.1.m1.1.1.cmml" xref="S1.SS1.p2.1.m1.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.1.m1.1c">\mathsf{outdeg}_{G}(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.1.m1.1d">sansserif_outdeg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math> and <math alttext="\mathsf{indeg}_{G}(v)" class="ltx_Math" display="inline" id="S1.SS1.p2.2.m2.1"><semantics id="S1.SS1.p2.2.m2.1a"><mrow id="S1.SS1.p2.2.m2.1.2" xref="S1.SS1.p2.2.m2.1.2.cmml"><msub id="S1.SS1.p2.2.m2.1.2.2" xref="S1.SS1.p2.2.m2.1.2.2.cmml"><mi id="S1.SS1.p2.2.m2.1.2.2.2" xref="S1.SS1.p2.2.m2.1.2.2.2.cmml">𝗂𝗇𝖽𝖾𝗀</mi><mi id="S1.SS1.p2.2.m2.1.2.2.3" xref="S1.SS1.p2.2.m2.1.2.2.3.cmml">G</mi></msub><mo id="S1.SS1.p2.2.m2.1.2.1" xref="S1.SS1.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.p2.2.m2.1.2.3.2" xref="S1.SS1.p2.2.m2.1.2.cmml"><mo id="S1.SS1.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S1.SS1.p2.2.m2.1.2.cmml">(</mo><mi id="S1.SS1.p2.2.m2.1.1" xref="S1.SS1.p2.2.m2.1.1.cmml">v</mi><mo id="S1.SS1.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S1.SS1.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.2.m2.1b"><apply id="S1.SS1.p2.2.m2.1.2.cmml" xref="S1.SS1.p2.2.m2.1.2"><times id="S1.SS1.p2.2.m2.1.2.1.cmml" xref="S1.SS1.p2.2.m2.1.2.1"></times><apply id="S1.SS1.p2.2.m2.1.2.2.cmml" xref="S1.SS1.p2.2.m2.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.p2.2.m2.1.2.2.1.cmml" xref="S1.SS1.p2.2.m2.1.2.2">subscript</csymbol><ci id="S1.SS1.p2.2.m2.1.2.2.2.cmml" xref="S1.SS1.p2.2.m2.1.2.2.2">𝗂𝗇𝖽𝖾𝗀</ci><ci id="S1.SS1.p2.2.m2.1.2.2.3.cmml" xref="S1.SS1.p2.2.m2.1.2.2.3">𝐺</ci></apply><ci id="S1.SS1.p2.2.m2.1.1.cmml" xref="S1.SS1.p2.2.m2.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.2.m2.1c">\mathsf{indeg}_{G}(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.2.m2.1d">sansserif_indeg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math> are respectively the out- and in-degrees of <math alttext="v" class="ltx_Math" display="inline" id="S1.SS1.p2.3.m3.1"><semantics id="S1.SS1.p2.3.m3.1a"><mi id="S1.SS1.p2.3.m3.1.1" xref="S1.SS1.p2.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.3.m3.1b"><ci id="S1.SS1.p2.3.m3.1.1.cmml" xref="S1.SS1.p2.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.3.m3.1d">italic_v</annotation></semantics></math> in <math alttext="G" class="ltx_Math" display="inline" id="S1.SS1.p2.4.m4.1"><semantics id="S1.SS1.p2.4.m4.1a"><mi id="S1.SS1.p2.4.m4.1.1" xref="S1.SS1.p2.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.4.m4.1b"><ci id="S1.SS1.p2.4.m4.1.1.cmml" xref="S1.SS1.p2.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.4.m4.1d">italic_G</annotation></semantics></math>. Note that <math alttext="\mathsf{bias}_{G}(v)" class="ltx_Math" display="inline" id="S1.SS1.p2.5.m5.1"><semantics id="S1.SS1.p2.5.m5.1a"><mrow id="S1.SS1.p2.5.m5.1.2" xref="S1.SS1.p2.5.m5.1.2.cmml"><msub id="S1.SS1.p2.5.m5.1.2.2" xref="S1.SS1.p2.5.m5.1.2.2.cmml"><mi id="S1.SS1.p2.5.m5.1.2.2.2" xref="S1.SS1.p2.5.m5.1.2.2.2.cmml">𝖻𝗂𝖺𝗌</mi><mi id="S1.SS1.p2.5.m5.1.2.2.3" xref="S1.SS1.p2.5.m5.1.2.2.3.cmml">G</mi></msub><mo id="S1.SS1.p2.5.m5.1.2.1" xref="S1.SS1.p2.5.m5.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.p2.5.m5.1.2.3.2" xref="S1.SS1.p2.5.m5.1.2.cmml"><mo id="S1.SS1.p2.5.m5.1.2.3.2.1" stretchy="false" xref="S1.SS1.p2.5.m5.1.2.cmml">(</mo><mi id="S1.SS1.p2.5.m5.1.1" xref="S1.SS1.p2.5.m5.1.1.cmml">v</mi><mo id="S1.SS1.p2.5.m5.1.2.3.2.2" stretchy="false" xref="S1.SS1.p2.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.5.m5.1b"><apply id="S1.SS1.p2.5.m5.1.2.cmml" xref="S1.SS1.p2.5.m5.1.2"><times id="S1.SS1.p2.5.m5.1.2.1.cmml" xref="S1.SS1.p2.5.m5.1.2.1"></times><apply id="S1.SS1.p2.5.m5.1.2.2.cmml" xref="S1.SS1.p2.5.m5.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.p2.5.m5.1.2.2.1.cmml" xref="S1.SS1.p2.5.m5.1.2.2">subscript</csymbol><ci id="S1.SS1.p2.5.m5.1.2.2.2.cmml" xref="S1.SS1.p2.5.m5.1.2.2.2">𝖻𝗂𝖺𝗌</ci><ci id="S1.SS1.p2.5.m5.1.2.2.3.cmml" xref="S1.SS1.p2.5.m5.1.2.2.3">𝐺</ci></apply><ci id="S1.SS1.p2.5.m5.1.1.cmml" xref="S1.SS1.p2.5.m5.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.5.m5.1c">\mathsf{bias}_{G}(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.5.m5.1d">sansserif_bias start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math> ranges from <math alttext="+1" class="ltx_Math" display="inline" id="S1.SS1.p2.6.m6.1"><semantics id="S1.SS1.p2.6.m6.1a"><mrow id="S1.SS1.p2.6.m6.1.1" xref="S1.SS1.p2.6.m6.1.1.cmml"><mo id="S1.SS1.p2.6.m6.1.1a" xref="S1.SS1.p2.6.m6.1.1.cmml">+</mo><mn id="S1.SS1.p2.6.m6.1.1.2" xref="S1.SS1.p2.6.m6.1.1.2.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.6.m6.1b"><apply id="S1.SS1.p2.6.m6.1.1.cmml" xref="S1.SS1.p2.6.m6.1.1"><plus id="S1.SS1.p2.6.m6.1.1.1.cmml" xref="S1.SS1.p2.6.m6.1.1"></plus><cn id="S1.SS1.p2.6.m6.1.1.2.cmml" type="integer" xref="S1.SS1.p2.6.m6.1.1.2">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.6.m6.1c">+1</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.6.m6.1d">+ 1</annotation></semantics></math> (<math alttext="v" class="ltx_Math" display="inline" id="S1.SS1.p2.7.m7.1"><semantics id="S1.SS1.p2.7.m7.1a"><mi id="S1.SS1.p2.7.m7.1.1" xref="S1.SS1.p2.7.m7.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.7.m7.1b"><ci id="S1.SS1.p2.7.m7.1.1.cmml" xref="S1.SS1.p2.7.m7.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.7.m7.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.7.m7.1d">italic_v</annotation></semantics></math> has only out-edges) to <math alttext="-1" class="ltx_Math" display="inline" id="S1.SS1.p2.8.m8.1"><semantics id="S1.SS1.p2.8.m8.1a"><mrow id="S1.SS1.p2.8.m8.1.1" xref="S1.SS1.p2.8.m8.1.1.cmml"><mo id="S1.SS1.p2.8.m8.1.1a" xref="S1.SS1.p2.8.m8.1.1.cmml">−</mo><mn id="S1.SS1.p2.8.m8.1.1.2" xref="S1.SS1.p2.8.m8.1.1.2.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.8.m8.1b"><apply id="S1.SS1.p2.8.m8.1.1.cmml" xref="S1.SS1.p2.8.m8.1.1"><minus id="S1.SS1.p2.8.m8.1.1.1.cmml" xref="S1.SS1.p2.8.m8.1.1"></minus><cn id="S1.SS1.p2.8.m8.1.1.2.cmml" type="integer" xref="S1.SS1.p2.8.m8.1.1.2">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.8.m8.1c">-1</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.8.m8.1d">- 1</annotation></semantics></math> (<math alttext="v" class="ltx_Math" display="inline" id="S1.SS1.p2.9.m9.1"><semantics id="S1.SS1.p2.9.m9.1a"><mi id="S1.SS1.p2.9.m9.1.1" xref="S1.SS1.p2.9.m9.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.9.m9.1b"><ci id="S1.SS1.p2.9.m9.1.1.cmml" xref="S1.SS1.p2.9.m9.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.9.m9.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.9.m9.1d">italic_v</annotation></semantics></math> has only in-edges).</p> </div> <div class="ltx_para" id="S1.SS1.p3"> <p class="ltx_p" id="S1.SS1.p3.11">Oblivious algorithms are a class of randomized algorithms for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS1.p3.11.1">Max-DiCut</span> which “only know” about the bias of each vertex. More formally, an oblivious algorithm is defined by a so-called <em class="ltx_emph ltx_font_italic" id="S1.SS1.p3.11.2">selection function</em> <math alttext="\mathsf{S}:[-1,+1]\to[0,1]" class="ltx_Math" display="inline" id="S1.SS1.p3.2.m2.4"><semantics id="S1.SS1.p3.2.m2.4a"><mrow id="S1.SS1.p3.2.m2.4.4" xref="S1.SS1.p3.2.m2.4.4.cmml"><mi id="S1.SS1.p3.2.m2.4.4.4" xref="S1.SS1.p3.2.m2.4.4.4.cmml">𝖲</mi><mo id="S1.SS1.p3.2.m2.4.4.3" lspace="0.278em" rspace="0.278em" xref="S1.SS1.p3.2.m2.4.4.3.cmml">:</mo><mrow id="S1.SS1.p3.2.m2.4.4.2" xref="S1.SS1.p3.2.m2.4.4.2.cmml"><mrow id="S1.SS1.p3.2.m2.4.4.2.2.2" xref="S1.SS1.p3.2.m2.4.4.2.2.3.cmml"><mo id="S1.SS1.p3.2.m2.4.4.2.2.2.3" stretchy="false" xref="S1.SS1.p3.2.m2.4.4.2.2.3.cmml">[</mo><mrow id="S1.SS1.p3.2.m2.3.3.1.1.1.1" xref="S1.SS1.p3.2.m2.3.3.1.1.1.1.cmml"><mo id="S1.SS1.p3.2.m2.3.3.1.1.1.1a" xref="S1.SS1.p3.2.m2.3.3.1.1.1.1.cmml">−</mo><mn id="S1.SS1.p3.2.m2.3.3.1.1.1.1.2" xref="S1.SS1.p3.2.m2.3.3.1.1.1.1.2.cmml">1</mn></mrow><mo id="S1.SS1.p3.2.m2.4.4.2.2.2.4" xref="S1.SS1.p3.2.m2.4.4.2.2.3.cmml">,</mo><mrow id="S1.SS1.p3.2.m2.4.4.2.2.2.2" xref="S1.SS1.p3.2.m2.4.4.2.2.2.2.cmml"><mo id="S1.SS1.p3.2.m2.4.4.2.2.2.2a" xref="S1.SS1.p3.2.m2.4.4.2.2.2.2.cmml">+</mo><mn id="S1.SS1.p3.2.m2.4.4.2.2.2.2.2" xref="S1.SS1.p3.2.m2.4.4.2.2.2.2.2.cmml">1</mn></mrow><mo id="S1.SS1.p3.2.m2.4.4.2.2.2.5" stretchy="false" xref="S1.SS1.p3.2.m2.4.4.2.2.3.cmml">]</mo></mrow><mo id="S1.SS1.p3.2.m2.4.4.2.3" stretchy="false" xref="S1.SS1.p3.2.m2.4.4.2.3.cmml">→</mo><mrow id="S1.SS1.p3.2.m2.4.4.2.4.2" xref="S1.SS1.p3.2.m2.4.4.2.4.1.cmml"><mo id="S1.SS1.p3.2.m2.4.4.2.4.2.1" stretchy="false" xref="S1.SS1.p3.2.m2.4.4.2.4.1.cmml">[</mo><mn id="S1.SS1.p3.2.m2.1.1" xref="S1.SS1.p3.2.m2.1.1.cmml">0</mn><mo id="S1.SS1.p3.2.m2.4.4.2.4.2.2" xref="S1.SS1.p3.2.m2.4.4.2.4.1.cmml">,</mo><mn id="S1.SS1.p3.2.m2.2.2" xref="S1.SS1.p3.2.m2.2.2.cmml">1</mn><mo id="S1.SS1.p3.2.m2.4.4.2.4.2.3" stretchy="false" xref="S1.SS1.p3.2.m2.4.4.2.4.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.2.m2.4b"><apply id="S1.SS1.p3.2.m2.4.4.cmml" xref="S1.SS1.p3.2.m2.4.4"><ci id="S1.SS1.p3.2.m2.4.4.3.cmml" xref="S1.SS1.p3.2.m2.4.4.3">:</ci><ci id="S1.SS1.p3.2.m2.4.4.4.cmml" xref="S1.SS1.p3.2.m2.4.4.4">𝖲</ci><apply id="S1.SS1.p3.2.m2.4.4.2.cmml" xref="S1.SS1.p3.2.m2.4.4.2"><ci id="S1.SS1.p3.2.m2.4.4.2.3.cmml" xref="S1.SS1.p3.2.m2.4.4.2.3">→</ci><interval closure="closed" id="S1.SS1.p3.2.m2.4.4.2.2.3.cmml" xref="S1.SS1.p3.2.m2.4.4.2.2.2"><apply id="S1.SS1.p3.2.m2.3.3.1.1.1.1.cmml" xref="S1.SS1.p3.2.m2.3.3.1.1.1.1"><minus id="S1.SS1.p3.2.m2.3.3.1.1.1.1.1.cmml" xref="S1.SS1.p3.2.m2.3.3.1.1.1.1"></minus><cn id="S1.SS1.p3.2.m2.3.3.1.1.1.1.2.cmml" type="integer" xref="S1.SS1.p3.2.m2.3.3.1.1.1.1.2">1</cn></apply><apply id="S1.SS1.p3.2.m2.4.4.2.2.2.2.cmml" xref="S1.SS1.p3.2.m2.4.4.2.2.2.2"><plus id="S1.SS1.p3.2.m2.4.4.2.2.2.2.1.cmml" xref="S1.SS1.p3.2.m2.4.4.2.2.2.2"></plus><cn id="S1.SS1.p3.2.m2.4.4.2.2.2.2.2.cmml" type="integer" xref="S1.SS1.p3.2.m2.4.4.2.2.2.2.2">1</cn></apply></interval><interval closure="closed" id="S1.SS1.p3.2.m2.4.4.2.4.1.cmml" xref="S1.SS1.p3.2.m2.4.4.2.4.2"><cn id="S1.SS1.p3.2.m2.1.1.cmml" type="integer" xref="S1.SS1.p3.2.m2.1.1">0</cn><cn id="S1.SS1.p3.2.m2.2.2.cmml" type="integer" xref="S1.SS1.p3.2.m2.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.2.m2.4c">\mathsf{S}:[-1,+1]\to[0,1]</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.2.m2.4d">sansserif_S : [ - 1 , + 1 ] → [ 0 , 1 ]</annotation></semantics></math>. The corresponding algorithm <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S1.SS1.p3.3.m3.1"><semantics id="S1.SS1.p3.3.m3.1a"><msub id="S1.SS1.p3.3.m3.1.1" xref="S1.SS1.p3.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS1.p3.3.m3.1.1.2" xref="S1.SS1.p3.3.m3.1.1.2.cmml">𝒪</mi><mi id="S1.SS1.p3.3.m3.1.1.3" xref="S1.SS1.p3.3.m3.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.3.m3.1b"><apply id="S1.SS1.p3.3.m3.1.1.cmml" xref="S1.SS1.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.3.m3.1.1.1.cmml" xref="S1.SS1.p3.3.m3.1.1">subscript</csymbol><ci id="S1.SS1.p3.3.m3.1.1.2.cmml" xref="S1.SS1.p3.3.m3.1.1.2">𝒪</ci><ci id="S1.SS1.p3.3.m3.1.1.3.cmml" xref="S1.SS1.p3.3.m3.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.3.m3.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.3.m3.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math>, given a graph <math alttext="G" class="ltx_Math" display="inline" id="S1.SS1.p3.4.m4.1"><semantics id="S1.SS1.p3.4.m4.1a"><mi id="S1.SS1.p3.4.m4.1.1" xref="S1.SS1.p3.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.4.m4.1b"><ci id="S1.SS1.p3.4.m4.1.1.cmml" xref="S1.SS1.p3.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.4.m4.1d">italic_G</annotation></semantics></math>, produces an assignment by independently setting each vertex <math alttext="v" class="ltx_Math" display="inline" id="S1.SS1.p3.5.m5.1"><semantics id="S1.SS1.p3.5.m5.1a"><mi id="S1.SS1.p3.5.m5.1.1" xref="S1.SS1.p3.5.m5.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.5.m5.1b"><ci id="S1.SS1.p3.5.m5.1.1.cmml" xref="S1.SS1.p3.5.m5.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.5.m5.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.5.m5.1d">italic_v</annotation></semantics></math> to equal <math alttext="1" class="ltx_Math" display="inline" id="S1.SS1.p3.6.m6.1"><semantics id="S1.SS1.p3.6.m6.1a"><mn id="S1.SS1.p3.6.m6.1.1" xref="S1.SS1.p3.6.m6.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.6.m6.1b"><cn id="S1.SS1.p3.6.m6.1.1.cmml" type="integer" xref="S1.SS1.p3.6.m6.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.6.m6.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.6.m6.1d">1</annotation></semantics></math> w.p. <math alttext="\mathsf{S}(\mathsf{bias}_{G}(v))" class="ltx_Math" display="inline" id="S1.SS1.p3.7.m7.2"><semantics id="S1.SS1.p3.7.m7.2a"><mrow id="S1.SS1.p3.7.m7.2.2" xref="S1.SS1.p3.7.m7.2.2.cmml"><mi id="S1.SS1.p3.7.m7.2.2.3" xref="S1.SS1.p3.7.m7.2.2.3.cmml">𝖲</mi><mo id="S1.SS1.p3.7.m7.2.2.2" xref="S1.SS1.p3.7.m7.2.2.2.cmml">⁢</mo><mrow id="S1.SS1.p3.7.m7.2.2.1.1" xref="S1.SS1.p3.7.m7.2.2.1.1.1.cmml"><mo id="S1.SS1.p3.7.m7.2.2.1.1.2" stretchy="false" xref="S1.SS1.p3.7.m7.2.2.1.1.1.cmml">(</mo><mrow id="S1.SS1.p3.7.m7.2.2.1.1.1" xref="S1.SS1.p3.7.m7.2.2.1.1.1.cmml"><msub id="S1.SS1.p3.7.m7.2.2.1.1.1.2" xref="S1.SS1.p3.7.m7.2.2.1.1.1.2.cmml"><mi id="S1.SS1.p3.7.m7.2.2.1.1.1.2.2" xref="S1.SS1.p3.7.m7.2.2.1.1.1.2.2.cmml">𝖻𝗂𝖺𝗌</mi><mi id="S1.SS1.p3.7.m7.2.2.1.1.1.2.3" xref="S1.SS1.p3.7.m7.2.2.1.1.1.2.3.cmml">G</mi></msub><mo id="S1.SS1.p3.7.m7.2.2.1.1.1.1" xref="S1.SS1.p3.7.m7.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S1.SS1.p3.7.m7.2.2.1.1.1.3.2" xref="S1.SS1.p3.7.m7.2.2.1.1.1.cmml"><mo id="S1.SS1.p3.7.m7.2.2.1.1.1.3.2.1" stretchy="false" xref="S1.SS1.p3.7.m7.2.2.1.1.1.cmml">(</mo><mi id="S1.SS1.p3.7.m7.1.1" xref="S1.SS1.p3.7.m7.1.1.cmml">v</mi><mo id="S1.SS1.p3.7.m7.2.2.1.1.1.3.2.2" stretchy="false" xref="S1.SS1.p3.7.m7.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.SS1.p3.7.m7.2.2.1.1.3" stretchy="false" xref="S1.SS1.p3.7.m7.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.7.m7.2b"><apply id="S1.SS1.p3.7.m7.2.2.cmml" xref="S1.SS1.p3.7.m7.2.2"><times id="S1.SS1.p3.7.m7.2.2.2.cmml" xref="S1.SS1.p3.7.m7.2.2.2"></times><ci id="S1.SS1.p3.7.m7.2.2.3.cmml" xref="S1.SS1.p3.7.m7.2.2.3">𝖲</ci><apply id="S1.SS1.p3.7.m7.2.2.1.1.1.cmml" xref="S1.SS1.p3.7.m7.2.2.1.1"><times id="S1.SS1.p3.7.m7.2.2.1.1.1.1.cmml" xref="S1.SS1.p3.7.m7.2.2.1.1.1.1"></times><apply id="S1.SS1.p3.7.m7.2.2.1.1.1.2.cmml" xref="S1.SS1.p3.7.m7.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p3.7.m7.2.2.1.1.1.2.1.cmml" xref="S1.SS1.p3.7.m7.2.2.1.1.1.2">subscript</csymbol><ci id="S1.SS1.p3.7.m7.2.2.1.1.1.2.2.cmml" xref="S1.SS1.p3.7.m7.2.2.1.1.1.2.2">𝖻𝗂𝖺𝗌</ci><ci id="S1.SS1.p3.7.m7.2.2.1.1.1.2.3.cmml" xref="S1.SS1.p3.7.m7.2.2.1.1.1.2.3">𝐺</ci></apply><ci id="S1.SS1.p3.7.m7.1.1.cmml" xref="S1.SS1.p3.7.m7.1.1">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.7.m7.2c">\mathsf{S}(\mathsf{bias}_{G}(v))</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.7.m7.2d">sansserif_S ( sansserif_bias start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) )</annotation></semantics></math> and <math alttext="0" class="ltx_Math" display="inline" id="S1.SS1.p3.8.m8.1"><semantics id="S1.SS1.p3.8.m8.1a"><mn id="S1.SS1.p3.8.m8.1.1" xref="S1.SS1.p3.8.m8.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.8.m8.1b"><cn id="S1.SS1.p3.8.m8.1.1.cmml" type="integer" xref="S1.SS1.p3.8.m8.1.1">0</cn></annotation-xml></semantics></math> otherwise. The <em class="ltx_emph ltx_font_italic" id="S1.SS1.p3.11.3">approximation ratio</em> of <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S1.SS1.p3.9.m9.1"><semantics id="S1.SS1.p3.9.m9.1a"><msub id="S1.SS1.p3.9.m9.1.1" xref="S1.SS1.p3.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS1.p3.9.m9.1.1.2" xref="S1.SS1.p3.9.m9.1.1.2.cmml">𝒪</mi><mi id="S1.SS1.p3.9.m9.1.1.3" xref="S1.SS1.p3.9.m9.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.9.m9.1b"><apply id="S1.SS1.p3.9.m9.1.1.cmml" xref="S1.SS1.p3.9.m9.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.9.m9.1.1.1.cmml" xref="S1.SS1.p3.9.m9.1.1">subscript</csymbol><ci id="S1.SS1.p3.9.m9.1.1.2.cmml" xref="S1.SS1.p3.9.m9.1.1.2">𝒪</ci><ci id="S1.SS1.p3.9.m9.1.1.3.cmml" xref="S1.SS1.p3.9.m9.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.9.m9.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.9.m9.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math>, denoted <math alttext="\alpha(\mathcal{O}_{\mathsf{S}})" class="ltx_Math" display="inline" id="S1.SS1.p3.10.m10.1"><semantics id="S1.SS1.p3.10.m10.1a"><mrow id="S1.SS1.p3.10.m10.1.1" xref="S1.SS1.p3.10.m10.1.1.cmml"><mi id="S1.SS1.p3.10.m10.1.1.3" xref="S1.SS1.p3.10.m10.1.1.3.cmml">α</mi><mo id="S1.SS1.p3.10.m10.1.1.2" xref="S1.SS1.p3.10.m10.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.p3.10.m10.1.1.1.1" xref="S1.SS1.p3.10.m10.1.1.1.1.1.cmml"><mo id="S1.SS1.p3.10.m10.1.1.1.1.2" stretchy="false" xref="S1.SS1.p3.10.m10.1.1.1.1.1.cmml">(</mo><msub id="S1.SS1.p3.10.m10.1.1.1.1.1" xref="S1.SS1.p3.10.m10.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS1.p3.10.m10.1.1.1.1.1.2" xref="S1.SS1.p3.10.m10.1.1.1.1.1.2.cmml">𝒪</mi><mi id="S1.SS1.p3.10.m10.1.1.1.1.1.3" xref="S1.SS1.p3.10.m10.1.1.1.1.1.3.cmml">𝖲</mi></msub><mo id="S1.SS1.p3.10.m10.1.1.1.1.3" stretchy="false" xref="S1.SS1.p3.10.m10.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.10.m10.1b"><apply id="S1.SS1.p3.10.m10.1.1.cmml" xref="S1.SS1.p3.10.m10.1.1"><times id="S1.SS1.p3.10.m10.1.1.2.cmml" xref="S1.SS1.p3.10.m10.1.1.2"></times><ci id="S1.SS1.p3.10.m10.1.1.3.cmml" xref="S1.SS1.p3.10.m10.1.1.3">𝛼</ci><apply id="S1.SS1.p3.10.m10.1.1.1.1.1.cmml" xref="S1.SS1.p3.10.m10.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.10.m10.1.1.1.1.1.1.cmml" xref="S1.SS1.p3.10.m10.1.1.1.1">subscript</csymbol><ci id="S1.SS1.p3.10.m10.1.1.1.1.1.2.cmml" xref="S1.SS1.p3.10.m10.1.1.1.1.1.2">𝒪</ci><ci id="S1.SS1.p3.10.m10.1.1.1.1.1.3.cmml" xref="S1.SS1.p3.10.m10.1.1.1.1.1.3">𝖲</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.10.m10.1c">\alpha(\mathcal{O}_{\mathsf{S}})</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.10.m10.1d">italic_α ( caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT )</annotation></semantics></math>, is the ratio of the expected value of this “oblivious assignment” to the value of the best assignment, minimized over all graphs <math alttext="G" class="ltx_Math" display="inline" id="S1.SS1.p3.11.m11.1"><semantics id="S1.SS1.p3.11.m11.1a"><mi id="S1.SS1.p3.11.m11.1.1" xref="S1.SS1.p3.11.m11.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.11.m11.1b"><ci id="S1.SS1.p3.11.m11.1.1.cmml" xref="S1.SS1.p3.11.m11.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.11.m11.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.11.m11.1d">italic_G</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SS1.p4"> <p class="ltx_p" id="S1.SS1.p4.2">Oblivious algorithms for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS1.p4.2.1">Max-DiCut</span> were introduced by <span class="ltx_ERROR undefined" id="S1.SS1.p4.2.2">\textcite</span>FJ15, who proved both upper and lower bounds<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>Note on language: In this paper, we use the usual convention that upper bounds are algorithms and lower bounds are hardness results. Confusingly, an “upper bound” actually lower-bounds the maximum ratio achievable by any oblivious algorithm, while a “lower bound” upper-bounds this ratio.</span></span></span> on their capacity to approximate <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS1.p4.2.3">Max-DiCut</span>:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.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="S1.Thmtheorem1.1.1.1">Theorem 1.1</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.2.2"> </span>(Prior upper bound, <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">FJ15</span>, Thm. 1.3]</cite>)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.3.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem1.p1"> <p class="ltx_p" id="S1.Thmtheorem1.p1.2"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem1.p1.2.2">There exists an oblivious algorithm <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.1.1.m1.1"><semantics id="S1.Thmtheorem1.p1.1.1.m1.1a"><msub id="S1.Thmtheorem1.p1.1.1.m1.1.1" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem1.p1.1.1.m1.1.1.2" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.2.cmml">𝒪</mi><mi id="S1.Thmtheorem1.p1.1.1.m1.1.1.3" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.1.1.m1.1b"><apply id="S1.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.1.1">subscript</csymbol><ci id="S1.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.2">𝒪</ci><ci id="S1.Thmtheorem1.p1.1.1.m1.1.1.3.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.1.1.m1.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.1.1.m1.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math> achieving an approximation ratio <math alttext="\alpha(\mathcal{O}_{\mathsf{S}})\geq 0.483" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.2.2.m2.1"><semantics id="S1.Thmtheorem1.p1.2.2.m2.1a"><mrow id="S1.Thmtheorem1.p1.2.2.m2.1.1" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.cmml"><mrow id="S1.Thmtheorem1.p1.2.2.m2.1.1.1" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.cmml"><mi id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.3" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.3.cmml">α</mi><mo id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.2" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.cmml"><mo id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.2" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.2.cmml">𝒪</mi><mi id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.3" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.3.cmml">𝖲</mi></msub><mo id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem1.p1.2.2.m2.1.1.2" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.2.cmml">≥</mo><mn id="S1.Thmtheorem1.p1.2.2.m2.1.1.3" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.3.cmml">0.483</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.2.2.m2.1b"><apply id="S1.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.1"><geq id="S1.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.2"></geq><apply id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1"><times id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.2.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.2"></times><ci id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.3.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.3">𝛼</ci><apply id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.2.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.2">𝒪</ci><ci id="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.3">𝖲</ci></apply></apply><cn id="S1.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" type="float" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.3">0.483</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.2.2.m2.1c">\alpha(\mathcal{O}_{\mathsf{S}})\geq 0.483</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.2.2.m2.1d">italic_α ( caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT ) ≥ 0.483</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.SS1.p5"> <p class="ltx_p" id="S1.SS1.p5.3">A selection function <math alttext="\mathsf{S}:[-1,+1]\to[0,1]" class="ltx_Math" display="inline" id="S1.SS1.p5.1.m1.4"><semantics id="S1.SS1.p5.1.m1.4a"><mrow id="S1.SS1.p5.1.m1.4.4" xref="S1.SS1.p5.1.m1.4.4.cmml"><mi id="S1.SS1.p5.1.m1.4.4.4" xref="S1.SS1.p5.1.m1.4.4.4.cmml">𝖲</mi><mo id="S1.SS1.p5.1.m1.4.4.3" lspace="0.278em" rspace="0.278em" xref="S1.SS1.p5.1.m1.4.4.3.cmml">:</mo><mrow id="S1.SS1.p5.1.m1.4.4.2" xref="S1.SS1.p5.1.m1.4.4.2.cmml"><mrow id="S1.SS1.p5.1.m1.4.4.2.2.2" xref="S1.SS1.p5.1.m1.4.4.2.2.3.cmml"><mo id="S1.SS1.p5.1.m1.4.4.2.2.2.3" stretchy="false" xref="S1.SS1.p5.1.m1.4.4.2.2.3.cmml">[</mo><mrow id="S1.SS1.p5.1.m1.3.3.1.1.1.1" xref="S1.SS1.p5.1.m1.3.3.1.1.1.1.cmml"><mo id="S1.SS1.p5.1.m1.3.3.1.1.1.1a" xref="S1.SS1.p5.1.m1.3.3.1.1.1.1.cmml">−</mo><mn id="S1.SS1.p5.1.m1.3.3.1.1.1.1.2" xref="S1.SS1.p5.1.m1.3.3.1.1.1.1.2.cmml">1</mn></mrow><mo id="S1.SS1.p5.1.m1.4.4.2.2.2.4" xref="S1.SS1.p5.1.m1.4.4.2.2.3.cmml">,</mo><mrow id="S1.SS1.p5.1.m1.4.4.2.2.2.2" xref="S1.SS1.p5.1.m1.4.4.2.2.2.2.cmml"><mo id="S1.SS1.p5.1.m1.4.4.2.2.2.2a" xref="S1.SS1.p5.1.m1.4.4.2.2.2.2.cmml">+</mo><mn id="S1.SS1.p5.1.m1.4.4.2.2.2.2.2" xref="S1.SS1.p5.1.m1.4.4.2.2.2.2.2.cmml">1</mn></mrow><mo id="S1.SS1.p5.1.m1.4.4.2.2.2.5" stretchy="false" xref="S1.SS1.p5.1.m1.4.4.2.2.3.cmml">]</mo></mrow><mo id="S1.SS1.p5.1.m1.4.4.2.3" stretchy="false" xref="S1.SS1.p5.1.m1.4.4.2.3.cmml">→</mo><mrow id="S1.SS1.p5.1.m1.4.4.2.4.2" xref="S1.SS1.p5.1.m1.4.4.2.4.1.cmml"><mo id="S1.SS1.p5.1.m1.4.4.2.4.2.1" stretchy="false" xref="S1.SS1.p5.1.m1.4.4.2.4.1.cmml">[</mo><mn id="S1.SS1.p5.1.m1.1.1" xref="S1.SS1.p5.1.m1.1.1.cmml">0</mn><mo id="S1.SS1.p5.1.m1.4.4.2.4.2.2" xref="S1.SS1.p5.1.m1.4.4.2.4.1.cmml">,</mo><mn id="S1.SS1.p5.1.m1.2.2" xref="S1.SS1.p5.1.m1.2.2.cmml">1</mn><mo id="S1.SS1.p5.1.m1.4.4.2.4.2.3" stretchy="false" xref="S1.SS1.p5.1.m1.4.4.2.4.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.1.m1.4b"><apply id="S1.SS1.p5.1.m1.4.4.cmml" xref="S1.SS1.p5.1.m1.4.4"><ci id="S1.SS1.p5.1.m1.4.4.3.cmml" xref="S1.SS1.p5.1.m1.4.4.3">:</ci><ci id="S1.SS1.p5.1.m1.4.4.4.cmml" xref="S1.SS1.p5.1.m1.4.4.4">𝖲</ci><apply id="S1.SS1.p5.1.m1.4.4.2.cmml" xref="S1.SS1.p5.1.m1.4.4.2"><ci id="S1.SS1.p5.1.m1.4.4.2.3.cmml" xref="S1.SS1.p5.1.m1.4.4.2.3">→</ci><interval closure="closed" id="S1.SS1.p5.1.m1.4.4.2.2.3.cmml" xref="S1.SS1.p5.1.m1.4.4.2.2.2"><apply id="S1.SS1.p5.1.m1.3.3.1.1.1.1.cmml" xref="S1.SS1.p5.1.m1.3.3.1.1.1.1"><minus id="S1.SS1.p5.1.m1.3.3.1.1.1.1.1.cmml" xref="S1.SS1.p5.1.m1.3.3.1.1.1.1"></minus><cn id="S1.SS1.p5.1.m1.3.3.1.1.1.1.2.cmml" type="integer" xref="S1.SS1.p5.1.m1.3.3.1.1.1.1.2">1</cn></apply><apply id="S1.SS1.p5.1.m1.4.4.2.2.2.2.cmml" xref="S1.SS1.p5.1.m1.4.4.2.2.2.2"><plus id="S1.SS1.p5.1.m1.4.4.2.2.2.2.1.cmml" xref="S1.SS1.p5.1.m1.4.4.2.2.2.2"></plus><cn id="S1.SS1.p5.1.m1.4.4.2.2.2.2.2.cmml" type="integer" xref="S1.SS1.p5.1.m1.4.4.2.2.2.2.2">1</cn></apply></interval><interval closure="closed" id="S1.SS1.p5.1.m1.4.4.2.4.1.cmml" xref="S1.SS1.p5.1.m1.4.4.2.4.2"><cn id="S1.SS1.p5.1.m1.1.1.cmml" type="integer" xref="S1.SS1.p5.1.m1.1.1">0</cn><cn id="S1.SS1.p5.1.m1.2.2.cmml" type="integer" xref="S1.SS1.p5.1.m1.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.1.m1.4c">\mathsf{S}:[-1,+1]\to[0,1]</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.1.m1.4d">sansserif_S : [ - 1 , + 1 ] → [ 0 , 1 ]</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S1.SS1.p5.3.1">antisymmetric</em> if for all <math alttext="b\in[-1,+1]" class="ltx_Math" display="inline" id="S1.SS1.p5.2.m2.2"><semantics id="S1.SS1.p5.2.m2.2a"><mrow id="S1.SS1.p5.2.m2.2.2" xref="S1.SS1.p5.2.m2.2.2.cmml"><mi id="S1.SS1.p5.2.m2.2.2.4" xref="S1.SS1.p5.2.m2.2.2.4.cmml">b</mi><mo id="S1.SS1.p5.2.m2.2.2.3" xref="S1.SS1.p5.2.m2.2.2.3.cmml">∈</mo><mrow id="S1.SS1.p5.2.m2.2.2.2.2" xref="S1.SS1.p5.2.m2.2.2.2.3.cmml"><mo id="S1.SS1.p5.2.m2.2.2.2.2.3" stretchy="false" xref="S1.SS1.p5.2.m2.2.2.2.3.cmml">[</mo><mrow id="S1.SS1.p5.2.m2.1.1.1.1.1" xref="S1.SS1.p5.2.m2.1.1.1.1.1.cmml"><mo id="S1.SS1.p5.2.m2.1.1.1.1.1a" xref="S1.SS1.p5.2.m2.1.1.1.1.1.cmml">−</mo><mn id="S1.SS1.p5.2.m2.1.1.1.1.1.2" xref="S1.SS1.p5.2.m2.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S1.SS1.p5.2.m2.2.2.2.2.4" xref="S1.SS1.p5.2.m2.2.2.2.3.cmml">,</mo><mrow id="S1.SS1.p5.2.m2.2.2.2.2.2" xref="S1.SS1.p5.2.m2.2.2.2.2.2.cmml"><mo id="S1.SS1.p5.2.m2.2.2.2.2.2a" xref="S1.SS1.p5.2.m2.2.2.2.2.2.cmml">+</mo><mn id="S1.SS1.p5.2.m2.2.2.2.2.2.2" xref="S1.SS1.p5.2.m2.2.2.2.2.2.2.cmml">1</mn></mrow><mo id="S1.SS1.p5.2.m2.2.2.2.2.5" stretchy="false" xref="S1.SS1.p5.2.m2.2.2.2.3.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.2.m2.2b"><apply id="S1.SS1.p5.2.m2.2.2.cmml" xref="S1.SS1.p5.2.m2.2.2"><in id="S1.SS1.p5.2.m2.2.2.3.cmml" xref="S1.SS1.p5.2.m2.2.2.3"></in><ci id="S1.SS1.p5.2.m2.2.2.4.cmml" xref="S1.SS1.p5.2.m2.2.2.4">𝑏</ci><interval closure="closed" id="S1.SS1.p5.2.m2.2.2.2.3.cmml" xref="S1.SS1.p5.2.m2.2.2.2.2"><apply id="S1.SS1.p5.2.m2.1.1.1.1.1.cmml" xref="S1.SS1.p5.2.m2.1.1.1.1.1"><minus id="S1.SS1.p5.2.m2.1.1.1.1.1.1.cmml" xref="S1.SS1.p5.2.m2.1.1.1.1.1"></minus><cn id="S1.SS1.p5.2.m2.1.1.1.1.1.2.cmml" type="integer" xref="S1.SS1.p5.2.m2.1.1.1.1.1.2">1</cn></apply><apply id="S1.SS1.p5.2.m2.2.2.2.2.2.cmml" xref="S1.SS1.p5.2.m2.2.2.2.2.2"><plus id="S1.SS1.p5.2.m2.2.2.2.2.2.1.cmml" xref="S1.SS1.p5.2.m2.2.2.2.2.2"></plus><cn id="S1.SS1.p5.2.m2.2.2.2.2.2.2.cmml" type="integer" xref="S1.SS1.p5.2.m2.2.2.2.2.2.2">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.2.m2.2c">b\in[-1,+1]</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.2.m2.2d">italic_b ∈ [ - 1 , + 1 ]</annotation></semantics></math>, <math alttext="\mathsf{S}(-b)=1-\mathsf{S}(b)" class="ltx_Math" display="inline" id="S1.SS1.p5.3.m3.2"><semantics id="S1.SS1.p5.3.m3.2a"><mrow id="S1.SS1.p5.3.m3.2.2" xref="S1.SS1.p5.3.m3.2.2.cmml"><mrow id="S1.SS1.p5.3.m3.2.2.1" xref="S1.SS1.p5.3.m3.2.2.1.cmml"><mi id="S1.SS1.p5.3.m3.2.2.1.3" xref="S1.SS1.p5.3.m3.2.2.1.3.cmml">𝖲</mi><mo id="S1.SS1.p5.3.m3.2.2.1.2" xref="S1.SS1.p5.3.m3.2.2.1.2.cmml">⁢</mo><mrow id="S1.SS1.p5.3.m3.2.2.1.1.1" xref="S1.SS1.p5.3.m3.2.2.1.1.1.1.cmml"><mo id="S1.SS1.p5.3.m3.2.2.1.1.1.2" stretchy="false" xref="S1.SS1.p5.3.m3.2.2.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.p5.3.m3.2.2.1.1.1.1" xref="S1.SS1.p5.3.m3.2.2.1.1.1.1.cmml"><mo id="S1.SS1.p5.3.m3.2.2.1.1.1.1a" xref="S1.SS1.p5.3.m3.2.2.1.1.1.1.cmml">−</mo><mi id="S1.SS1.p5.3.m3.2.2.1.1.1.1.2" xref="S1.SS1.p5.3.m3.2.2.1.1.1.1.2.cmml">b</mi></mrow><mo id="S1.SS1.p5.3.m3.2.2.1.1.1.3" stretchy="false" xref="S1.SS1.p5.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.SS1.p5.3.m3.2.2.2" xref="S1.SS1.p5.3.m3.2.2.2.cmml">=</mo><mrow id="S1.SS1.p5.3.m3.2.2.3" xref="S1.SS1.p5.3.m3.2.2.3.cmml"><mn id="S1.SS1.p5.3.m3.2.2.3.2" xref="S1.SS1.p5.3.m3.2.2.3.2.cmml">1</mn><mo id="S1.SS1.p5.3.m3.2.2.3.1" xref="S1.SS1.p5.3.m3.2.2.3.1.cmml">−</mo><mrow id="S1.SS1.p5.3.m3.2.2.3.3" xref="S1.SS1.p5.3.m3.2.2.3.3.cmml"><mi id="S1.SS1.p5.3.m3.2.2.3.3.2" xref="S1.SS1.p5.3.m3.2.2.3.3.2.cmml">𝖲</mi><mo id="S1.SS1.p5.3.m3.2.2.3.3.1" xref="S1.SS1.p5.3.m3.2.2.3.3.1.cmml">⁢</mo><mrow id="S1.SS1.p5.3.m3.2.2.3.3.3.2" xref="S1.SS1.p5.3.m3.2.2.3.3.cmml"><mo id="S1.SS1.p5.3.m3.2.2.3.3.3.2.1" stretchy="false" xref="S1.SS1.p5.3.m3.2.2.3.3.cmml">(</mo><mi id="S1.SS1.p5.3.m3.1.1" xref="S1.SS1.p5.3.m3.1.1.cmml">b</mi><mo id="S1.SS1.p5.3.m3.2.2.3.3.3.2.2" stretchy="false" xref="S1.SS1.p5.3.m3.2.2.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.3.m3.2b"><apply id="S1.SS1.p5.3.m3.2.2.cmml" xref="S1.SS1.p5.3.m3.2.2"><eq id="S1.SS1.p5.3.m3.2.2.2.cmml" xref="S1.SS1.p5.3.m3.2.2.2"></eq><apply id="S1.SS1.p5.3.m3.2.2.1.cmml" xref="S1.SS1.p5.3.m3.2.2.1"><times id="S1.SS1.p5.3.m3.2.2.1.2.cmml" xref="S1.SS1.p5.3.m3.2.2.1.2"></times><ci id="S1.SS1.p5.3.m3.2.2.1.3.cmml" xref="S1.SS1.p5.3.m3.2.2.1.3">𝖲</ci><apply id="S1.SS1.p5.3.m3.2.2.1.1.1.1.cmml" xref="S1.SS1.p5.3.m3.2.2.1.1.1"><minus id="S1.SS1.p5.3.m3.2.2.1.1.1.1.1.cmml" xref="S1.SS1.p5.3.m3.2.2.1.1.1"></minus><ci id="S1.SS1.p5.3.m3.2.2.1.1.1.1.2.cmml" xref="S1.SS1.p5.3.m3.2.2.1.1.1.1.2">𝑏</ci></apply></apply><apply id="S1.SS1.p5.3.m3.2.2.3.cmml" xref="S1.SS1.p5.3.m3.2.2.3"><minus id="S1.SS1.p5.3.m3.2.2.3.1.cmml" xref="S1.SS1.p5.3.m3.2.2.3.1"></minus><cn id="S1.SS1.p5.3.m3.2.2.3.2.cmml" type="integer" xref="S1.SS1.p5.3.m3.2.2.3.2">1</cn><apply id="S1.SS1.p5.3.m3.2.2.3.3.cmml" xref="S1.SS1.p5.3.m3.2.2.3.3"><times id="S1.SS1.p5.3.m3.2.2.3.3.1.cmml" xref="S1.SS1.p5.3.m3.2.2.3.3.1"></times><ci id="S1.SS1.p5.3.m3.2.2.3.3.2.cmml" xref="S1.SS1.p5.3.m3.2.2.3.3.2">𝖲</ci><ci id="S1.SS1.p5.3.m3.1.1.cmml" xref="S1.SS1.p5.3.m3.1.1">𝑏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.3.m3.2c">\mathsf{S}(-b)=1-\mathsf{S}(b)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.3.m3.2d">sansserif_S ( - italic_b ) = 1 - sansserif_S ( italic_b )</annotation></semantics></math> (see also <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S2.E7" title="Definition 2.7 (Antisymmetry). ‣ 2.2 Max-DiCut and oblivious algorithms ‣ 2 Preliminaries ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Definition</span> <span class="ltx_text ltx_ref_tag">2.7</span></a>). Most of the selection functions which have been studied have this property.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.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="S1.Thmtheorem2.1.1.1">Theorem 1.2</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem2.2.2"> </span>(Prior lower bound for antisymmetric selection, <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">FJ15</span>, Thm. 1.4]</cite>)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem2.3.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem2.p1"> <p class="ltx_p" id="S1.Thmtheorem2.p1.3"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem2.p1.3.3">If <math alttext="\mathsf{S}:[-1,+1]\to[0,1]" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.1.1.m1.4"><semantics id="S1.Thmtheorem2.p1.1.1.m1.4a"><mrow id="S1.Thmtheorem2.p1.1.1.m1.4.4" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.cmml"><mi id="S1.Thmtheorem2.p1.1.1.m1.4.4.4" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.4.cmml">𝖲</mi><mo id="S1.Thmtheorem2.p1.1.1.m1.4.4.3" lspace="0.278em" rspace="0.278em" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.3.cmml">:</mo><mrow id="S1.Thmtheorem2.p1.1.1.m1.4.4.2" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.cmml"><mrow id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.3.cmml"><mo id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.3" stretchy="false" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.3.cmml">[</mo><mrow id="S1.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1" xref="S1.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.cmml"><mo id="S1.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1a" xref="S1.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.cmml">−</mo><mn id="S1.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.2" xref="S1.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.2.cmml">1</mn></mrow><mo id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.4" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.3.cmml">,</mo><mrow id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2.cmml"><mo id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2a" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2.cmml">+</mo><mn id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2.2" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2.2.cmml">1</mn></mrow><mo id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.5" stretchy="false" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.3.cmml">]</mo></mrow><mo id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.3" stretchy="false" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.3.cmml">→</mo><mrow id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.4.2" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.4.1.cmml"><mo id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.4.2.1" stretchy="false" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.4.1.cmml">[</mo><mn id="S1.Thmtheorem2.p1.1.1.m1.1.1" xref="S1.Thmtheorem2.p1.1.1.m1.1.1.cmml">0</mn><mo id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.4.2.2" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.4.1.cmml">,</mo><mn id="S1.Thmtheorem2.p1.1.1.m1.2.2" xref="S1.Thmtheorem2.p1.1.1.m1.2.2.cmml">1</mn><mo id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.4.2.3" stretchy="false" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.4.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.1.1.m1.4b"><apply id="S1.Thmtheorem2.p1.1.1.m1.4.4.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.4.4"><ci id="S1.Thmtheorem2.p1.1.1.m1.4.4.3.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.3">:</ci><ci id="S1.Thmtheorem2.p1.1.1.m1.4.4.4.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.4">𝖲</ci><apply id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2"><ci id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.3.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.3">→</ci><interval closure="closed" id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.3.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2"><apply id="S1.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1"><minus id="S1.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1"></minus><cn id="S1.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.2.cmml" type="integer" xref="S1.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.2">1</cn></apply><apply id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2"><plus id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2.1.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2"></plus><cn id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2.2.cmml" type="integer" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.2.2.2.2">1</cn></apply></interval><interval closure="closed" id="S1.Thmtheorem2.p1.1.1.m1.4.4.2.4.1.cmml" xref="S1.Thmtheorem2.p1.1.1.m1.4.4.2.4.2"><cn id="S1.Thmtheorem2.p1.1.1.m1.1.1.cmml" type="integer" xref="S1.Thmtheorem2.p1.1.1.m1.1.1">0</cn><cn id="S1.Thmtheorem2.p1.1.1.m1.2.2.cmml" type="integer" xref="S1.Thmtheorem2.p1.1.1.m1.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.1.1.m1.4c">\mathsf{S}:[-1,+1]\to[0,1]</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.1.1.m1.4d">sansserif_S : [ - 1 , + 1 ] → [ 0 , 1 ]</annotation></semantics></math> is an antisymmetric selection function, then the oblivious algorithm <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.2.2.m2.1"><semantics id="S1.Thmtheorem2.p1.2.2.m2.1a"><msub id="S1.Thmtheorem2.p1.2.2.m2.1.1" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem2.p1.2.2.m2.1.1.2" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.2.cmml">𝒪</mi><mi id="S1.Thmtheorem2.p1.2.2.m2.1.1.3" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.2.2.m2.1b"><apply id="S1.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.1">subscript</csymbol><ci id="S1.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.2">𝒪</ci><ci id="S1.Thmtheorem2.p1.2.2.m2.1.1.3.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.2.2.m2.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.2.2.m2.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math> achieves an approximation ratio <math alttext="\alpha(\mathcal{O}_{\mathsf{S}})\leq 0.4899" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.3.3.m3.1"><semantics id="S1.Thmtheorem2.p1.3.3.m3.1a"><mrow id="S1.Thmtheorem2.p1.3.3.m3.1.1" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.cmml"><mrow id="S1.Thmtheorem2.p1.3.3.m3.1.1.1" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.cmml"><mi id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.3" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.3.cmml">α</mi><mo id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.2" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.cmml"><mo id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.2" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.2.cmml">𝒪</mi><mi id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.3" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.3.cmml">𝖲</mi></msub><mo id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem2.p1.3.3.m3.1.1.2" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.2.cmml">≤</mo><mn id="S1.Thmtheorem2.p1.3.3.m3.1.1.3" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.cmml">0.4899</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.3.3.m3.1b"><apply id="S1.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1"><leq id="S1.Thmtheorem2.p1.3.3.m3.1.1.2.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.2"></leq><apply id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1"><times id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.2.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.2"></times><ci id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.3.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.3">𝛼</ci><apply id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.2.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.2">𝒪</ci><ci id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.1.1.1.3">𝖲</ci></apply></apply><cn id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.cmml" type="float" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3">0.4899</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.3.3.m3.1c">\alpha(\mathcal{O}_{\mathsf{S}})\leq 0.4899</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.3.3.m3.1d">italic_α ( caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT ) ≤ 0.4899</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.SS1.p6"> <p class="ltx_p" id="S1.SS1.p6.1">We include a brief proof of <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem2" title="Theorem 1.2 (Prior lower bound for antisymmetric selection, [FJ15, Thm. 1.4]). ‣ 1.1 Background: Directed cuts, bias, and oblivious algorithms ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.2</span></a> in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#A1" title="Appendix A Recap: The prior lower bound of \textciteFJ15 (Theorem 1.2) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Appendix</span> <span class="ltx_text ltx_ref_tag">A</span></a> to facilitate comparison with our new lower bounds.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.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="S1.Thmtheorem3.1.1.1">Theorem 1.3</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem3.2.2"> </span>(Prior lower bound for general selection, <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">FJ15</span>, Thm. 1.5]</cite>)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem3.3.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem3.p1"> <p class="ltx_p" id="S1.Thmtheorem3.p1.3"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem3.p1.3.3">If <math alttext="\mathsf{S}:[-1,+1]\to[0,1]" class="ltx_Math" display="inline" id="S1.Thmtheorem3.p1.1.1.m1.4"><semantics id="S1.Thmtheorem3.p1.1.1.m1.4a"><mrow id="S1.Thmtheorem3.p1.1.1.m1.4.4" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.cmml"><mi id="S1.Thmtheorem3.p1.1.1.m1.4.4.4" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.4.cmml">𝖲</mi><mo id="S1.Thmtheorem3.p1.1.1.m1.4.4.3" lspace="0.278em" rspace="0.278em" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.3.cmml">:</mo><mrow id="S1.Thmtheorem3.p1.1.1.m1.4.4.2" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.cmml"><mrow id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.3.cmml"><mo id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2.3" stretchy="false" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.3.cmml">[</mo><mrow id="S1.Thmtheorem3.p1.1.1.m1.3.3.1.1.1.1" xref="S1.Thmtheorem3.p1.1.1.m1.3.3.1.1.1.1.cmml"><mo id="S1.Thmtheorem3.p1.1.1.m1.3.3.1.1.1.1a" xref="S1.Thmtheorem3.p1.1.1.m1.3.3.1.1.1.1.cmml">−</mo><mn id="S1.Thmtheorem3.p1.1.1.m1.3.3.1.1.1.1.2" xref="S1.Thmtheorem3.p1.1.1.m1.3.3.1.1.1.1.2.cmml">1</mn></mrow><mo id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2.4" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.3.cmml">,</mo><mrow id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2.2" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2.2.cmml"><mo id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2.2a" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2.2.cmml">+</mo><mn id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2.2.2" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2.2.2.cmml">1</mn></mrow><mo id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2.5" stretchy="false" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.3.cmml">]</mo></mrow><mo id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.3" stretchy="false" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.3.cmml">→</mo><mrow id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.4.2" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.4.1.cmml"><mo id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.4.2.1" stretchy="false" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.4.1.cmml">[</mo><mn id="S1.Thmtheorem3.p1.1.1.m1.1.1" xref="S1.Thmtheorem3.p1.1.1.m1.1.1.cmml">0</mn><mo id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.4.2.2" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.4.1.cmml">,</mo><mn id="S1.Thmtheorem3.p1.1.1.m1.2.2" xref="S1.Thmtheorem3.p1.1.1.m1.2.2.cmml">1</mn><mo id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.4.2.3" stretchy="false" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.4.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem3.p1.1.1.m1.4b"><apply id="S1.Thmtheorem3.p1.1.1.m1.4.4.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.4.4"><ci id="S1.Thmtheorem3.p1.1.1.m1.4.4.3.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.3">:</ci><ci id="S1.Thmtheorem3.p1.1.1.m1.4.4.4.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.4">𝖲</ci><apply id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2"><ci id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.3.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.3">→</ci><interval closure="closed" id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.3.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2"><apply id="S1.Thmtheorem3.p1.1.1.m1.3.3.1.1.1.1.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.3.3.1.1.1.1"><minus id="S1.Thmtheorem3.p1.1.1.m1.3.3.1.1.1.1.1.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.3.3.1.1.1.1"></minus><cn id="S1.Thmtheorem3.p1.1.1.m1.3.3.1.1.1.1.2.cmml" type="integer" xref="S1.Thmtheorem3.p1.1.1.m1.3.3.1.1.1.1.2">1</cn></apply><apply id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2.2.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2.2"><plus id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2.2.1.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2.2"></plus><cn id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2.2.2.cmml" type="integer" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.2.2.2.2">1</cn></apply></interval><interval closure="closed" id="S1.Thmtheorem3.p1.1.1.m1.4.4.2.4.1.cmml" xref="S1.Thmtheorem3.p1.1.1.m1.4.4.2.4.2"><cn id="S1.Thmtheorem3.p1.1.1.m1.1.1.cmml" type="integer" xref="S1.Thmtheorem3.p1.1.1.m1.1.1">0</cn><cn id="S1.Thmtheorem3.p1.1.1.m1.2.2.cmml" type="integer" xref="S1.Thmtheorem3.p1.1.1.m1.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem3.p1.1.1.m1.4c">\mathsf{S}:[-1,+1]\to[0,1]</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem3.p1.1.1.m1.4d">sansserif_S : [ - 1 , + 1 ] → [ 0 , 1 ]</annotation></semantics></math> is any selection function, then the oblivious algorithm <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S1.Thmtheorem3.p1.2.2.m2.1"><semantics id="S1.Thmtheorem3.p1.2.2.m2.1a"><msub id="S1.Thmtheorem3.p1.2.2.m2.1.1" xref="S1.Thmtheorem3.p1.2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem3.p1.2.2.m2.1.1.2" xref="S1.Thmtheorem3.p1.2.2.m2.1.1.2.cmml">𝒪</mi><mi id="S1.Thmtheorem3.p1.2.2.m2.1.1.3" xref="S1.Thmtheorem3.p1.2.2.m2.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem3.p1.2.2.m2.1b"><apply id="S1.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem3.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem3.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmtheorem3.p1.2.2.m2.1.1">subscript</csymbol><ci id="S1.Thmtheorem3.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmtheorem3.p1.2.2.m2.1.1.2">𝒪</ci><ci id="S1.Thmtheorem3.p1.2.2.m2.1.1.3.cmml" xref="S1.Thmtheorem3.p1.2.2.m2.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem3.p1.2.2.m2.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem3.p1.2.2.m2.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math> achieves an approximation ratio <math alttext="\alpha(\mathcal{O}_{\mathsf{S}})\leq 0.4998" class="ltx_Math" display="inline" id="S1.Thmtheorem3.p1.3.3.m3.1"><semantics id="S1.Thmtheorem3.p1.3.3.m3.1a"><mrow id="S1.Thmtheorem3.p1.3.3.m3.1.1" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.cmml"><mrow id="S1.Thmtheorem3.p1.3.3.m3.1.1.1" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.cmml"><mi id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.3" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.3.cmml">α</mi><mo id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.2" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.cmml"><mo id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.2" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.2.cmml">𝒪</mi><mi id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.cmml">𝖲</mi></msub><mo id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem3.p1.3.3.m3.1.1.2" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.2.cmml">≤</mo><mn id="S1.Thmtheorem3.p1.3.3.m3.1.1.3" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.3.cmml">0.4998</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem3.p1.3.3.m3.1b"><apply id="S1.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S1.Thmtheorem3.p1.3.3.m3.1.1"><leq id="S1.Thmtheorem3.p1.3.3.m3.1.1.2.cmml" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.2"></leq><apply id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.cmml" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1"><times id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.2.cmml" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.2"></times><ci id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.3.cmml" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.3">𝛼</ci><apply id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.2.cmml" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.2">𝒪</ci><ci id="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3">𝖲</ci></apply></apply><cn id="S1.Thmtheorem3.p1.3.3.m3.1.1.3.cmml" type="float" xref="S1.Thmtheorem3.p1.3.3.m3.1.1.3">0.4998</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem3.p1.3.3.m3.1c">\alpha(\mathcal{O}_{\mathsf{S}})\leq 0.4998</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem3.p1.3.3.m3.1d">italic_α ( caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT ) ≤ 0.4998</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_remark" id="Thmremarkx1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmremarkx1.1.1.1">Remark</span></span><span class="ltx_text ltx_font_bold" id="Thmremarkx1.2.2">.</span> </h6> <div class="ltx_para" id="Thmremarkx1.p1"> <p class="ltx_p" id="Thmremarkx1.p1.18"><span class="ltx_text ltx_font_italic" id="Thmremarkx1.p1.18.18">The constant in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem3" title="Theorem 1.3 (Prior lower bound for general selection, [FJ15, Thm. 1.5]). ‣ 1.1 Background: Directed cuts, bias, and oblivious algorithms ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.3</span></a> is not optimized in the work of <span class="ltx_ERROR undefined" id="Thmremarkx1.p1.18.18.1">\textcite</span>FJ15. However, it is straightforward to compute the best possible constant achievable with their proof technique. Indeed, the proof of <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">FJ15</span>, Thm. 1.5]</cite> considers two graphs <math alttext="G" class="ltx_Math" display="inline" id="Thmremarkx1.p1.1.1.m1.1"><semantics id="Thmremarkx1.p1.1.1.m1.1a"><mi id="Thmremarkx1.p1.1.1.m1.1.1" xref="Thmremarkx1.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.1.1.m1.1b"><ci id="Thmremarkx1.p1.1.1.m1.1.1.cmml" xref="Thmremarkx1.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.1.1.m1.1d">italic_G</annotation></semantics></math> and <math alttext="L" class="ltx_Math" display="inline" id="Thmremarkx1.p1.2.2.m2.1"><semantics id="Thmremarkx1.p1.2.2.m2.1a"><mi id="Thmremarkx1.p1.2.2.m2.1.1" xref="Thmremarkx1.p1.2.2.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.2.2.m2.1b"><ci id="Thmremarkx1.p1.2.2.m2.1.1.cmml" xref="Thmremarkx1.p1.2.2.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.2.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.2.2.m2.1d">italic_L</annotation></semantics></math>, and shows that if <math alttext="\mathsf{S}" class="ltx_Math" display="inline" id="Thmremarkx1.p1.3.3.m3.1"><semantics id="Thmremarkx1.p1.3.3.m3.1a"><mi id="Thmremarkx1.p1.3.3.m3.1.1" xref="Thmremarkx1.p1.3.3.m3.1.1.cmml">𝖲</mi><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.3.3.m3.1b"><ci id="Thmremarkx1.p1.3.3.m3.1.1.cmml" xref="Thmremarkx1.p1.3.3.m3.1.1">𝖲</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.3.3.m3.1c">\mathsf{S}</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.3.3.m3.1d">sansserif_S</annotation></semantics></math> is any selection function with <math alttext="\mathsf{S}(\frac{1}{2})=\frac{1}{2}+\delta" class="ltx_Math" display="inline" id="Thmremarkx1.p1.4.4.m4.1"><semantics id="Thmremarkx1.p1.4.4.m4.1a"><mrow id="Thmremarkx1.p1.4.4.m4.1.2" xref="Thmremarkx1.p1.4.4.m4.1.2.cmml"><mrow id="Thmremarkx1.p1.4.4.m4.1.2.2" xref="Thmremarkx1.p1.4.4.m4.1.2.2.cmml"><mi id="Thmremarkx1.p1.4.4.m4.1.2.2.2" xref="Thmremarkx1.p1.4.4.m4.1.2.2.2.cmml">𝖲</mi><mo id="Thmremarkx1.p1.4.4.m4.1.2.2.1" xref="Thmremarkx1.p1.4.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="Thmremarkx1.p1.4.4.m4.1.2.2.3.2" xref="Thmremarkx1.p1.4.4.m4.1.1.cmml"><mo id="Thmremarkx1.p1.4.4.m4.1.2.2.3.2.1" stretchy="false" xref="Thmremarkx1.p1.4.4.m4.1.1.cmml">(</mo><mfrac id="Thmremarkx1.p1.4.4.m4.1.1" xref="Thmremarkx1.p1.4.4.m4.1.1.cmml"><mn id="Thmremarkx1.p1.4.4.m4.1.1.2" xref="Thmremarkx1.p1.4.4.m4.1.1.2.cmml">1</mn><mn id="Thmremarkx1.p1.4.4.m4.1.1.3" xref="Thmremarkx1.p1.4.4.m4.1.1.3.cmml">2</mn></mfrac><mo id="Thmremarkx1.p1.4.4.m4.1.2.2.3.2.2" stretchy="false" xref="Thmremarkx1.p1.4.4.m4.1.1.cmml">)</mo></mrow></mrow><mo id="Thmremarkx1.p1.4.4.m4.1.2.1" xref="Thmremarkx1.p1.4.4.m4.1.2.1.cmml">=</mo><mrow id="Thmremarkx1.p1.4.4.m4.1.2.3" xref="Thmremarkx1.p1.4.4.m4.1.2.3.cmml"><mfrac id="Thmremarkx1.p1.4.4.m4.1.2.3.2" xref="Thmremarkx1.p1.4.4.m4.1.2.3.2.cmml"><mn id="Thmremarkx1.p1.4.4.m4.1.2.3.2.2" xref="Thmremarkx1.p1.4.4.m4.1.2.3.2.2.cmml">1</mn><mn id="Thmremarkx1.p1.4.4.m4.1.2.3.2.3" xref="Thmremarkx1.p1.4.4.m4.1.2.3.2.3.cmml">2</mn></mfrac><mo id="Thmremarkx1.p1.4.4.m4.1.2.3.1" xref="Thmremarkx1.p1.4.4.m4.1.2.3.1.cmml">+</mo><mi id="Thmremarkx1.p1.4.4.m4.1.2.3.3" xref="Thmremarkx1.p1.4.4.m4.1.2.3.3.cmml">δ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.4.4.m4.1b"><apply id="Thmremarkx1.p1.4.4.m4.1.2.cmml" xref="Thmremarkx1.p1.4.4.m4.1.2"><eq id="Thmremarkx1.p1.4.4.m4.1.2.1.cmml" xref="Thmremarkx1.p1.4.4.m4.1.2.1"></eq><apply id="Thmremarkx1.p1.4.4.m4.1.2.2.cmml" xref="Thmremarkx1.p1.4.4.m4.1.2.2"><times id="Thmremarkx1.p1.4.4.m4.1.2.2.1.cmml" xref="Thmremarkx1.p1.4.4.m4.1.2.2.1"></times><ci id="Thmremarkx1.p1.4.4.m4.1.2.2.2.cmml" xref="Thmremarkx1.p1.4.4.m4.1.2.2.2">𝖲</ci><apply id="Thmremarkx1.p1.4.4.m4.1.1.cmml" xref="Thmremarkx1.p1.4.4.m4.1.2.2.3.2"><divide id="Thmremarkx1.p1.4.4.m4.1.1.1.cmml" xref="Thmremarkx1.p1.4.4.m4.1.2.2.3.2"></divide><cn id="Thmremarkx1.p1.4.4.m4.1.1.2.cmml" type="integer" xref="Thmremarkx1.p1.4.4.m4.1.1.2">1</cn><cn id="Thmremarkx1.p1.4.4.m4.1.1.3.cmml" type="integer" xref="Thmremarkx1.p1.4.4.m4.1.1.3">2</cn></apply></apply><apply id="Thmremarkx1.p1.4.4.m4.1.2.3.cmml" xref="Thmremarkx1.p1.4.4.m4.1.2.3"><plus id="Thmremarkx1.p1.4.4.m4.1.2.3.1.cmml" xref="Thmremarkx1.p1.4.4.m4.1.2.3.1"></plus><apply id="Thmremarkx1.p1.4.4.m4.1.2.3.2.cmml" xref="Thmremarkx1.p1.4.4.m4.1.2.3.2"><divide id="Thmremarkx1.p1.4.4.m4.1.2.3.2.1.cmml" xref="Thmremarkx1.p1.4.4.m4.1.2.3.2"></divide><cn id="Thmremarkx1.p1.4.4.m4.1.2.3.2.2.cmml" type="integer" xref="Thmremarkx1.p1.4.4.m4.1.2.3.2.2">1</cn><cn id="Thmremarkx1.p1.4.4.m4.1.2.3.2.3.cmml" type="integer" xref="Thmremarkx1.p1.4.4.m4.1.2.3.2.3">2</cn></apply><ci id="Thmremarkx1.p1.4.4.m4.1.2.3.3.cmml" xref="Thmremarkx1.p1.4.4.m4.1.2.3.3">𝛿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.4.4.m4.1c">\mathsf{S}(\frac{1}{2})=\frac{1}{2}+\delta</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.4.4.m4.1d">sansserif_S ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG + italic_δ</annotation></semantics></math>, then <math alttext="\mathsf{S}" class="ltx_Math" display="inline" id="Thmremarkx1.p1.5.5.m5.1"><semantics id="Thmremarkx1.p1.5.5.m5.1a"><mi id="Thmremarkx1.p1.5.5.m5.1.1" xref="Thmremarkx1.p1.5.5.m5.1.1.cmml">𝖲</mi><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.5.5.m5.1b"><ci id="Thmremarkx1.p1.5.5.m5.1.1.cmml" xref="Thmremarkx1.p1.5.5.m5.1.1">𝖲</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.5.5.m5.1c">\mathsf{S}</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.5.5.m5.1d">sansserif_S</annotation></semantics></math> achieves ratio at most <math alttext="0.4899+\delta" class="ltx_Math" display="inline" id="Thmremarkx1.p1.6.6.m6.1"><semantics id="Thmremarkx1.p1.6.6.m6.1a"><mrow id="Thmremarkx1.p1.6.6.m6.1.1" xref="Thmremarkx1.p1.6.6.m6.1.1.cmml"><mn id="Thmremarkx1.p1.6.6.m6.1.1.2" xref="Thmremarkx1.p1.6.6.m6.1.1.2.cmml">0.4899</mn><mo id="Thmremarkx1.p1.6.6.m6.1.1.1" xref="Thmremarkx1.p1.6.6.m6.1.1.1.cmml">+</mo><mi id="Thmremarkx1.p1.6.6.m6.1.1.3" xref="Thmremarkx1.p1.6.6.m6.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.6.6.m6.1b"><apply id="Thmremarkx1.p1.6.6.m6.1.1.cmml" xref="Thmremarkx1.p1.6.6.m6.1.1"><plus id="Thmremarkx1.p1.6.6.m6.1.1.1.cmml" xref="Thmremarkx1.p1.6.6.m6.1.1.1"></plus><cn id="Thmremarkx1.p1.6.6.m6.1.1.2.cmml" type="float" xref="Thmremarkx1.p1.6.6.m6.1.1.2">0.4899</cn><ci id="Thmremarkx1.p1.6.6.m6.1.1.3.cmml" xref="Thmremarkx1.p1.6.6.m6.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.6.6.m6.1c">0.4899+\delta</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.6.6.m6.1d">0.4899 + italic_δ</annotation></semantics></math> on <math alttext="G" class="ltx_Math" display="inline" id="Thmremarkx1.p1.7.7.m7.1"><semantics id="Thmremarkx1.p1.7.7.m7.1a"><mi id="Thmremarkx1.p1.7.7.m7.1.1" xref="Thmremarkx1.p1.7.7.m7.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.7.7.m7.1b"><ci id="Thmremarkx1.p1.7.7.m7.1.1.cmml" xref="Thmremarkx1.p1.7.7.m7.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.7.7.m7.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.7.7.m7.1d">italic_G</annotation></semantics></math> and <math alttext="\frac{1}{2}-2\delta^{2}" class="ltx_Math" display="inline" id="Thmremarkx1.p1.8.8.m8.1"><semantics id="Thmremarkx1.p1.8.8.m8.1a"><mrow id="Thmremarkx1.p1.8.8.m8.1.1" xref="Thmremarkx1.p1.8.8.m8.1.1.cmml"><mfrac id="Thmremarkx1.p1.8.8.m8.1.1.2" xref="Thmremarkx1.p1.8.8.m8.1.1.2.cmml"><mn id="Thmremarkx1.p1.8.8.m8.1.1.2.2" xref="Thmremarkx1.p1.8.8.m8.1.1.2.2.cmml">1</mn><mn id="Thmremarkx1.p1.8.8.m8.1.1.2.3" xref="Thmremarkx1.p1.8.8.m8.1.1.2.3.cmml">2</mn></mfrac><mo id="Thmremarkx1.p1.8.8.m8.1.1.1" xref="Thmremarkx1.p1.8.8.m8.1.1.1.cmml">−</mo><mrow id="Thmremarkx1.p1.8.8.m8.1.1.3" xref="Thmremarkx1.p1.8.8.m8.1.1.3.cmml"><mn id="Thmremarkx1.p1.8.8.m8.1.1.3.2" xref="Thmremarkx1.p1.8.8.m8.1.1.3.2.cmml">2</mn><mo id="Thmremarkx1.p1.8.8.m8.1.1.3.1" xref="Thmremarkx1.p1.8.8.m8.1.1.3.1.cmml">⁢</mo><msup id="Thmremarkx1.p1.8.8.m8.1.1.3.3" xref="Thmremarkx1.p1.8.8.m8.1.1.3.3.cmml"><mi id="Thmremarkx1.p1.8.8.m8.1.1.3.3.2" xref="Thmremarkx1.p1.8.8.m8.1.1.3.3.2.cmml">δ</mi><mn id="Thmremarkx1.p1.8.8.m8.1.1.3.3.3" xref="Thmremarkx1.p1.8.8.m8.1.1.3.3.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.8.8.m8.1b"><apply id="Thmremarkx1.p1.8.8.m8.1.1.cmml" xref="Thmremarkx1.p1.8.8.m8.1.1"><minus id="Thmremarkx1.p1.8.8.m8.1.1.1.cmml" xref="Thmremarkx1.p1.8.8.m8.1.1.1"></minus><apply id="Thmremarkx1.p1.8.8.m8.1.1.2.cmml" xref="Thmremarkx1.p1.8.8.m8.1.1.2"><divide id="Thmremarkx1.p1.8.8.m8.1.1.2.1.cmml" xref="Thmremarkx1.p1.8.8.m8.1.1.2"></divide><cn id="Thmremarkx1.p1.8.8.m8.1.1.2.2.cmml" type="integer" xref="Thmremarkx1.p1.8.8.m8.1.1.2.2">1</cn><cn id="Thmremarkx1.p1.8.8.m8.1.1.2.3.cmml" type="integer" xref="Thmremarkx1.p1.8.8.m8.1.1.2.3">2</cn></apply><apply id="Thmremarkx1.p1.8.8.m8.1.1.3.cmml" xref="Thmremarkx1.p1.8.8.m8.1.1.3"><times id="Thmremarkx1.p1.8.8.m8.1.1.3.1.cmml" xref="Thmremarkx1.p1.8.8.m8.1.1.3.1"></times><cn id="Thmremarkx1.p1.8.8.m8.1.1.3.2.cmml" type="integer" xref="Thmremarkx1.p1.8.8.m8.1.1.3.2">2</cn><apply id="Thmremarkx1.p1.8.8.m8.1.1.3.3.cmml" xref="Thmremarkx1.p1.8.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="Thmremarkx1.p1.8.8.m8.1.1.3.3.1.cmml" xref="Thmremarkx1.p1.8.8.m8.1.1.3.3">superscript</csymbol><ci id="Thmremarkx1.p1.8.8.m8.1.1.3.3.2.cmml" xref="Thmremarkx1.p1.8.8.m8.1.1.3.3.2">𝛿</ci><cn id="Thmremarkx1.p1.8.8.m8.1.1.3.3.3.cmml" type="integer" xref="Thmremarkx1.p1.8.8.m8.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.8.8.m8.1c">\frac{1}{2}-2\delta^{2}</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.8.8.m8.1d">divide start_ARG 1 end_ARG start_ARG 2 end_ARG - 2 italic_δ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> on <math alttext="L" class="ltx_Math" display="inline" id="Thmremarkx1.p1.9.9.m9.1"><semantics id="Thmremarkx1.p1.9.9.m9.1a"><mi id="Thmremarkx1.p1.9.9.m9.1.1" xref="Thmremarkx1.p1.9.9.m9.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.9.9.m9.1b"><ci id="Thmremarkx1.p1.9.9.m9.1.1.cmml" xref="Thmremarkx1.p1.9.9.m9.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.9.9.m9.1c">L</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.9.9.m9.1d">italic_L</annotation></semantics></math>; hence, <math alttext="\mathsf{S}" class="ltx_Math" display="inline" id="Thmremarkx1.p1.10.10.m10.1"><semantics id="Thmremarkx1.p1.10.10.m10.1a"><mi id="Thmremarkx1.p1.10.10.m10.1.1" xref="Thmremarkx1.p1.10.10.m10.1.1.cmml">𝖲</mi><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.10.10.m10.1b"><ci id="Thmremarkx1.p1.10.10.m10.1.1.cmml" xref="Thmremarkx1.p1.10.10.m10.1.1">𝖲</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.10.10.m10.1c">\mathsf{S}</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.10.10.m10.1d">sansserif_S</annotation></semantics></math> achieves ratio strictly below <math alttext="\frac{1}{2}" class="ltx_Math" display="inline" id="Thmremarkx1.p1.11.11.m11.1"><semantics id="Thmremarkx1.p1.11.11.m11.1a"><mfrac id="Thmremarkx1.p1.11.11.m11.1.1" xref="Thmremarkx1.p1.11.11.m11.1.1.cmml"><mn id="Thmremarkx1.p1.11.11.m11.1.1.2" xref="Thmremarkx1.p1.11.11.m11.1.1.2.cmml">1</mn><mn id="Thmremarkx1.p1.11.11.m11.1.1.3" xref="Thmremarkx1.p1.11.11.m11.1.1.3.cmml">2</mn></mfrac><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.11.11.m11.1b"><apply id="Thmremarkx1.p1.11.11.m11.1.1.cmml" xref="Thmremarkx1.p1.11.11.m11.1.1"><divide id="Thmremarkx1.p1.11.11.m11.1.1.1.cmml" xref="Thmremarkx1.p1.11.11.m11.1.1"></divide><cn id="Thmremarkx1.p1.11.11.m11.1.1.2.cmml" type="integer" xref="Thmremarkx1.p1.11.11.m11.1.1.2">1</cn><cn id="Thmremarkx1.p1.11.11.m11.1.1.3.cmml" type="integer" xref="Thmremarkx1.p1.11.11.m11.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.11.11.m11.1c">\frac{1}{2}</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.11.11.m11.1d">divide start_ARG 1 end_ARG start_ARG 2 end_ARG</annotation></semantics></math> on either <math alttext="G" class="ltx_Math" display="inline" id="Thmremarkx1.p1.12.12.m12.1"><semantics id="Thmremarkx1.p1.12.12.m12.1a"><mi id="Thmremarkx1.p1.12.12.m12.1.1" xref="Thmremarkx1.p1.12.12.m12.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.12.12.m12.1b"><ci id="Thmremarkx1.p1.12.12.m12.1.1.cmml" xref="Thmremarkx1.p1.12.12.m12.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.12.12.m12.1c">G</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.12.12.m12.1d">italic_G</annotation></semantics></math> or <math alttext="L" class="ltx_Math" display="inline" id="Thmremarkx1.p1.13.13.m13.1"><semantics id="Thmremarkx1.p1.13.13.m13.1a"><mi id="Thmremarkx1.p1.13.13.m13.1.1" xref="Thmremarkx1.p1.13.13.m13.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.13.13.m13.1b"><ci id="Thmremarkx1.p1.13.13.m13.1.1.cmml" xref="Thmremarkx1.p1.13.13.m13.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.13.13.m13.1c">L</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.13.13.m13.1d">italic_L</annotation></semantics></math>. But equating these two quantities and solving for <math alttext="\delta" class="ltx_Math" display="inline" id="Thmremarkx1.p1.14.14.m14.1"><semantics id="Thmremarkx1.p1.14.14.m14.1a"><mi id="Thmremarkx1.p1.14.14.m14.1.1" xref="Thmremarkx1.p1.14.14.m14.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.14.14.m14.1b"><ci id="Thmremarkx1.p1.14.14.m14.1.1.cmml" xref="Thmremarkx1.p1.14.14.m14.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.14.14.m14.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.14.14.m14.1d">italic_δ</annotation></semantics></math> yields <math alttext="\delta=0.0099" class="ltx_Math" display="inline" id="Thmremarkx1.p1.15.15.m15.1"><semantics id="Thmremarkx1.p1.15.15.m15.1a"><mrow id="Thmremarkx1.p1.15.15.m15.1.1" xref="Thmremarkx1.p1.15.15.m15.1.1.cmml"><mi id="Thmremarkx1.p1.15.15.m15.1.1.2" xref="Thmremarkx1.p1.15.15.m15.1.1.2.cmml">δ</mi><mo id="Thmremarkx1.p1.15.15.m15.1.1.1" xref="Thmremarkx1.p1.15.15.m15.1.1.1.cmml">=</mo><mn id="Thmremarkx1.p1.15.15.m15.1.1.3" xref="Thmremarkx1.p1.15.15.m15.1.1.3.cmml">0.0099</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.15.15.m15.1b"><apply id="Thmremarkx1.p1.15.15.m15.1.1.cmml" xref="Thmremarkx1.p1.15.15.m15.1.1"><eq id="Thmremarkx1.p1.15.15.m15.1.1.1.cmml" xref="Thmremarkx1.p1.15.15.m15.1.1.1"></eq><ci id="Thmremarkx1.p1.15.15.m15.1.1.2.cmml" xref="Thmremarkx1.p1.15.15.m15.1.1.2">𝛿</ci><cn id="Thmremarkx1.p1.15.15.m15.1.1.3.cmml" type="float" xref="Thmremarkx1.p1.15.15.m15.1.1.3">0.0099</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.15.15.m15.1c">\delta=0.0099</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.15.15.m15.1d">italic_δ = 0.0099</annotation></semantics></math>; at this point, both quantities are roughly <math alttext="0.4998" class="ltx_Math" display="inline" id="Thmremarkx1.p1.16.16.m16.1"><semantics id="Thmremarkx1.p1.16.16.m16.1a"><mn id="Thmremarkx1.p1.16.16.m16.1.1" xref="Thmremarkx1.p1.16.16.m16.1.1.cmml">0.4998</mn><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.16.16.m16.1b"><cn id="Thmremarkx1.p1.16.16.m16.1.1.cmml" type="float" xref="Thmremarkx1.p1.16.16.m16.1.1">0.4998</cn></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.16.16.m16.1c">0.4998</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.16.16.m16.1d">0.4998</annotation></semantics></math>. At any other value of <math alttext="\delta" class="ltx_Math" display="inline" id="Thmremarkx1.p1.17.17.m17.1"><semantics id="Thmremarkx1.p1.17.17.m17.1a"><mi id="Thmremarkx1.p1.17.17.m17.1.1" xref="Thmremarkx1.p1.17.17.m17.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.17.17.m17.1b"><ci id="Thmremarkx1.p1.17.17.m17.1.1.cmml" xref="Thmremarkx1.p1.17.17.m17.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.17.17.m17.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.17.17.m17.1d">italic_δ</annotation></semantics></math>, at least one of the quantities will be larger, and therefore, <math alttext="0.4998" class="ltx_Math" display="inline" id="Thmremarkx1.p1.18.18.m18.1"><semantics id="Thmremarkx1.p1.18.18.m18.1a"><mn id="Thmremarkx1.p1.18.18.m18.1.1" xref="Thmremarkx1.p1.18.18.m18.1.1.cmml">0.4998</mn><annotation-xml encoding="MathML-Content" id="Thmremarkx1.p1.18.18.m18.1b"><cn id="Thmremarkx1.p1.18.18.m18.1.1.cmml" type="float" xref="Thmremarkx1.p1.18.18.m18.1.1">0.4998</cn></annotation-xml><annotation encoding="application/x-tex" id="Thmremarkx1.p1.18.18.m18.1c">0.4998</annotation><annotation encoding="application/x-llamapun" id="Thmremarkx1.p1.18.18.m18.1d">0.4998</annotation></semantics></math> is the optimal bound.</span></p> </div> </div> <div class="ltx_para" id="S1.SS1.p7"> <p class="ltx_p" id="S1.SS1.p7.2">One particularly nice feature of oblivious algorithms described by <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">FJ15</span>]</cite> is that (if the selection function <math alttext="\mathsf{S}" class="ltx_Math" display="inline" id="S1.SS1.p7.1.m1.1"><semantics id="S1.SS1.p7.1.m1.1a"><mi id="S1.SS1.p7.1.m1.1.1" xref="S1.SS1.p7.1.m1.1.1.cmml">𝖲</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p7.1.m1.1b"><ci id="S1.SS1.p7.1.m1.1.1.cmml" xref="S1.SS1.p7.1.m1.1.1">𝖲</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p7.1.m1.1c">\mathsf{S}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p7.1.m1.1d">sansserif_S</annotation></semantics></math> is piecewise constant) the approximation ratio of <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S1.SS1.p7.2.m2.1"><semantics id="S1.SS1.p7.2.m2.1a"><msub id="S1.SS1.p7.2.m2.1.1" xref="S1.SS1.p7.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS1.p7.2.m2.1.1.2" xref="S1.SS1.p7.2.m2.1.1.2.cmml">𝒪</mi><mi id="S1.SS1.p7.2.m2.1.1.3" xref="S1.SS1.p7.2.m2.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p7.2.m2.1b"><apply id="S1.SS1.p7.2.m2.1.1.cmml" xref="S1.SS1.p7.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS1.p7.2.m2.1.1.1.cmml" xref="S1.SS1.p7.2.m2.1.1">subscript</csymbol><ci id="S1.SS1.p7.2.m2.1.1.2.cmml" xref="S1.SS1.p7.2.m2.1.1.2">𝒪</ci><ci id="S1.SS1.p7.2.m2.1.1.3.cmml" xref="S1.SS1.p7.2.m2.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p7.2.m2.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p7.2.m2.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math> can be calculated by a simple <em class="ltx_emph ltx_font_italic" id="S1.SS1.p7.2.1">linear program</em> (LP); indeed, this LP essentially “encodes” the process of minimizing the approximation ratio over all possible graphs to find the worst-case input (see <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S2.Thmtheorem1" title="Theorem 2.1 (LP for antisymmetric selection functions). ‣ 2.3 Linear program ‣ 2 Preliminaries ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2.1</span></a> below). A more recent work of <span class="ltx_ERROR undefined" id="S1.SS1.p7.2.2">\textcite</span>Sin23-kand provided an open-source <span class="ltx_text ltx_font_typewriter" id="S1.SS1.p7.2.3">python</span> implementation of the ratio-calculating LP, and used it to improve <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem1" title="Theorem 1.1 (Prior upper bound, [FJ15, Thm. 1.3]). ‣ 1.1 Background: Directed cuts, bias, and oblivious algorithms ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.1</span></a> via a more refined analysis of a similar oblivious algorithm:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.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="S1.Thmtheorem4.1.1.1">Theorem 1.4</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem4.2.2"> </span>(Improved prior upper bound, <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">Sin23-kand</span>]</cite>)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem4.3.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem4.p1"> <p class="ltx_p" id="S1.Thmtheorem4.p1.2"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem4.p1.2.2">There exists an oblivious algorithm <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.1.1.m1.1"><semantics id="S1.Thmtheorem4.p1.1.1.m1.1a"><msub id="S1.Thmtheorem4.p1.1.1.m1.1.1" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem4.p1.1.1.m1.1.1.2" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.2.cmml">𝒪</mi><mi id="S1.Thmtheorem4.p1.1.1.m1.1.1.3" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.1.1.m1.1b"><apply id="S1.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.1.1.m1.1.1.1.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1">subscript</csymbol><ci id="S1.Thmtheorem4.p1.1.1.m1.1.1.2.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.2">𝒪</ci><ci id="S1.Thmtheorem4.p1.1.1.m1.1.1.3.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.1.1.m1.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.1.1.m1.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math> achieving an approximation ratio <math alttext="\alpha(\mathcal{O}_{\mathsf{S}})\geq 0.484" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.2.2.m2.1"><semantics id="S1.Thmtheorem4.p1.2.2.m2.1a"><mrow id="S1.Thmtheorem4.p1.2.2.m2.1.1" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mrow id="S1.Thmtheorem4.p1.2.2.m2.1.1.1" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.cmml"><mi id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.3" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.3.cmml">α</mi><mo id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.2" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.cmml"><mo id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.2" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.2.cmml">𝒪</mi><mi id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.3" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.3.cmml">𝖲</mi></msub><mo id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">≥</mo><mn id="S1.Thmtheorem4.p1.2.2.m2.1.1.3" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.3.cmml">0.484</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.2.2.m2.1b"><apply id="S1.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1"><geq id="S1.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.2"></geq><apply id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1"><times id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.2.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.2"></times><ci id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.3.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.3">𝛼</ci><apply id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.2.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.2">𝒪</ci><ci id="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.3">𝖲</ci></apply></apply><cn id="S1.Thmtheorem4.p1.2.2.m2.1.1.3.cmml" type="float" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.3">0.484</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.2.2.m2.1c">\alpha(\mathcal{O}_{\mathsf{S}})\geq 0.484</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.2.2.m2.1d">italic_α ( caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT ) ≥ 0.484</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.SS1.p8"> <p class="ltx_p" id="S1.SS1.p8.1">However, both the works <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">FJ15</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">Sin23-kand</span>]</cite> left a significant open question:</p> </div> <div class="ltx_theorem ltx_theorem_question" id="Thmquestionx1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmquestionx1.1.1.1">Question</span></span><span class="ltx_text ltx_font_bold" id="Thmquestionx1.2.2">.</span> </h6> <div class="ltx_para" id="Thmquestionx1.p1"> <p class="ltx_p" id="Thmquestionx1.p1.3"><span class="ltx_text ltx_font_italic" id="Thmquestionx1.p1.3.3">What is the best possible approximation ratio <math alttext="\alpha(\mathcal{O}_{\mathsf{S}})" class="ltx_Math" display="inline" id="Thmquestionx1.p1.1.1.m1.1"><semantics id="Thmquestionx1.p1.1.1.m1.1a"><mrow id="Thmquestionx1.p1.1.1.m1.1.1" xref="Thmquestionx1.p1.1.1.m1.1.1.cmml"><mi id="Thmquestionx1.p1.1.1.m1.1.1.3" xref="Thmquestionx1.p1.1.1.m1.1.1.3.cmml">α</mi><mo id="Thmquestionx1.p1.1.1.m1.1.1.2" xref="Thmquestionx1.p1.1.1.m1.1.1.2.cmml">⁢</mo><mrow id="Thmquestionx1.p1.1.1.m1.1.1.1.1" xref="Thmquestionx1.p1.1.1.m1.1.1.1.1.1.cmml"><mo id="Thmquestionx1.p1.1.1.m1.1.1.1.1.2" stretchy="false" xref="Thmquestionx1.p1.1.1.m1.1.1.1.1.1.cmml">(</mo><msub id="Thmquestionx1.p1.1.1.m1.1.1.1.1.1" xref="Thmquestionx1.p1.1.1.m1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmquestionx1.p1.1.1.m1.1.1.1.1.1.2" xref="Thmquestionx1.p1.1.1.m1.1.1.1.1.1.2.cmml">𝒪</mi><mi id="Thmquestionx1.p1.1.1.m1.1.1.1.1.1.3" xref="Thmquestionx1.p1.1.1.m1.1.1.1.1.1.3.cmml">𝖲</mi></msub><mo id="Thmquestionx1.p1.1.1.m1.1.1.1.1.3" stretchy="false" xref="Thmquestionx1.p1.1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmquestionx1.p1.1.1.m1.1b"><apply id="Thmquestionx1.p1.1.1.m1.1.1.cmml" xref="Thmquestionx1.p1.1.1.m1.1.1"><times id="Thmquestionx1.p1.1.1.m1.1.1.2.cmml" xref="Thmquestionx1.p1.1.1.m1.1.1.2"></times><ci id="Thmquestionx1.p1.1.1.m1.1.1.3.cmml" xref="Thmquestionx1.p1.1.1.m1.1.1.3">𝛼</ci><apply id="Thmquestionx1.p1.1.1.m1.1.1.1.1.1.cmml" xref="Thmquestionx1.p1.1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="Thmquestionx1.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="Thmquestionx1.p1.1.1.m1.1.1.1.1">subscript</csymbol><ci id="Thmquestionx1.p1.1.1.m1.1.1.1.1.1.2.cmml" xref="Thmquestionx1.p1.1.1.m1.1.1.1.1.1.2">𝒪</ci><ci id="Thmquestionx1.p1.1.1.m1.1.1.1.1.1.3.cmml" xref="Thmquestionx1.p1.1.1.m1.1.1.1.1.1.3">𝖲</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmquestionx1.p1.1.1.m1.1c">\alpha(\mathcal{O}_{\mathsf{S}})</annotation><annotation encoding="application/x-llamapun" id="Thmquestionx1.p1.1.1.m1.1d">italic_α ( caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT )</annotation></semantics></math> which can be achieved by any oblivious algorithm <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="Thmquestionx1.p1.2.2.m2.1"><semantics id="Thmquestionx1.p1.2.2.m2.1a"><msub id="Thmquestionx1.p1.2.2.m2.1.1" xref="Thmquestionx1.p1.2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmquestionx1.p1.2.2.m2.1.1.2" xref="Thmquestionx1.p1.2.2.m2.1.1.2.cmml">𝒪</mi><mi id="Thmquestionx1.p1.2.2.m2.1.1.3" xref="Thmquestionx1.p1.2.2.m2.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="Thmquestionx1.p1.2.2.m2.1b"><apply id="Thmquestionx1.p1.2.2.m2.1.1.cmml" xref="Thmquestionx1.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="Thmquestionx1.p1.2.2.m2.1.1.1.cmml" xref="Thmquestionx1.p1.2.2.m2.1.1">subscript</csymbol><ci id="Thmquestionx1.p1.2.2.m2.1.1.2.cmml" xref="Thmquestionx1.p1.2.2.m2.1.1.2">𝒪</ci><ci id="Thmquestionx1.p1.2.2.m2.1.1.3.cmml" xref="Thmquestionx1.p1.2.2.m2.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmquestionx1.p1.2.2.m2.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="Thmquestionx1.p1.2.2.m2.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math> for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="Thmquestionx1.p1.3.3.1">Max-DiCut</span>?</span></p> </div> </div> <div class="ltx_para" id="S1.SS1.p9"> <p class="ltx_p" id="S1.SS1.p9.1">As we describe in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.SS3" title="1.3 Motivations ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">1.3</span></a> below, oblivious algorithms for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS1.p9.1.1">Max-DiCut</span> are known to imply algorithms achieving (arbitrarily close to) the same ratio in several different streaming models <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">SSSV23-dicut</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">SSSV23-random-ordering</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">kallaugher2023exponential</span>]</cite>. Resolving this question would therefore characterize the best approximation ratios achievable via these streaming techniques. In this work, we make progress on the question and tighter upper and lower bounds on oblivious algorithms via intricate computer searches.</p> </div> </section> <section class="ltx_subsection" id="S1.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.2 </span>Results</h3> <div class="ltx_para" id="S1.SS2.p1"> <p class="ltx_p" id="S1.SS2.p1.4">We define a class of selection functions, which we call <em class="ltx_emph ltx_font_italic" id="S1.SS2.p1.4.1">piecewise linear <em class="ltx_emph ltx_font_upright" id="S1.SS2.p1.4.1.1">(PL)</em> sigmoid</em> functions (see <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S3.E1" title="Definition 3.1 (PL sigmoid functions). ‣ 3 Improved oblivious algorithms (Theorem 1.5) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Definition</span> <span class="ltx_text ltx_ref_tag">3.1</span></a> below). These functions are denoted <math alttext="\mathsf{PLSigmoid}_{b}" class="ltx_Math" display="inline" id="S1.SS2.p1.1.m1.1"><semantics id="S1.SS2.p1.1.m1.1a"><msub id="S1.SS2.p1.1.m1.1.1" xref="S1.SS2.p1.1.m1.1.1.cmml"><mi id="S1.SS2.p1.1.m1.1.1.2" xref="S1.SS2.p1.1.m1.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S1.SS2.p1.1.m1.1.1.3" xref="S1.SS2.p1.1.m1.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.1.m1.1b"><apply id="S1.SS2.p1.1.m1.1.1.cmml" xref="S1.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS2.p1.1.m1.1.1.1.cmml" xref="S1.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S1.SS2.p1.1.m1.1.1.2.cmml" xref="S1.SS2.p1.1.m1.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S1.SS2.p1.1.m1.1.1.3.cmml" xref="S1.SS2.p1.1.m1.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.1.m1.1c">\mathsf{PLSigmoid}_{b}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.1.m1.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="b\in[0,1]" class="ltx_Math" display="inline" id="S1.SS2.p1.2.m2.2"><semantics id="S1.SS2.p1.2.m2.2a"><mrow id="S1.SS2.p1.2.m2.2.3" xref="S1.SS2.p1.2.m2.2.3.cmml"><mi id="S1.SS2.p1.2.m2.2.3.2" xref="S1.SS2.p1.2.m2.2.3.2.cmml">b</mi><mo id="S1.SS2.p1.2.m2.2.3.1" xref="S1.SS2.p1.2.m2.2.3.1.cmml">∈</mo><mrow id="S1.SS2.p1.2.m2.2.3.3.2" xref="S1.SS2.p1.2.m2.2.3.3.1.cmml"><mo id="S1.SS2.p1.2.m2.2.3.3.2.1" stretchy="false" xref="S1.SS2.p1.2.m2.2.3.3.1.cmml">[</mo><mn id="S1.SS2.p1.2.m2.1.1" xref="S1.SS2.p1.2.m2.1.1.cmml">0</mn><mo id="S1.SS2.p1.2.m2.2.3.3.2.2" xref="S1.SS2.p1.2.m2.2.3.3.1.cmml">,</mo><mn id="S1.SS2.p1.2.m2.2.2" xref="S1.SS2.p1.2.m2.2.2.cmml">1</mn><mo id="S1.SS2.p1.2.m2.2.3.3.2.3" stretchy="false" xref="S1.SS2.p1.2.m2.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.2.m2.2b"><apply id="S1.SS2.p1.2.m2.2.3.cmml" xref="S1.SS2.p1.2.m2.2.3"><in id="S1.SS2.p1.2.m2.2.3.1.cmml" xref="S1.SS2.p1.2.m2.2.3.1"></in><ci id="S1.SS2.p1.2.m2.2.3.2.cmml" xref="S1.SS2.p1.2.m2.2.3.2">𝑏</ci><interval closure="closed" id="S1.SS2.p1.2.m2.2.3.3.1.cmml" xref="S1.SS2.p1.2.m2.2.3.3.2"><cn id="S1.SS2.p1.2.m2.1.1.cmml" type="integer" xref="S1.SS2.p1.2.m2.1.1">0</cn><cn id="S1.SS2.p1.2.m2.2.2.cmml" type="integer" xref="S1.SS2.p1.2.m2.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.2.m2.2c">b\in[0,1]</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.2.m2.2d">italic_b ∈ [ 0 , 1 ]</annotation></semantics></math> is an “intercept” parameter. <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem1" title="Theorem 1.1 (Prior upper bound, [FJ15, Thm. 1.3]). ‣ 1.1 Background: Directed cuts, bias, and oblivious algorithms ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorems</span> <span class="ltx_text ltx_ref_tag">1.1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem4" title="Theorem 1.4 (Improved prior upper bound, [Sin23-kand]). ‣ 1.1 Background: Directed cuts, bias, and oblivious algorithms ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">1.4</span></a> were proven by analyzing discretizations (i.e., piecewise-constant versions) of <math alttext="\mathsf{PLSigmoid}_{1/2}" class="ltx_Math" display="inline" id="S1.SS2.p1.3.m3.1"><semantics id="S1.SS2.p1.3.m3.1a"><msub id="S1.SS2.p1.3.m3.1.1" xref="S1.SS2.p1.3.m3.1.1.cmml"><mi id="S1.SS2.p1.3.m3.1.1.2" xref="S1.SS2.p1.3.m3.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S1.SS2.p1.3.m3.1.1.3" xref="S1.SS2.p1.3.m3.1.1.3.cmml"><mn id="S1.SS2.p1.3.m3.1.1.3.2" xref="S1.SS2.p1.3.m3.1.1.3.2.cmml">1</mn><mo id="S1.SS2.p1.3.m3.1.1.3.1" xref="S1.SS2.p1.3.m3.1.1.3.1.cmml">/</mo><mn id="S1.SS2.p1.3.m3.1.1.3.3" xref="S1.SS2.p1.3.m3.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.3.m3.1b"><apply id="S1.SS2.p1.3.m3.1.1.cmml" xref="S1.SS2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SS2.p1.3.m3.1.1.1.cmml" xref="S1.SS2.p1.3.m3.1.1">subscript</csymbol><ci id="S1.SS2.p1.3.m3.1.1.2.cmml" xref="S1.SS2.p1.3.m3.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S1.SS2.p1.3.m3.1.1.3.cmml" xref="S1.SS2.p1.3.m3.1.1.3"><divide id="S1.SS2.p1.3.m3.1.1.3.1.cmml" xref="S1.SS2.p1.3.m3.1.1.3.1"></divide><cn id="S1.SS2.p1.3.m3.1.1.3.2.cmml" type="integer" xref="S1.SS2.p1.3.m3.1.1.3.2">1</cn><cn id="S1.SS2.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S1.SS2.p1.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.3.m3.1c">\mathsf{PLSigmoid}_{1/2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.3.m3.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math>. We demonstrate a strictly better oblivious algorithm, using a discretization of a PL sigmoid function with a different intercept, namely, <math alttext="\mathsf{PLSigmoid}_{149/309}" class="ltx_Math" display="inline" id="S1.SS2.p1.4.m4.1"><semantics id="S1.SS2.p1.4.m4.1a"><msub id="S1.SS2.p1.4.m4.1.1" xref="S1.SS2.p1.4.m4.1.1.cmml"><mi id="S1.SS2.p1.4.m4.1.1.2" xref="S1.SS2.p1.4.m4.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S1.SS2.p1.4.m4.1.1.3" xref="S1.SS2.p1.4.m4.1.1.3.cmml"><mn id="S1.SS2.p1.4.m4.1.1.3.2" xref="S1.SS2.p1.4.m4.1.1.3.2.cmml">149</mn><mo id="S1.SS2.p1.4.m4.1.1.3.1" xref="S1.SS2.p1.4.m4.1.1.3.1.cmml">/</mo><mn id="S1.SS2.p1.4.m4.1.1.3.3" xref="S1.SS2.p1.4.m4.1.1.3.3.cmml">309</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.4.m4.1b"><apply id="S1.SS2.p1.4.m4.1.1.cmml" xref="S1.SS2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S1.SS2.p1.4.m4.1.1.1.cmml" xref="S1.SS2.p1.4.m4.1.1">subscript</csymbol><ci id="S1.SS2.p1.4.m4.1.1.2.cmml" xref="S1.SS2.p1.4.m4.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S1.SS2.p1.4.m4.1.1.3.cmml" xref="S1.SS2.p1.4.m4.1.1.3"><divide id="S1.SS2.p1.4.m4.1.1.3.1.cmml" xref="S1.SS2.p1.4.m4.1.1.3.1"></divide><cn id="S1.SS2.p1.4.m4.1.1.3.2.cmml" type="integer" xref="S1.SS2.p1.4.m4.1.1.3.2">149</cn><cn id="S1.SS2.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S1.SS2.p1.4.m4.1.1.3.3">309</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.4.m4.1c">\mathsf{PLSigmoid}_{149/309}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.4.m4.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT 149 / 309 end_POSTSUBSCRIPT</annotation></semantics></math>:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.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="S1.Thmtheorem5.1.1.1">Theorem 1.5</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem5.2.2"> </span>(New upper bound)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem5.3.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem5.p1"> <p class="ltx_p" id="S1.Thmtheorem5.p1.2"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem5.p1.2.2">There exists an oblivious algorithm <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S1.Thmtheorem5.p1.1.1.m1.1"><semantics id="S1.Thmtheorem5.p1.1.1.m1.1a"><msub id="S1.Thmtheorem5.p1.1.1.m1.1.1" xref="S1.Thmtheorem5.p1.1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem5.p1.1.1.m1.1.1.2" xref="S1.Thmtheorem5.p1.1.1.m1.1.1.2.cmml">𝒪</mi><mi id="S1.Thmtheorem5.p1.1.1.m1.1.1.3" xref="S1.Thmtheorem5.p1.1.1.m1.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem5.p1.1.1.m1.1b"><apply id="S1.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem5.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem5.p1.1.1.m1.1.1.1.cmml" xref="S1.Thmtheorem5.p1.1.1.m1.1.1">subscript</csymbol><ci id="S1.Thmtheorem5.p1.1.1.m1.1.1.2.cmml" xref="S1.Thmtheorem5.p1.1.1.m1.1.1.2">𝒪</ci><ci id="S1.Thmtheorem5.p1.1.1.m1.1.1.3.cmml" xref="S1.Thmtheorem5.p1.1.1.m1.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem5.p1.1.1.m1.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem5.p1.1.1.m1.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math> achieving an approximation ratio <math alttext="\alpha(\mathcal{O}_{\mathsf{S}})\geq 0.485359" class="ltx_Math" display="inline" id="S1.Thmtheorem5.p1.2.2.m2.1"><semantics id="S1.Thmtheorem5.p1.2.2.m2.1a"><mrow id="S1.Thmtheorem5.p1.2.2.m2.1.1" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.cmml"><mrow id="S1.Thmtheorem5.p1.2.2.m2.1.1.1" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1.cmml"><mi id="S1.Thmtheorem5.p1.2.2.m2.1.1.1.3" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1.3.cmml">α</mi><mo id="S1.Thmtheorem5.p1.2.2.m2.1.1.1.2" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1.cmml"><mo id="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1.2" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1.2.cmml">𝒪</mi><mi id="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1.3" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1.3.cmml">𝖲</mi></msub><mo id="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem5.p1.2.2.m2.1.1.2" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.2.cmml">≥</mo><mn id="S1.Thmtheorem5.p1.2.2.m2.1.1.3" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.3.cmml">0.485359</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem5.p1.2.2.m2.1b"><apply id="S1.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem5.p1.2.2.m2.1.1"><geq id="S1.Thmtheorem5.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.2"></geq><apply id="S1.Thmtheorem5.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1"><times id="S1.Thmtheorem5.p1.2.2.m2.1.1.1.2.cmml" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1.2"></times><ci id="S1.Thmtheorem5.p1.2.2.m2.1.1.1.3.cmml" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1.3">𝛼</ci><apply id="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1.2.cmml" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1.2">𝒪</ci><ci id="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.1.1.1.1.3">𝖲</ci></apply></apply><cn id="S1.Thmtheorem5.p1.2.2.m2.1.1.3.cmml" type="float" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.3">0.485359</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem5.p1.2.2.m2.1c">\alpha(\mathcal{O}_{\mathsf{S}})\geq 0.485359</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem5.p1.2.2.m2.1d">italic_α ( caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT ) ≥ 0.485359</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.SS2.p2"> <p class="ltx_p" id="S1.SS2.p2.1">We also complement this theorem with lower bounds, both for <math alttext="\mathsf{PLSigmoid}_{1/2}" class="ltx_Math" display="inline" id="S1.SS2.p2.1.m1.1"><semantics id="S1.SS2.p2.1.m1.1a"><msub id="S1.SS2.p2.1.m1.1.1" xref="S1.SS2.p2.1.m1.1.1.cmml"><mi id="S1.SS2.p2.1.m1.1.1.2" xref="S1.SS2.p2.1.m1.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S1.SS2.p2.1.m1.1.1.3" xref="S1.SS2.p2.1.m1.1.1.3.cmml"><mn id="S1.SS2.p2.1.m1.1.1.3.2" xref="S1.SS2.p2.1.m1.1.1.3.2.cmml">1</mn><mo id="S1.SS2.p2.1.m1.1.1.3.1" xref="S1.SS2.p2.1.m1.1.1.3.1.cmml">/</mo><mn id="S1.SS2.p2.1.m1.1.1.3.3" xref="S1.SS2.p2.1.m1.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.1.m1.1b"><apply id="S1.SS2.p2.1.m1.1.1.cmml" xref="S1.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS2.p2.1.m1.1.1.1.cmml" xref="S1.SS2.p2.1.m1.1.1">subscript</csymbol><ci id="S1.SS2.p2.1.m1.1.1.2.cmml" xref="S1.SS2.p2.1.m1.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S1.SS2.p2.1.m1.1.1.3.cmml" xref="S1.SS2.p2.1.m1.1.1.3"><divide id="S1.SS2.p2.1.m1.1.1.3.1.cmml" xref="S1.SS2.p2.1.m1.1.1.3.1"></divide><cn id="S1.SS2.p2.1.m1.1.1.3.2.cmml" type="integer" xref="S1.SS2.p2.1.m1.1.1.3.2">1</cn><cn id="S1.SS2.p2.1.m1.1.1.3.3.cmml" type="integer" xref="S1.SS2.p2.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.1.m1.1c">\mathsf{PLSigmoid}_{1/2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.1.m1.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math> itself and for arbitrary PL sigmoid functions:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.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="S1.Thmtheorem6.2.1.1">Theorem 1.6</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem6.3.2"> </span>(Lower bound for PL sigmoid selection with <math alttext="b=1/2" class="ltx_Math" display="inline" id="S1.Thmtheorem6.1.m1.1"><semantics id="S1.Thmtheorem6.1.m1.1b"><mrow id="S1.Thmtheorem6.1.m1.1.1" xref="S1.Thmtheorem6.1.m1.1.1.cmml"><mi id="S1.Thmtheorem6.1.m1.1.1.2" xref="S1.Thmtheorem6.1.m1.1.1.2.cmml">b</mi><mo id="S1.Thmtheorem6.1.m1.1.1.1" xref="S1.Thmtheorem6.1.m1.1.1.1.cmml">=</mo><mrow id="S1.Thmtheorem6.1.m1.1.1.3" xref="S1.Thmtheorem6.1.m1.1.1.3.cmml"><mn id="S1.Thmtheorem6.1.m1.1.1.3.2" xref="S1.Thmtheorem6.1.m1.1.1.3.2.cmml">1</mn><mo id="S1.Thmtheorem6.1.m1.1.1.3.1" xref="S1.Thmtheorem6.1.m1.1.1.3.1.cmml">/</mo><mn id="S1.Thmtheorem6.1.m1.1.1.3.3" xref="S1.Thmtheorem6.1.m1.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem6.1.m1.1c"><apply id="S1.Thmtheorem6.1.m1.1.1.cmml" xref="S1.Thmtheorem6.1.m1.1.1"><eq id="S1.Thmtheorem6.1.m1.1.1.1.cmml" xref="S1.Thmtheorem6.1.m1.1.1.1"></eq><ci id="S1.Thmtheorem6.1.m1.1.1.2.cmml" xref="S1.Thmtheorem6.1.m1.1.1.2">𝑏</ci><apply id="S1.Thmtheorem6.1.m1.1.1.3.cmml" xref="S1.Thmtheorem6.1.m1.1.1.3"><divide id="S1.Thmtheorem6.1.m1.1.1.3.1.cmml" xref="S1.Thmtheorem6.1.m1.1.1.3.1"></divide><cn id="S1.Thmtheorem6.1.m1.1.1.3.2.cmml" type="integer" xref="S1.Thmtheorem6.1.m1.1.1.3.2">1</cn><cn id="S1.Thmtheorem6.1.m1.1.1.3.3.cmml" type="integer" xref="S1.Thmtheorem6.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem6.1.m1.1d">b=1/2</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem6.1.m1.1e">italic_b = 1 / 2</annotation></semantics></math> intercept)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem6.4.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem6.p1"> <p class="ltx_p" id="S1.Thmtheorem6.p1.1"><math alttext="\alpha(\mathcal{O}_{\mathsf{PLSigmoid}_{1/2}})\leq 0.485282" class="ltx_Math" display="inline" id="S1.Thmtheorem6.p1.1.m1.1"><semantics id="S1.Thmtheorem6.p1.1.m1.1a"><mrow id="S1.Thmtheorem6.p1.1.m1.1.1" xref="S1.Thmtheorem6.p1.1.m1.1.1.cmml"><mrow id="S1.Thmtheorem6.p1.1.m1.1.1.1" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.cmml"><mi id="S1.Thmtheorem6.p1.1.m1.1.1.1.3" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.3.cmml">α</mi><mo id="S1.Thmtheorem6.p1.1.m1.1.1.1.2" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.2" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.2.cmml">𝒪</mi><msub id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.cmml"><mi id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.2" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3.cmml"><mn id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3.2" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3.1" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3.1.cmml">/</mo><mn id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3.3" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3.3.cmml">2</mn></mrow></msub></msub><mo id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem6.p1.1.m1.1.1.2" xref="S1.Thmtheorem6.p1.1.m1.1.1.2.cmml">≤</mo><mn id="S1.Thmtheorem6.p1.1.m1.1.1.3" xref="S1.Thmtheorem6.p1.1.m1.1.1.3.cmml">0.485282</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem6.p1.1.m1.1b"><apply id="S1.Thmtheorem6.p1.1.m1.1.1.cmml" xref="S1.Thmtheorem6.p1.1.m1.1.1"><leq id="S1.Thmtheorem6.p1.1.m1.1.1.2.cmml" xref="S1.Thmtheorem6.p1.1.m1.1.1.2"></leq><apply id="S1.Thmtheorem6.p1.1.m1.1.1.1.cmml" xref="S1.Thmtheorem6.p1.1.m1.1.1.1"><times id="S1.Thmtheorem6.p1.1.m1.1.1.1.2.cmml" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.2"></times><ci id="S1.Thmtheorem6.p1.1.m1.1.1.1.3.cmml" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.3">𝛼</ci><apply id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.2">𝒪</ci><apply id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.1.cmml" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.2.cmml" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3.cmml" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3"><divide id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3.1.cmml" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3.1"></divide><cn id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3.2">1</cn><cn id="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S1.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.3.3">2</cn></apply></apply></apply></apply><cn id="S1.Thmtheorem6.p1.1.m1.1.1.3.cmml" type="float" xref="S1.Thmtheorem6.p1.1.m1.1.1.3">0.485282</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem6.p1.1.m1.1c">\alpha(\mathcal{O}_{\mathsf{PLSigmoid}_{1/2}})\leq 0.485282</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem6.p1.1.m1.1d">italic_α ( caligraphic_O start_POSTSUBSCRIPT sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ≤ 0.485282</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem6.p1.1.1">.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.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="S1.Thmtheorem7.1.1.1">Theorem 1.7</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem7.2.2"> </span>(Lower bound for PL sigmoid selection with arbitrary intercept)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem7.3.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem7.p1"> <p class="ltx_p" id="S1.Thmtheorem7.p1.2"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem7.p1.2.2">For every <math alttext="b\in[0,1]" class="ltx_Math" display="inline" id="S1.Thmtheorem7.p1.1.1.m1.2"><semantics id="S1.Thmtheorem7.p1.1.1.m1.2a"><mrow id="S1.Thmtheorem7.p1.1.1.m1.2.3" xref="S1.Thmtheorem7.p1.1.1.m1.2.3.cmml"><mi id="S1.Thmtheorem7.p1.1.1.m1.2.3.2" xref="S1.Thmtheorem7.p1.1.1.m1.2.3.2.cmml">b</mi><mo id="S1.Thmtheorem7.p1.1.1.m1.2.3.1" xref="S1.Thmtheorem7.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S1.Thmtheorem7.p1.1.1.m1.2.3.3.2" xref="S1.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml"><mo id="S1.Thmtheorem7.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S1.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml">[</mo><mn id="S1.Thmtheorem7.p1.1.1.m1.1.1" xref="S1.Thmtheorem7.p1.1.1.m1.1.1.cmml">0</mn><mo id="S1.Thmtheorem7.p1.1.1.m1.2.3.3.2.2" xref="S1.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml">,</mo><mn id="S1.Thmtheorem7.p1.1.1.m1.2.2" xref="S1.Thmtheorem7.p1.1.1.m1.2.2.cmml">1</mn><mo id="S1.Thmtheorem7.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S1.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem7.p1.1.1.m1.2b"><apply id="S1.Thmtheorem7.p1.1.1.m1.2.3.cmml" xref="S1.Thmtheorem7.p1.1.1.m1.2.3"><in id="S1.Thmtheorem7.p1.1.1.m1.2.3.1.cmml" xref="S1.Thmtheorem7.p1.1.1.m1.2.3.1"></in><ci id="S1.Thmtheorem7.p1.1.1.m1.2.3.2.cmml" xref="S1.Thmtheorem7.p1.1.1.m1.2.3.2">𝑏</ci><interval closure="closed" id="S1.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml" xref="S1.Thmtheorem7.p1.1.1.m1.2.3.3.2"><cn id="S1.Thmtheorem7.p1.1.1.m1.1.1.cmml" type="integer" xref="S1.Thmtheorem7.p1.1.1.m1.1.1">0</cn><cn id="S1.Thmtheorem7.p1.1.1.m1.2.2.cmml" type="integer" xref="S1.Thmtheorem7.p1.1.1.m1.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem7.p1.1.1.m1.2c">b\in[0,1]</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem7.p1.1.1.m1.2d">italic_b ∈ [ 0 , 1 ]</annotation></semantics></math>, <math alttext="\alpha(\mathcal{O}_{\mathsf{PLSigmoid}_{b}})\leq 0.486" class="ltx_Math" display="inline" id="S1.Thmtheorem7.p1.2.2.m2.1"><semantics id="S1.Thmtheorem7.p1.2.2.m2.1a"><mrow id="S1.Thmtheorem7.p1.2.2.m2.1.1" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.cmml"><mrow id="S1.Thmtheorem7.p1.2.2.m2.1.1.1" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.cmml"><mi id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.3" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.3.cmml">α</mi><mo id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.2" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.cmml"><mo id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.2" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.2.cmml">𝒪</mi><msub id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.3" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.3.cmml"><mi id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.3.2" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.3.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.3.3" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.3.3.cmml">b</mi></msub></msub><mo id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem7.p1.2.2.m2.1.1.2" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.2.cmml">≤</mo><mn id="S1.Thmtheorem7.p1.2.2.m2.1.1.3" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.3.cmml">0.486</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem7.p1.2.2.m2.1b"><apply id="S1.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem7.p1.2.2.m2.1.1"><leq id="S1.Thmtheorem7.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.2"></leq><apply id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1"><times id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.2.cmml" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.2"></times><ci id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.3.cmml" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.3">𝛼</ci><apply id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.2.cmml" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.2">𝒪</ci><apply id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.3.1.cmml" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.3">subscript</csymbol><ci id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.3.2.cmml" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.3.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.3.3.cmml" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.1.1.1.1.3.3">𝑏</ci></apply></apply></apply><cn id="S1.Thmtheorem7.p1.2.2.m2.1.1.3.cmml" type="float" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.3">0.486</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem7.p1.2.2.m2.1c">\alpha(\mathcal{O}_{\mathsf{PLSigmoid}_{b}})\leq 0.486</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem7.p1.2.2.m2.1d">italic_α ( caligraphic_O start_POSTSUBSCRIPT sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ≤ 0.486</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.SS2.p3"> <p class="ltx_p" id="S1.SS2.p3.4">Note that these two lower bounds hold for PL sigmoid functions themselves (which are continuous), while the upper bound <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem5" title="Theorem 1.5 (New upper bound). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.5</span></a> used a discretization of such a function, so they are not formally comparable. In particular, we observe that given the approximation ratio seems to increase and converge to a limit during finer discretization (see <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S3.F2" title="In 3 Improved oblivious algorithms (Theorem 1.5) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Fig.</span> <span class="ltx_text ltx_ref_tag">2</span></a> below). In light of this, we note that the ratio in this theorem, <math alttext="0.485282" class="ltx_Math" display="inline" id="S1.SS2.p3.1.m1.1"><semantics id="S1.SS2.p3.1.m1.1a"><mn id="S1.SS2.p3.1.m1.1.1" xref="S1.SS2.p3.1.m1.1.1.cmml">0.485282</mn><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.1.m1.1b"><cn id="S1.SS2.p3.1.m1.1.1.cmml" type="float" xref="S1.SS2.p3.1.m1.1.1">0.485282</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.1.m1.1c">0.485282</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.1.m1.1d">0.485282</annotation></semantics></math>, is only slightly larger than the ratio from <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem5" title="Theorem 1.5 (New upper bound). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.5</span></a>, <math alttext="0.485275" class="ltx_Math" display="inline" id="S1.SS2.p3.2.m2.1"><semantics id="S1.SS2.p3.2.m2.1a"><mn id="S1.SS2.p3.2.m2.1.1" xref="S1.SS2.p3.2.m2.1.1.cmml">0.485275</mn><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.2.m2.1b"><cn id="S1.SS2.p3.2.m2.1.1.cmml" type="float" xref="S1.SS2.p3.2.m2.1.1">0.485275</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.2.m2.1c">0.485275</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.2.m2.1d">0.485275</annotation></semantics></math>. This is heuristic evidence that the selection function <math alttext="\mathsf{PLSigmoid}_{149/309}" class="ltx_Math" display="inline" id="S1.SS2.p3.3.m3.1"><semantics id="S1.SS2.p3.3.m3.1a"><msub id="S1.SS2.p3.3.m3.1.1" xref="S1.SS2.p3.3.m3.1.1.cmml"><mi id="S1.SS2.p3.3.m3.1.1.2" xref="S1.SS2.p3.3.m3.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S1.SS2.p3.3.m3.1.1.3" xref="S1.SS2.p3.3.m3.1.1.3.cmml"><mn id="S1.SS2.p3.3.m3.1.1.3.2" xref="S1.SS2.p3.3.m3.1.1.3.2.cmml">149</mn><mo id="S1.SS2.p3.3.m3.1.1.3.1" xref="S1.SS2.p3.3.m3.1.1.3.1.cmml">/</mo><mn id="S1.SS2.p3.3.m3.1.1.3.3" xref="S1.SS2.p3.3.m3.1.1.3.3.cmml">309</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.3.m3.1b"><apply id="S1.SS2.p3.3.m3.1.1.cmml" xref="S1.SS2.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.3.m3.1.1.1.cmml" xref="S1.SS2.p3.3.m3.1.1">subscript</csymbol><ci id="S1.SS2.p3.3.m3.1.1.2.cmml" xref="S1.SS2.p3.3.m3.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S1.SS2.p3.3.m3.1.1.3.cmml" xref="S1.SS2.p3.3.m3.1.1.3"><divide id="S1.SS2.p3.3.m3.1.1.3.1.cmml" xref="S1.SS2.p3.3.m3.1.1.3.1"></divide><cn id="S1.SS2.p3.3.m3.1.1.3.2.cmml" type="integer" xref="S1.SS2.p3.3.m3.1.1.3.2">149</cn><cn id="S1.SS2.p3.3.m3.1.1.3.3.cmml" type="integer" xref="S1.SS2.p3.3.m3.1.1.3.3">309</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.3.m3.1c">\mathsf{PLSigmoid}_{149/309}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.3.m3.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT 149 / 309 end_POSTSUBSCRIPT</annotation></semantics></math> has a strictly higher approximation ratio than the selection function <math alttext="\mathsf{PLSigmoid}_{1/2}" class="ltx_Math" display="inline" id="S1.SS2.p3.4.m4.1"><semantics id="S1.SS2.p3.4.m4.1a"><msub id="S1.SS2.p3.4.m4.1.1" xref="S1.SS2.p3.4.m4.1.1.cmml"><mi id="S1.SS2.p3.4.m4.1.1.2" xref="S1.SS2.p3.4.m4.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S1.SS2.p3.4.m4.1.1.3" xref="S1.SS2.p3.4.m4.1.1.3.cmml"><mn id="S1.SS2.p3.4.m4.1.1.3.2" xref="S1.SS2.p3.4.m4.1.1.3.2.cmml">1</mn><mo id="S1.SS2.p3.4.m4.1.1.3.1" xref="S1.SS2.p3.4.m4.1.1.3.1.cmml">/</mo><mn id="S1.SS2.p3.4.m4.1.1.3.3" xref="S1.SS2.p3.4.m4.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.4.m4.1b"><apply id="S1.SS2.p3.4.m4.1.1.cmml" xref="S1.SS2.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.4.m4.1.1.1.cmml" xref="S1.SS2.p3.4.m4.1.1">subscript</csymbol><ci id="S1.SS2.p3.4.m4.1.1.2.cmml" xref="S1.SS2.p3.4.m4.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S1.SS2.p3.4.m4.1.1.3.cmml" xref="S1.SS2.p3.4.m4.1.1.3"><divide id="S1.SS2.p3.4.m4.1.1.3.1.cmml" xref="S1.SS2.p3.4.m4.1.1.3.1"></divide><cn id="S1.SS2.p3.4.m4.1.1.3.2.cmml" type="integer" xref="S1.SS2.p3.4.m4.1.1.3.2">1</cn><cn id="S1.SS2.p3.4.m4.1.1.3.3.cmml" type="integer" xref="S1.SS2.p3.4.m4.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.4.m4.1c">\mathsf{PLSigmoid}_{1/2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.4.m4.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SS2.p4"> <p class="ltx_p" id="S1.SS2.p4.1">We prove another, more general, lower bound that holds against antisymmetric selection functions:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.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="S1.Thmtheorem8.1.1.1">Theorem 1.8</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem8.2.2"> </span>(Lower bound for symmetric selection)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem8.3.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem8.p1"> <p class="ltx_p" id="S1.Thmtheorem8.p1.3"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem8.p1.3.3">For every <em class="ltx_emph ltx_font_upright" id="S1.Thmtheorem8.p1.3.3.1">antisymmetric</em> selection function <math alttext="\mathsf{S}" class="ltx_Math" display="inline" id="S1.Thmtheorem8.p1.1.1.m1.1"><semantics id="S1.Thmtheorem8.p1.1.1.m1.1a"><mi id="S1.Thmtheorem8.p1.1.1.m1.1.1" xref="S1.Thmtheorem8.p1.1.1.m1.1.1.cmml">𝖲</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem8.p1.1.1.m1.1b"><ci id="S1.Thmtheorem8.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem8.p1.1.1.m1.1.1">𝖲</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem8.p1.1.1.m1.1c">\mathsf{S}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem8.p1.1.1.m1.1d">sansserif_S</annotation></semantics></math>, <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S1.Thmtheorem8.p1.2.2.m2.1"><semantics id="S1.Thmtheorem8.p1.2.2.m2.1a"><msub id="S1.Thmtheorem8.p1.2.2.m2.1.1" xref="S1.Thmtheorem8.p1.2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem8.p1.2.2.m2.1.1.2" xref="S1.Thmtheorem8.p1.2.2.m2.1.1.2.cmml">𝒪</mi><mi id="S1.Thmtheorem8.p1.2.2.m2.1.1.3" xref="S1.Thmtheorem8.p1.2.2.m2.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem8.p1.2.2.m2.1b"><apply id="S1.Thmtheorem8.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem8.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem8.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmtheorem8.p1.2.2.m2.1.1">subscript</csymbol><ci id="S1.Thmtheorem8.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmtheorem8.p1.2.2.m2.1.1.2">𝒪</ci><ci id="S1.Thmtheorem8.p1.2.2.m2.1.1.3.cmml" xref="S1.Thmtheorem8.p1.2.2.m2.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem8.p1.2.2.m2.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem8.p1.2.2.m2.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math> achieves an approximation ratio <math alttext="\alpha(\mathcal{O}_{\mathsf{S}})\leq 0.4889" class="ltx_Math" display="inline" id="S1.Thmtheorem8.p1.3.3.m3.1"><semantics id="S1.Thmtheorem8.p1.3.3.m3.1a"><mrow id="S1.Thmtheorem8.p1.3.3.m3.1.1" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.cmml"><mrow id="S1.Thmtheorem8.p1.3.3.m3.1.1.1" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1.cmml"><mi id="S1.Thmtheorem8.p1.3.3.m3.1.1.1.3" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1.3.cmml">α</mi><mo id="S1.Thmtheorem8.p1.3.3.m3.1.1.1.2" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.cmml"><mo id="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.2" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.2.cmml">𝒪</mi><mi id="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.3" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.3.cmml">𝖲</mi></msub><mo id="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem8.p1.3.3.m3.1.1.2" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.2.cmml">≤</mo><mn id="S1.Thmtheorem8.p1.3.3.m3.1.1.3" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.3.cmml">0.4889</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem8.p1.3.3.m3.1b"><apply id="S1.Thmtheorem8.p1.3.3.m3.1.1.cmml" xref="S1.Thmtheorem8.p1.3.3.m3.1.1"><leq id="S1.Thmtheorem8.p1.3.3.m3.1.1.2.cmml" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.2"></leq><apply id="S1.Thmtheorem8.p1.3.3.m3.1.1.1.cmml" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1"><times id="S1.Thmtheorem8.p1.3.3.m3.1.1.1.2.cmml" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1.2"></times><ci id="S1.Thmtheorem8.p1.3.3.m3.1.1.1.3.cmml" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1.3">𝛼</ci><apply id="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.2.cmml" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.2">𝒪</ci><ci id="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.3">𝖲</ci></apply></apply><cn id="S1.Thmtheorem8.p1.3.3.m3.1.1.3.cmml" type="float" xref="S1.Thmtheorem8.p1.3.3.m3.1.1.3">0.4889</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem8.p1.3.3.m3.1c">\alpha(\mathcal{O}_{\mathsf{S}})\leq 0.4889</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem8.p1.3.3.m3.1d">italic_α ( caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT ) ≤ 0.4889</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.SS2.p5"> <p class="ltx_p" id="S1.SS2.p5.1"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem8" title="Theorem 1.8 (Lower bound for symmetric selection). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.8</span></a> should be compared against <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem2" title="Theorem 1.2 (Prior lower bound for antisymmetric selection, [FJ15, Thm. 1.4]). ‣ 1.1 Background: Directed cuts, bias, and oblivious algorithms ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.2</span></a>, which it improves by roughly <math alttext="0.001" class="ltx_Math" display="inline" id="S1.SS2.p5.1.m1.1"><semantics id="S1.SS2.p5.1.m1.1a"><mn id="S1.SS2.p5.1.m1.1.1" xref="S1.SS2.p5.1.m1.1.1.cmml">0.001</mn><annotation-xml encoding="MathML-Content" id="S1.SS2.p5.1.m1.1b"><cn id="S1.SS2.p5.1.m1.1.1.cmml" type="float" xref="S1.SS2.p5.1.m1.1.1">0.001</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p5.1.m1.1c">0.001</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p5.1.m1.1d">0.001</annotation></semantics></math>. Finally, we prove a lower bound against arbitrary (not necessarily antisymmetric) selection functions:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.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="S1.Thmtheorem9.1.1.1">Theorem 1.9</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem9.2.2"> </span>(Lower bound for general selection)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem9.3.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem9.p1"> <p class="ltx_p" id="S1.Thmtheorem9.p1.3"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem9.p1.3.3">For <em class="ltx_emph ltx_font_upright" id="S1.Thmtheorem9.p1.3.3.1">every</em> selection function <math alttext="\mathsf{S}" class="ltx_Math" display="inline" id="S1.Thmtheorem9.p1.1.1.m1.1"><semantics id="S1.Thmtheorem9.p1.1.1.m1.1a"><mi id="S1.Thmtheorem9.p1.1.1.m1.1.1" xref="S1.Thmtheorem9.p1.1.1.m1.1.1.cmml">𝖲</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem9.p1.1.1.m1.1b"><ci id="S1.Thmtheorem9.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem9.p1.1.1.m1.1.1">𝖲</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem9.p1.1.1.m1.1c">\mathsf{S}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem9.p1.1.1.m1.1d">sansserif_S</annotation></semantics></math>, <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S1.Thmtheorem9.p1.2.2.m2.1"><semantics id="S1.Thmtheorem9.p1.2.2.m2.1a"><msub id="S1.Thmtheorem9.p1.2.2.m2.1.1" xref="S1.Thmtheorem9.p1.2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem9.p1.2.2.m2.1.1.2" xref="S1.Thmtheorem9.p1.2.2.m2.1.1.2.cmml">𝒪</mi><mi id="S1.Thmtheorem9.p1.2.2.m2.1.1.3" xref="S1.Thmtheorem9.p1.2.2.m2.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem9.p1.2.2.m2.1b"><apply id="S1.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem9.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem9.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmtheorem9.p1.2.2.m2.1.1">subscript</csymbol><ci id="S1.Thmtheorem9.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmtheorem9.p1.2.2.m2.1.1.2">𝒪</ci><ci id="S1.Thmtheorem9.p1.2.2.m2.1.1.3.cmml" xref="S1.Thmtheorem9.p1.2.2.m2.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem9.p1.2.2.m2.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem9.p1.2.2.m2.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math> achieves an approximation ratio <math alttext="\alpha(\mathcal{O}_{\mathsf{S}})\leq 0.4955" class="ltx_Math" display="inline" id="S1.Thmtheorem9.p1.3.3.m3.1"><semantics id="S1.Thmtheorem9.p1.3.3.m3.1a"><mrow id="S1.Thmtheorem9.p1.3.3.m3.1.1" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.cmml"><mrow id="S1.Thmtheorem9.p1.3.3.m3.1.1.1" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1.cmml"><mi id="S1.Thmtheorem9.p1.3.3.m3.1.1.1.3" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1.3.cmml">α</mi><mo id="S1.Thmtheorem9.p1.3.3.m3.1.1.1.2" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.1.cmml"><mo id="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.1" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.1.2" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.1.2.cmml">𝒪</mi><mi id="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.1.3" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.1.3.cmml">𝖲</mi></msub><mo id="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem9.p1.3.3.m3.1.1.2" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.2.cmml">≤</mo><mn id="S1.Thmtheorem9.p1.3.3.m3.1.1.3" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.3.cmml">0.4955</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem9.p1.3.3.m3.1b"><apply id="S1.Thmtheorem9.p1.3.3.m3.1.1.cmml" xref="S1.Thmtheorem9.p1.3.3.m3.1.1"><leq id="S1.Thmtheorem9.p1.3.3.m3.1.1.2.cmml" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.2"></leq><apply id="S1.Thmtheorem9.p1.3.3.m3.1.1.1.cmml" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1"><times id="S1.Thmtheorem9.p1.3.3.m3.1.1.1.2.cmml" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1.2"></times><ci id="S1.Thmtheorem9.p1.3.3.m3.1.1.1.3.cmml" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1.3">𝛼</ci><apply id="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.1.2.cmml" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.1.2">𝒪</ci><ci id="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.1.1.1.1.3">𝖲</ci></apply></apply><cn id="S1.Thmtheorem9.p1.3.3.m3.1.1.3.cmml" type="float" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.3">0.4955</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem9.p1.3.3.m3.1c">\alpha(\mathcal{O}_{\mathsf{S}})\leq 0.4955</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem9.p1.3.3.m3.1d">italic_α ( caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT ) ≤ 0.4955</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.SS2.p6"> <p class="ltx_p" id="S1.SS2.p6.1">The proof of this theorem is itself a conceptual contribution: The lower bound is witnessed by a single, simple graph, with only one bias (up to sign). In contrast, <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem2" title="Theorem 1.2 (Prior lower bound for antisymmetric selection, [FJ15, Thm. 1.4]). ‣ 1.1 Background: Directed cuts, bias, and oblivious algorithms ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.2</span></a> was weaker (by <math alttext="0.003" class="ltx_Math" display="inline" id="S1.SS2.p6.1.m1.1"><semantics id="S1.SS2.p6.1.m1.1a"><mn id="S1.SS2.p6.1.m1.1.1" xref="S1.SS2.p6.1.m1.1.1.cmml">0.003</mn><annotation-xml encoding="MathML-Content" id="S1.SS2.p6.1.m1.1b"><cn id="S1.SS2.p6.1.m1.1.1.cmml" type="float" xref="S1.SS2.p6.1.m1.1.1">0.003</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p6.1.m1.1c">0.003</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p6.1.m1.1d">0.003</annotation></semantics></math>), and its proof used multiple graphs, one of which had two distinct biases appearing (up to sign).</p> </div> </section> <section class="ltx_subsection" id="S1.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.3 </span>Motivations</h3> <section class="ltx_paragraph" id="S1.SS3.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Downstream applications.</h4> <div class="ltx_para" id="S1.SS3.SSS0.Px1.p1"> <span class="ltx_ERROR undefined" id="S1.SS3.SSS0.Px1.p1.9">\textcite</span> <p class="ltx_p" id="S1.SS3.SSS0.Px1.p1.8">FJ15 were interested in oblivious algorithms both in their own right as a nontrivial class of combinatorial algorithms for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS3.SSS0.Px1.p1.8.1">Max-DiCut</span> and because of connections with “local” and “distributed” models of computation. But more recently, several works have established that the existence of good oblivious algorithms for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS3.SSS0.Px1.p1.8.2">Max-DiCut</span> implies the existence of certain kinds of good streaming algorithms for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS3.SSS0.Px1.p1.8.3">Max-DiCut</span>. These algorithms are given a list of the graph’s directed edges as input and must output an estimate of its <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS3.SSS0.Px1.p1.8.4">Max-DiCut</span> value. In particular, the existence of <math alttext="\alpha" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p1.5.m5.1"><semantics id="S1.SS3.SSS0.Px1.p1.5.m5.1a"><mi id="S1.SS3.SSS0.Px1.p1.5.m5.1.1" xref="S1.SS3.SSS0.Px1.p1.5.m5.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.5.m5.1b"><ci id="S1.SS3.SSS0.Px1.p1.5.m5.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.5.m5.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.5.m5.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.5.m5.1d">italic_α</annotation></semantics></math>-approximation oblivious algorithms for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS3.SSS0.Px1.p1.8.5">Max-DiCut</span> is known to imply <math alttext="(\alpha-\epsilon)" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p1.7.m7.1"><semantics id="S1.SS3.SSS0.Px1.p1.7.m7.1a"><mrow id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.cmml"><mo id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.2" stretchy="false" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.cmml">(</mo><mrow id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.cmml"><mi id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.2" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.2.cmml">α</mi><mo id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.cmml">−</mo><mi id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.3" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.3" stretchy="false" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.7.m7.1b"><apply id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1"><minus id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.1"></minus><ci id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.2">𝛼</ci><ci id="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.3.cmml" xref="S1.SS3.SSS0.Px1.p1.7.m7.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.7.m7.1c">(\alpha-\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.7.m7.1d">( italic_α - italic_ϵ )</annotation></semantics></math>-approximation algorithms for all <math alttext="\epsilon>0" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p1.8.m8.1"><semantics id="S1.SS3.SSS0.Px1.p1.8.m8.1a"><mrow id="S1.SS3.SSS0.Px1.p1.8.m8.1.1" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.cmml"><mi id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.2" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.2.cmml">ϵ</mi><mo id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.cmml">></mo><mn id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.3" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p1.8.m8.1b"><apply id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1"><gt id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.1"></gt><ci id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.2">italic-ϵ</ci><cn id="S1.SS3.SSS0.Px1.p1.8.m8.1.1.3.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p1.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p1.8.m8.1c">\epsilon>0</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p1.8.m8.1d">italic_ϵ > 0</annotation></semantics></math> in the following models:</p> </div> <figure class="ltx_table" id="S1.T1"> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S1.T1.8"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S1.T1.8.9.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S1.T1.8.9.1.1"><span class="ltx_text ltx_font_bold" id="S1.T1.8.9.1.1.1">Space</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S1.T1.8.9.1.2"><span class="ltx_text ltx_font_bold" id="S1.T1.8.9.1.2.1">Input ordering</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S1.T1.8.9.1.3"><span class="ltx_text ltx_font_bold" id="S1.T1.8.9.1.3.1"># passes</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S1.T1.8.9.1.4"><span class="ltx_text ltx_font_bold" id="S1.T1.8.9.1.4.1">Setting</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S1.T1.8.9.1.5"><span class="ltx_text ltx_font_bold" id="S1.T1.8.9.1.5.1">Citation</span></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S1.T1.2.2"> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T1.1.1.1"><math alttext="O(\log n)" class="ltx_Math" display="inline" id="S1.T1.1.1.1.m1.1"><semantics id="S1.T1.1.1.1.m1.1a"><mrow id="S1.T1.1.1.1.m1.1.1" xref="S1.T1.1.1.1.m1.1.1.cmml"><mi id="S1.T1.1.1.1.m1.1.1.3" xref="S1.T1.1.1.1.m1.1.1.3.cmml">O</mi><mo id="S1.T1.1.1.1.m1.1.1.2" xref="S1.T1.1.1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T1.1.1.1.m1.1.1.1.1" xref="S1.T1.1.1.1.m1.1.1.1.1.1.cmml"><mo id="S1.T1.1.1.1.m1.1.1.1.1.2" stretchy="false" xref="S1.T1.1.1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T1.1.1.1.m1.1.1.1.1.1" xref="S1.T1.1.1.1.m1.1.1.1.1.1.cmml"><mi id="S1.T1.1.1.1.m1.1.1.1.1.1.1" xref="S1.T1.1.1.1.m1.1.1.1.1.1.1.cmml">log</mi><mo id="S1.T1.1.1.1.m1.1.1.1.1.1a" lspace="0.167em" xref="S1.T1.1.1.1.m1.1.1.1.1.1.cmml">⁡</mo><mi id="S1.T1.1.1.1.m1.1.1.1.1.1.2" xref="S1.T1.1.1.1.m1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S1.T1.1.1.1.m1.1.1.1.1.3" stretchy="false" xref="S1.T1.1.1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T1.1.1.1.m1.1b"><apply id="S1.T1.1.1.1.m1.1.1.cmml" xref="S1.T1.1.1.1.m1.1.1"><times id="S1.T1.1.1.1.m1.1.1.2.cmml" xref="S1.T1.1.1.1.m1.1.1.2"></times><ci id="S1.T1.1.1.1.m1.1.1.3.cmml" xref="S1.T1.1.1.1.m1.1.1.3">𝑂</ci><apply id="S1.T1.1.1.1.m1.1.1.1.1.1.cmml" xref="S1.T1.1.1.1.m1.1.1.1.1"><log id="S1.T1.1.1.1.m1.1.1.1.1.1.1.cmml" xref="S1.T1.1.1.1.m1.1.1.1.1.1.1"></log><ci id="S1.T1.1.1.1.m1.1.1.1.1.1.2.cmml" xref="S1.T1.1.1.1.m1.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.1.1.1.m1.1c">O(\log n)</annotation><annotation encoding="application/x-llamapun" id="S1.T1.1.1.1.m1.1d">italic_O ( roman_log italic_n )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T1.2.2.3">Random</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T1.2.2.2"><math alttext="1" class="ltx_Math" display="inline" id="S1.T1.2.2.2.m1.1"><semantics id="S1.T1.2.2.2.m1.1a"><mn id="S1.T1.2.2.2.m1.1.1" xref="S1.T1.2.2.2.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.T1.2.2.2.m1.1b"><cn id="S1.T1.2.2.2.m1.1.1.cmml" type="integer" xref="S1.T1.2.2.2.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.2.2.2.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.T1.2.2.2.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T1.2.2.4">Classical</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T1.2.2.5"><cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">SSSV23-random-ordering</span>]</cite></td> </tr> <tr class="ltx_tr" id="S1.T1.4.4"> <td class="ltx_td ltx_align_center" id="S1.T1.3.3.1"><math alttext="O(\log n)" class="ltx_Math" display="inline" id="S1.T1.3.3.1.m1.1"><semantics id="S1.T1.3.3.1.m1.1a"><mrow id="S1.T1.3.3.1.m1.1.1" xref="S1.T1.3.3.1.m1.1.1.cmml"><mi id="S1.T1.3.3.1.m1.1.1.3" xref="S1.T1.3.3.1.m1.1.1.3.cmml">O</mi><mo id="S1.T1.3.3.1.m1.1.1.2" xref="S1.T1.3.3.1.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T1.3.3.1.m1.1.1.1.1" 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xref="S1.T1.3.3.1.m1.1.1.1.1.1.1"></log><ci id="S1.T1.3.3.1.m1.1.1.1.1.1.2.cmml" xref="S1.T1.3.3.1.m1.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.3.3.1.m1.1c">O(\log n)</annotation><annotation encoding="application/x-llamapun" id="S1.T1.3.3.1.m1.1d">italic_O ( roman_log italic_n )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S1.T1.4.4.3">Adversarial</td> <td class="ltx_td ltx_align_center" id="S1.T1.4.4.2"><math alttext="2" class="ltx_Math" display="inline" id="S1.T1.4.4.2.m1.1"><semantics id="S1.T1.4.4.2.m1.1a"><mn id="S1.T1.4.4.2.m1.1.1" xref="S1.T1.4.4.2.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.T1.4.4.2.m1.1b"><cn id="S1.T1.4.4.2.m1.1.1.cmml" type="integer" xref="S1.T1.4.4.2.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.4.4.2.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.T1.4.4.2.m1.1d">2</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S1.T1.4.4.4">Classical</td> <td class="ltx_td ltx_align_center" id="S1.T1.4.4.5"><cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">SSSV23-random-ordering</span>]</cite></td> </tr> <tr class="ltx_tr" id="S1.T1.6.6"> <td class="ltx_td ltx_align_center" id="S1.T1.5.5.1"><math alttext="O(\sqrt{n}\operatorname{polylog}n)" class="ltx_Math" display="inline" id="S1.T1.5.5.1.m1.1"><semantics id="S1.T1.5.5.1.m1.1a"><mrow id="S1.T1.5.5.1.m1.1.1" xref="S1.T1.5.5.1.m1.1.1.cmml"><mi id="S1.T1.5.5.1.m1.1.1.3" xref="S1.T1.5.5.1.m1.1.1.3.cmml">O</mi><mo id="S1.T1.5.5.1.m1.1.1.2" xref="S1.T1.5.5.1.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T1.5.5.1.m1.1.1.1.1" xref="S1.T1.5.5.1.m1.1.1.1.1.1.cmml"><mo id="S1.T1.5.5.1.m1.1.1.1.1.2" stretchy="false" xref="S1.T1.5.5.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T1.5.5.1.m1.1.1.1.1.1" 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id="S1.T1.5.5.1.m1.1.1.3.cmml" xref="S1.T1.5.5.1.m1.1.1.3">𝑂</ci><apply id="S1.T1.5.5.1.m1.1.1.1.1.1.cmml" xref="S1.T1.5.5.1.m1.1.1.1.1"><times id="S1.T1.5.5.1.m1.1.1.1.1.1.1.cmml" xref="S1.T1.5.5.1.m1.1.1.1.1.1.1"></times><apply id="S1.T1.5.5.1.m1.1.1.1.1.1.2.cmml" xref="S1.T1.5.5.1.m1.1.1.1.1.1.2"><root id="S1.T1.5.5.1.m1.1.1.1.1.1.2a.cmml" xref="S1.T1.5.5.1.m1.1.1.1.1.1.2"></root><ci id="S1.T1.5.5.1.m1.1.1.1.1.1.2.2.cmml" xref="S1.T1.5.5.1.m1.1.1.1.1.1.2.2">𝑛</ci></apply><apply id="S1.T1.5.5.1.m1.1.1.1.1.1.3.cmml" xref="S1.T1.5.5.1.m1.1.1.1.1.1.3"><ci id="S1.T1.5.5.1.m1.1.1.1.1.1.3.1.cmml" xref="S1.T1.5.5.1.m1.1.1.1.1.1.3.1">polylog</ci><ci id="S1.T1.5.5.1.m1.1.1.1.1.1.3.2.cmml" xref="S1.T1.5.5.1.m1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.5.5.1.m1.1c">O(\sqrt{n}\operatorname{polylog}n)</annotation><annotation encoding="application/x-llamapun" id="S1.T1.5.5.1.m1.1d">italic_O ( square-root start_ARG italic_n end_ARG roman_polylog italic_n )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S1.T1.6.6.3">Adversarial</td> <td class="ltx_td ltx_align_center" id="S1.T1.6.6.2"><math alttext="1" class="ltx_Math" display="inline" id="S1.T1.6.6.2.m1.1"><semantics id="S1.T1.6.6.2.m1.1a"><mn id="S1.T1.6.6.2.m1.1.1" xref="S1.T1.6.6.2.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.T1.6.6.2.m1.1b"><cn id="S1.T1.6.6.2.m1.1.1.cmml" type="integer" xref="S1.T1.6.6.2.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.6.6.2.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.T1.6.6.2.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S1.T1.6.6.4">Classical</td> <td class="ltx_td ltx_align_center" id="S1.T1.6.6.5"><cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">SSSV23-dicut</span>]</cite></td> </tr> <tr class="ltx_tr" id="S1.T1.8.8"> <td class="ltx_td ltx_align_center" id="S1.T1.7.7.1"><math alttext="O(\operatorname{polylog}n)" class="ltx_Math" display="inline" id="S1.T1.7.7.1.m1.1"><semantics id="S1.T1.7.7.1.m1.1a"><mrow id="S1.T1.7.7.1.m1.1.1" xref="S1.T1.7.7.1.m1.1.1.cmml"><mi id="S1.T1.7.7.1.m1.1.1.3" xref="S1.T1.7.7.1.m1.1.1.3.cmml">O</mi><mo id="S1.T1.7.7.1.m1.1.1.2" xref="S1.T1.7.7.1.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T1.7.7.1.m1.1.1.1.1" xref="S1.T1.7.7.1.m1.1.1.1.1.1.cmml"><mo id="S1.T1.7.7.1.m1.1.1.1.1.2" stretchy="false" xref="S1.T1.7.7.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T1.7.7.1.m1.1.1.1.1.1" xref="S1.T1.7.7.1.m1.1.1.1.1.1.cmml"><mi id="S1.T1.7.7.1.m1.1.1.1.1.1.1" xref="S1.T1.7.7.1.m1.1.1.1.1.1.1.cmml">polylog</mi><mo id="S1.T1.7.7.1.m1.1.1.1.1.1a" lspace="0.167em" xref="S1.T1.7.7.1.m1.1.1.1.1.1.cmml">⁡</mo><mi id="S1.T1.7.7.1.m1.1.1.1.1.1.2" xref="S1.T1.7.7.1.m1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S1.T1.7.7.1.m1.1.1.1.1.3" stretchy="false" xref="S1.T1.7.7.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T1.7.7.1.m1.1b"><apply id="S1.T1.7.7.1.m1.1.1.cmml" xref="S1.T1.7.7.1.m1.1.1"><times id="S1.T1.7.7.1.m1.1.1.2.cmml" xref="S1.T1.7.7.1.m1.1.1.2"></times><ci id="S1.T1.7.7.1.m1.1.1.3.cmml" xref="S1.T1.7.7.1.m1.1.1.3">𝑂</ci><apply id="S1.T1.7.7.1.m1.1.1.1.1.1.cmml" xref="S1.T1.7.7.1.m1.1.1.1.1"><ci id="S1.T1.7.7.1.m1.1.1.1.1.1.1.cmml" xref="S1.T1.7.7.1.m1.1.1.1.1.1.1">polylog</ci><ci id="S1.T1.7.7.1.m1.1.1.1.1.1.2.cmml" xref="S1.T1.7.7.1.m1.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.7.7.1.m1.1c">O(\operatorname{polylog}n)</annotation><annotation encoding="application/x-llamapun" id="S1.T1.7.7.1.m1.1d">italic_O ( roman_polylog italic_n )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S1.T1.8.8.3">Adversarial</td> <td class="ltx_td ltx_align_center" id="S1.T1.8.8.2"><math alttext="1" class="ltx_Math" display="inline" id="S1.T1.8.8.2.m1.1"><semantics id="S1.T1.8.8.2.m1.1a"><mn id="S1.T1.8.8.2.m1.1.1" xref="S1.T1.8.8.2.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.T1.8.8.2.m1.1b"><cn id="S1.T1.8.8.2.m1.1.1.cmml" type="integer" xref="S1.T1.8.8.2.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.8.8.2.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.T1.8.8.2.m1.1d">1</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S1.T1.8.8.4">Quantum</td> <td class="ltx_td ltx_align_center" id="S1.T1.8.8.5"><cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">kallaugher2023exponential</span>]</cite></td> </tr> </tbody> </table> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S1.T1.10.1.1" style="font-size:90%;">Table 1</span>: </span><span class="ltx_text" id="S1.T1.11.2" style="font-size:90%;">Streaming models into which oblivious algorithms are known to “translate”, achieving the same approximation ratio up to arbitrarily small constants.</span></figcaption> </figure> <div class="ltx_para" id="S1.SS3.SSS0.Px1.p2"> <p class="ltx_p" id="S1.SS3.SSS0.Px1.p2.1">Thus, further improved oblivious algorithms, like we provide in this paper, imply further improvements in the state-of-the-art for all of these streaming models.</p> </div> <div class="ltx_para" id="S1.SS3.SSS0.Px1.p3"> <p class="ltx_p" id="S1.SS3.SSS0.Px1.p3.12">The fact that oblivious algorithms can be “implemented” as streaming algorithms in these models is motivated by lower bounds known in some related models. In particular, it was known by a previous result of <span class="ltx_ERROR undefined" id="S1.SS3.SSS0.Px1.p3.12.1">\textcite</span>CGV20 that <math alttext="(4/9-\epsilon)" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p3.1.m1.1"><semantics id="S1.SS3.SSS0.Px1.p3.1.m1.1a"><mrow id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.cmml"><mo id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.2" stretchy="false" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.cmml">(</mo><mrow id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.cmml"><mrow id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2.cmml"><mn id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2.2" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2.2.cmml">4</mn><mo id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2.1" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2.1.cmml">/</mo><mn id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2.3" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2.3.cmml">9</mn></mrow><mo id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.1" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.1.cmml">−</mo><mi id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.3" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.3" stretchy="false" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p3.1.m1.1b"><apply id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1"><minus id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.1"></minus><apply id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2"><divide id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2.1.cmml" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2.1"></divide><cn id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2.2.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2.2">4</cn><cn id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2.3.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.2.3">9</cn></apply><ci id="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.3.cmml" xref="S1.SS3.SSS0.Px1.p3.1.m1.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p3.1.m1.1c">(4/9-\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p3.1.m1.1d">( 4 / 9 - italic_ϵ )</annotation></semantics></math>-approximations to <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS3.SSS0.Px1.p3.12.2">Max-DiCut</span> can be computed using <math alttext="O(\log n)" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p3.3.m3.1"><semantics id="S1.SS3.SSS0.Px1.p3.3.m3.1a"><mrow id="S1.SS3.SSS0.Px1.p3.3.m3.1.1" xref="S1.SS3.SSS0.Px1.p3.3.m3.1.1.cmml"><mi id="S1.SS3.SSS0.Px1.p3.3.m3.1.1.3" xref="S1.SS3.SSS0.Px1.p3.3.m3.1.1.3.cmml">O</mi><mo id="S1.SS3.SSS0.Px1.p3.3.m3.1.1.2" xref="S1.SS3.SSS0.Px1.p3.3.m3.1.1.2.cmml">⁢</mo><mrow id="S1.SS3.SSS0.Px1.p3.3.m3.1.1.1.1" xref="S1.SS3.SSS0.Px1.p3.3.m3.1.1.1.1.1.cmml"><mo 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id="S1.SS3.SSS0.Px1.p3.3.m3.1.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p3.3.m3.1.1.1.1"><log id="S1.SS3.SSS0.Px1.p3.3.m3.1.1.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p3.3.m3.1.1.1.1.1.1"></log><ci id="S1.SS3.SSS0.Px1.p3.3.m3.1.1.1.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p3.3.m3.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p3.3.m3.1c">O(\log n)</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p3.3.m3.1d">italic_O ( roman_log italic_n )</annotation></semantics></math> space in a single, classical, adversarially-ordered pass, while <math alttext="(4/9+\epsilon)" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p3.4.m4.1"><semantics id="S1.SS3.SSS0.Px1.p3.4.m4.1a"><mrow id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.cmml"><mo id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.2" stretchy="false" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.cmml">(</mo><mrow id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.cmml"><mrow id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2.cmml"><mn id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2.2" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2.2.cmml">4</mn><mo id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2.1" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2.1.cmml">/</mo><mn id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2.3" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2.3.cmml">9</mn></mrow><mo id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.cmml">+</mo><mi id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.3" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.3" stretchy="false" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p3.4.m4.1b"><apply id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1"><plus id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.1"></plus><apply id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2"><divide id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2.1.cmml" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2.1"></divide><cn id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2.2.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2.2">4</cn><cn id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2.3.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.2.3">9</cn></apply><ci id="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.3.cmml" xref="S1.SS3.SSS0.Px1.p3.4.m4.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p3.4.m4.1c">(4/9+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p3.4.m4.1d">( 4 / 9 + italic_ϵ )</annotation></semantics></math>-approximations require <math alttext="\Omega(\sqrt{n})" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p3.5.m5.1"><semantics id="S1.SS3.SSS0.Px1.p3.5.m5.1a"><mrow id="S1.SS3.SSS0.Px1.p3.5.m5.1.2" xref="S1.SS3.SSS0.Px1.p3.5.m5.1.2.cmml"><mi id="S1.SS3.SSS0.Px1.p3.5.m5.1.2.2" mathvariant="normal" xref="S1.SS3.SSS0.Px1.p3.5.m5.1.2.2.cmml">Ω</mi><mo id="S1.SS3.SSS0.Px1.p3.5.m5.1.2.1" xref="S1.SS3.SSS0.Px1.p3.5.m5.1.2.1.cmml">⁢</mo><mrow id="S1.SS3.SSS0.Px1.p3.5.m5.1.2.3.2" xref="S1.SS3.SSS0.Px1.p3.5.m5.1.1.cmml"><mo id="S1.SS3.SSS0.Px1.p3.5.m5.1.2.3.2.1" stretchy="false" xref="S1.SS3.SSS0.Px1.p3.5.m5.1.1.cmml">(</mo><msqrt id="S1.SS3.SSS0.Px1.p3.5.m5.1.1" xref="S1.SS3.SSS0.Px1.p3.5.m5.1.1.cmml"><mi id="S1.SS3.SSS0.Px1.p3.5.m5.1.1.2" xref="S1.SS3.SSS0.Px1.p3.5.m5.1.1.2.cmml">n</mi></msqrt><mo id="S1.SS3.SSS0.Px1.p3.5.m5.1.2.3.2.2" stretchy="false" xref="S1.SS3.SSS0.Px1.p3.5.m5.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p3.5.m5.1b"><apply id="S1.SS3.SSS0.Px1.p3.5.m5.1.2.cmml" xref="S1.SS3.SSS0.Px1.p3.5.m5.1.2"><times id="S1.SS3.SSS0.Px1.p3.5.m5.1.2.1.cmml" xref="S1.SS3.SSS0.Px1.p3.5.m5.1.2.1"></times><ci id="S1.SS3.SSS0.Px1.p3.5.m5.1.2.2.cmml" xref="S1.SS3.SSS0.Px1.p3.5.m5.1.2.2">Ω</ci><apply id="S1.SS3.SSS0.Px1.p3.5.m5.1.1.cmml" xref="S1.SS3.SSS0.Px1.p3.5.m5.1.2.3.2"><root id="S1.SS3.SSS0.Px1.p3.5.m5.1.1a.cmml" xref="S1.SS3.SSS0.Px1.p3.5.m5.1.2.3.2"></root><ci id="S1.SS3.SSS0.Px1.p3.5.m5.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p3.5.m5.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p3.5.m5.1c">\Omega(\sqrt{n})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p3.5.m5.1d">roman_Ω ( square-root start_ARG italic_n end_ARG )</annotation></semantics></math> space for a single, classical, adversarially-ordered pass. Thus, the fact that <span class="ltx_ERROR undefined" id="S1.SS3.SSS0.Px1.p3.12.3">\textcite</span>FJ15 constructed oblivious algorithms achieving an approximation ratio <math alttext="0.483>4/9" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p3.6.m6.1"><semantics id="S1.SS3.SSS0.Px1.p3.6.m6.1a"><mrow id="S1.SS3.SSS0.Px1.p3.6.m6.1.1" xref="S1.SS3.SSS0.Px1.p3.6.m6.1.1.cmml"><mn id="S1.SS3.SSS0.Px1.p3.6.m6.1.1.2" xref="S1.SS3.SSS0.Px1.p3.6.m6.1.1.2.cmml">0.483</mn><mo id="S1.SS3.SSS0.Px1.p3.6.m6.1.1.1" xref="S1.SS3.SSS0.Px1.p3.6.m6.1.1.1.cmml">></mo><mrow id="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3" xref="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3.cmml"><mn id="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3.2" xref="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3.2.cmml">4</mn><mo id="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3.1" xref="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3.1.cmml">/</mo><mn id="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3.3" xref="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3.3.cmml">9</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p3.6.m6.1b"><apply id="S1.SS3.SSS0.Px1.p3.6.m6.1.1.cmml" xref="S1.SS3.SSS0.Px1.p3.6.m6.1.1"><gt id="S1.SS3.SSS0.Px1.p3.6.m6.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p3.6.m6.1.1.1"></gt><cn id="S1.SS3.SSS0.Px1.p3.6.m6.1.1.2.cmml" type="float" xref="S1.SS3.SSS0.Px1.p3.6.m6.1.1.2">0.483</cn><apply id="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3.cmml" xref="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3"><divide id="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3.1.cmml" xref="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3.1"></divide><cn id="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3.2.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3.2">4</cn><cn id="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3.3.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p3.6.m6.1.1.3.3">9</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p3.6.m6.1c">0.483>4/9</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p3.6.m6.1d">0.483 > 4 / 9</annotation></semantics></math> implied that the <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">CGV20</span>]</cite> lower bound was “tight”: Adjusting the model to add either polylogarithmically more space, random ordering, a second pass, or access to quantum bits yields strictly better approximations! In contrast, for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS3.SSS0.Px1.p3.12.4">Max-DiCut</span>’s “undirected cousin” <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS3.SSS0.Px1.p3.12.5">Max-Cut</span>, optimal lower bounds are known even for <math alttext="O(\sqrt{n})" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p3.9.m9.1"><semantics id="S1.SS3.SSS0.Px1.p3.9.m9.1a"><mrow id="S1.SS3.SSS0.Px1.p3.9.m9.1.2" xref="S1.SS3.SSS0.Px1.p3.9.m9.1.2.cmml"><mi id="S1.SS3.SSS0.Px1.p3.9.m9.1.2.2" xref="S1.SS3.SSS0.Px1.p3.9.m9.1.2.2.cmml">O</mi><mo id="S1.SS3.SSS0.Px1.p3.9.m9.1.2.1" xref="S1.SS3.SSS0.Px1.p3.9.m9.1.2.1.cmml">⁢</mo><mrow id="S1.SS3.SSS0.Px1.p3.9.m9.1.2.3.2" xref="S1.SS3.SSS0.Px1.p3.9.m9.1.1.cmml"><mo id="S1.SS3.SSS0.Px1.p3.9.m9.1.2.3.2.1" stretchy="false" xref="S1.SS3.SSS0.Px1.p3.9.m9.1.1.cmml">(</mo><msqrt id="S1.SS3.SSS0.Px1.p3.9.m9.1.1" xref="S1.SS3.SSS0.Px1.p3.9.m9.1.1.cmml"><mi id="S1.SS3.SSS0.Px1.p3.9.m9.1.1.2" xref="S1.SS3.SSS0.Px1.p3.9.m9.1.1.2.cmml">n</mi></msqrt><mo id="S1.SS3.SSS0.Px1.p3.9.m9.1.2.3.2.2" stretchy="false" xref="S1.SS3.SSS0.Px1.p3.9.m9.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p3.9.m9.1b"><apply id="S1.SS3.SSS0.Px1.p3.9.m9.1.2.cmml" xref="S1.SS3.SSS0.Px1.p3.9.m9.1.2"><times id="S1.SS3.SSS0.Px1.p3.9.m9.1.2.1.cmml" xref="S1.SS3.SSS0.Px1.p3.9.m9.1.2.1"></times><ci id="S1.SS3.SSS0.Px1.p3.9.m9.1.2.2.cmml" xref="S1.SS3.SSS0.Px1.p3.9.m9.1.2.2">𝑂</ci><apply id="S1.SS3.SSS0.Px1.p3.9.m9.1.1.cmml" xref="S1.SS3.SSS0.Px1.p3.9.m9.1.2.3.2"><root id="S1.SS3.SSS0.Px1.p3.9.m9.1.1a.cmml" xref="S1.SS3.SSS0.Px1.p3.9.m9.1.2.3.2"></root><ci id="S1.SS3.SSS0.Px1.p3.9.m9.1.1.2.cmml" xref="S1.SS3.SSS0.Px1.p3.9.m9.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p3.9.m9.1c">O(\sqrt{n})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p3.9.m9.1d">italic_O ( square-root start_ARG italic_n end_ARG )</annotation></semantics></math>-space random-ordering algorithms and <math alttext="o(n)" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p3.10.m10.1"><semantics id="S1.SS3.SSS0.Px1.p3.10.m10.1a"><mrow id="S1.SS3.SSS0.Px1.p3.10.m10.1.2" xref="S1.SS3.SSS0.Px1.p3.10.m10.1.2.cmml"><mi id="S1.SS3.SSS0.Px1.p3.10.m10.1.2.2" xref="S1.SS3.SSS0.Px1.p3.10.m10.1.2.2.cmml">o</mi><mo id="S1.SS3.SSS0.Px1.p3.10.m10.1.2.1" xref="S1.SS3.SSS0.Px1.p3.10.m10.1.2.1.cmml">⁢</mo><mrow id="S1.SS3.SSS0.Px1.p3.10.m10.1.2.3.2" xref="S1.SS3.SSS0.Px1.p3.10.m10.1.2.cmml"><mo id="S1.SS3.SSS0.Px1.p3.10.m10.1.2.3.2.1" stretchy="false" xref="S1.SS3.SSS0.Px1.p3.10.m10.1.2.cmml">(</mo><mi id="S1.SS3.SSS0.Px1.p3.10.m10.1.1" xref="S1.SS3.SSS0.Px1.p3.10.m10.1.1.cmml">n</mi><mo id="S1.SS3.SSS0.Px1.p3.10.m10.1.2.3.2.2" stretchy="false" xref="S1.SS3.SSS0.Px1.p3.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p3.10.m10.1b"><apply id="S1.SS3.SSS0.Px1.p3.10.m10.1.2.cmml" xref="S1.SS3.SSS0.Px1.p3.10.m10.1.2"><times id="S1.SS3.SSS0.Px1.p3.10.m10.1.2.1.cmml" xref="S1.SS3.SSS0.Px1.p3.10.m10.1.2.1"></times><ci id="S1.SS3.SSS0.Px1.p3.10.m10.1.2.2.cmml" xref="S1.SS3.SSS0.Px1.p3.10.m10.1.2.2">𝑜</ci><ci id="S1.SS3.SSS0.Px1.p3.10.m10.1.1.cmml" xref="S1.SS3.SSS0.Px1.p3.10.m10.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p3.10.m10.1c">o(n)</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p3.10.m10.1d">italic_o ( italic_n )</annotation></semantics></math>-space adversarial-ordering algorithms <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">KKS15</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">KK19</span>]</cite>. In the quantum setting, <span class="ltx_ERROR undefined" id="S1.SS3.SSS0.Px1.p3.12.6">\textcite</span>kallaugher2023exponential claim that <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS3.SSS0.Px1.p3.12.7">Max-DiCut</span> is the first discrete optimization problem with a provable exponential separation in complexity between classical and quantum algorithms. These separations are all powered by the existence of oblivious algorithms achieving a ratio strictly above <math alttext="4/9" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px1.p3.12.m12.1"><semantics id="S1.SS3.SSS0.Px1.p3.12.m12.1a"><mrow id="S1.SS3.SSS0.Px1.p3.12.m12.1.1" xref="S1.SS3.SSS0.Px1.p3.12.m12.1.1.cmml"><mn id="S1.SS3.SSS0.Px1.p3.12.m12.1.1.2" xref="S1.SS3.SSS0.Px1.p3.12.m12.1.1.2.cmml">4</mn><mo id="S1.SS3.SSS0.Px1.p3.12.m12.1.1.1" xref="S1.SS3.SSS0.Px1.p3.12.m12.1.1.1.cmml">/</mo><mn id="S1.SS3.SSS0.Px1.p3.12.m12.1.1.3" xref="S1.SS3.SSS0.Px1.p3.12.m12.1.1.3.cmml">9</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px1.p3.12.m12.1b"><apply id="S1.SS3.SSS0.Px1.p3.12.m12.1.1.cmml" xref="S1.SS3.SSS0.Px1.p3.12.m12.1.1"><divide id="S1.SS3.SSS0.Px1.p3.12.m12.1.1.1.cmml" xref="S1.SS3.SSS0.Px1.p3.12.m12.1.1.1"></divide><cn id="S1.SS3.SSS0.Px1.p3.12.m12.1.1.2.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p3.12.m12.1.1.2">4</cn><cn id="S1.SS3.SSS0.Px1.p3.12.m12.1.1.3.cmml" type="integer" xref="S1.SS3.SSS0.Px1.p3.12.m12.1.1.3">9</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px1.p3.12.m12.1c">4/9</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px1.p3.12.m12.1d">4 / 9</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="S1.SS3.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Oblivious algorithms for other problems.</h4> <div class="ltx_para" id="S1.SS3.SSS0.Px2.p1"> <p class="ltx_p" id="S1.SS3.SSS0.Px2.p1.2">Oblivious algorithms for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS3.SSS0.Px2.p1.2.1">Max-DiCut</span> have also been studied in several other areas. In mechanism design, <span class="ltx_ERROR undefined" id="S1.SS3.SSS0.Px2.p1.2.2">\textcite</span>Luk14 showed that any <em class="ltx_emph ltx_font_italic" id="S1.SS3.SSS0.Px2.p1.2.3">monotone</em><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>An oblivious algorithm is said to be monotone if its corresponding selection function <math alttext="\mathsf{S}" class="ltx_Math" display="inline" id="footnote2.m1.1"><semantics id="footnote2.m1.1b"><mi id="footnote2.m1.1.1" xref="footnote2.m1.1.1.cmml">𝖲</mi><annotation-xml encoding="MathML-Content" id="footnote2.m1.1c"><ci id="footnote2.m1.1.1.cmml" xref="footnote2.m1.1.1">𝖲</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m1.1d">\mathsf{S}</annotation><annotation encoding="application/x-llamapun" id="footnote2.m1.1e">sansserif_S</annotation></semantics></math> is monotone, i.e., <math alttext="\mathsf{S}(x)\leq\mathsf{S}(y)" class="ltx_Math" display="inline" id="footnote2.m2.2"><semantics id="footnote2.m2.2b"><mrow id="footnote2.m2.2.3" xref="footnote2.m2.2.3.cmml"><mrow id="footnote2.m2.2.3.2" xref="footnote2.m2.2.3.2.cmml"><mi id="footnote2.m2.2.3.2.2" xref="footnote2.m2.2.3.2.2.cmml">𝖲</mi><mo id="footnote2.m2.2.3.2.1" xref="footnote2.m2.2.3.2.1.cmml">⁢</mo><mrow id="footnote2.m2.2.3.2.3.2" xref="footnote2.m2.2.3.2.cmml"><mo id="footnote2.m2.2.3.2.3.2.1" stretchy="false" xref="footnote2.m2.2.3.2.cmml">(</mo><mi id="footnote2.m2.1.1" xref="footnote2.m2.1.1.cmml">x</mi><mo id="footnote2.m2.2.3.2.3.2.2" stretchy="false" xref="footnote2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="footnote2.m2.2.3.1" xref="footnote2.m2.2.3.1.cmml">≤</mo><mrow id="footnote2.m2.2.3.3" xref="footnote2.m2.2.3.3.cmml"><mi id="footnote2.m2.2.3.3.2" xref="footnote2.m2.2.3.3.2.cmml">𝖲</mi><mo id="footnote2.m2.2.3.3.1" xref="footnote2.m2.2.3.3.1.cmml">⁢</mo><mrow id="footnote2.m2.2.3.3.3.2" xref="footnote2.m2.2.3.3.cmml"><mo id="footnote2.m2.2.3.3.3.2.1" stretchy="false" xref="footnote2.m2.2.3.3.cmml">(</mo><mi id="footnote2.m2.2.2" xref="footnote2.m2.2.2.cmml">y</mi><mo id="footnote2.m2.2.3.3.3.2.2" stretchy="false" xref="footnote2.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m2.2c"><apply id="footnote2.m2.2.3.cmml" xref="footnote2.m2.2.3"><leq id="footnote2.m2.2.3.1.cmml" xref="footnote2.m2.2.3.1"></leq><apply id="footnote2.m2.2.3.2.cmml" xref="footnote2.m2.2.3.2"><times id="footnote2.m2.2.3.2.1.cmml" xref="footnote2.m2.2.3.2.1"></times><ci id="footnote2.m2.2.3.2.2.cmml" xref="footnote2.m2.2.3.2.2">𝖲</ci><ci id="footnote2.m2.1.1.cmml" xref="footnote2.m2.1.1">𝑥</ci></apply><apply id="footnote2.m2.2.3.3.cmml" xref="footnote2.m2.2.3.3"><times id="footnote2.m2.2.3.3.1.cmml" xref="footnote2.m2.2.3.3.1"></times><ci id="footnote2.m2.2.3.3.2.cmml" xref="footnote2.m2.2.3.3.2">𝖲</ci><ci id="footnote2.m2.2.2.cmml" xref="footnote2.m2.2.2">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m2.2d">\mathsf{S}(x)\leq\mathsf{S}(y)</annotation><annotation encoding="application/x-llamapun" id="footnote2.m2.2e">sansserif_S ( italic_x ) ≤ sansserif_S ( italic_y )</annotation></semantics></math> if and only if <math alttext="x\leq y" class="ltx_Math" display="inline" id="footnote2.m3.1"><semantics id="footnote2.m3.1b"><mrow id="footnote2.m3.1.1" xref="footnote2.m3.1.1.cmml"><mi id="footnote2.m3.1.1.2" xref="footnote2.m3.1.1.2.cmml">x</mi><mo id="footnote2.m3.1.1.1" xref="footnote2.m3.1.1.1.cmml">≤</mo><mi id="footnote2.m3.1.1.3" xref="footnote2.m3.1.1.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m3.1c"><apply id="footnote2.m3.1.1.cmml" xref="footnote2.m3.1.1"><leq id="footnote2.m3.1.1.1.cmml" xref="footnote2.m3.1.1.1"></leq><ci id="footnote2.m3.1.1.2.cmml" xref="footnote2.m3.1.1.2">𝑥</ci><ci id="footnote2.m3.1.1.3.cmml" xref="footnote2.m3.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m3.1d">x\leq y</annotation><annotation encoding="application/x-llamapun" id="footnote2.m3.1e">italic_x ≤ italic_y</annotation></semantics></math>.</span></span></span> oblivious algorithm is a <em class="ltx_emph ltx_font_italic" id="S1.SS3.SSS0.Px2.p1.2.4">strategy-proof mechanism</em><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>A mechanism for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="footnote3.1">Max-DiCut</span> is defined as follows. There are <math alttext="m" class="ltx_Math" display="inline" id="footnote3.m2.1"><semantics id="footnote3.m2.1b"><mi id="footnote3.m2.1.1" xref="footnote3.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="footnote3.m2.1c"><ci id="footnote3.m2.1.1.cmml" xref="footnote3.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote3.m2.1d">m</annotation><annotation encoding="application/x-llamapun" id="footnote3.m2.1e">italic_m</annotation></semantics></math> players, each given a unique edge of the directed graph. A mechanism is a function that asks each player to reveal their edge and then chooses a (random) assignment of the vertices. The player may or may not reveal their edge truthfully. The utility of each player is the expected probability that their edge is satisfied by this assignment. A mechanism is said to be strategy-proof if the optimal strategy for every player is to be truthful.</span></span></span> for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS3.SSS0.Px2.p1.2.5">Max-DiCut</span>. <span class="ltx_ERROR undefined" id="S1.SS3.SSS0.Px2.p1.2.6">\textcite</span>BFS19 showed applications of oblivious algorithms to the online submodular optimization problem.</p> </div> <div class="ltx_para" id="S1.SS3.SSS0.Px2.p2"> <span class="ltx_ERROR undefined" id="S1.SS3.SSS0.Px2.p2.10">\textcite</span> <p class="ltx_p" id="S1.SS3.SSS0.Px2.p2.9">Sin23-kand recently extended the definition of oblivious algorithms to a more general version of <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS3.SSS0.Px2.p2.9.1">Max-DiCut</span> called <math alttext="\textsc{Max-}k\textsc{And}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p2.2.m2.1"><semantics id="S1.SS3.SSS0.Px2.p2.2.m2.1a"><mrow id="S1.SS3.SSS0.Px2.p2.2.m2.1.1" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.cmml"><mtext class="ltx_font_smallcaps" id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.2" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.2a.cmml">Max-</mtext><mo id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.cmml">⁢</mo><mi id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.3" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.3.cmml">k</mi><mo id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1a" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.cmml">⁢</mo><mtext class="ltx_font_smallcaps" id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.4" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.4a.cmml">And</mtext></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p2.2.m2.1b"><apply id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1"><times id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.1"></times><ci id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.2a.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.2"><mtext class="ltx_font_smallcaps" id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.2">Max-</mtext></ci><ci id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.3">𝑘</ci><ci id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.4a.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.4"><mtext class="ltx_font_smallcaps" id="S1.SS3.SSS0.Px2.p2.2.m2.1.1.4.cmml" xref="S1.SS3.SSS0.Px2.p2.2.m2.1.1.4">And</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p2.2.m2.1c">\textsc{Max-}k\textsc{And}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p2.2.m2.1d">Max- italic_k And</annotation></semantics></math>, where each constraint applies to <math alttext="k" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p2.3.m3.1"><semantics id="S1.SS3.SSS0.Px2.p2.3.m3.1a"><mi id="S1.SS3.SSS0.Px2.p2.3.m3.1.1" xref="S1.SS3.SSS0.Px2.p2.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p2.3.m3.1b"><ci id="S1.SS3.SSS0.Px2.p2.3.m3.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p2.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p2.3.m3.1d">italic_k</annotation></semantics></math> variables and specifies a single required bit for each variable. (<span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS3.SSS0.Px2.p2.9.2">Max-DiCut</span> is the special case where <math alttext="k=2" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p2.5.m5.1"><semantics id="S1.SS3.SSS0.Px2.p2.5.m5.1a"><mrow id="S1.SS3.SSS0.Px2.p2.5.m5.1.1" xref="S1.SS3.SSS0.Px2.p2.5.m5.1.1.cmml"><mi id="S1.SS3.SSS0.Px2.p2.5.m5.1.1.2" xref="S1.SS3.SSS0.Px2.p2.5.m5.1.1.2.cmml">k</mi><mo id="S1.SS3.SSS0.Px2.p2.5.m5.1.1.1" xref="S1.SS3.SSS0.Px2.p2.5.m5.1.1.1.cmml">=</mo><mn id="S1.SS3.SSS0.Px2.p2.5.m5.1.1.3" xref="S1.SS3.SSS0.Px2.p2.5.m5.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p2.5.m5.1b"><apply id="S1.SS3.SSS0.Px2.p2.5.m5.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.5.m5.1.1"><eq id="S1.SS3.SSS0.Px2.p2.5.m5.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.5.m5.1.1.1"></eq><ci id="S1.SS3.SSS0.Px2.p2.5.m5.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.p2.5.m5.1.1.2">𝑘</ci><cn id="S1.SS3.SSS0.Px2.p2.5.m5.1.1.3.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p2.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p2.5.m5.1c">k=2</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p2.5.m5.1d">italic_k = 2</annotation></semantics></math> and in each constraint, exactly one variable needs to be assigned <math alttext="1" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p2.6.m6.1"><semantics id="S1.SS3.SSS0.Px2.p2.6.m6.1a"><mn id="S1.SS3.SSS0.Px2.p2.6.m6.1.1" xref="S1.SS3.SSS0.Px2.p2.6.m6.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p2.6.m6.1b"><cn id="S1.SS3.SSS0.Px2.p2.6.m6.1.1.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p2.6.m6.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p2.6.m6.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p2.6.m6.1d">1</annotation></semantics></math> and the other <math alttext="0" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p2.7.m7.1"><semantics id="S1.SS3.SSS0.Px2.p2.7.m7.1a"><mn id="S1.SS3.SSS0.Px2.p2.7.m7.1.1" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p2.7.m7.1b"><cn id="S1.SS3.SSS0.Px2.p2.7.m7.1.1.cmml" type="integer" xref="S1.SS3.SSS0.Px2.p2.7.m7.1.1">0</cn></annotation-xml></semantics></math>.) He also generalized the LP of <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">FJ15</span>]</cite> for calculating the ratio of a piecewise-constant oblivious <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S1.SS3.SSS0.Px2.p2.9.3">Max-DiCut</span> algorithm to <math alttext="\textsc{Max-}k\textsc{And}" class="ltx_Math" display="inline" id="S1.SS3.SSS0.Px2.p2.9.m9.1"><semantics id="S1.SS3.SSS0.Px2.p2.9.m9.1a"><mrow id="S1.SS3.SSS0.Px2.p2.9.m9.1.1" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.cmml"><mtext class="ltx_font_smallcaps" id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.2" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.2a.cmml">Max-</mtext><mo id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.1" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.1.cmml">⁢</mo><mi id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.3" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.3.cmml">k</mi><mo id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.1a" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.1.cmml">⁢</mo><mtext class="ltx_font_smallcaps" id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.4" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.4a.cmml">And</mtext></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.SSS0.Px2.p2.9.m9.1b"><apply id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1"><times id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.1.cmml" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.1"></times><ci id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.2a.cmml" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.2"><mtext class="ltx_font_smallcaps" id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.2.cmml" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.2">Max-</mtext></ci><ci id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.3.cmml" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.3">𝑘</ci><ci id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.4a.cmml" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.4"><mtext class="ltx_font_smallcaps" id="S1.SS3.SSS0.Px2.p2.9.m9.1.1.4.cmml" xref="S1.SS3.SSS0.Px2.p2.9.m9.1.1.4">And</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.SSS0.Px2.p2.9.m9.1c">\textsc{Max-}k\textsc{And}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.SSS0.Px2.p2.9.m9.1d">Max- italic_k And</annotation></semantics></math>, and used this to achieve streaming separation results à la <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">SSSV23-random-ordering</span>]</cite>.</p> </div> </section> </section> <section class="ltx_subsection" id="S1.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.4 </span>Structure of rest of the paper</h3> <div class="ltx_para" id="S1.SS4.p1"> <p class="ltx_p" id="S1.SS4.p1.4"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S2" title="2 Preliminaries ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">2</span></a> contains some of the preliminary background used in the rest of the paper. In <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S3" title="3 Improved oblivious algorithms (Theorem 1.5) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3</span></a>, we discuss our new selection function that achieves an approximation ratio of at least 0.485275. In <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S4.SS2" title="4.2 Bounds against 𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽_{1/2} (Theorem 1.6) ‣ 4 Lower bounds ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.2</span></a>, we prove <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem6" title="Theorem 1.6 (Lower bound for PL sigmoid selection with 𝑏=1/2 intercept). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.6</span></a> by explicitly constructing a graph (only on two vertices!) for which the <math alttext="\mathsf{PLSigmoid}_{1/2}" class="ltx_Math" display="inline" id="S1.SS4.p1.1.m1.1"><semantics id="S1.SS4.p1.1.m1.1a"><msub id="S1.SS4.p1.1.m1.1.1" xref="S1.SS4.p1.1.m1.1.1.cmml"><mi id="S1.SS4.p1.1.m1.1.1.2" xref="S1.SS4.p1.1.m1.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S1.SS4.p1.1.m1.1.1.3" xref="S1.SS4.p1.1.m1.1.1.3.cmml"><mn id="S1.SS4.p1.1.m1.1.1.3.2" xref="S1.SS4.p1.1.m1.1.1.3.2.cmml">1</mn><mo id="S1.SS4.p1.1.m1.1.1.3.1" xref="S1.SS4.p1.1.m1.1.1.3.1.cmml">/</mo><mn id="S1.SS4.p1.1.m1.1.1.3.3" xref="S1.SS4.p1.1.m1.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.SS4.p1.1.m1.1b"><apply id="S1.SS4.p1.1.m1.1.1.cmml" xref="S1.SS4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS4.p1.1.m1.1.1.1.cmml" xref="S1.SS4.p1.1.m1.1.1">subscript</csymbol><ci id="S1.SS4.p1.1.m1.1.1.2.cmml" xref="S1.SS4.p1.1.m1.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S1.SS4.p1.1.m1.1.1.3.cmml" xref="S1.SS4.p1.1.m1.1.1.3"><divide id="S1.SS4.p1.1.m1.1.1.3.1.cmml" xref="S1.SS4.p1.1.m1.1.1.3.1"></divide><cn id="S1.SS4.p1.1.m1.1.1.3.2.cmml" type="integer" xref="S1.SS4.p1.1.m1.1.1.3.2">1</cn><cn id="S1.SS4.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S1.SS4.p1.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS4.p1.1.m1.1c">\mathsf{PLSigmoid}_{1/2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS4.p1.1.m1.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math> function achieves a strictly weaker than <math alttext="0.485282" class="ltx_Math" display="inline" id="S1.SS4.p1.2.m2.1"><semantics id="S1.SS4.p1.2.m2.1a"><mn id="S1.SS4.p1.2.m2.1.1" xref="S1.SS4.p1.2.m2.1.1.cmml">0.485282</mn><annotation-xml encoding="MathML-Content" id="S1.SS4.p1.2.m2.1b"><cn id="S1.SS4.p1.2.m2.1.1.cmml" type="float" xref="S1.SS4.p1.2.m2.1.1">0.485282</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS4.p1.2.m2.1c">0.485282</annotation><annotation encoding="application/x-llamapun" id="S1.SS4.p1.2.m2.1d">0.485282</annotation></semantics></math> approximation. In <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S4.SS3" title="4.3 Lower bound for PL sigmoid functions (Theorem 1.7) ‣ 4 Lower bounds ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.3</span></a>, we prove <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem7" title="Theorem 1.7 (Lower bound for PL sigmoid selection with arbitrary intercept). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.7</span></a> by constructing three graphs (one of which is on forty vertices!) and showing that every <math alttext="\mathsf{PLSigmoid}" class="ltx_Math" display="inline" id="S1.SS4.p1.3.m3.1"><semantics id="S1.SS4.p1.3.m3.1a"><mi id="S1.SS4.p1.3.m3.1.1" xref="S1.SS4.p1.3.m3.1.1.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><annotation-xml encoding="MathML-Content" id="S1.SS4.p1.3.m3.1b"><ci id="S1.SS4.p1.3.m3.1.1.cmml" xref="S1.SS4.p1.3.m3.1.1">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS4.p1.3.m3.1c">\mathsf{PLSigmoid}</annotation><annotation encoding="application/x-llamapun" id="S1.SS4.p1.3.m3.1d">sansserif_PLSigmoid</annotation></semantics></math> function has an approximation ratio of at most <math alttext="0.486" class="ltx_Math" display="inline" id="S1.SS4.p1.4.m4.1"><semantics id="S1.SS4.p1.4.m4.1a"><mn id="S1.SS4.p1.4.m4.1.1" xref="S1.SS4.p1.4.m4.1.1.cmml">0.486</mn><annotation-xml encoding="MathML-Content" id="S1.SS4.p1.4.m4.1b"><cn id="S1.SS4.p1.4.m4.1.1.cmml" type="float" xref="S1.SS4.p1.4.m4.1.1">0.486</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS4.p1.4.m4.1c">0.486</annotation><annotation encoding="application/x-llamapun" id="S1.SS4.p1.4.m4.1d">0.486</annotation></semantics></math> on at least one of these graphs. We also give a detailed description of the linear program that we use to generate these graphs. In <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S5" title="5 Lower bound for arbitrary selection functions (Theorem 1.9) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">5</span></a>, we prove <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem9" title="Theorem 1.9 (Lower bound for general selection). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.9</span></a> using a pair of graphs (a two-vertex and a four-vertex graph). Finally, in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S6" title="6 Lower bounds for antisymmetric selection functions (Theorem 1.8) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">6</span></a>, we prove <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem8" title="Theorem 1.8 (Lower bound for symmetric selection). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.8</span></a> and give a detailed description of the methodology we use to construct the lower bound instance.</p> </div> </section> <section class="ltx_subsection" id="S1.SS5"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.5 </span>Code</h3> <div class="ltx_para" id="S1.SS5.p1"> <p class="ltx_p" id="S1.SS5.p1.1">All code for this paper is available on GitHub at <a class="ltx_ref ltx_url ltx_font_typewriter" href="https://github.com/singerng/oblivious-csps/" title="">https://github.com/singerng/oblivious-csps/</a>.</p> </div> </section> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Preliminaries</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">We begin with some basic notations for directed graphs and for oblivious algorithms.</p> </div> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1 </span>Directed graphs</h3> <div class="ltx_theorem ltx_theorem_definition" id="S2.E1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.E1.1.1.1">Definition 2.1</span></span><span class="ltx_text ltx_font_bold" id="S2.E1.2.2"> </span>(Directed graphs)<span class="ltx_text ltx_font_bold" id="S2.E1.3.3">.</span> </h6> <div class="ltx_para" id="S2.E1.p1"> <p class="ltx_p" id="S2.E1.p1.6"><span class="ltx_text ltx_font_italic" id="S2.E1.p1.6.6">A <em class="ltx_emph ltx_font_upright" id="S2.E1.p1.6.6.1">(weighted) directed graph</em> <math alttext="G" class="ltx_Math" display="inline" id="S2.E1.p1.1.1.m1.1"><semantics id="S2.E1.p1.1.1.m1.1a"><mi id="S2.E1.p1.1.1.m1.1.1" xref="S2.E1.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.E1.p1.1.1.m1.1b"><ci id="S2.E1.p1.1.1.m1.1.1.cmml" xref="S2.E1.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.E1.p1.1.1.m1.1d">italic_G</annotation></semantics></math> is a pair <math alttext="(V,w)" class="ltx_Math" display="inline" id="S2.E1.p1.2.2.m2.2"><semantics id="S2.E1.p1.2.2.m2.2a"><mrow id="S2.E1.p1.2.2.m2.2.3.2" xref="S2.E1.p1.2.2.m2.2.3.1.cmml"><mo id="S2.E1.p1.2.2.m2.2.3.2.1" stretchy="false" xref="S2.E1.p1.2.2.m2.2.3.1.cmml">(</mo><mi id="S2.E1.p1.2.2.m2.1.1" xref="S2.E1.p1.2.2.m2.1.1.cmml">V</mi><mo id="S2.E1.p1.2.2.m2.2.3.2.2" xref="S2.E1.p1.2.2.m2.2.3.1.cmml">,</mo><mi id="S2.E1.p1.2.2.m2.2.2" xref="S2.E1.p1.2.2.m2.2.2.cmml">w</mi><mo id="S2.E1.p1.2.2.m2.2.3.2.3" stretchy="false" xref="S2.E1.p1.2.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E1.p1.2.2.m2.2b"><interval closure="open" id="S2.E1.p1.2.2.m2.2.3.1.cmml" xref="S2.E1.p1.2.2.m2.2.3.2"><ci id="S2.E1.p1.2.2.m2.1.1.cmml" xref="S2.E1.p1.2.2.m2.1.1">𝑉</ci><ci id="S2.E1.p1.2.2.m2.2.2.cmml" xref="S2.E1.p1.2.2.m2.2.2">𝑤</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.p1.2.2.m2.2c">(V,w)</annotation><annotation encoding="application/x-llamapun" id="S2.E1.p1.2.2.m2.2d">( italic_V , italic_w )</annotation></semantics></math> where <math alttext="V" class="ltx_Math" display="inline" id="S2.E1.p1.3.3.m3.1"><semantics id="S2.E1.p1.3.3.m3.1a"><mi id="S2.E1.p1.3.3.m3.1.1" xref="S2.E1.p1.3.3.m3.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S2.E1.p1.3.3.m3.1b"><ci id="S2.E1.p1.3.3.m3.1.1.cmml" xref="S2.E1.p1.3.3.m3.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.p1.3.3.m3.1c">V</annotation><annotation encoding="application/x-llamapun" id="S2.E1.p1.3.3.m3.1d">italic_V</annotation></semantics></math> is a set of <em class="ltx_emph ltx_font_upright" id="S2.E1.p1.6.6.2">vertices</em> and <math alttext="w:V\times V\to\mathbb{R}_{\geq 0}" class="ltx_Math" display="inline" id="S2.E1.p1.4.4.m4.1"><semantics id="S2.E1.p1.4.4.m4.1a"><mrow id="S2.E1.p1.4.4.m4.1.1" xref="S2.E1.p1.4.4.m4.1.1.cmml"><mi id="S2.E1.p1.4.4.m4.1.1.2" xref="S2.E1.p1.4.4.m4.1.1.2.cmml">w</mi><mo id="S2.E1.p1.4.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.E1.p1.4.4.m4.1.1.1.cmml">:</mo><mrow id="S2.E1.p1.4.4.m4.1.1.3" xref="S2.E1.p1.4.4.m4.1.1.3.cmml"><mrow id="S2.E1.p1.4.4.m4.1.1.3.2" xref="S2.E1.p1.4.4.m4.1.1.3.2.cmml"><mi id="S2.E1.p1.4.4.m4.1.1.3.2.2" xref="S2.E1.p1.4.4.m4.1.1.3.2.2.cmml">V</mi><mo id="S2.E1.p1.4.4.m4.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S2.E1.p1.4.4.m4.1.1.3.2.1.cmml">×</mo><mi id="S2.E1.p1.4.4.m4.1.1.3.2.3" xref="S2.E1.p1.4.4.m4.1.1.3.2.3.cmml">V</mi></mrow><mo id="S2.E1.p1.4.4.m4.1.1.3.1" stretchy="false" xref="S2.E1.p1.4.4.m4.1.1.3.1.cmml">→</mo><msub id="S2.E1.p1.4.4.m4.1.1.3.3" xref="S2.E1.p1.4.4.m4.1.1.3.3.cmml"><mi id="S2.E1.p1.4.4.m4.1.1.3.3.2" xref="S2.E1.p1.4.4.m4.1.1.3.3.2.cmml">ℝ</mi><mrow id="S2.E1.p1.4.4.m4.1.1.3.3.3" xref="S2.E1.p1.4.4.m4.1.1.3.3.3.cmml"><mi id="S2.E1.p1.4.4.m4.1.1.3.3.3.2" xref="S2.E1.p1.4.4.m4.1.1.3.3.3.2.cmml"></mi><mo id="S2.E1.p1.4.4.m4.1.1.3.3.3.1" xref="S2.E1.p1.4.4.m4.1.1.3.3.3.1.cmml">≥</mo><mn id="S2.E1.p1.4.4.m4.1.1.3.3.3.3" xref="S2.E1.p1.4.4.m4.1.1.3.3.3.3.cmml">0</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E1.p1.4.4.m4.1b"><apply id="S2.E1.p1.4.4.m4.1.1.cmml" xref="S2.E1.p1.4.4.m4.1.1"><ci id="S2.E1.p1.4.4.m4.1.1.1.cmml" xref="S2.E1.p1.4.4.m4.1.1.1">:</ci><ci id="S2.E1.p1.4.4.m4.1.1.2.cmml" xref="S2.E1.p1.4.4.m4.1.1.2">𝑤</ci><apply id="S2.E1.p1.4.4.m4.1.1.3.cmml" xref="S2.E1.p1.4.4.m4.1.1.3"><ci id="S2.E1.p1.4.4.m4.1.1.3.1.cmml" xref="S2.E1.p1.4.4.m4.1.1.3.1">→</ci><apply id="S2.E1.p1.4.4.m4.1.1.3.2.cmml" xref="S2.E1.p1.4.4.m4.1.1.3.2"><times id="S2.E1.p1.4.4.m4.1.1.3.2.1.cmml" xref="S2.E1.p1.4.4.m4.1.1.3.2.1"></times><ci id="S2.E1.p1.4.4.m4.1.1.3.2.2.cmml" xref="S2.E1.p1.4.4.m4.1.1.3.2.2">𝑉</ci><ci id="S2.E1.p1.4.4.m4.1.1.3.2.3.cmml" xref="S2.E1.p1.4.4.m4.1.1.3.2.3">𝑉</ci></apply><apply id="S2.E1.p1.4.4.m4.1.1.3.3.cmml" xref="S2.E1.p1.4.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S2.E1.p1.4.4.m4.1.1.3.3.1.cmml" xref="S2.E1.p1.4.4.m4.1.1.3.3">subscript</csymbol><ci id="S2.E1.p1.4.4.m4.1.1.3.3.2.cmml" xref="S2.E1.p1.4.4.m4.1.1.3.3.2">ℝ</ci><apply id="S2.E1.p1.4.4.m4.1.1.3.3.3.cmml" xref="S2.E1.p1.4.4.m4.1.1.3.3.3"><geq id="S2.E1.p1.4.4.m4.1.1.3.3.3.1.cmml" xref="S2.E1.p1.4.4.m4.1.1.3.3.3.1"></geq><csymbol cd="latexml" id="S2.E1.p1.4.4.m4.1.1.3.3.3.2.cmml" xref="S2.E1.p1.4.4.m4.1.1.3.3.3.2">absent</csymbol><cn id="S2.E1.p1.4.4.m4.1.1.3.3.3.3.cmml" type="integer" xref="S2.E1.p1.4.4.m4.1.1.3.3.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.p1.4.4.m4.1c">w:V\times V\to\mathbb{R}_{\geq 0}</annotation><annotation encoding="application/x-llamapun" id="S2.E1.p1.4.4.m4.1d">italic_w : italic_V × italic_V → blackboard_R start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT</annotation></semantics></math> is a <em class="ltx_emph ltx_font_upright" id="S2.E1.p1.6.6.3">weight function</em> satisfying <math alttext="w(v,v)=0" class="ltx_Math" display="inline" id="S2.E1.p1.5.5.m5.2"><semantics id="S2.E1.p1.5.5.m5.2a"><mrow id="S2.E1.p1.5.5.m5.2.3" xref="S2.E1.p1.5.5.m5.2.3.cmml"><mrow id="S2.E1.p1.5.5.m5.2.3.2" xref="S2.E1.p1.5.5.m5.2.3.2.cmml"><mi id="S2.E1.p1.5.5.m5.2.3.2.2" xref="S2.E1.p1.5.5.m5.2.3.2.2.cmml">w</mi><mo id="S2.E1.p1.5.5.m5.2.3.2.1" xref="S2.E1.p1.5.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S2.E1.p1.5.5.m5.2.3.2.3.2" xref="S2.E1.p1.5.5.m5.2.3.2.3.1.cmml"><mo id="S2.E1.p1.5.5.m5.2.3.2.3.2.1" stretchy="false" xref="S2.E1.p1.5.5.m5.2.3.2.3.1.cmml">(</mo><mi id="S2.E1.p1.5.5.m5.1.1" xref="S2.E1.p1.5.5.m5.1.1.cmml">v</mi><mo id="S2.E1.p1.5.5.m5.2.3.2.3.2.2" xref="S2.E1.p1.5.5.m5.2.3.2.3.1.cmml">,</mo><mi id="S2.E1.p1.5.5.m5.2.2" xref="S2.E1.p1.5.5.m5.2.2.cmml">v</mi><mo id="S2.E1.p1.5.5.m5.2.3.2.3.2.3" stretchy="false" xref="S2.E1.p1.5.5.m5.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.p1.5.5.m5.2.3.1" xref="S2.E1.p1.5.5.m5.2.3.1.cmml">=</mo><mn id="S2.E1.p1.5.5.m5.2.3.3" xref="S2.E1.p1.5.5.m5.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.E1.p1.5.5.m5.2b"><apply id="S2.E1.p1.5.5.m5.2.3.cmml" xref="S2.E1.p1.5.5.m5.2.3"><eq id="S2.E1.p1.5.5.m5.2.3.1.cmml" xref="S2.E1.p1.5.5.m5.2.3.1"></eq><apply id="S2.E1.p1.5.5.m5.2.3.2.cmml" xref="S2.E1.p1.5.5.m5.2.3.2"><times id="S2.E1.p1.5.5.m5.2.3.2.1.cmml" xref="S2.E1.p1.5.5.m5.2.3.2.1"></times><ci id="S2.E1.p1.5.5.m5.2.3.2.2.cmml" xref="S2.E1.p1.5.5.m5.2.3.2.2">𝑤</ci><interval closure="open" id="S2.E1.p1.5.5.m5.2.3.2.3.1.cmml" xref="S2.E1.p1.5.5.m5.2.3.2.3.2"><ci id="S2.E1.p1.5.5.m5.1.1.cmml" xref="S2.E1.p1.5.5.m5.1.1">𝑣</ci><ci id="S2.E1.p1.5.5.m5.2.2.cmml" xref="S2.E1.p1.5.5.m5.2.2">𝑣</ci></interval></apply><cn id="S2.E1.p1.5.5.m5.2.3.3.cmml" type="integer" xref="S2.E1.p1.5.5.m5.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.p1.5.5.m5.2c">w(v,v)=0</annotation><annotation encoding="application/x-llamapun" id="S2.E1.p1.5.5.m5.2d">italic_w ( italic_v , italic_v ) = 0</annotation></semantics></math> for all <math alttext="v\in V" class="ltx_Math" display="inline" id="S2.E1.p1.6.6.m6.1"><semantics id="S2.E1.p1.6.6.m6.1a"><mrow id="S2.E1.p1.6.6.m6.1.1" xref="S2.E1.p1.6.6.m6.1.1.cmml"><mi id="S2.E1.p1.6.6.m6.1.1.2" xref="S2.E1.p1.6.6.m6.1.1.2.cmml">v</mi><mo id="S2.E1.p1.6.6.m6.1.1.1" xref="S2.E1.p1.6.6.m6.1.1.1.cmml">∈</mo><mi id="S2.E1.p1.6.6.m6.1.1.3" xref="S2.E1.p1.6.6.m6.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.E1.p1.6.6.m6.1b"><apply id="S2.E1.p1.6.6.m6.1.1.cmml" xref="S2.E1.p1.6.6.m6.1.1"><in id="S2.E1.p1.6.6.m6.1.1.1.cmml" xref="S2.E1.p1.6.6.m6.1.1.1"></in><ci id="S2.E1.p1.6.6.m6.1.1.2.cmml" xref="S2.E1.p1.6.6.m6.1.1.2">𝑣</ci><ci id="S2.E1.p1.6.6.m6.1.1.3.cmml" xref="S2.E1.p1.6.6.m6.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.p1.6.6.m6.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S2.E1.p1.6.6.m6.1d">italic_v ∈ italic_V</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.8">Oftentimes, we are interested in weighted graphs with integer weights. In this setting, it is useful to think of (multi)graphs as pairs <math alttext="(V,E)" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.2"><semantics id="S2.SS1.p1.1.m1.2a"><mrow id="S2.SS1.p1.1.m1.2.3.2" xref="S2.SS1.p1.1.m1.2.3.1.cmml"><mo id="S2.SS1.p1.1.m1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.1.m1.2.3.1.cmml">(</mo><mi id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml">V</mi><mo id="S2.SS1.p1.1.m1.2.3.2.2" xref="S2.SS1.p1.1.m1.2.3.1.cmml">,</mo><mi id="S2.SS1.p1.1.m1.2.2" xref="S2.SS1.p1.1.m1.2.2.cmml">E</mi><mo id="S2.SS1.p1.1.m1.2.3.2.3" stretchy="false" xref="S2.SS1.p1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.2b"><interval closure="open" id="S2.SS1.p1.1.m1.2.3.1.cmml" xref="S2.SS1.p1.1.m1.2.3.2"><ci id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">𝑉</ci><ci id="S2.SS1.p1.1.m1.2.2.cmml" xref="S2.SS1.p1.1.m1.2.2">𝐸</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.2c">(V,E)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.2d">( italic_V , italic_E )</annotation></semantics></math>, where <math alttext="E\subseteq V\times V" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.1"><semantics id="S2.SS1.p1.2.m2.1a"><mrow id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml"><mi id="S2.SS1.p1.2.m2.1.1.2" xref="S2.SS1.p1.2.m2.1.1.2.cmml">E</mi><mo id="S2.SS1.p1.2.m2.1.1.1" xref="S2.SS1.p1.2.m2.1.1.1.cmml">⊆</mo><mrow id="S2.SS1.p1.2.m2.1.1.3" xref="S2.SS1.p1.2.m2.1.1.3.cmml"><mi id="S2.SS1.p1.2.m2.1.1.3.2" xref="S2.SS1.p1.2.m2.1.1.3.2.cmml">V</mi><mo id="S2.SS1.p1.2.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p1.2.m2.1.1.3.1.cmml">×</mo><mi id="S2.SS1.p1.2.m2.1.1.3.3" xref="S2.SS1.p1.2.m2.1.1.3.3.cmml">V</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m2.1b"><apply id="S2.SS1.p1.2.m2.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1"><subset id="S2.SS1.p1.2.m2.1.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1.1"></subset><ci id="S2.SS1.p1.2.m2.1.1.2.cmml" xref="S2.SS1.p1.2.m2.1.1.2">𝐸</ci><apply id="S2.SS1.p1.2.m2.1.1.3.cmml" xref="S2.SS1.p1.2.m2.1.1.3"><times id="S2.SS1.p1.2.m2.1.1.3.1.cmml" xref="S2.SS1.p1.2.m2.1.1.3.1"></times><ci id="S2.SS1.p1.2.m2.1.1.3.2.cmml" xref="S2.SS1.p1.2.m2.1.1.3.2">𝑉</ci><ci id="S2.SS1.p1.2.m2.1.1.3.3.cmml" xref="S2.SS1.p1.2.m2.1.1.3.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m2.1c">E\subseteq V\times V</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m2.1d">italic_E ⊆ italic_V × italic_V</annotation></semantics></math> is a multiset of <em class="ltx_emph ltx_font_italic" id="S2.SS1.p1.8.1">edges</em> (again, <math alttext="(v,v)\not\in V" class="ltx_Math" display="inline" id="S2.SS1.p1.3.m3.2"><semantics id="S2.SS1.p1.3.m3.2a"><mrow id="S2.SS1.p1.3.m3.2.3" xref="S2.SS1.p1.3.m3.2.3.cmml"><mrow id="S2.SS1.p1.3.m3.2.3.2.2" xref="S2.SS1.p1.3.m3.2.3.2.1.cmml"><mo id="S2.SS1.p1.3.m3.2.3.2.2.1" stretchy="false" xref="S2.SS1.p1.3.m3.2.3.2.1.cmml">(</mo><mi id="S2.SS1.p1.3.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml">v</mi><mo id="S2.SS1.p1.3.m3.2.3.2.2.2" xref="S2.SS1.p1.3.m3.2.3.2.1.cmml">,</mo><mi id="S2.SS1.p1.3.m3.2.2" xref="S2.SS1.p1.3.m3.2.2.cmml">v</mi><mo id="S2.SS1.p1.3.m3.2.3.2.2.3" stretchy="false" xref="S2.SS1.p1.3.m3.2.3.2.1.cmml">)</mo></mrow><mo id="S2.SS1.p1.3.m3.2.3.1" xref="S2.SS1.p1.3.m3.2.3.1.cmml">∉</mo><mi id="S2.SS1.p1.3.m3.2.3.3" xref="S2.SS1.p1.3.m3.2.3.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m3.2b"><apply id="S2.SS1.p1.3.m3.2.3.cmml" xref="S2.SS1.p1.3.m3.2.3"><notin id="S2.SS1.p1.3.m3.2.3.1.cmml" xref="S2.SS1.p1.3.m3.2.3.1"></notin><interval closure="open" id="S2.SS1.p1.3.m3.2.3.2.1.cmml" xref="S2.SS1.p1.3.m3.2.3.2.2"><ci id="S2.SS1.p1.3.m3.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1">𝑣</ci><ci id="S2.SS1.p1.3.m3.2.2.cmml" xref="S2.SS1.p1.3.m3.2.2">𝑣</ci></interval><ci id="S2.SS1.p1.3.m3.2.3.3.cmml" xref="S2.SS1.p1.3.m3.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.2c">(v,v)\not\in V</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m3.2d">( italic_v , italic_v ) ∉ italic_V</annotation></semantics></math> for all <math alttext="v\in V" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m4.1"><semantics id="S2.SS1.p1.4.m4.1a"><mrow id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml"><mi id="S2.SS1.p1.4.m4.1.1.2" xref="S2.SS1.p1.4.m4.1.1.2.cmml">v</mi><mo id="S2.SS1.p1.4.m4.1.1.1" xref="S2.SS1.p1.4.m4.1.1.1.cmml">∈</mo><mi id="S2.SS1.p1.4.m4.1.1.3" xref="S2.SS1.p1.4.m4.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m4.1b"><apply id="S2.SS1.p1.4.m4.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1"><in id="S2.SS1.p1.4.m4.1.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1.1"></in><ci id="S2.SS1.p1.4.m4.1.1.2.cmml" xref="S2.SS1.p1.4.m4.1.1.2">𝑣</ci><ci id="S2.SS1.p1.4.m4.1.1.3.cmml" xref="S2.SS1.p1.4.m4.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m4.1d">italic_v ∈ italic_V</annotation></semantics></math>); the set <math alttext="E" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m5.1"><semantics id="S2.SS1.p1.5.m5.1a"><mi id="S2.SS1.p1.5.m5.1.1" xref="S2.SS1.p1.5.m5.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m5.1b"><ci id="S2.SS1.p1.5.m5.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m5.1c">E</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.5.m5.1d">italic_E</annotation></semantics></math> induces the weighted graph where <math alttext="w(v_{1},v_{2})" class="ltx_Math" display="inline" id="S2.SS1.p1.6.m6.2"><semantics id="S2.SS1.p1.6.m6.2a"><mrow id="S2.SS1.p1.6.m6.2.2" xref="S2.SS1.p1.6.m6.2.2.cmml"><mi id="S2.SS1.p1.6.m6.2.2.4" xref="S2.SS1.p1.6.m6.2.2.4.cmml">w</mi><mo id="S2.SS1.p1.6.m6.2.2.3" xref="S2.SS1.p1.6.m6.2.2.3.cmml">⁢</mo><mrow id="S2.SS1.p1.6.m6.2.2.2.2" xref="S2.SS1.p1.6.m6.2.2.2.3.cmml"><mo id="S2.SS1.p1.6.m6.2.2.2.2.3" stretchy="false" xref="S2.SS1.p1.6.m6.2.2.2.3.cmml">(</mo><msub id="S2.SS1.p1.6.m6.1.1.1.1.1" xref="S2.SS1.p1.6.m6.1.1.1.1.1.cmml"><mi id="S2.SS1.p1.6.m6.1.1.1.1.1.2" xref="S2.SS1.p1.6.m6.1.1.1.1.1.2.cmml">v</mi><mn id="S2.SS1.p1.6.m6.1.1.1.1.1.3" xref="S2.SS1.p1.6.m6.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p1.6.m6.2.2.2.2.4" xref="S2.SS1.p1.6.m6.2.2.2.3.cmml">,</mo><msub id="S2.SS1.p1.6.m6.2.2.2.2.2" xref="S2.SS1.p1.6.m6.2.2.2.2.2.cmml"><mi id="S2.SS1.p1.6.m6.2.2.2.2.2.2" xref="S2.SS1.p1.6.m6.2.2.2.2.2.2.cmml">v</mi><mn id="S2.SS1.p1.6.m6.2.2.2.2.2.3" xref="S2.SS1.p1.6.m6.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.SS1.p1.6.m6.2.2.2.2.5" stretchy="false" xref="S2.SS1.p1.6.m6.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.6.m6.2b"><apply id="S2.SS1.p1.6.m6.2.2.cmml" xref="S2.SS1.p1.6.m6.2.2"><times id="S2.SS1.p1.6.m6.2.2.3.cmml" xref="S2.SS1.p1.6.m6.2.2.3"></times><ci id="S2.SS1.p1.6.m6.2.2.4.cmml" xref="S2.SS1.p1.6.m6.2.2.4">𝑤</ci><interval closure="open" id="S2.SS1.p1.6.m6.2.2.2.3.cmml" xref="S2.SS1.p1.6.m6.2.2.2.2"><apply id="S2.SS1.p1.6.m6.1.1.1.1.1.cmml" xref="S2.SS1.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.1.1.1.1.cmml" xref="S2.SS1.p1.6.m6.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p1.6.m6.1.1.1.1.1.2.cmml" xref="S2.SS1.p1.6.m6.1.1.1.1.1.2">𝑣</ci><cn id="S2.SS1.p1.6.m6.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.p1.6.m6.1.1.1.1.1.3">1</cn></apply><apply id="S2.SS1.p1.6.m6.2.2.2.2.2.cmml" xref="S2.SS1.p1.6.m6.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.2.2.2.2.2.1.cmml" xref="S2.SS1.p1.6.m6.2.2.2.2.2">subscript</csymbol><ci id="S2.SS1.p1.6.m6.2.2.2.2.2.2.cmml" xref="S2.SS1.p1.6.m6.2.2.2.2.2.2">𝑣</ci><cn id="S2.SS1.p1.6.m6.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS1.p1.6.m6.2.2.2.2.2.3">2</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.6.m6.2c">w(v_{1},v_{2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.6.m6.2d">italic_w ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> is the multiplicity of <math alttext="(v_{1},v_{2})" class="ltx_Math" display="inline" id="S2.SS1.p1.7.m7.2"><semantics id="S2.SS1.p1.7.m7.2a"><mrow id="S2.SS1.p1.7.m7.2.2.2" xref="S2.SS1.p1.7.m7.2.2.3.cmml"><mo id="S2.SS1.p1.7.m7.2.2.2.3" stretchy="false" xref="S2.SS1.p1.7.m7.2.2.3.cmml">(</mo><msub id="S2.SS1.p1.7.m7.1.1.1.1" xref="S2.SS1.p1.7.m7.1.1.1.1.cmml"><mi id="S2.SS1.p1.7.m7.1.1.1.1.2" xref="S2.SS1.p1.7.m7.1.1.1.1.2.cmml">v</mi><mn id="S2.SS1.p1.7.m7.1.1.1.1.3" xref="S2.SS1.p1.7.m7.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p1.7.m7.2.2.2.4" xref="S2.SS1.p1.7.m7.2.2.3.cmml">,</mo><msub id="S2.SS1.p1.7.m7.2.2.2.2" xref="S2.SS1.p1.7.m7.2.2.2.2.cmml"><mi id="S2.SS1.p1.7.m7.2.2.2.2.2" xref="S2.SS1.p1.7.m7.2.2.2.2.2.cmml">v</mi><mn id="S2.SS1.p1.7.m7.2.2.2.2.3" xref="S2.SS1.p1.7.m7.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.SS1.p1.7.m7.2.2.2.5" stretchy="false" xref="S2.SS1.p1.7.m7.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.7.m7.2b"><interval closure="open" id="S2.SS1.p1.7.m7.2.2.3.cmml" xref="S2.SS1.p1.7.m7.2.2.2"><apply id="S2.SS1.p1.7.m7.1.1.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.7.m7.1.1.1.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p1.7.m7.1.1.1.1.2.cmml" xref="S2.SS1.p1.7.m7.1.1.1.1.2">𝑣</ci><cn id="S2.SS1.p1.7.m7.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.p1.7.m7.1.1.1.1.3">1</cn></apply><apply id="S2.SS1.p1.7.m7.2.2.2.2.cmml" xref="S2.SS1.p1.7.m7.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.7.m7.2.2.2.2.1.cmml" xref="S2.SS1.p1.7.m7.2.2.2.2">subscript</csymbol><ci id="S2.SS1.p1.7.m7.2.2.2.2.2.cmml" xref="S2.SS1.p1.7.m7.2.2.2.2.2">𝑣</ci><cn id="S2.SS1.p1.7.m7.2.2.2.2.3.cmml" type="integer" xref="S2.SS1.p1.7.m7.2.2.2.2.3">2</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.7.m7.2c">(v_{1},v_{2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.7.m7.2d">( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> in <math alttext="E" class="ltx_Math" display="inline" id="S2.SS1.p1.8.m8.1"><semantics id="S2.SS1.p1.8.m8.1a"><mi id="S2.SS1.p1.8.m8.1.1" xref="S2.SS1.p1.8.m8.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.8.m8.1b"><ci id="S2.SS1.p1.8.m8.1.1.cmml" xref="S2.SS1.p1.8.m8.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.8.m8.1c">E</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.8.m8.1d">italic_E</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.E2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.E2.1.1.1">Definition 2.2</span></span><span class="ltx_text ltx_font_bold" id="S2.E2.2.2"> </span>(Degree)<span class="ltx_text ltx_font_bold" id="S2.E2.3.3">.</span> </h6> <div class="ltx_para" id="S2.E2.p1"> <p class="ltx_p" id="S2.E2.p1.3"><span class="ltx_text ltx_font_italic" id="S2.E2.p1.3.3">Let <math alttext="G=(V,w)" class="ltx_Math" display="inline" id="S2.E2.p1.1.1.m1.2"><semantics id="S2.E2.p1.1.1.m1.2a"><mrow id="S2.E2.p1.1.1.m1.2.3" xref="S2.E2.p1.1.1.m1.2.3.cmml"><mi id="S2.E2.p1.1.1.m1.2.3.2" xref="S2.E2.p1.1.1.m1.2.3.2.cmml">G</mi><mo id="S2.E2.p1.1.1.m1.2.3.1" xref="S2.E2.p1.1.1.m1.2.3.1.cmml">=</mo><mrow id="S2.E2.p1.1.1.m1.2.3.3.2" xref="S2.E2.p1.1.1.m1.2.3.3.1.cmml"><mo id="S2.E2.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.E2.p1.1.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.E2.p1.1.1.m1.1.1" xref="S2.E2.p1.1.1.m1.1.1.cmml">V</mi><mo id="S2.E2.p1.1.1.m1.2.3.3.2.2" xref="S2.E2.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.E2.p1.1.1.m1.2.2" xref="S2.E2.p1.1.1.m1.2.2.cmml">w</mi><mo id="S2.E2.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.E2.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E2.p1.1.1.m1.2b"><apply id="S2.E2.p1.1.1.m1.2.3.cmml" xref="S2.E2.p1.1.1.m1.2.3"><eq id="S2.E2.p1.1.1.m1.2.3.1.cmml" xref="S2.E2.p1.1.1.m1.2.3.1"></eq><ci id="S2.E2.p1.1.1.m1.2.3.2.cmml" xref="S2.E2.p1.1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S2.E2.p1.1.1.m1.2.3.3.1.cmml" xref="S2.E2.p1.1.1.m1.2.3.3.2"><ci id="S2.E2.p1.1.1.m1.1.1.cmml" xref="S2.E2.p1.1.1.m1.1.1">𝑉</ci><ci id="S2.E2.p1.1.1.m1.2.2.cmml" xref="S2.E2.p1.1.1.m1.2.2">𝑤</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.p1.1.1.m1.2c">G=(V,w)</annotation><annotation encoding="application/x-llamapun" id="S2.E2.p1.1.1.m1.2d">italic_G = ( italic_V , italic_w )</annotation></semantics></math> be a directed graph. For a vertex <math alttext="v\in V" class="ltx_Math" display="inline" id="S2.E2.p1.2.2.m2.1"><semantics id="S2.E2.p1.2.2.m2.1a"><mrow id="S2.E2.p1.2.2.m2.1.1" xref="S2.E2.p1.2.2.m2.1.1.cmml"><mi id="S2.E2.p1.2.2.m2.1.1.2" xref="S2.E2.p1.2.2.m2.1.1.2.cmml">v</mi><mo id="S2.E2.p1.2.2.m2.1.1.1" xref="S2.E2.p1.2.2.m2.1.1.1.cmml">∈</mo><mi id="S2.E2.p1.2.2.m2.1.1.3" xref="S2.E2.p1.2.2.m2.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.E2.p1.2.2.m2.1b"><apply id="S2.E2.p1.2.2.m2.1.1.cmml" xref="S2.E2.p1.2.2.m2.1.1"><in id="S2.E2.p1.2.2.m2.1.1.1.cmml" xref="S2.E2.p1.2.2.m2.1.1.1"></in><ci id="S2.E2.p1.2.2.m2.1.1.2.cmml" xref="S2.E2.p1.2.2.m2.1.1.2">𝑣</ci><ci id="S2.E2.p1.2.2.m2.1.1.3.cmml" xref="S2.E2.p1.2.2.m2.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.p1.2.2.m2.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S2.E2.p1.2.2.m2.1d">italic_v ∈ italic_V</annotation></semantics></math>, the <em class="ltx_emph ltx_font_upright" id="S2.E2.p1.3.3.1">out-, in-</em>, and <em class="ltx_emph ltx_font_upright" id="S2.E2.p1.3.3.2">total degrees</em> of <math alttext="v" class="ltx_Math" display="inline" id="S2.E2.p1.3.3.m3.1"><semantics id="S2.E2.p1.3.3.m3.1a"><mi id="S2.E2.p1.3.3.m3.1.1" xref="S2.E2.p1.3.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.E2.p1.3.3.m3.1b"><ci id="S2.E2.p1.3.3.m3.1.1.cmml" xref="S2.E2.p1.3.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.p1.3.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.E2.p1.3.3.m3.1d">italic_v</annotation></semantics></math> are</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.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="\mathsf{outdeg}_{G}(v)\stackrel{{\scriptstyle\mathrm{\small def}}}{{=}}\sum_{u% \in V}w(v,u),\mathsf{indeg}_{G}(v)\stackrel{{\scriptstyle\mathrm{\small def}}}% {{=}}\sum_{u\in V}w(u,v),\text{ and }\mathsf{deg}_{G}(v)\stackrel{{% \scriptstyle\mathrm{\small def}}}{{=}}\mathsf{outdeg}_{G}(v)+\mathsf{indeg}_{G% }(v)." class="ltx_Math" display="block" id="S2.Ex1.m1.10"><semantics id="S2.Ex1.m1.10a"><mrow id="S2.Ex1.m1.10.10.1"><mrow id="S2.Ex1.m1.10.10.1.1.2" xref="S2.Ex1.m1.10.10.1.1.3.cmml"><mrow id="S2.Ex1.m1.10.10.1.1.1.1" xref="S2.Ex1.m1.10.10.1.1.1.1.cmml"><mrow id="S2.Ex1.m1.10.10.1.1.1.1.2" xref="S2.Ex1.m1.10.10.1.1.1.1.2.cmml"><msub id="S2.Ex1.m1.10.10.1.1.1.1.2.2" xref="S2.Ex1.m1.10.10.1.1.1.1.2.2.cmml"><mi id="S2.Ex1.m1.10.10.1.1.1.1.2.2.2" xref="S2.Ex1.m1.10.10.1.1.1.1.2.2.2.cmml">𝗈𝗎𝗍𝖽𝖾𝗀</mi><mi id="S2.Ex1.m1.10.10.1.1.1.1.2.2.3" xref="S2.Ex1.m1.10.10.1.1.1.1.2.2.3.cmml">G</mi></msub><mo id="S2.Ex1.m1.10.10.1.1.1.1.2.1" xref="S2.Ex1.m1.10.10.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.10.10.1.1.1.1.2.3.2" xref="S2.Ex1.m1.10.10.1.1.1.1.2.cmml"><mo id="S2.Ex1.m1.10.10.1.1.1.1.2.3.2.1" stretchy="false" xref="S2.Ex1.m1.10.10.1.1.1.1.2.cmml">(</mo><mi id="S2.Ex1.m1.1.1" xref="S2.Ex1.m1.1.1.cmml">v</mi><mo id="S2.Ex1.m1.10.10.1.1.1.1.2.3.2.2" stretchy="false" xref="S2.Ex1.m1.10.10.1.1.1.1.2.cmml">)</mo></mrow></mrow><mover id="S2.Ex1.m1.10.10.1.1.1.1.1" xref="S2.Ex1.m1.10.10.1.1.1.1.1.cmml"><mo id="S2.Ex1.m1.10.10.1.1.1.1.1.2" rspace="0.111em" xref="S2.Ex1.m1.10.10.1.1.1.1.1.2.cmml">=</mo><mi id="S2.Ex1.m1.10.10.1.1.1.1.1.3" mathsize="128%" xref="S2.Ex1.m1.10.10.1.1.1.1.1.3.cmml">def</mi></mover><mrow id="S2.Ex1.m1.10.10.1.1.1.1.3" xref="S2.Ex1.m1.10.10.1.1.1.1.3.cmml"><munder id="S2.Ex1.m1.10.10.1.1.1.1.3.1" xref="S2.Ex1.m1.10.10.1.1.1.1.3.1.cmml"><mo id="S2.Ex1.m1.10.10.1.1.1.1.3.1.2" movablelimits="false" xref="S2.Ex1.m1.10.10.1.1.1.1.3.1.2.cmml">∑</mo><mrow id="S2.Ex1.m1.10.10.1.1.1.1.3.1.3" xref="S2.Ex1.m1.10.10.1.1.1.1.3.1.3.cmml"><mi id="S2.Ex1.m1.10.10.1.1.1.1.3.1.3.2" xref="S2.Ex1.m1.10.10.1.1.1.1.3.1.3.2.cmml">u</mi><mo id="S2.Ex1.m1.10.10.1.1.1.1.3.1.3.1" xref="S2.Ex1.m1.10.10.1.1.1.1.3.1.3.1.cmml">∈</mo><mi id="S2.Ex1.m1.10.10.1.1.1.1.3.1.3.3" xref="S2.Ex1.m1.10.10.1.1.1.1.3.1.3.3.cmml">V</mi></mrow></munder><mrow id="S2.Ex1.m1.10.10.1.1.1.1.3.2" xref="S2.Ex1.m1.10.10.1.1.1.1.3.2.cmml"><mi id="S2.Ex1.m1.10.10.1.1.1.1.3.2.2" xref="S2.Ex1.m1.10.10.1.1.1.1.3.2.2.cmml">w</mi><mo id="S2.Ex1.m1.10.10.1.1.1.1.3.2.1" xref="S2.Ex1.m1.10.10.1.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.10.10.1.1.1.1.3.2.3.2" xref="S2.Ex1.m1.10.10.1.1.1.1.3.2.3.1.cmml"><mo id="S2.Ex1.m1.10.10.1.1.1.1.3.2.3.2.1" stretchy="false" xref="S2.Ex1.m1.10.10.1.1.1.1.3.2.3.1.cmml">(</mo><mi id="S2.Ex1.m1.2.2" xref="S2.Ex1.m1.2.2.cmml">v</mi><mo id="S2.Ex1.m1.10.10.1.1.1.1.3.2.3.2.2" xref="S2.Ex1.m1.10.10.1.1.1.1.3.2.3.1.cmml">,</mo><mi id="S2.Ex1.m1.3.3" xref="S2.Ex1.m1.3.3.cmml">u</mi><mo id="S2.Ex1.m1.10.10.1.1.1.1.3.2.3.2.3" stretchy="false" xref="S2.Ex1.m1.10.10.1.1.1.1.3.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.Ex1.m1.10.10.1.1.2.3" xref="S2.Ex1.m1.10.10.1.1.3a.cmml">,</mo><mrow id="S2.Ex1.m1.10.10.1.1.2.2.2" xref="S2.Ex1.m1.10.10.1.1.2.2.3.cmml"><mrow id="S2.Ex1.m1.10.10.1.1.2.2.1.1" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.cmml"><mrow id="S2.Ex1.m1.10.10.1.1.2.2.1.1.2" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.2.cmml"><msub id="S2.Ex1.m1.10.10.1.1.2.2.1.1.2.2" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.2.2.cmml"><mi id="S2.Ex1.m1.10.10.1.1.2.2.1.1.2.2.2" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.2.2.2.cmml">𝗂𝗇𝖽𝖾𝗀</mi><mi id="S2.Ex1.m1.10.10.1.1.2.2.1.1.2.2.3" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.2.2.3.cmml">G</mi></msub><mo id="S2.Ex1.m1.10.10.1.1.2.2.1.1.2.1" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.10.10.1.1.2.2.1.1.2.3.2" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.2.cmml"><mo id="S2.Ex1.m1.10.10.1.1.2.2.1.1.2.3.2.1" stretchy="false" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.2.cmml">(</mo><mi id="S2.Ex1.m1.4.4" xref="S2.Ex1.m1.4.4.cmml">v</mi><mo id="S2.Ex1.m1.10.10.1.1.2.2.1.1.2.3.2.2" stretchy="false" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.2.cmml">)</mo></mrow></mrow><mover id="S2.Ex1.m1.10.10.1.1.2.2.1.1.1" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.1.cmml"><mo id="S2.Ex1.m1.10.10.1.1.2.2.1.1.1.2" rspace="0.111em" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.1.2.cmml">=</mo><mi id="S2.Ex1.m1.10.10.1.1.2.2.1.1.1.3" mathsize="128%" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.1.3.cmml">def</mi></mover><mrow id="S2.Ex1.m1.10.10.1.1.2.2.1.1.3" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.cmml"><munder id="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.1" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.1.cmml"><mo id="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.1.2" movablelimits="false" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.1.2.cmml">∑</mo><mrow id="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.1.3" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.1.3.cmml"><mi id="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.1.3.2" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.1.3.2.cmml">u</mi><mo id="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.1.3.1" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.1.3.1.cmml">∈</mo><mi id="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.1.3.3" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.1.3.3.cmml">V</mi></mrow></munder><mrow id="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.2" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.2.cmml"><mi id="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.2.2" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.2.2.cmml">w</mi><mo id="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.2.1" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.2.3.2" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.2.3.1.cmml"><mo id="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.2.3.2.1" stretchy="false" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.2.3.1.cmml">(</mo><mi id="S2.Ex1.m1.5.5" xref="S2.Ex1.m1.5.5.cmml">u</mi><mo id="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.2.3.2.2" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.2.3.1.cmml">,</mo><mi id="S2.Ex1.m1.6.6" xref="S2.Ex1.m1.6.6.cmml">v</mi><mo id="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.2.3.2.3" stretchy="false" xref="S2.Ex1.m1.10.10.1.1.2.2.1.1.3.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.Ex1.m1.10.10.1.1.2.2.2.3" xref="S2.Ex1.m1.10.10.1.1.2.2.3a.cmml">,</mo><mrow id="S2.Ex1.m1.10.10.1.1.2.2.2.2" xref="S2.Ex1.m1.10.10.1.1.2.2.2.2.cmml"><mrow id="S2.Ex1.m1.10.10.1.1.2.2.2.2.2" xref="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.2" xref="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.2a.cmml"> and </mtext><mo id="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.1" xref="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.1.cmml">⁢</mo><msub id="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.3" xref="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.3.cmml"><mi id="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.3.2" xref="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.3.2.cmml">𝖽𝖾𝗀</mi><mi id="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.3.3" xref="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.3.3.cmml">G</mi></msub><mo id="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.1a" xref="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.4.2" xref="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.cmml"><mo id="S2.Ex1.m1.10.10.1.1.2.2.2.2.2.4.2.1" stretchy="false" 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id="S2.Ex1.m1.10c">\mathsf{outdeg}_{G}(v)\stackrel{{\scriptstyle\mathrm{\small def}}}{{=}}\sum_{u% \in V}w(v,u),\mathsf{indeg}_{G}(v)\stackrel{{\scriptstyle\mathrm{\small def}}}% {{=}}\sum_{u\in V}w(u,v),\text{ and }\mathsf{deg}_{G}(v)\stackrel{{% \scriptstyle\mathrm{\small def}}}{{=}}\mathsf{outdeg}_{G}(v)+\mathsf{indeg}_{G% }(v).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.10d">sansserif_outdeg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_u ∈ italic_V end_POSTSUBSCRIPT italic_w ( italic_v , italic_u ) , sansserif_indeg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_u ∈ italic_V end_POSTSUBSCRIPT italic_w ( italic_u , italic_v ) , and sansserif_deg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP sansserif_outdeg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) + sansserif_indeg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.E2.p1.5"><span class="ltx_text ltx_font_italic" id="S2.E2.p1.5.2">A vertex <math alttext="v" class="ltx_Math" display="inline" id="S2.E2.p1.4.1.m1.1"><semantics id="S2.E2.p1.4.1.m1.1a"><mi id="S2.E2.p1.4.1.m1.1.1" xref="S2.E2.p1.4.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.E2.p1.4.1.m1.1b"><ci id="S2.E2.p1.4.1.m1.1.1.cmml" xref="S2.E2.p1.4.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.p1.4.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.E2.p1.4.1.m1.1d">italic_v</annotation></semantics></math> is <em class="ltx_emph ltx_font_upright" id="S2.E2.p1.5.2.1">isolated</em> if <math alttext="\mathsf{deg}_{G}(v)=0" class="ltx_Math" display="inline" id="S2.E2.p1.5.2.m2.1"><semantics id="S2.E2.p1.5.2.m2.1a"><mrow id="S2.E2.p1.5.2.m2.1.2" xref="S2.E2.p1.5.2.m2.1.2.cmml"><mrow id="S2.E2.p1.5.2.m2.1.2.2" xref="S2.E2.p1.5.2.m2.1.2.2.cmml"><msub id="S2.E2.p1.5.2.m2.1.2.2.2" xref="S2.E2.p1.5.2.m2.1.2.2.2.cmml"><mi id="S2.E2.p1.5.2.m2.1.2.2.2.2" xref="S2.E2.p1.5.2.m2.1.2.2.2.2.cmml">𝖽𝖾𝗀</mi><mi id="S2.E2.p1.5.2.m2.1.2.2.2.3" xref="S2.E2.p1.5.2.m2.1.2.2.2.3.cmml">G</mi></msub><mo id="S2.E2.p1.5.2.m2.1.2.2.1" xref="S2.E2.p1.5.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S2.E2.p1.5.2.m2.1.2.2.3.2" xref="S2.E2.p1.5.2.m2.1.2.2.cmml"><mo id="S2.E2.p1.5.2.m2.1.2.2.3.2.1" stretchy="false" xref="S2.E2.p1.5.2.m2.1.2.2.cmml">(</mo><mi id="S2.E2.p1.5.2.m2.1.1" xref="S2.E2.p1.5.2.m2.1.1.cmml">v</mi><mo id="S2.E2.p1.5.2.m2.1.2.2.3.2.2" stretchy="false" xref="S2.E2.p1.5.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.E2.p1.5.2.m2.1.2.1" xref="S2.E2.p1.5.2.m2.1.2.1.cmml">=</mo><mn id="S2.E2.p1.5.2.m2.1.2.3" xref="S2.E2.p1.5.2.m2.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.E2.p1.5.2.m2.1b"><apply id="S2.E2.p1.5.2.m2.1.2.cmml" xref="S2.E2.p1.5.2.m2.1.2"><eq id="S2.E2.p1.5.2.m2.1.2.1.cmml" xref="S2.E2.p1.5.2.m2.1.2.1"></eq><apply id="S2.E2.p1.5.2.m2.1.2.2.cmml" xref="S2.E2.p1.5.2.m2.1.2.2"><times id="S2.E2.p1.5.2.m2.1.2.2.1.cmml" xref="S2.E2.p1.5.2.m2.1.2.2.1"></times><apply id="S2.E2.p1.5.2.m2.1.2.2.2.cmml" xref="S2.E2.p1.5.2.m2.1.2.2.2"><csymbol cd="ambiguous" id="S2.E2.p1.5.2.m2.1.2.2.2.1.cmml" xref="S2.E2.p1.5.2.m2.1.2.2.2">subscript</csymbol><ci id="S2.E2.p1.5.2.m2.1.2.2.2.2.cmml" xref="S2.E2.p1.5.2.m2.1.2.2.2.2">𝖽𝖾𝗀</ci><ci id="S2.E2.p1.5.2.m2.1.2.2.2.3.cmml" xref="S2.E2.p1.5.2.m2.1.2.2.2.3">𝐺</ci></apply><ci id="S2.E2.p1.5.2.m2.1.1.cmml" xref="S2.E2.p1.5.2.m2.1.1">𝑣</ci></apply><cn id="S2.E2.p1.5.2.m2.1.2.3.cmml" type="integer" xref="S2.E2.p1.5.2.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.p1.5.2.m2.1c">\mathsf{deg}_{G}(v)=0</annotation><annotation encoding="application/x-llamapun" id="S2.E2.p1.5.2.m2.1d">sansserif_deg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) = 0</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.2">We also let <math alttext="m_{G}\stackrel{{\scriptstyle\mathrm{\small def}}}{{=}}\sum_{v_{1}\neq v_{2}\in V% }w(v_{1},v_{2})" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.2"><semantics id="S2.SS1.p2.1.m1.2a"><mrow id="S2.SS1.p2.1.m1.2.2" xref="S2.SS1.p2.1.m1.2.2.cmml"><msub id="S2.SS1.p2.1.m1.2.2.4" xref="S2.SS1.p2.1.m1.2.2.4.cmml"><mi id="S2.SS1.p2.1.m1.2.2.4.2" xref="S2.SS1.p2.1.m1.2.2.4.2.cmml">m</mi><mi id="S2.SS1.p2.1.m1.2.2.4.3" xref="S2.SS1.p2.1.m1.2.2.4.3.cmml">G</mi></msub><mover id="S2.SS1.p2.1.m1.2.2.3" 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cd="ambiguous" id="S2.SS1.p2.1.m1.2.2.2.2.2.2.2.1.cmml" xref="S2.SS1.p2.1.m1.2.2.2.2.2.2.2">subscript</csymbol><ci id="S2.SS1.p2.1.m1.2.2.2.2.2.2.2.2.cmml" xref="S2.SS1.p2.1.m1.2.2.2.2.2.2.2.2">𝑣</ci><cn id="S2.SS1.p2.1.m1.2.2.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS1.p2.1.m1.2.2.2.2.2.2.2.3">2</cn></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.2c">m_{G}\stackrel{{\scriptstyle\mathrm{\small def}}}{{=}}\sum_{v_{1}\neq v_{2}\in V% }w(v_{1},v_{2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.2d">italic_m start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP ∑ start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≠ italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_V end_POSTSUBSCRIPT italic_w ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> denote the total weight in <math alttext="G" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.1"><semantics id="S2.SS1.p2.2.m2.1a"><mi id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.1b"><ci id="S2.SS1.p2.2.m2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.1d">italic_G</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.E3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.E3.1.1.1">Definition 2.3</span></span><span class="ltx_text ltx_font_bold" id="S2.E3.2.2"> </span>(Bias)<span class="ltx_text ltx_font_bold" id="S2.E3.3.3">.</span> </h6> <div class="ltx_para" id="S2.E3.p1"> <p class="ltx_p" id="S2.E3.p1.3"><span class="ltx_text ltx_font_italic" id="S2.E3.p1.3.3">Let <math alttext="G=(V,w)" class="ltx_Math" display="inline" id="S2.E3.p1.1.1.m1.2"><semantics id="S2.E3.p1.1.1.m1.2a"><mrow id="S2.E3.p1.1.1.m1.2.3" xref="S2.E3.p1.1.1.m1.2.3.cmml"><mi id="S2.E3.p1.1.1.m1.2.3.2" xref="S2.E3.p1.1.1.m1.2.3.2.cmml">G</mi><mo id="S2.E3.p1.1.1.m1.2.3.1" xref="S2.E3.p1.1.1.m1.2.3.1.cmml">=</mo><mrow id="S2.E3.p1.1.1.m1.2.3.3.2" xref="S2.E3.p1.1.1.m1.2.3.3.1.cmml"><mo id="S2.E3.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.E3.p1.1.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.E3.p1.1.1.m1.1.1" xref="S2.E3.p1.1.1.m1.1.1.cmml">V</mi><mo id="S2.E3.p1.1.1.m1.2.3.3.2.2" xref="S2.E3.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.E3.p1.1.1.m1.2.2" xref="S2.E3.p1.1.1.m1.2.2.cmml">w</mi><mo id="S2.E3.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.E3.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E3.p1.1.1.m1.2b"><apply id="S2.E3.p1.1.1.m1.2.3.cmml" xref="S2.E3.p1.1.1.m1.2.3"><eq id="S2.E3.p1.1.1.m1.2.3.1.cmml" xref="S2.E3.p1.1.1.m1.2.3.1"></eq><ci id="S2.E3.p1.1.1.m1.2.3.2.cmml" xref="S2.E3.p1.1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S2.E3.p1.1.1.m1.2.3.3.1.cmml" xref="S2.E3.p1.1.1.m1.2.3.3.2"><ci id="S2.E3.p1.1.1.m1.1.1.cmml" xref="S2.E3.p1.1.1.m1.1.1">𝑉</ci><ci id="S2.E3.p1.1.1.m1.2.2.cmml" xref="S2.E3.p1.1.1.m1.2.2">𝑤</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.p1.1.1.m1.2c">G=(V,w)</annotation><annotation encoding="application/x-llamapun" id="S2.E3.p1.1.1.m1.2d">italic_G = ( italic_V , italic_w )</annotation></semantics></math> be a graph, and <math alttext="v\in V" class="ltx_Math" display="inline" id="S2.E3.p1.2.2.m2.1"><semantics id="S2.E3.p1.2.2.m2.1a"><mrow id="S2.E3.p1.2.2.m2.1.1" xref="S2.E3.p1.2.2.m2.1.1.cmml"><mi id="S2.E3.p1.2.2.m2.1.1.2" xref="S2.E3.p1.2.2.m2.1.1.2.cmml">v</mi><mo id="S2.E3.p1.2.2.m2.1.1.1" xref="S2.E3.p1.2.2.m2.1.1.1.cmml">∈</mo><mi id="S2.E3.p1.2.2.m2.1.1.3" xref="S2.E3.p1.2.2.m2.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.E3.p1.2.2.m2.1b"><apply id="S2.E3.p1.2.2.m2.1.1.cmml" xref="S2.E3.p1.2.2.m2.1.1"><in id="S2.E3.p1.2.2.m2.1.1.1.cmml" xref="S2.E3.p1.2.2.m2.1.1.1"></in><ci id="S2.E3.p1.2.2.m2.1.1.2.cmml" xref="S2.E3.p1.2.2.m2.1.1.2">𝑣</ci><ci id="S2.E3.p1.2.2.m2.1.1.3.cmml" xref="S2.E3.p1.2.2.m2.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.p1.2.2.m2.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S2.E3.p1.2.2.m2.1d">italic_v ∈ italic_V</annotation></semantics></math> a nonisolated vertex. Then the <em class="ltx_emph ltx_font_upright" id="S2.E3.p1.3.3.1">bias</em> of <math alttext="v" class="ltx_Math" display="inline" id="S2.E3.p1.3.3.m3.1"><semantics id="S2.E3.p1.3.3.m3.1a"><mi id="S2.E3.p1.3.3.m3.1.1" xref="S2.E3.p1.3.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.E3.p1.3.3.m3.1b"><ci id="S2.E3.p1.3.3.m3.1.1.cmml" xref="S2.E3.p1.3.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.p1.3.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.E3.p1.3.3.m3.1d">italic_v</annotation></semantics></math> is</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.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="\mathsf{bias}_{G}(v)\stackrel{{\scriptstyle\mathrm{\small def}}}{{=}}\frac{% \mathsf{outdeg}_{G}(v)-\mathsf{indeg}_{G}(v)}{\mathsf{deg}_{G}(v)}." class="ltx_Math" display="block" id="S2.Ex2.m1.5"><semantics id="S2.Ex2.m1.5a"><mrow id="S2.Ex2.m1.5.5.1" xref="S2.Ex2.m1.5.5.1.1.cmml"><mrow id="S2.Ex2.m1.5.5.1.1" xref="S2.Ex2.m1.5.5.1.1.cmml"><mrow id="S2.Ex2.m1.5.5.1.1.2" xref="S2.Ex2.m1.5.5.1.1.2.cmml"><msub id="S2.Ex2.m1.5.5.1.1.2.2" xref="S2.Ex2.m1.5.5.1.1.2.2.cmml"><mi id="S2.Ex2.m1.5.5.1.1.2.2.2" xref="S2.Ex2.m1.5.5.1.1.2.2.2.cmml">𝖻𝗂𝖺𝗌</mi><mi id="S2.Ex2.m1.5.5.1.1.2.2.3" xref="S2.Ex2.m1.5.5.1.1.2.2.3.cmml">G</mi></msub><mo id="S2.Ex2.m1.5.5.1.1.2.1" xref="S2.Ex2.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.5.5.1.1.2.3.2" xref="S2.Ex2.m1.5.5.1.1.2.cmml"><mo id="S2.Ex2.m1.5.5.1.1.2.3.2.1" stretchy="false" xref="S2.Ex2.m1.5.5.1.1.2.cmml">(</mo><mi id="S2.Ex2.m1.4.4" xref="S2.Ex2.m1.4.4.cmml">v</mi><mo id="S2.Ex2.m1.5.5.1.1.2.3.2.2" stretchy="false" xref="S2.Ex2.m1.5.5.1.1.2.cmml">)</mo></mrow></mrow><mover id="S2.Ex2.m1.5.5.1.1.1" xref="S2.Ex2.m1.5.5.1.1.1.cmml"><mo id="S2.Ex2.m1.5.5.1.1.1.2" xref="S2.Ex2.m1.5.5.1.1.1.2.cmml">=</mo><mi id="S2.Ex2.m1.5.5.1.1.1.3" mathsize="128%" xref="S2.Ex2.m1.5.5.1.1.1.3.cmml">def</mi></mover><mfrac id="S2.Ex2.m1.3.3" xref="S2.Ex2.m1.3.3.cmml"><mrow id="S2.Ex2.m1.2.2.2" xref="S2.Ex2.m1.2.2.2.cmml"><mrow id="S2.Ex2.m1.2.2.2.4" xref="S2.Ex2.m1.2.2.2.4.cmml"><msub id="S2.Ex2.m1.2.2.2.4.2" xref="S2.Ex2.m1.2.2.2.4.2.cmml"><mi id="S2.Ex2.m1.2.2.2.4.2.2" xref="S2.Ex2.m1.2.2.2.4.2.2.cmml">𝗈𝗎𝗍𝖽𝖾𝗀</mi><mi id="S2.Ex2.m1.2.2.2.4.2.3" xref="S2.Ex2.m1.2.2.2.4.2.3.cmml">G</mi></msub><mo id="S2.Ex2.m1.2.2.2.4.1" xref="S2.Ex2.m1.2.2.2.4.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.2.2.2.4.3.2" xref="S2.Ex2.m1.2.2.2.4.cmml"><mo id="S2.Ex2.m1.2.2.2.4.3.2.1" stretchy="false" xref="S2.Ex2.m1.2.2.2.4.cmml">(</mo><mi id="S2.Ex2.m1.1.1.1.1" xref="S2.Ex2.m1.1.1.1.1.cmml">v</mi><mo id="S2.Ex2.m1.2.2.2.4.3.2.2" stretchy="false" xref="S2.Ex2.m1.2.2.2.4.cmml">)</mo></mrow></mrow><mo id="S2.Ex2.m1.2.2.2.3" xref="S2.Ex2.m1.2.2.2.3.cmml">−</mo><mrow id="S2.Ex2.m1.2.2.2.5" xref="S2.Ex2.m1.2.2.2.5.cmml"><msub id="S2.Ex2.m1.2.2.2.5.2" xref="S2.Ex2.m1.2.2.2.5.2.cmml"><mi id="S2.Ex2.m1.2.2.2.5.2.2" xref="S2.Ex2.m1.2.2.2.5.2.2.cmml">𝗂𝗇𝖽𝖾𝗀</mi><mi id="S2.Ex2.m1.2.2.2.5.2.3" xref="S2.Ex2.m1.2.2.2.5.2.3.cmml">G</mi></msub><mo id="S2.Ex2.m1.2.2.2.5.1" xref="S2.Ex2.m1.2.2.2.5.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.2.2.2.5.3.2" xref="S2.Ex2.m1.2.2.2.5.cmml"><mo id="S2.Ex2.m1.2.2.2.5.3.2.1" stretchy="false" xref="S2.Ex2.m1.2.2.2.5.cmml">(</mo><mi id="S2.Ex2.m1.2.2.2.2" xref="S2.Ex2.m1.2.2.2.2.cmml">v</mi><mo id="S2.Ex2.m1.2.2.2.5.3.2.2" stretchy="false" xref="S2.Ex2.m1.2.2.2.5.cmml">)</mo></mrow></mrow></mrow><mrow id="S2.Ex2.m1.3.3.3" xref="S2.Ex2.m1.3.3.3.cmml"><msub id="S2.Ex2.m1.3.3.3.3" xref="S2.Ex2.m1.3.3.3.3.cmml"><mi id="S2.Ex2.m1.3.3.3.3.2" xref="S2.Ex2.m1.3.3.3.3.2.cmml">𝖽𝖾𝗀</mi><mi id="S2.Ex2.m1.3.3.3.3.3" xref="S2.Ex2.m1.3.3.3.3.3.cmml">G</mi></msub><mo id="S2.Ex2.m1.3.3.3.2" xref="S2.Ex2.m1.3.3.3.2.cmml">⁢</mo><mrow id="S2.Ex2.m1.3.3.3.4.2" xref="S2.Ex2.m1.3.3.3.cmml"><mo id="S2.Ex2.m1.3.3.3.4.2.1" stretchy="false" xref="S2.Ex2.m1.3.3.3.cmml">(</mo><mi id="S2.Ex2.m1.3.3.3.1" xref="S2.Ex2.m1.3.3.3.1.cmml">v</mi><mo id="S2.Ex2.m1.3.3.3.4.2.2" stretchy="false" xref="S2.Ex2.m1.3.3.3.cmml">)</mo></mrow></mrow></mfrac></mrow><mo id="S2.Ex2.m1.5.5.1.2" lspace="0em" xref="S2.Ex2.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex2.m1.5b"><apply id="S2.Ex2.m1.5.5.1.1.cmml" xref="S2.Ex2.m1.5.5.1"><apply id="S2.Ex2.m1.5.5.1.1.1.cmml" xref="S2.Ex2.m1.5.5.1.1.1"><csymbol cd="ambiguous" id="S2.Ex2.m1.5.5.1.1.1.1.cmml" xref="S2.Ex2.m1.5.5.1.1.1">superscript</csymbol><eq id="S2.Ex2.m1.5.5.1.1.1.2.cmml" xref="S2.Ex2.m1.5.5.1.1.1.2"></eq><ci id="S2.Ex2.m1.5.5.1.1.1.3.cmml" xref="S2.Ex2.m1.5.5.1.1.1.3">def</ci></apply><apply id="S2.Ex2.m1.5.5.1.1.2.cmml" xref="S2.Ex2.m1.5.5.1.1.2"><times id="S2.Ex2.m1.5.5.1.1.2.1.cmml" xref="S2.Ex2.m1.5.5.1.1.2.1"></times><apply id="S2.Ex2.m1.5.5.1.1.2.2.cmml" xref="S2.Ex2.m1.5.5.1.1.2.2"><csymbol cd="ambiguous" id="S2.Ex2.m1.5.5.1.1.2.2.1.cmml" xref="S2.Ex2.m1.5.5.1.1.2.2">subscript</csymbol><ci id="S2.Ex2.m1.5.5.1.1.2.2.2.cmml" xref="S2.Ex2.m1.5.5.1.1.2.2.2">𝖻𝗂𝖺𝗌</ci><ci id="S2.Ex2.m1.5.5.1.1.2.2.3.cmml" xref="S2.Ex2.m1.5.5.1.1.2.2.3">𝐺</ci></apply><ci id="S2.Ex2.m1.4.4.cmml" xref="S2.Ex2.m1.4.4">𝑣</ci></apply><apply id="S2.Ex2.m1.3.3.cmml" xref="S2.Ex2.m1.3.3"><divide id="S2.Ex2.m1.3.3.4.cmml" xref="S2.Ex2.m1.3.3"></divide><apply id="S2.Ex2.m1.2.2.2.cmml" xref="S2.Ex2.m1.2.2.2"><minus id="S2.Ex2.m1.2.2.2.3.cmml" xref="S2.Ex2.m1.2.2.2.3"></minus><apply id="S2.Ex2.m1.2.2.2.4.cmml" xref="S2.Ex2.m1.2.2.2.4"><times id="S2.Ex2.m1.2.2.2.4.1.cmml" xref="S2.Ex2.m1.2.2.2.4.1"></times><apply id="S2.Ex2.m1.2.2.2.4.2.cmml" xref="S2.Ex2.m1.2.2.2.4.2"><csymbol cd="ambiguous" id="S2.Ex2.m1.2.2.2.4.2.1.cmml" xref="S2.Ex2.m1.2.2.2.4.2">subscript</csymbol><ci id="S2.Ex2.m1.2.2.2.4.2.2.cmml" xref="S2.Ex2.m1.2.2.2.4.2.2">𝗈𝗎𝗍𝖽𝖾𝗀</ci><ci id="S2.Ex2.m1.2.2.2.4.2.3.cmml" xref="S2.Ex2.m1.2.2.2.4.2.3">𝐺</ci></apply><ci id="S2.Ex2.m1.1.1.1.1.cmml" xref="S2.Ex2.m1.1.1.1.1">𝑣</ci></apply><apply id="S2.Ex2.m1.2.2.2.5.cmml" xref="S2.Ex2.m1.2.2.2.5"><times id="S2.Ex2.m1.2.2.2.5.1.cmml" xref="S2.Ex2.m1.2.2.2.5.1"></times><apply id="S2.Ex2.m1.2.2.2.5.2.cmml" xref="S2.Ex2.m1.2.2.2.5.2"><csymbol cd="ambiguous" id="S2.Ex2.m1.2.2.2.5.2.1.cmml" xref="S2.Ex2.m1.2.2.2.5.2">subscript</csymbol><ci id="S2.Ex2.m1.2.2.2.5.2.2.cmml" xref="S2.Ex2.m1.2.2.2.5.2.2">𝗂𝗇𝖽𝖾𝗀</ci><ci id="S2.Ex2.m1.2.2.2.5.2.3.cmml" xref="S2.Ex2.m1.2.2.2.5.2.3">𝐺</ci></apply><ci id="S2.Ex2.m1.2.2.2.2.cmml" xref="S2.Ex2.m1.2.2.2.2">𝑣</ci></apply></apply><apply id="S2.Ex2.m1.3.3.3.cmml" xref="S2.Ex2.m1.3.3.3"><times id="S2.Ex2.m1.3.3.3.2.cmml" xref="S2.Ex2.m1.3.3.3.2"></times><apply id="S2.Ex2.m1.3.3.3.3.cmml" xref="S2.Ex2.m1.3.3.3.3"><csymbol cd="ambiguous" id="S2.Ex2.m1.3.3.3.3.1.cmml" xref="S2.Ex2.m1.3.3.3.3">subscript</csymbol><ci id="S2.Ex2.m1.3.3.3.3.2.cmml" xref="S2.Ex2.m1.3.3.3.3.2">𝖽𝖾𝗀</ci><ci id="S2.Ex2.m1.3.3.3.3.3.cmml" xref="S2.Ex2.m1.3.3.3.3.3">𝐺</ci></apply><ci id="S2.Ex2.m1.3.3.3.1.cmml" xref="S2.Ex2.m1.3.3.3.1">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m1.5c">\mathsf{bias}_{G}(v)\stackrel{{\scriptstyle\mathrm{\small def}}}{{=}}\frac{% \mathsf{outdeg}_{G}(v)-\mathsf{indeg}_{G}(v)}{\mathsf{deg}_{G}(v)}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m1.5d">sansserif_bias start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP divide start_ARG sansserif_outdeg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) - sansserif_indeg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) end_ARG start_ARG sansserif_deg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.1">Observe that <math alttext="-1\leq\mathsf{bias}_{G}(v)\leq+1" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><mrow id="S2.SS1.p3.1.m1.1.2" xref="S2.SS1.p3.1.m1.1.2.cmml"><mrow id="S2.SS1.p3.1.m1.1.2.2" xref="S2.SS1.p3.1.m1.1.2.2.cmml"><mo id="S2.SS1.p3.1.m1.1.2.2a" xref="S2.SS1.p3.1.m1.1.2.2.cmml">−</mo><mn id="S2.SS1.p3.1.m1.1.2.2.2" xref="S2.SS1.p3.1.m1.1.2.2.2.cmml">1</mn></mrow><mo id="S2.SS1.p3.1.m1.1.2.3" xref="S2.SS1.p3.1.m1.1.2.3.cmml">≤</mo><mrow id="S2.SS1.p3.1.m1.1.2.4" xref="S2.SS1.p3.1.m1.1.2.4.cmml"><msub id="S2.SS1.p3.1.m1.1.2.4.2" xref="S2.SS1.p3.1.m1.1.2.4.2.cmml"><mi id="S2.SS1.p3.1.m1.1.2.4.2.2" xref="S2.SS1.p3.1.m1.1.2.4.2.2.cmml">𝖻𝗂𝖺𝗌</mi><mi id="S2.SS1.p3.1.m1.1.2.4.2.3" xref="S2.SS1.p3.1.m1.1.2.4.2.3.cmml">G</mi></msub><mo id="S2.SS1.p3.1.m1.1.2.4.1" xref="S2.SS1.p3.1.m1.1.2.4.1.cmml">⁢</mo><mrow id="S2.SS1.p3.1.m1.1.2.4.3.2" xref="S2.SS1.p3.1.m1.1.2.4.cmml"><mo id="S2.SS1.p3.1.m1.1.2.4.3.2.1" stretchy="false" xref="S2.SS1.p3.1.m1.1.2.4.cmml">(</mo><mi id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml">v</mi><mo id="S2.SS1.p3.1.m1.1.2.4.3.2.2" stretchy="false" xref="S2.SS1.p3.1.m1.1.2.4.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p3.1.m1.1.2.5" xref="S2.SS1.p3.1.m1.1.2.5.cmml">≤</mo><mrow id="S2.SS1.p3.1.m1.1.2.6" xref="S2.SS1.p3.1.m1.1.2.6.cmml"><mo id="S2.SS1.p3.1.m1.1.2.6a" xref="S2.SS1.p3.1.m1.1.2.6.cmml">+</mo><mn id="S2.SS1.p3.1.m1.1.2.6.2" xref="S2.SS1.p3.1.m1.1.2.6.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><apply id="S2.SS1.p3.1.m1.1.2.cmml" xref="S2.SS1.p3.1.m1.1.2"><and id="S2.SS1.p3.1.m1.1.2a.cmml" xref="S2.SS1.p3.1.m1.1.2"></and><apply id="S2.SS1.p3.1.m1.1.2b.cmml" xref="S2.SS1.p3.1.m1.1.2"><leq id="S2.SS1.p3.1.m1.1.2.3.cmml" xref="S2.SS1.p3.1.m1.1.2.3"></leq><apply id="S2.SS1.p3.1.m1.1.2.2.cmml" xref="S2.SS1.p3.1.m1.1.2.2"><minus id="S2.SS1.p3.1.m1.1.2.2.1.cmml" xref="S2.SS1.p3.1.m1.1.2.2"></minus><cn id="S2.SS1.p3.1.m1.1.2.2.2.cmml" type="integer" xref="S2.SS1.p3.1.m1.1.2.2.2">1</cn></apply><apply id="S2.SS1.p3.1.m1.1.2.4.cmml" xref="S2.SS1.p3.1.m1.1.2.4"><times id="S2.SS1.p3.1.m1.1.2.4.1.cmml" xref="S2.SS1.p3.1.m1.1.2.4.1"></times><apply id="S2.SS1.p3.1.m1.1.2.4.2.cmml" xref="S2.SS1.p3.1.m1.1.2.4.2"><csymbol cd="ambiguous" id="S2.SS1.p3.1.m1.1.2.4.2.1.cmml" xref="S2.SS1.p3.1.m1.1.2.4.2">subscript</csymbol><ci id="S2.SS1.p3.1.m1.1.2.4.2.2.cmml" xref="S2.SS1.p3.1.m1.1.2.4.2.2">𝖻𝗂𝖺𝗌</ci><ci id="S2.SS1.p3.1.m1.1.2.4.2.3.cmml" xref="S2.SS1.p3.1.m1.1.2.4.2.3">𝐺</ci></apply><ci id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1">𝑣</ci></apply></apply><apply id="S2.SS1.p3.1.m1.1.2c.cmml" xref="S2.SS1.p3.1.m1.1.2"><leq id="S2.SS1.p3.1.m1.1.2.5.cmml" xref="S2.SS1.p3.1.m1.1.2.5"></leq><share href="https://arxiv.org/html/2411.12976v1#S2.SS1.p3.1.m1.1.2.4.cmml" id="S2.SS1.p3.1.m1.1.2d.cmml" xref="S2.SS1.p3.1.m1.1.2"></share><apply id="S2.SS1.p3.1.m1.1.2.6.cmml" xref="S2.SS1.p3.1.m1.1.2.6"><plus id="S2.SS1.p3.1.m1.1.2.6.1.cmml" xref="S2.SS1.p3.1.m1.1.2.6"></plus><cn id="S2.SS1.p3.1.m1.1.2.6.2.cmml" type="integer" xref="S2.SS1.p3.1.m1.1.2.6.2">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">-1\leq\mathsf{bias}_{G}(v)\leq+1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">- 1 ≤ sansserif_bias start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) ≤ + 1</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2 </span><span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S2.SS2.2.1">Max-DiCut</span> and oblivious algorithms</h3> <div class="ltx_theorem ltx_theorem_definition" id="S2.E4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.E4.2.1.1">Definition 2.4</span></span><span class="ltx_text ltx_font_bold" id="S2.E4.3.2"> </span>(<span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S2.E4.4.3">Max-DiCut</span>)<span class="ltx_text ltx_font_bold" id="S2.E4.5.4">.</span> </h6> <div class="ltx_para" id="S2.E4.p1"> <p class="ltx_p" id="S2.E4.p1.5"><span class="ltx_text ltx_font_italic" id="S2.E4.p1.5.5">The <em class="ltx_emph ltx_font_upright" id="S2.E4.p1.5.5.2">maximum directed cut</em> (<span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S2.E4.p1.5.5.3">Max-DiCut</span>) problem is defined as follows: The input is a directed graph <math alttext="G=(V,w)" class="ltx_Math" display="inline" id="S2.E4.p1.2.2.m2.2"><semantics id="S2.E4.p1.2.2.m2.2a"><mrow id="S2.E4.p1.2.2.m2.2.3" xref="S2.E4.p1.2.2.m2.2.3.cmml"><mi id="S2.E4.p1.2.2.m2.2.3.2" xref="S2.E4.p1.2.2.m2.2.3.2.cmml">G</mi><mo id="S2.E4.p1.2.2.m2.2.3.1" xref="S2.E4.p1.2.2.m2.2.3.1.cmml">=</mo><mrow id="S2.E4.p1.2.2.m2.2.3.3.2" xref="S2.E4.p1.2.2.m2.2.3.3.1.cmml"><mo id="S2.E4.p1.2.2.m2.2.3.3.2.1" stretchy="false" xref="S2.E4.p1.2.2.m2.2.3.3.1.cmml">(</mo><mi id="S2.E4.p1.2.2.m2.1.1" xref="S2.E4.p1.2.2.m2.1.1.cmml">V</mi><mo id="S2.E4.p1.2.2.m2.2.3.3.2.2" xref="S2.E4.p1.2.2.m2.2.3.3.1.cmml">,</mo><mi id="S2.E4.p1.2.2.m2.2.2" xref="S2.E4.p1.2.2.m2.2.2.cmml">w</mi><mo id="S2.E4.p1.2.2.m2.2.3.3.2.3" stretchy="false" xref="S2.E4.p1.2.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E4.p1.2.2.m2.2b"><apply id="S2.E4.p1.2.2.m2.2.3.cmml" xref="S2.E4.p1.2.2.m2.2.3"><eq id="S2.E4.p1.2.2.m2.2.3.1.cmml" xref="S2.E4.p1.2.2.m2.2.3.1"></eq><ci id="S2.E4.p1.2.2.m2.2.3.2.cmml" xref="S2.E4.p1.2.2.m2.2.3.2">𝐺</ci><interval closure="open" id="S2.E4.p1.2.2.m2.2.3.3.1.cmml" xref="S2.E4.p1.2.2.m2.2.3.3.2"><ci id="S2.E4.p1.2.2.m2.1.1.cmml" xref="S2.E4.p1.2.2.m2.1.1">𝑉</ci><ci id="S2.E4.p1.2.2.m2.2.2.cmml" xref="S2.E4.p1.2.2.m2.2.2">𝑤</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.p1.2.2.m2.2c">G=(V,w)</annotation><annotation encoding="application/x-llamapun" id="S2.E4.p1.2.2.m2.2d">italic_G = ( italic_V , italic_w )</annotation></semantics></math>. For any <em class="ltx_emph ltx_font_upright" id="S2.E4.p1.5.5.4">assignment</em> (a.k.a. <em class="ltx_emph ltx_font_upright" id="S2.E4.p1.5.5.5">cut</em>) <math alttext="\boldsymbol{x}=(x_{v})_{v\in V}\in\{0,1\}^{V}" class="ltx_Math" display="inline" id="S2.E4.p1.3.3.m3.3"><semantics id="S2.E4.p1.3.3.m3.3a"><mrow id="S2.E4.p1.3.3.m3.3.3" xref="S2.E4.p1.3.3.m3.3.3.cmml"><mi id="S2.E4.p1.3.3.m3.3.3.3" xref="S2.E4.p1.3.3.m3.3.3.3.cmml">𝐱</mi><mo id="S2.E4.p1.3.3.m3.3.3.4" xref="S2.E4.p1.3.3.m3.3.3.4.cmml">=</mo><msub id="S2.E4.p1.3.3.m3.3.3.1" xref="S2.E4.p1.3.3.m3.3.3.1.cmml"><mrow id="S2.E4.p1.3.3.m3.3.3.1.1.1" xref="S2.E4.p1.3.3.m3.3.3.1.1.1.1.cmml"><mo id="S2.E4.p1.3.3.m3.3.3.1.1.1.2" stretchy="false" xref="S2.E4.p1.3.3.m3.3.3.1.1.1.1.cmml">(</mo><msub id="S2.E4.p1.3.3.m3.3.3.1.1.1.1" xref="S2.E4.p1.3.3.m3.3.3.1.1.1.1.cmml"><mi id="S2.E4.p1.3.3.m3.3.3.1.1.1.1.2" xref="S2.E4.p1.3.3.m3.3.3.1.1.1.1.2.cmml">x</mi><mi id="S2.E4.p1.3.3.m3.3.3.1.1.1.1.3" xref="S2.E4.p1.3.3.m3.3.3.1.1.1.1.3.cmml">v</mi></msub><mo id="S2.E4.p1.3.3.m3.3.3.1.1.1.3" stretchy="false" xref="S2.E4.p1.3.3.m3.3.3.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.E4.p1.3.3.m3.3.3.1.3" xref="S2.E4.p1.3.3.m3.3.3.1.3.cmml"><mi id="S2.E4.p1.3.3.m3.3.3.1.3.2" xref="S2.E4.p1.3.3.m3.3.3.1.3.2.cmml">v</mi><mo id="S2.E4.p1.3.3.m3.3.3.1.3.1" xref="S2.E4.p1.3.3.m3.3.3.1.3.1.cmml">∈</mo><mi id="S2.E4.p1.3.3.m3.3.3.1.3.3" xref="S2.E4.p1.3.3.m3.3.3.1.3.3.cmml">V</mi></mrow></msub><mo id="S2.E4.p1.3.3.m3.3.3.5" xref="S2.E4.p1.3.3.m3.3.3.5.cmml">∈</mo><msup id="S2.E4.p1.3.3.m3.3.3.6" xref="S2.E4.p1.3.3.m3.3.3.6.cmml"><mrow id="S2.E4.p1.3.3.m3.3.3.6.2.2" xref="S2.E4.p1.3.3.m3.3.3.6.2.1.cmml"><mo id="S2.E4.p1.3.3.m3.3.3.6.2.2.1" stretchy="false" xref="S2.E4.p1.3.3.m3.3.3.6.2.1.cmml">{</mo><mn id="S2.E4.p1.3.3.m3.1.1" xref="S2.E4.p1.3.3.m3.1.1.cmml">0</mn><mo id="S2.E4.p1.3.3.m3.3.3.6.2.2.2" xref="S2.E4.p1.3.3.m3.3.3.6.2.1.cmml">,</mo><mn id="S2.E4.p1.3.3.m3.2.2" xref="S2.E4.p1.3.3.m3.2.2.cmml">1</mn><mo id="S2.E4.p1.3.3.m3.3.3.6.2.2.3" stretchy="false" xref="S2.E4.p1.3.3.m3.3.3.6.2.1.cmml">}</mo></mrow><mi id="S2.E4.p1.3.3.m3.3.3.6.3" xref="S2.E4.p1.3.3.m3.3.3.6.3.cmml">V</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.E4.p1.3.3.m3.3b"><apply id="S2.E4.p1.3.3.m3.3.3.cmml" xref="S2.E4.p1.3.3.m3.3.3"><and id="S2.E4.p1.3.3.m3.3.3a.cmml" xref="S2.E4.p1.3.3.m3.3.3"></and><apply id="S2.E4.p1.3.3.m3.3.3b.cmml" xref="S2.E4.p1.3.3.m3.3.3"><eq id="S2.E4.p1.3.3.m3.3.3.4.cmml" xref="S2.E4.p1.3.3.m3.3.3.4"></eq><ci id="S2.E4.p1.3.3.m3.3.3.3.cmml" xref="S2.E4.p1.3.3.m3.3.3.3">𝐱</ci><apply id="S2.E4.p1.3.3.m3.3.3.1.cmml" xref="S2.E4.p1.3.3.m3.3.3.1"><csymbol cd="ambiguous" id="S2.E4.p1.3.3.m3.3.3.1.2.cmml" xref="S2.E4.p1.3.3.m3.3.3.1">subscript</csymbol><apply id="S2.E4.p1.3.3.m3.3.3.1.1.1.1.cmml" xref="S2.E4.p1.3.3.m3.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.E4.p1.3.3.m3.3.3.1.1.1.1.1.cmml" xref="S2.E4.p1.3.3.m3.3.3.1.1.1">subscript</csymbol><ci id="S2.E4.p1.3.3.m3.3.3.1.1.1.1.2.cmml" xref="S2.E4.p1.3.3.m3.3.3.1.1.1.1.2">𝑥</ci><ci id="S2.E4.p1.3.3.m3.3.3.1.1.1.1.3.cmml" xref="S2.E4.p1.3.3.m3.3.3.1.1.1.1.3">𝑣</ci></apply><apply id="S2.E4.p1.3.3.m3.3.3.1.3.cmml" xref="S2.E4.p1.3.3.m3.3.3.1.3"><in id="S2.E4.p1.3.3.m3.3.3.1.3.1.cmml" xref="S2.E4.p1.3.3.m3.3.3.1.3.1"></in><ci id="S2.E4.p1.3.3.m3.3.3.1.3.2.cmml" xref="S2.E4.p1.3.3.m3.3.3.1.3.2">𝑣</ci><ci id="S2.E4.p1.3.3.m3.3.3.1.3.3.cmml" xref="S2.E4.p1.3.3.m3.3.3.1.3.3">𝑉</ci></apply></apply></apply><apply id="S2.E4.p1.3.3.m3.3.3c.cmml" xref="S2.E4.p1.3.3.m3.3.3"><in id="S2.E4.p1.3.3.m3.3.3.5.cmml" xref="S2.E4.p1.3.3.m3.3.3.5"></in><share href="https://arxiv.org/html/2411.12976v1#S2.E4.p1.3.3.m3.3.3.1.cmml" id="S2.E4.p1.3.3.m3.3.3d.cmml" xref="S2.E4.p1.3.3.m3.3.3"></share><apply id="S2.E4.p1.3.3.m3.3.3.6.cmml" xref="S2.E4.p1.3.3.m3.3.3.6"><csymbol cd="ambiguous" id="S2.E4.p1.3.3.m3.3.3.6.1.cmml" xref="S2.E4.p1.3.3.m3.3.3.6">superscript</csymbol><set id="S2.E4.p1.3.3.m3.3.3.6.2.1.cmml" xref="S2.E4.p1.3.3.m3.3.3.6.2.2"><cn id="S2.E4.p1.3.3.m3.1.1.cmml" type="integer" xref="S2.E4.p1.3.3.m3.1.1">0</cn><cn id="S2.E4.p1.3.3.m3.2.2.cmml" type="integer" xref="S2.E4.p1.3.3.m3.2.2">1</cn></set><ci id="S2.E4.p1.3.3.m3.3.3.6.3.cmml" xref="S2.E4.p1.3.3.m3.3.3.6.3">𝑉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.p1.3.3.m3.3c">\boldsymbol{x}=(x_{v})_{v\in V}\in\{0,1\}^{V}</annotation><annotation encoding="application/x-llamapun" id="S2.E4.p1.3.3.m3.3d">bold_italic_x = ( italic_x start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_V end_POSTSUPERSCRIPT</annotation></semantics></math>, the <em class="ltx_emph ltx_font_upright" id="S2.E4.p1.4.4.1"><span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S2.E4.p1.4.4.1.1">Max-DiCut</span> value</em> of <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S2.E4.p1.5.5.m4.1"><semantics id="S2.E4.p1.5.5.m4.1a"><mi id="S2.E4.p1.5.5.m4.1.1" xref="S2.E4.p1.5.5.m4.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S2.E4.p1.5.5.m4.1b"><ci id="S2.E4.p1.5.5.m4.1.1.cmml" xref="S2.E4.p1.5.5.m4.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.p1.5.5.m4.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S2.E4.p1.5.5.m4.1d">bold_italic_x</annotation></semantics></math> is</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.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="\mathsf{val}_{G}(\boldsymbol{x})\stackrel{{\scriptstyle\mathrm{\small def}}}{{% =}}\frac{1}{m_{G}}\sum_{v_{1}\neq v_{2}\in V}w(v_{1},v_{2})\cdot\mathbbm{1}[x_% {v_{1}}=1\text{ and }x_{v_{2}}=0]." class="ltx_Math" display="block" id="S2.Ex3.m1.2"><semantics id="S2.Ex3.m1.2a"><mrow id="S2.Ex3.m1.2.2.1" xref="S2.Ex3.m1.2.2.1.1.cmml"><mrow id="S2.Ex3.m1.2.2.1.1" xref="S2.Ex3.m1.2.2.1.1.cmml"><mrow id="S2.Ex3.m1.2.2.1.1.5" xref="S2.Ex3.m1.2.2.1.1.5.cmml"><msub id="S2.Ex3.m1.2.2.1.1.5.2" xref="S2.Ex3.m1.2.2.1.1.5.2.cmml"><mi id="S2.Ex3.m1.2.2.1.1.5.2.2" xref="S2.Ex3.m1.2.2.1.1.5.2.2.cmml">𝗏𝖺𝗅</mi><mi id="S2.Ex3.m1.2.2.1.1.5.2.3" xref="S2.Ex3.m1.2.2.1.1.5.2.3.cmml">G</mi></msub><mo id="S2.Ex3.m1.2.2.1.1.5.1" xref="S2.Ex3.m1.2.2.1.1.5.1.cmml">⁢</mo><mrow id="S2.Ex3.m1.2.2.1.1.5.3.2" xref="S2.Ex3.m1.2.2.1.1.5.cmml"><mo id="S2.Ex3.m1.2.2.1.1.5.3.2.1" stretchy="false" xref="S2.Ex3.m1.2.2.1.1.5.cmml">(</mo><mi id="S2.Ex3.m1.1.1" xref="S2.Ex3.m1.1.1.cmml">𝒙</mi><mo id="S2.Ex3.m1.2.2.1.1.5.3.2.2" stretchy="false" xref="S2.Ex3.m1.2.2.1.1.5.cmml">)</mo></mrow></mrow><mover id="S2.Ex3.m1.2.2.1.1.4" xref="S2.Ex3.m1.2.2.1.1.4.cmml"><mo id="S2.Ex3.m1.2.2.1.1.4.2" xref="S2.Ex3.m1.2.2.1.1.4.2.cmml">=</mo><mi id="S2.Ex3.m1.2.2.1.1.4.3" mathsize="128%" xref="S2.Ex3.m1.2.2.1.1.4.3.cmml">def</mi></mover><mrow id="S2.Ex3.m1.2.2.1.1.3" xref="S2.Ex3.m1.2.2.1.1.3.cmml"><mfrac id="S2.Ex3.m1.2.2.1.1.3.5" xref="S2.Ex3.m1.2.2.1.1.3.5.cmml"><mn id="S2.Ex3.m1.2.2.1.1.3.5.2" xref="S2.Ex3.m1.2.2.1.1.3.5.2.cmml">1</mn><msub id="S2.Ex3.m1.2.2.1.1.3.5.3" xref="S2.Ex3.m1.2.2.1.1.3.5.3.cmml"><mi id="S2.Ex3.m1.2.2.1.1.3.5.3.2" xref="S2.Ex3.m1.2.2.1.1.3.5.3.2.cmml">m</mi><mi id="S2.Ex3.m1.2.2.1.1.3.5.3.3" xref="S2.Ex3.m1.2.2.1.1.3.5.3.3.cmml">G</mi></msub></mfrac><mo id="S2.Ex3.m1.2.2.1.1.3.4" xref="S2.Ex3.m1.2.2.1.1.3.4.cmml">⁢</mo><mrow id="S2.Ex3.m1.2.2.1.1.3.3" xref="S2.Ex3.m1.2.2.1.1.3.3.cmml"><munder id="S2.Ex3.m1.2.2.1.1.3.3.4" xref="S2.Ex3.m1.2.2.1.1.3.3.4.cmml"><mo id="S2.Ex3.m1.2.2.1.1.3.3.4.2" movablelimits="false" xref="S2.Ex3.m1.2.2.1.1.3.3.4.2.cmml">∑</mo><mrow id="S2.Ex3.m1.2.2.1.1.3.3.4.3" xref="S2.Ex3.m1.2.2.1.1.3.3.4.3.cmml"><msub 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start_ARG italic_m start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≠ italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_V end_POSTSUBSCRIPT italic_w ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ⋅ blackboard_1 [ italic_x start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 1 and italic_x start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0 ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.E4.p1.7"><span class="ltx_text ltx_font_italic" id="S2.E4.p1.7.2">The <em class="ltx_emph ltx_font_upright" id="S2.E4.p1.6.1.1"><span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S2.E4.p1.6.1.1.1">Max-DiCut</span> value</em> of <math alttext="G" class="ltx_Math" display="inline" 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xref="S2.Ex4.m1.2.2.2.5">superscript</csymbol><set id="S2.Ex4.m1.2.2.2.5.2.1.cmml" xref="S2.Ex4.m1.2.2.2.5.2.2"><cn id="S2.Ex4.m1.1.1.1.1.cmml" type="integer" xref="S2.Ex4.m1.1.1.1.1">0</cn><cn id="S2.Ex4.m1.2.2.2.2.cmml" type="integer" xref="S2.Ex4.m1.2.2.2.2">1</cn></set><ci id="S2.Ex4.m1.2.2.2.5.3.cmml" xref="S2.Ex4.m1.2.2.2.5.3">𝑉</ci></apply></apply></apply><apply id="S2.Ex4.m1.4.4.1.1.3.2.2.cmml" xref="S2.Ex4.m1.4.4.1.1.3.2.2"><csymbol cd="ambiguous" id="S2.Ex4.m1.4.4.1.1.3.2.2.1.cmml" xref="S2.Ex4.m1.4.4.1.1.3.2.2">subscript</csymbol><ci id="S2.Ex4.m1.4.4.1.1.3.2.2.2.cmml" xref="S2.Ex4.m1.4.4.1.1.3.2.2.2">𝗏𝖺𝗅</ci><ci id="S2.Ex4.m1.4.4.1.1.3.2.2.3.cmml" xref="S2.Ex4.m1.4.4.1.1.3.2.2.3">𝐺</ci></apply></apply><ci id="S2.Ex4.m1.3.3.cmml" xref="S2.Ex4.m1.3.3">𝒙</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m1.4c">\mathsf{val}_{G}\stackrel{{\scriptstyle\mathrm{\small def}}}{{=}}\max_{% \boldsymbol{x}\in\{0,1\}^{V}}\mathsf{val}_{G}(\boldsymbol{x}).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m1.4d">sansserif_val start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP roman_max start_POSTSUBSCRIPT bold_italic_x ∈ { 0 , 1 } start_POSTSUPERSCRIPT italic_V end_POSTSUPERSCRIPT end_POSTSUBSCRIPT sansserif_val start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( bold_italic_x ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.E4.p1.10"><span class="ltx_text ltx_font_italic" id="S2.E4.p1.10.3">The goal of the <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S2.E4.p1.10.3.1">Max-DiCut</span> problem is to approximate <math alttext="\mathsf{val}_{G}" class="ltx_Math" display="inline" id="S2.E4.p1.9.2.m2.1"><semantics id="S2.E4.p1.9.2.m2.1a"><msub id="S2.E4.p1.9.2.m2.1.1" xref="S2.E4.p1.9.2.m2.1.1.cmml"><mi id="S2.E4.p1.9.2.m2.1.1.2" xref="S2.E4.p1.9.2.m2.1.1.2.cmml">𝗏𝖺𝗅</mi><mi id="S2.E4.p1.9.2.m2.1.1.3" xref="S2.E4.p1.9.2.m2.1.1.3.cmml">G</mi></msub><annotation-xml encoding="MathML-Content" id="S2.E4.p1.9.2.m2.1b"><apply id="S2.E4.p1.9.2.m2.1.1.cmml" xref="S2.E4.p1.9.2.m2.1.1"><csymbol cd="ambiguous" id="S2.E4.p1.9.2.m2.1.1.1.cmml" xref="S2.E4.p1.9.2.m2.1.1">subscript</csymbol><ci id="S2.E4.p1.9.2.m2.1.1.2.cmml" xref="S2.E4.p1.9.2.m2.1.1.2">𝗏𝖺𝗅</ci><ci id="S2.E4.p1.9.2.m2.1.1.3.cmml" xref="S2.E4.p1.9.2.m2.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.p1.9.2.m2.1c">\mathsf{val}_{G}</annotation><annotation encoding="application/x-llamapun" id="S2.E4.p1.9.2.m2.1d">sansserif_val start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math> given <math alttext="G" class="ltx_Math" display="inline" id="S2.E4.p1.10.3.m3.1"><semantics id="S2.E4.p1.10.3.m3.1a"><mi id="S2.E4.p1.10.3.m3.1.1" xref="S2.E4.p1.10.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.E4.p1.10.3.m3.1b"><ci id="S2.E4.p1.10.3.m3.1.1.cmml" xref="S2.E4.p1.10.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.p1.10.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.E4.p1.10.3.m3.1d">italic_G</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.3">Now, we consider algorithms for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S2.SS2.p1.3.1">Max-DiCut</span> which estimate <math alttext="\mathsf{val}_{G}" class="ltx_Math" display="inline" id="S2.SS2.p1.2.m2.1"><semantics id="S2.SS2.p1.2.m2.1a"><msub id="S2.SS2.p1.2.m2.1.1" xref="S2.SS2.p1.2.m2.1.1.cmml"><mi id="S2.SS2.p1.2.m2.1.1.2" xref="S2.SS2.p1.2.m2.1.1.2.cmml">𝗏𝖺𝗅</mi><mi id="S2.SS2.p1.2.m2.1.1.3" xref="S2.SS2.p1.2.m2.1.1.3.cmml">G</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.2.m2.1b"><apply id="S2.SS2.p1.2.m2.1.1.cmml" xref="S2.SS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.2.m2.1.1.1.cmml" xref="S2.SS2.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS2.p1.2.m2.1.1.2.cmml" xref="S2.SS2.p1.2.m2.1.1.2">𝗏𝖺𝗅</ci><ci id="S2.SS2.p1.2.m2.1.1.3.cmml" xref="S2.SS2.p1.2.m2.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.2.m2.1c">\mathsf{val}_{G}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.2.m2.1d">sansserif_val start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math> using only the biases of vertices in <math alttext="G" class="ltx_Math" display="inline" id="S2.SS2.p1.3.m3.1"><semantics id="S2.SS2.p1.3.m3.1a"><mi id="S2.SS2.p1.3.m3.1.1" xref="S2.SS2.p1.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.3.m3.1b"><ci id="S2.SS2.p1.3.m3.1.1.cmml" xref="S2.SS2.p1.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.3.m3.1d">italic_G</annotation></semantics></math>:</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.E5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.E5.1.1.1">Definition 2.5</span></span><span class="ltx_text ltx_font_bold" id="S2.E5.2.2">.</span> </h6> <div class="ltx_para" id="S2.E5.p1"> <p class="ltx_p" id="S2.E5.p1.5"><span class="ltx_text ltx_font_italic" id="S2.E5.p1.5.5">An <em class="ltx_emph ltx_font_upright" id="S2.E5.p1.5.5.1">oblivious algorithm</em> for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S2.E5.p1.5.5.2">Max-DiCut</span> is defined by a <em class="ltx_emph ltx_font_upright" id="S2.E5.p1.5.5.3">selection function</em> <math alttext="\mathsf{S}:[-1,+1]\to[0,1]" class="ltx_Math" display="inline" id="S2.E5.p1.2.2.m2.4"><semantics id="S2.E5.p1.2.2.m2.4a"><mrow id="S2.E5.p1.2.2.m2.4.4" xref="S2.E5.p1.2.2.m2.4.4.cmml"><mi id="S2.E5.p1.2.2.m2.4.4.4" xref="S2.E5.p1.2.2.m2.4.4.4.cmml">𝖲</mi><mo id="S2.E5.p1.2.2.m2.4.4.3" lspace="0.278em" rspace="0.278em" xref="S2.E5.p1.2.2.m2.4.4.3.cmml">:</mo><mrow id="S2.E5.p1.2.2.m2.4.4.2" xref="S2.E5.p1.2.2.m2.4.4.2.cmml"><mrow id="S2.E5.p1.2.2.m2.4.4.2.2.2" xref="S2.E5.p1.2.2.m2.4.4.2.2.3.cmml"><mo id="S2.E5.p1.2.2.m2.4.4.2.2.2.3" stretchy="false" xref="S2.E5.p1.2.2.m2.4.4.2.2.3.cmml">[</mo><mrow id="S2.E5.p1.2.2.m2.3.3.1.1.1.1" xref="S2.E5.p1.2.2.m2.3.3.1.1.1.1.cmml"><mo id="S2.E5.p1.2.2.m2.3.3.1.1.1.1a" xref="S2.E5.p1.2.2.m2.3.3.1.1.1.1.cmml">−</mo><mn id="S2.E5.p1.2.2.m2.3.3.1.1.1.1.2" xref="S2.E5.p1.2.2.m2.3.3.1.1.1.1.2.cmml">1</mn></mrow><mo id="S2.E5.p1.2.2.m2.4.4.2.2.2.4" xref="S2.E5.p1.2.2.m2.4.4.2.2.3.cmml">,</mo><mrow id="S2.E5.p1.2.2.m2.4.4.2.2.2.2" xref="S2.E5.p1.2.2.m2.4.4.2.2.2.2.cmml"><mo id="S2.E5.p1.2.2.m2.4.4.2.2.2.2a" xref="S2.E5.p1.2.2.m2.4.4.2.2.2.2.cmml">+</mo><mn id="S2.E5.p1.2.2.m2.4.4.2.2.2.2.2" xref="S2.E5.p1.2.2.m2.4.4.2.2.2.2.2.cmml">1</mn></mrow><mo id="S2.E5.p1.2.2.m2.4.4.2.2.2.5" stretchy="false" xref="S2.E5.p1.2.2.m2.4.4.2.2.3.cmml">]</mo></mrow><mo id="S2.E5.p1.2.2.m2.4.4.2.3" stretchy="false" xref="S2.E5.p1.2.2.m2.4.4.2.3.cmml">→</mo><mrow id="S2.E5.p1.2.2.m2.4.4.2.4.2" xref="S2.E5.p1.2.2.m2.4.4.2.4.1.cmml"><mo id="S2.E5.p1.2.2.m2.4.4.2.4.2.1" stretchy="false" xref="S2.E5.p1.2.2.m2.4.4.2.4.1.cmml">[</mo><mn id="S2.E5.p1.2.2.m2.1.1" xref="S2.E5.p1.2.2.m2.1.1.cmml">0</mn><mo id="S2.E5.p1.2.2.m2.4.4.2.4.2.2" xref="S2.E5.p1.2.2.m2.4.4.2.4.1.cmml">,</mo><mn id="S2.E5.p1.2.2.m2.2.2" xref="S2.E5.p1.2.2.m2.2.2.cmml">1</mn><mo id="S2.E5.p1.2.2.m2.4.4.2.4.2.3" stretchy="false" xref="S2.E5.p1.2.2.m2.4.4.2.4.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E5.p1.2.2.m2.4b"><apply id="S2.E5.p1.2.2.m2.4.4.cmml" xref="S2.E5.p1.2.2.m2.4.4"><ci id="S2.E5.p1.2.2.m2.4.4.3.cmml" xref="S2.E5.p1.2.2.m2.4.4.3">:</ci><ci id="S2.E5.p1.2.2.m2.4.4.4.cmml" xref="S2.E5.p1.2.2.m2.4.4.4">𝖲</ci><apply id="S2.E5.p1.2.2.m2.4.4.2.cmml" xref="S2.E5.p1.2.2.m2.4.4.2"><ci id="S2.E5.p1.2.2.m2.4.4.2.3.cmml" xref="S2.E5.p1.2.2.m2.4.4.2.3">→</ci><interval closure="closed" id="S2.E5.p1.2.2.m2.4.4.2.2.3.cmml" xref="S2.E5.p1.2.2.m2.4.4.2.2.2"><apply id="S2.E5.p1.2.2.m2.3.3.1.1.1.1.cmml" xref="S2.E5.p1.2.2.m2.3.3.1.1.1.1"><minus id="S2.E5.p1.2.2.m2.3.3.1.1.1.1.1.cmml" xref="S2.E5.p1.2.2.m2.3.3.1.1.1.1"></minus><cn id="S2.E5.p1.2.2.m2.3.3.1.1.1.1.2.cmml" type="integer" xref="S2.E5.p1.2.2.m2.3.3.1.1.1.1.2">1</cn></apply><apply id="S2.E5.p1.2.2.m2.4.4.2.2.2.2.cmml" xref="S2.E5.p1.2.2.m2.4.4.2.2.2.2"><plus id="S2.E5.p1.2.2.m2.4.4.2.2.2.2.1.cmml" xref="S2.E5.p1.2.2.m2.4.4.2.2.2.2"></plus><cn id="S2.E5.p1.2.2.m2.4.4.2.2.2.2.2.cmml" type="integer" xref="S2.E5.p1.2.2.m2.4.4.2.2.2.2.2">1</cn></apply></interval><interval closure="closed" id="S2.E5.p1.2.2.m2.4.4.2.4.1.cmml" xref="S2.E5.p1.2.2.m2.4.4.2.4.2"><cn id="S2.E5.p1.2.2.m2.1.1.cmml" type="integer" xref="S2.E5.p1.2.2.m2.1.1">0</cn><cn id="S2.E5.p1.2.2.m2.2.2.cmml" type="integer" xref="S2.E5.p1.2.2.m2.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.p1.2.2.m2.4c">\mathsf{S}:[-1,+1]\to[0,1]</annotation><annotation encoding="application/x-llamapun" id="S2.E5.p1.2.2.m2.4d">sansserif_S : [ - 1 , + 1 ] → [ 0 , 1 ]</annotation></semantics></math>. The corresponding algorithm, denoted <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S2.E5.p1.3.3.m3.1"><semantics id="S2.E5.p1.3.3.m3.1a"><msub id="S2.E5.p1.3.3.m3.1.1" xref="S2.E5.p1.3.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E5.p1.3.3.m3.1.1.2" xref="S2.E5.p1.3.3.m3.1.1.2.cmml">𝒪</mi><mi id="S2.E5.p1.3.3.m3.1.1.3" xref="S2.E5.p1.3.3.m3.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S2.E5.p1.3.3.m3.1b"><apply id="S2.E5.p1.3.3.m3.1.1.cmml" xref="S2.E5.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S2.E5.p1.3.3.m3.1.1.1.cmml" xref="S2.E5.p1.3.3.m3.1.1">subscript</csymbol><ci id="S2.E5.p1.3.3.m3.1.1.2.cmml" xref="S2.E5.p1.3.3.m3.1.1.2">𝒪</ci><ci id="S2.E5.p1.3.3.m3.1.1.3.cmml" xref="S2.E5.p1.3.3.m3.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.p1.3.3.m3.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.E5.p1.3.3.m3.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math>, behaves as follows. Given a directed graph <math alttext="G=(V,w)" class="ltx_Math" display="inline" id="S2.E5.p1.4.4.m4.2"><semantics id="S2.E5.p1.4.4.m4.2a"><mrow id="S2.E5.p1.4.4.m4.2.3" xref="S2.E5.p1.4.4.m4.2.3.cmml"><mi id="S2.E5.p1.4.4.m4.2.3.2" xref="S2.E5.p1.4.4.m4.2.3.2.cmml">G</mi><mo id="S2.E5.p1.4.4.m4.2.3.1" xref="S2.E5.p1.4.4.m4.2.3.1.cmml">=</mo><mrow id="S2.E5.p1.4.4.m4.2.3.3.2" xref="S2.E5.p1.4.4.m4.2.3.3.1.cmml"><mo id="S2.E5.p1.4.4.m4.2.3.3.2.1" stretchy="false" xref="S2.E5.p1.4.4.m4.2.3.3.1.cmml">(</mo><mi id="S2.E5.p1.4.4.m4.1.1" xref="S2.E5.p1.4.4.m4.1.1.cmml">V</mi><mo id="S2.E5.p1.4.4.m4.2.3.3.2.2" xref="S2.E5.p1.4.4.m4.2.3.3.1.cmml">,</mo><mi id="S2.E5.p1.4.4.m4.2.2" xref="S2.E5.p1.4.4.m4.2.2.cmml">w</mi><mo id="S2.E5.p1.4.4.m4.2.3.3.2.3" stretchy="false" xref="S2.E5.p1.4.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E5.p1.4.4.m4.2b"><apply id="S2.E5.p1.4.4.m4.2.3.cmml" xref="S2.E5.p1.4.4.m4.2.3"><eq id="S2.E5.p1.4.4.m4.2.3.1.cmml" xref="S2.E5.p1.4.4.m4.2.3.1"></eq><ci id="S2.E5.p1.4.4.m4.2.3.2.cmml" xref="S2.E5.p1.4.4.m4.2.3.2">𝐺</ci><interval closure="open" id="S2.E5.p1.4.4.m4.2.3.3.1.cmml" xref="S2.E5.p1.4.4.m4.2.3.3.2"><ci id="S2.E5.p1.4.4.m4.1.1.cmml" xref="S2.E5.p1.4.4.m4.1.1">𝑉</ci><ci id="S2.E5.p1.4.4.m4.2.2.cmml" xref="S2.E5.p1.4.4.m4.2.2">𝑤</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.p1.4.4.m4.2c">G=(V,w)</annotation><annotation encoding="application/x-llamapun" id="S2.E5.p1.4.4.m4.2d">italic_G = ( italic_V , italic_w )</annotation></semantics></math>, <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S2.E5.p1.5.5.m5.1"><semantics id="S2.E5.p1.5.5.m5.1a"><msub id="S2.E5.p1.5.5.m5.1.1" xref="S2.E5.p1.5.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E5.p1.5.5.m5.1.1.2" xref="S2.E5.p1.5.5.m5.1.1.2.cmml">𝒪</mi><mi id="S2.E5.p1.5.5.m5.1.1.3" xref="S2.E5.p1.5.5.m5.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S2.E5.p1.5.5.m5.1b"><apply id="S2.E5.p1.5.5.m5.1.1.cmml" xref="S2.E5.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="S2.E5.p1.5.5.m5.1.1.1.cmml" xref="S2.E5.p1.5.5.m5.1.1">subscript</csymbol><ci id="S2.E5.p1.5.5.m5.1.1.2.cmml" xref="S2.E5.p1.5.5.m5.1.1.2">𝒪</ci><ci id="S2.E5.p1.5.5.m5.1.1.3.cmml" xref="S2.E5.p1.5.5.m5.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.p1.5.5.m5.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.E5.p1.5.5.m5.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math> outputs</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.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="\mathcal{O}_{\mathsf{S}}(G):=\frac{1}{m}\sum_{v_{1}\neq v_{2}\in V}w(v_{1},v_{% 2})\cdot(\mathsf{S}(\mathsf{bias}_{G}(v_{1})))(1-\mathsf{S}(\mathsf{bias}_{G}(% v_{2})))." class="ltx_Math" display="block" id="S2.Ex5.m1.2"><semantics id="S2.Ex5.m1.2a"><mrow id="S2.Ex5.m1.2.2.1" xref="S2.Ex5.m1.2.2.1.1.cmml"><mrow id="S2.Ex5.m1.2.2.1.1" xref="S2.Ex5.m1.2.2.1.1.cmml"><mrow id="S2.Ex5.m1.2.2.1.1.6" xref="S2.Ex5.m1.2.2.1.1.6.cmml"><msub id="S2.Ex5.m1.2.2.1.1.6.2" xref="S2.Ex5.m1.2.2.1.1.6.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex5.m1.2.2.1.1.6.2.2" xref="S2.Ex5.m1.2.2.1.1.6.2.2.cmml">𝒪</mi><mi id="S2.Ex5.m1.2.2.1.1.6.2.3" xref="S2.Ex5.m1.2.2.1.1.6.2.3.cmml">𝖲</mi></msub><mo id="S2.Ex5.m1.2.2.1.1.6.1" xref="S2.Ex5.m1.2.2.1.1.6.1.cmml">⁢</mo><mrow id="S2.Ex5.m1.2.2.1.1.6.3.2" xref="S2.Ex5.m1.2.2.1.1.6.cmml"><mo id="S2.Ex5.m1.2.2.1.1.6.3.2.1" stretchy="false" xref="S2.Ex5.m1.2.2.1.1.6.cmml">(</mo><mi id="S2.Ex5.m1.1.1" xref="S2.Ex5.m1.1.1.cmml">G</mi><mo id="S2.Ex5.m1.2.2.1.1.6.3.2.2" rspace="0.278em" stretchy="false" xref="S2.Ex5.m1.2.2.1.1.6.cmml">)</mo></mrow></mrow><mo id="S2.Ex5.m1.2.2.1.1.5" rspace="0.278em" xref="S2.Ex5.m1.2.2.1.1.5.cmml">:=</mo><mrow id="S2.Ex5.m1.2.2.1.1.4" xref="S2.Ex5.m1.2.2.1.1.4.cmml"><mfrac id="S2.Ex5.m1.2.2.1.1.4.6" xref="S2.Ex5.m1.2.2.1.1.4.6.cmml"><mn id="S2.Ex5.m1.2.2.1.1.4.6.2" xref="S2.Ex5.m1.2.2.1.1.4.6.2.cmml">1</mn><mi id="S2.Ex5.m1.2.2.1.1.4.6.3" xref="S2.Ex5.m1.2.2.1.1.4.6.3.cmml">m</mi></mfrac><mo id="S2.Ex5.m1.2.2.1.1.4.5" xref="S2.Ex5.m1.2.2.1.1.4.5.cmml">⁢</mo><mrow id="S2.Ex5.m1.2.2.1.1.4.4" xref="S2.Ex5.m1.2.2.1.1.4.4.cmml"><munder id="S2.Ex5.m1.2.2.1.1.4.4.5" xref="S2.Ex5.m1.2.2.1.1.4.4.5.cmml"><mo id="S2.Ex5.m1.2.2.1.1.4.4.5.2" movablelimits="false" xref="S2.Ex5.m1.2.2.1.1.4.4.5.2.cmml">∑</mo><mrow id="S2.Ex5.m1.2.2.1.1.4.4.5.3" xref="S2.Ex5.m1.2.2.1.1.4.4.5.3.cmml"><msub id="S2.Ex5.m1.2.2.1.1.4.4.5.3.2" xref="S2.Ex5.m1.2.2.1.1.4.4.5.3.2.cmml"><mi id="S2.Ex5.m1.2.2.1.1.4.4.5.3.2.2" xref="S2.Ex5.m1.2.2.1.1.4.4.5.3.2.2.cmml">v</mi><mn id="S2.Ex5.m1.2.2.1.1.4.4.5.3.2.3" xref="S2.Ex5.m1.2.2.1.1.4.4.5.3.2.3.cmml">1</mn></msub><mo id="S2.Ex5.m1.2.2.1.1.4.4.5.3.3" xref="S2.Ex5.m1.2.2.1.1.4.4.5.3.3.cmml">≠</mo><msub id="S2.Ex5.m1.2.2.1.1.4.4.5.3.4" xref="S2.Ex5.m1.2.2.1.1.4.4.5.3.4.cmml"><mi id="S2.Ex5.m1.2.2.1.1.4.4.5.3.4.2" xref="S2.Ex5.m1.2.2.1.1.4.4.5.3.4.2.cmml">v</mi><mn id="S2.Ex5.m1.2.2.1.1.4.4.5.3.4.3" xref="S2.Ex5.m1.2.2.1.1.4.4.5.3.4.3.cmml">2</mn></msub><mo id="S2.Ex5.m1.2.2.1.1.4.4.5.3.5" xref="S2.Ex5.m1.2.2.1.1.4.4.5.3.5.cmml">∈</mo><mi id="S2.Ex5.m1.2.2.1.1.4.4.5.3.6" xref="S2.Ex5.m1.2.2.1.1.4.4.5.3.6.cmml">V</mi></mrow></munder><mrow id="S2.Ex5.m1.2.2.1.1.4.4.4" xref="S2.Ex5.m1.2.2.1.1.4.4.4.cmml"><mrow id="S2.Ex5.m1.2.2.1.1.3.3.3.3" xref="S2.Ex5.m1.2.2.1.1.3.3.3.3.cmml"><mrow id="S2.Ex5.m1.2.2.1.1.2.2.2.2.2" xref="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.cmml"><mi id="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.4" xref="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.4.cmml">w</mi><mo id="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.3" xref="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.3.cmml">⁢</mo><mrow id="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.2.2" xref="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.2.3.cmml"><mo id="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.2.2.3" stretchy="false" xref="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.2.3.cmml">(</mo><msub id="S2.Ex5.m1.2.2.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex5.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex5.m1.2.2.1.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex5.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.cmml">v</mi><mn id="S2.Ex5.m1.2.2.1.1.1.1.1.1.1.1.1.1.3" xref="S2.Ex5.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.2.2.4" xref="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.2.3.cmml">,</mo><msub id="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.2.2.2" xref="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.2.2.2.cmml"><mi id="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.2.2.2.2" xref="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.2.2.2.2.cmml">v</mi><mn id="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.2.2.2.3" xref="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.2.2.5" rspace="0.055em" stretchy="false" xref="S2.Ex5.m1.2.2.1.1.2.2.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.Ex5.m1.2.2.1.1.3.3.3.3.4" rspace="0.222em" xref="S2.Ex5.m1.2.2.1.1.3.3.3.3.4.cmml">⋅</mo><mrow id="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1" xref="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1.1.cmml"><mo id="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1.2" stretchy="false" xref="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1.1.cmml">(</mo><mrow id="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1.1" xref="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1.1.cmml"><mi id="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1.1.3" xref="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1.1.3.cmml">𝖲</mi><mo id="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1.1.2" xref="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1.1.2.cmml">⁢</mo><mrow id="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1.1.1.1" xref="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1.1.1.1.1.cmml"><mo id="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1.1.1.1.2" stretchy="false" xref="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1.1.1.1.1" xref="S2.Ex5.m1.2.2.1.1.3.3.3.3.3.1.1.1.1.1.cmml"><msub 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id="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.3.cmml" xref="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.3">𝖲</ci><apply id="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.cmml" xref="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1"><times id="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.2.cmml" xref="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.2"></times><apply id="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.3.cmml" xref="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.3.1.cmml" xref="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.3.2.cmml" xref="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.3.2">𝖻𝗂𝖺𝗌</ci><ci id="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.3.3.cmml" xref="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.3.3">𝐺</ci></apply><apply id="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.1.1.1.cmml" xref="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.1.1.1.2">𝑣</ci><cn id="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.Ex5.m1.2.2.1.1.4.4.4.4.1.1.1.1.1.1.1.1.1.3">2</cn></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex5.m1.2c">\mathcal{O}_{\mathsf{S}}(G):=\frac{1}{m}\sum_{v_{1}\neq v_{2}\in V}w(v_{1},v_{% 2})\cdot(\mathsf{S}(\mathsf{bias}_{G}(v_{1})))(1-\mathsf{S}(\mathsf{bias}_{G}(% v_{2}))).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m1.2d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT ( italic_G ) := divide start_ARG 1 end_ARG start_ARG italic_m end_ARG ∑ start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≠ italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_V end_POSTSUBSCRIPT italic_w ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ⋅ ( sansserif_S ( sansserif_bias start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ) ) ( 1 - sansserif_S ( sansserif_bias start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.4">Observe that <math alttext="\mathcal{A}^{\mathsf{S}}(G)" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><mrow id="S2.SS2.p2.1.m1.1.2" xref="S2.SS2.p2.1.m1.1.2.cmml"><msup id="S2.SS2.p2.1.m1.1.2.2" xref="S2.SS2.p2.1.m1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.1.m1.1.2.2.2" xref="S2.SS2.p2.1.m1.1.2.2.2.cmml">𝒜</mi><mi id="S2.SS2.p2.1.m1.1.2.2.3" xref="S2.SS2.p2.1.m1.1.2.2.3.cmml">𝖲</mi></msup><mo id="S2.SS2.p2.1.m1.1.2.1" xref="S2.SS2.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p2.1.m1.1.2.3.2" xref="S2.SS2.p2.1.m1.1.2.cmml"><mo id="S2.SS2.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS2.p2.1.m1.1.2.cmml">(</mo><mi id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml">G</mi><mo id="S2.SS2.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS2.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><apply id="S2.SS2.p2.1.m1.1.2.cmml" xref="S2.SS2.p2.1.m1.1.2"><times id="S2.SS2.p2.1.m1.1.2.1.cmml" xref="S2.SS2.p2.1.m1.1.2.1"></times><apply id="S2.SS2.p2.1.m1.1.2.2.cmml" xref="S2.SS2.p2.1.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p2.1.m1.1.2.2.1.cmml" xref="S2.SS2.p2.1.m1.1.2.2">superscript</csymbol><ci id="S2.SS2.p2.1.m1.1.2.2.2.cmml" xref="S2.SS2.p2.1.m1.1.2.2.2">𝒜</ci><ci id="S2.SS2.p2.1.m1.1.2.2.3.cmml" xref="S2.SS2.p2.1.m1.1.2.2.3">𝖲</ci></apply><ci id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">\mathcal{A}^{\mathsf{S}}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">caligraphic_A start_POSTSUPERSCRIPT sansserif_S end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math> equals the expected value of the random cut which assigns each (nonisolated) vertex <math alttext="v" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m2.1"><semantics id="S2.SS2.p2.2.m2.1a"><mi id="S2.SS2.p2.2.m2.1.1" xref="S2.SS2.p2.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.2.m2.1b"><ci id="S2.SS2.p2.2.m2.1.1.cmml" xref="S2.SS2.p2.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m2.1d">italic_v</annotation></semantics></math> as an independent <math 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id="S2.SS2.p2.3.m3.2.2.1.1.1.3.2.1" stretchy="false" xref="S2.SS2.p2.3.m3.2.2.1.1.1.cmml">(</mo><mi id="S2.SS2.p2.3.m3.1.1" xref="S2.SS2.p2.3.m3.1.1.cmml">v</mi><mo id="S2.SS2.p2.3.m3.2.2.1.1.1.3.2.2" stretchy="false" xref="S2.SS2.p2.3.m3.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p2.3.m3.2.2.1.1.3" stretchy="false" xref="S2.SS2.p2.3.m3.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m3.2b"><apply id="S2.SS2.p2.3.m3.2.2.cmml" xref="S2.SS2.p2.3.m3.2.2"><times id="S2.SS2.p2.3.m3.2.2.2.cmml" xref="S2.SS2.p2.3.m3.2.2.2"></times><ci id="S2.SS2.p2.3.m3.2.2.3.cmml" xref="S2.SS2.p2.3.m3.2.2.3">𝖡𝖾𝗋𝗇</ci><apply id="S2.SS2.p2.3.m3.2.2.1.1.1.cmml" xref="S2.SS2.p2.3.m3.2.2.1.1"><times id="S2.SS2.p2.3.m3.2.2.1.1.1.1.cmml" xref="S2.SS2.p2.3.m3.2.2.1.1.1.1"></times><apply id="S2.SS2.p2.3.m3.2.2.1.1.1.2.cmml" xref="S2.SS2.p2.3.m3.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p2.3.m3.2.2.1.1.1.2.1.cmml" xref="S2.SS2.p2.3.m3.2.2.1.1.1.2">subscript</csymbol><ci id="S2.SS2.p2.3.m3.2.2.1.1.1.2.2.cmml" xref="S2.SS2.p2.3.m3.2.2.1.1.1.2.2">𝖻𝗂𝖺𝗌</ci><ci id="S2.SS2.p2.3.m3.2.2.1.1.1.2.3.cmml" xref="S2.SS2.p2.3.m3.2.2.1.1.1.2.3">𝐺</ci></apply><ci id="S2.SS2.p2.3.m3.1.1.cmml" xref="S2.SS2.p2.3.m3.1.1">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m3.2c">\mathsf{Bern}(\mathsf{bias}_{G}(v))</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m3.2d">sansserif_Bern ( sansserif_bias start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) )</annotation></semantics></math> random variable. Thus, <math alttext="\mathcal{A}^{\mathsf{S}}(G)\leq\mathsf{val}_{G}" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m4.1"><semantics id="S2.SS2.p2.4.m4.1a"><mrow id="S2.SS2.p2.4.m4.1.2" xref="S2.SS2.p2.4.m4.1.2.cmml"><mrow id="S2.SS2.p2.4.m4.1.2.2" xref="S2.SS2.p2.4.m4.1.2.2.cmml"><msup id="S2.SS2.p2.4.m4.1.2.2.2" xref="S2.SS2.p2.4.m4.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.4.m4.1.2.2.2.2" xref="S2.SS2.p2.4.m4.1.2.2.2.2.cmml">𝒜</mi><mi id="S2.SS2.p2.4.m4.1.2.2.2.3" xref="S2.SS2.p2.4.m4.1.2.2.2.3.cmml">𝖲</mi></msup><mo id="S2.SS2.p2.4.m4.1.2.2.1" xref="S2.SS2.p2.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS2.p2.4.m4.1.2.2.3.2" xref="S2.SS2.p2.4.m4.1.2.2.cmml"><mo id="S2.SS2.p2.4.m4.1.2.2.3.2.1" stretchy="false" xref="S2.SS2.p2.4.m4.1.2.2.cmml">(</mo><mi id="S2.SS2.p2.4.m4.1.1" xref="S2.SS2.p2.4.m4.1.1.cmml">G</mi><mo id="S2.SS2.p2.4.m4.1.2.2.3.2.2" stretchy="false" xref="S2.SS2.p2.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p2.4.m4.1.2.1" xref="S2.SS2.p2.4.m4.1.2.1.cmml">≤</mo><msub id="S2.SS2.p2.4.m4.1.2.3" xref="S2.SS2.p2.4.m4.1.2.3.cmml"><mi id="S2.SS2.p2.4.m4.1.2.3.2" xref="S2.SS2.p2.4.m4.1.2.3.2.cmml">𝗏𝖺𝗅</mi><mi id="S2.SS2.p2.4.m4.1.2.3.3" xref="S2.SS2.p2.4.m4.1.2.3.3.cmml">G</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m4.1b"><apply id="S2.SS2.p2.4.m4.1.2.cmml" xref="S2.SS2.p2.4.m4.1.2"><leq id="S2.SS2.p2.4.m4.1.2.1.cmml" xref="S2.SS2.p2.4.m4.1.2.1"></leq><apply id="S2.SS2.p2.4.m4.1.2.2.cmml" xref="S2.SS2.p2.4.m4.1.2.2"><times id="S2.SS2.p2.4.m4.1.2.2.1.cmml" xref="S2.SS2.p2.4.m4.1.2.2.1"></times><apply id="S2.SS2.p2.4.m4.1.2.2.2.cmml" xref="S2.SS2.p2.4.m4.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p2.4.m4.1.2.2.2.1.cmml" xref="S2.SS2.p2.4.m4.1.2.2.2">superscript</csymbol><ci id="S2.SS2.p2.4.m4.1.2.2.2.2.cmml" xref="S2.SS2.p2.4.m4.1.2.2.2.2">𝒜</ci><ci id="S2.SS2.p2.4.m4.1.2.2.2.3.cmml" xref="S2.SS2.p2.4.m4.1.2.2.2.3">𝖲</ci></apply><ci id="S2.SS2.p2.4.m4.1.1.cmml" xref="S2.SS2.p2.4.m4.1.1">𝐺</ci></apply><apply id="S2.SS2.p2.4.m4.1.2.3.cmml" xref="S2.SS2.p2.4.m4.1.2.3"><csymbol cd="ambiguous" id="S2.SS2.p2.4.m4.1.2.3.1.cmml" xref="S2.SS2.p2.4.m4.1.2.3">subscript</csymbol><ci id="S2.SS2.p2.4.m4.1.2.3.2.cmml" xref="S2.SS2.p2.4.m4.1.2.3.2">𝗏𝖺𝗅</ci><ci id="S2.SS2.p2.4.m4.1.2.3.3.cmml" xref="S2.SS2.p2.4.m4.1.2.3.3">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m4.1c">\mathcal{A}^{\mathsf{S}}(G)\leq\mathsf{val}_{G}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m4.1d">caligraphic_A start_POSTSUPERSCRIPT sansserif_S end_POSTSUPERSCRIPT ( italic_G ) ≤ sansserif_val start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT</annotation></semantics></math>. We are interested in how good of an approximation the algorithm provides:</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.E6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.E6.1.1.1">Definition 2.6</span></span><span class="ltx_text ltx_font_bold" id="S2.E6.2.2"> </span>(Approximation)<span class="ltx_text ltx_font_bold" id="S2.E6.3.3">.</span> </h6> <div class="ltx_para" id="S2.E6.p1"> <p class="ltx_p" id="S2.E6.p1.4"><span class="ltx_text ltx_font_italic" id="S2.E6.p1.4.4">Let <math alttext="G" class="ltx_Math" display="inline" id="S2.E6.p1.1.1.m1.1"><semantics id="S2.E6.p1.1.1.m1.1a"><mi id="S2.E6.p1.1.1.m1.1.1" xref="S2.E6.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.E6.p1.1.1.m1.1b"><ci id="S2.E6.p1.1.1.m1.1.1.cmml" xref="S2.E6.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.E6.p1.1.1.m1.1d">italic_G</annotation></semantics></math> be a graph and <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S2.E6.p1.2.2.m2.1"><semantics id="S2.E6.p1.2.2.m2.1a"><msub id="S2.E6.p1.2.2.m2.1.1" xref="S2.E6.p1.2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E6.p1.2.2.m2.1.1.2" xref="S2.E6.p1.2.2.m2.1.1.2.cmml">𝒪</mi><mi id="S2.E6.p1.2.2.m2.1.1.3" xref="S2.E6.p1.2.2.m2.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S2.E6.p1.2.2.m2.1b"><apply id="S2.E6.p1.2.2.m2.1.1.cmml" xref="S2.E6.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.E6.p1.2.2.m2.1.1.1.cmml" xref="S2.E6.p1.2.2.m2.1.1">subscript</csymbol><ci id="S2.E6.p1.2.2.m2.1.1.2.cmml" xref="S2.E6.p1.2.2.m2.1.1.2">𝒪</ci><ci id="S2.E6.p1.2.2.m2.1.1.3.cmml" xref="S2.E6.p1.2.2.m2.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.p1.2.2.m2.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.E6.p1.2.2.m2.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math> an oblivious algorithm. The <em class="ltx_emph ltx_font_upright" id="S2.E6.p1.4.4.1">approximation ratio</em> of <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S2.E6.p1.3.3.m3.1"><semantics id="S2.E6.p1.3.3.m3.1a"><msub id="S2.E6.p1.3.3.m3.1.1" xref="S2.E6.p1.3.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E6.p1.3.3.m3.1.1.2" xref="S2.E6.p1.3.3.m3.1.1.2.cmml">𝒪</mi><mi id="S2.E6.p1.3.3.m3.1.1.3" xref="S2.E6.p1.3.3.m3.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S2.E6.p1.3.3.m3.1b"><apply id="S2.E6.p1.3.3.m3.1.1.cmml" xref="S2.E6.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S2.E6.p1.3.3.m3.1.1.1.cmml" xref="S2.E6.p1.3.3.m3.1.1">subscript</csymbol><ci id="S2.E6.p1.3.3.m3.1.1.2.cmml" xref="S2.E6.p1.3.3.m3.1.1.2">𝒪</ci><ci id="S2.E6.p1.3.3.m3.1.1.3.cmml" xref="S2.E6.p1.3.3.m3.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.p1.3.3.m3.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.E6.p1.3.3.m3.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math> on <math alttext="G" class="ltx_Math" display="inline" id="S2.E6.p1.4.4.m4.1"><semantics id="S2.E6.p1.4.4.m4.1a"><mi id="S2.E6.p1.4.4.m4.1.1" xref="S2.E6.p1.4.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.E6.p1.4.4.m4.1b"><ci id="S2.E6.p1.4.4.m4.1.1.cmml" xref="S2.E6.p1.4.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.p1.4.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.E6.p1.4.4.m4.1d">italic_G</annotation></semantics></math> is</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.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="\alpha(\mathcal{O}_{\mathsf{S}};G)\stackrel{{\scriptstyle\mathrm{\small def}}}% {{=}}\frac{\mathcal{O}_{\mathsf{S}}(G)}{\mathsf{val}_{G}}." class="ltx_Math" display="block" id="S2.Ex6.m1.3"><semantics id="S2.Ex6.m1.3a"><mrow id="S2.Ex6.m1.3.3.1" xref="S2.Ex6.m1.3.3.1.1.cmml"><mrow id="S2.Ex6.m1.3.3.1.1" xref="S2.Ex6.m1.3.3.1.1.cmml"><mrow id="S2.Ex6.m1.3.3.1.1.1" xref="S2.Ex6.m1.3.3.1.1.1.cmml"><mi id="S2.Ex6.m1.3.3.1.1.1.3" xref="S2.Ex6.m1.3.3.1.1.1.3.cmml">α</mi><mo id="S2.Ex6.m1.3.3.1.1.1.2" xref="S2.Ex6.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex6.m1.3.3.1.1.1.1.1" xref="S2.Ex6.m1.3.3.1.1.1.1.2.cmml"><mo id="S2.Ex6.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S2.Ex6.m1.3.3.1.1.1.1.2.cmml">(</mo><msub id="S2.Ex6.m1.3.3.1.1.1.1.1.1" xref="S2.Ex6.m1.3.3.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex6.m1.3.3.1.1.1.1.1.1.2" xref="S2.Ex6.m1.3.3.1.1.1.1.1.1.2.cmml">𝒪</mi><mi id="S2.Ex6.m1.3.3.1.1.1.1.1.1.3" xref="S2.Ex6.m1.3.3.1.1.1.1.1.1.3.cmml">𝖲</mi></msub><mo id="S2.Ex6.m1.3.3.1.1.1.1.1.3" xref="S2.Ex6.m1.3.3.1.1.1.1.2.cmml">;</mo><mi id="S2.Ex6.m1.2.2" xref="S2.Ex6.m1.2.2.cmml">G</mi><mo id="S2.Ex6.m1.3.3.1.1.1.1.1.4" stretchy="false" xref="S2.Ex6.m1.3.3.1.1.1.1.2.cmml">)</mo></mrow></mrow><mover id="S2.Ex6.m1.3.3.1.1.2" xref="S2.Ex6.m1.3.3.1.1.2.cmml"><mo id="S2.Ex6.m1.3.3.1.1.2.2" xref="S2.Ex6.m1.3.3.1.1.2.2.cmml">=</mo><mi id="S2.Ex6.m1.3.3.1.1.2.3" mathsize="128%" xref="S2.Ex6.m1.3.3.1.1.2.3.cmml">def</mi></mover><mfrac id="S2.Ex6.m1.1.1" xref="S2.Ex6.m1.1.1.cmml"><mrow id="S2.Ex6.m1.1.1.1" xref="S2.Ex6.m1.1.1.1.cmml"><msub id="S2.Ex6.m1.1.1.1.3" xref="S2.Ex6.m1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex6.m1.1.1.1.3.2" xref="S2.Ex6.m1.1.1.1.3.2.cmml">𝒪</mi><mi id="S2.Ex6.m1.1.1.1.3.3" xref="S2.Ex6.m1.1.1.1.3.3.cmml">𝖲</mi></msub><mo id="S2.Ex6.m1.1.1.1.2" xref="S2.Ex6.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex6.m1.1.1.1.4.2" xref="S2.Ex6.m1.1.1.1.cmml"><mo id="S2.Ex6.m1.1.1.1.4.2.1" stretchy="false" xref="S2.Ex6.m1.1.1.1.cmml">(</mo><mi id="S2.Ex6.m1.1.1.1.1" 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caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT ( italic_G ) end_ARG start_ARG sansserif_val start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.E6.p1.5"><span class="ltx_text ltx_font_italic" id="S2.E6.p1.5.1">The <em class="ltx_emph ltx_font_upright" id="S2.E6.p1.5.1.1">approximation ratio</em> of <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S2.E6.p1.5.1.m1.1"><semantics id="S2.E6.p1.5.1.m1.1a"><msub id="S2.E6.p1.5.1.m1.1.1" xref="S2.E6.p1.5.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E6.p1.5.1.m1.1.1.2" xref="S2.E6.p1.5.1.m1.1.1.2.cmml">𝒪</mi><mi id="S2.E6.p1.5.1.m1.1.1.3" xref="S2.E6.p1.5.1.m1.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S2.E6.p1.5.1.m1.1b"><apply id="S2.E6.p1.5.1.m1.1.1.cmml" xref="S2.E6.p1.5.1.m1.1.1"><csymbol cd="ambiguous" 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xref="S2.Ex7.m1.2.2.1.1.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.Ex7.m1.2.2.1.1.2.1.1.1.1.1.cmml" xref="S2.Ex7.m1.2.2.1.1.2.1.1.1.1">subscript</csymbol><ci id="S2.Ex7.m1.2.2.1.1.2.1.1.1.1.2.cmml" xref="S2.Ex7.m1.2.2.1.1.2.1.1.1.1.2">𝒪</ci><ci id="S2.Ex7.m1.2.2.1.1.2.1.1.1.1.3.cmml" xref="S2.Ex7.m1.2.2.1.1.2.1.1.1.1.3">𝖲</ci></apply><ci id="S2.Ex7.m1.1.1.cmml" xref="S2.Ex7.m1.1.1">𝐺</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex7.m1.2c">\alpha(\mathcal{O}_{\mathsf{S}})\stackrel{{\scriptstyle\mathrm{\small def}}}{{% =}}\inf_{\text{graph }G}\alpha(\mathcal{O}_{\mathsf{S}};G).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex7.m1.2d">italic_α ( caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG roman_def end_ARG end_RELOP roman_inf start_POSTSUBSCRIPT graph italic_G end_POSTSUBSCRIPT italic_α ( caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT ; italic_G ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S2.SS2.p3"> <p class="ltx_p" id="S2.SS2.p3.1">In this paper, we are interested in some specific types of selection functions.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.E7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.E7.1.1.1">Definition 2.7</span></span><span class="ltx_text ltx_font_bold" id="S2.E7.2.2"> </span>(Antisymmetry)<span class="ltx_text ltx_font_bold" id="S2.E7.3.3">.</span> </h6> <div class="ltx_para" id="S2.E7.p1"> <p class="ltx_p" id="S2.E7.p1.3"><span class="ltx_text ltx_font_italic" id="S2.E7.p1.3.3">A selection function <math alttext="\mathsf{S}:[-1,+1]\to[0,1]" class="ltx_Math" display="inline" id="S2.E7.p1.1.1.m1.4"><semantics id="S2.E7.p1.1.1.m1.4a"><mrow 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id="S2.E7.p1.1.1.m1.4.4.2.2.2.2.2" xref="S2.E7.p1.1.1.m1.4.4.2.2.2.2.2.cmml">1</mn></mrow><mo id="S2.E7.p1.1.1.m1.4.4.2.2.2.5" stretchy="false" xref="S2.E7.p1.1.1.m1.4.4.2.2.3.cmml">]</mo></mrow><mo id="S2.E7.p1.1.1.m1.4.4.2.3" stretchy="false" xref="S2.E7.p1.1.1.m1.4.4.2.3.cmml">→</mo><mrow id="S2.E7.p1.1.1.m1.4.4.2.4.2" xref="S2.E7.p1.1.1.m1.4.4.2.4.1.cmml"><mo id="S2.E7.p1.1.1.m1.4.4.2.4.2.1" stretchy="false" xref="S2.E7.p1.1.1.m1.4.4.2.4.1.cmml">[</mo><mn id="S2.E7.p1.1.1.m1.1.1" xref="S2.E7.p1.1.1.m1.1.1.cmml">0</mn><mo id="S2.E7.p1.1.1.m1.4.4.2.4.2.2" xref="S2.E7.p1.1.1.m1.4.4.2.4.1.cmml">,</mo><mn id="S2.E7.p1.1.1.m1.2.2" xref="S2.E7.p1.1.1.m1.2.2.cmml">1</mn><mo id="S2.E7.p1.1.1.m1.4.4.2.4.2.3" stretchy="false" xref="S2.E7.p1.1.1.m1.4.4.2.4.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E7.p1.1.1.m1.4b"><apply id="S2.E7.p1.1.1.m1.4.4.cmml" xref="S2.E7.p1.1.1.m1.4.4"><ci id="S2.E7.p1.1.1.m1.4.4.3.cmml" xref="S2.E7.p1.1.1.m1.4.4.3">:</ci><ci id="S2.E7.p1.1.1.m1.4.4.4.cmml" xref="S2.E7.p1.1.1.m1.4.4.4">𝖲</ci><apply id="S2.E7.p1.1.1.m1.4.4.2.cmml" xref="S2.E7.p1.1.1.m1.4.4.2"><ci id="S2.E7.p1.1.1.m1.4.4.2.3.cmml" xref="S2.E7.p1.1.1.m1.4.4.2.3">→</ci><interval closure="closed" id="S2.E7.p1.1.1.m1.4.4.2.2.3.cmml" xref="S2.E7.p1.1.1.m1.4.4.2.2.2"><apply id="S2.E7.p1.1.1.m1.3.3.1.1.1.1.cmml" xref="S2.E7.p1.1.1.m1.3.3.1.1.1.1"><minus id="S2.E7.p1.1.1.m1.3.3.1.1.1.1.1.cmml" xref="S2.E7.p1.1.1.m1.3.3.1.1.1.1"></minus><cn id="S2.E7.p1.1.1.m1.3.3.1.1.1.1.2.cmml" type="integer" xref="S2.E7.p1.1.1.m1.3.3.1.1.1.1.2">1</cn></apply><apply id="S2.E7.p1.1.1.m1.4.4.2.2.2.2.cmml" xref="S2.E7.p1.1.1.m1.4.4.2.2.2.2"><plus id="S2.E7.p1.1.1.m1.4.4.2.2.2.2.1.cmml" xref="S2.E7.p1.1.1.m1.4.4.2.2.2.2"></plus><cn id="S2.E7.p1.1.1.m1.4.4.2.2.2.2.2.cmml" type="integer" xref="S2.E7.p1.1.1.m1.4.4.2.2.2.2.2">1</cn></apply></interval><interval closure="closed" id="S2.E7.p1.1.1.m1.4.4.2.4.1.cmml" xref="S2.E7.p1.1.1.m1.4.4.2.4.2"><cn id="S2.E7.p1.1.1.m1.1.1.cmml" type="integer" xref="S2.E7.p1.1.1.m1.1.1">0</cn><cn id="S2.E7.p1.1.1.m1.2.2.cmml" type="integer" xref="S2.E7.p1.1.1.m1.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E7.p1.1.1.m1.4c">\mathsf{S}:[-1,+1]\to[0,1]</annotation><annotation encoding="application/x-llamapun" id="S2.E7.p1.1.1.m1.4d">sansserif_S : [ - 1 , + 1 ] → [ 0 , 1 ]</annotation></semantics></math> is <em class="ltx_emph ltx_font_upright" id="S2.E7.p1.3.3.1">antisymmetric</em> if <math alttext="\mathsf{S}(x)=1-\mathsf{S}(-x)" class="ltx_Math" display="inline" id="S2.E7.p1.2.2.m2.2"><semantics id="S2.E7.p1.2.2.m2.2a"><mrow id="S2.E7.p1.2.2.m2.2.2" xref="S2.E7.p1.2.2.m2.2.2.cmml"><mrow id="S2.E7.p1.2.2.m2.2.2.3" xref="S2.E7.p1.2.2.m2.2.2.3.cmml"><mi id="S2.E7.p1.2.2.m2.2.2.3.2" xref="S2.E7.p1.2.2.m2.2.2.3.2.cmml">𝖲</mi><mo id="S2.E7.p1.2.2.m2.2.2.3.1" xref="S2.E7.p1.2.2.m2.2.2.3.1.cmml">⁢</mo><mrow id="S2.E7.p1.2.2.m2.2.2.3.3.2" xref="S2.E7.p1.2.2.m2.2.2.3.cmml"><mo id="S2.E7.p1.2.2.m2.2.2.3.3.2.1" stretchy="false" xref="S2.E7.p1.2.2.m2.2.2.3.cmml">(</mo><mi id="S2.E7.p1.2.2.m2.1.1" xref="S2.E7.p1.2.2.m2.1.1.cmml">x</mi><mo id="S2.E7.p1.2.2.m2.2.2.3.3.2.2" stretchy="false" xref="S2.E7.p1.2.2.m2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.E7.p1.2.2.m2.2.2.2" xref="S2.E7.p1.2.2.m2.2.2.2.cmml">=</mo><mrow id="S2.E7.p1.2.2.m2.2.2.1" xref="S2.E7.p1.2.2.m2.2.2.1.cmml"><mn id="S2.E7.p1.2.2.m2.2.2.1.3" xref="S2.E7.p1.2.2.m2.2.2.1.3.cmml">1</mn><mo id="S2.E7.p1.2.2.m2.2.2.1.2" xref="S2.E7.p1.2.2.m2.2.2.1.2.cmml">−</mo><mrow id="S2.E7.p1.2.2.m2.2.2.1.1" xref="S2.E7.p1.2.2.m2.2.2.1.1.cmml"><mi id="S2.E7.p1.2.2.m2.2.2.1.1.3" xref="S2.E7.p1.2.2.m2.2.2.1.1.3.cmml">𝖲</mi><mo id="S2.E7.p1.2.2.m2.2.2.1.1.2" xref="S2.E7.p1.2.2.m2.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.E7.p1.2.2.m2.2.2.1.1.1.1" xref="S2.E7.p1.2.2.m2.2.2.1.1.1.1.1.cmml"><mo id="S2.E7.p1.2.2.m2.2.2.1.1.1.1.2" stretchy="false" xref="S2.E7.p1.2.2.m2.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S2.E7.p1.2.2.m2.2.2.1.1.1.1.1" xref="S2.E7.p1.2.2.m2.2.2.1.1.1.1.1.cmml"><mo id="S2.E7.p1.2.2.m2.2.2.1.1.1.1.1a" xref="S2.E7.p1.2.2.m2.2.2.1.1.1.1.1.cmml">−</mo><mi id="S2.E7.p1.2.2.m2.2.2.1.1.1.1.1.2" xref="S2.E7.p1.2.2.m2.2.2.1.1.1.1.1.2.cmml">x</mi></mrow><mo id="S2.E7.p1.2.2.m2.2.2.1.1.1.1.3" stretchy="false" xref="S2.E7.p1.2.2.m2.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E7.p1.2.2.m2.2b"><apply id="S2.E7.p1.2.2.m2.2.2.cmml" xref="S2.E7.p1.2.2.m2.2.2"><eq id="S2.E7.p1.2.2.m2.2.2.2.cmml" xref="S2.E7.p1.2.2.m2.2.2.2"></eq><apply id="S2.E7.p1.2.2.m2.2.2.3.cmml" xref="S2.E7.p1.2.2.m2.2.2.3"><times id="S2.E7.p1.2.2.m2.2.2.3.1.cmml" xref="S2.E7.p1.2.2.m2.2.2.3.1"></times><ci id="S2.E7.p1.2.2.m2.2.2.3.2.cmml" xref="S2.E7.p1.2.2.m2.2.2.3.2">𝖲</ci><ci id="S2.E7.p1.2.2.m2.1.1.cmml" xref="S2.E7.p1.2.2.m2.1.1">𝑥</ci></apply><apply id="S2.E7.p1.2.2.m2.2.2.1.cmml" xref="S2.E7.p1.2.2.m2.2.2.1"><minus id="S2.E7.p1.2.2.m2.2.2.1.2.cmml" xref="S2.E7.p1.2.2.m2.2.2.1.2"></minus><cn id="S2.E7.p1.2.2.m2.2.2.1.3.cmml" type="integer" xref="S2.E7.p1.2.2.m2.2.2.1.3">1</cn><apply id="S2.E7.p1.2.2.m2.2.2.1.1.cmml" xref="S2.E7.p1.2.2.m2.2.2.1.1"><times id="S2.E7.p1.2.2.m2.2.2.1.1.2.cmml" xref="S2.E7.p1.2.2.m2.2.2.1.1.2"></times><ci id="S2.E7.p1.2.2.m2.2.2.1.1.3.cmml" xref="S2.E7.p1.2.2.m2.2.2.1.1.3">𝖲</ci><apply id="S2.E7.p1.2.2.m2.2.2.1.1.1.1.1.cmml" xref="S2.E7.p1.2.2.m2.2.2.1.1.1.1"><minus id="S2.E7.p1.2.2.m2.2.2.1.1.1.1.1.1.cmml" xref="S2.E7.p1.2.2.m2.2.2.1.1.1.1"></minus><ci id="S2.E7.p1.2.2.m2.2.2.1.1.1.1.1.2.cmml" xref="S2.E7.p1.2.2.m2.2.2.1.1.1.1.1.2">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E7.p1.2.2.m2.2c">\mathsf{S}(x)=1-\mathsf{S}(-x)</annotation><annotation encoding="application/x-llamapun" id="S2.E7.p1.2.2.m2.2d">sansserif_S ( italic_x ) = 1 - sansserif_S ( - italic_x )</annotation></semantics></math> for all <math alttext="x\in[-1,+1]." class="ltx_Math" display="inline" id="S2.E7.p1.3.3.m3.1"><semantics id="S2.E7.p1.3.3.m3.1a"><mrow id="S2.E7.p1.3.3.m3.1.1.1" xref="S2.E7.p1.3.3.m3.1.1.1.1.cmml"><mrow id="S2.E7.p1.3.3.m3.1.1.1.1" xref="S2.E7.p1.3.3.m3.1.1.1.1.cmml"><mi id="S2.E7.p1.3.3.m3.1.1.1.1.4" xref="S2.E7.p1.3.3.m3.1.1.1.1.4.cmml">x</mi><mo id="S2.E7.p1.3.3.m3.1.1.1.1.3" xref="S2.E7.p1.3.3.m3.1.1.1.1.3.cmml">∈</mo><mrow id="S2.E7.p1.3.3.m3.1.1.1.1.2.2" xref="S2.E7.p1.3.3.m3.1.1.1.1.2.3.cmml"><mo id="S2.E7.p1.3.3.m3.1.1.1.1.2.2.3" stretchy="false" xref="S2.E7.p1.3.3.m3.1.1.1.1.2.3.cmml">[</mo><mrow id="S2.E7.p1.3.3.m3.1.1.1.1.1.1.1" xref="S2.E7.p1.3.3.m3.1.1.1.1.1.1.1.cmml"><mo id="S2.E7.p1.3.3.m3.1.1.1.1.1.1.1a" xref="S2.E7.p1.3.3.m3.1.1.1.1.1.1.1.cmml">−</mo><mn id="S2.E7.p1.3.3.m3.1.1.1.1.1.1.1.2" xref="S2.E7.p1.3.3.m3.1.1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S2.E7.p1.3.3.m3.1.1.1.1.2.2.4" xref="S2.E7.p1.3.3.m3.1.1.1.1.2.3.cmml">,</mo><mrow id="S2.E7.p1.3.3.m3.1.1.1.1.2.2.2" xref="S2.E7.p1.3.3.m3.1.1.1.1.2.2.2.cmml"><mo id="S2.E7.p1.3.3.m3.1.1.1.1.2.2.2a" xref="S2.E7.p1.3.3.m3.1.1.1.1.2.2.2.cmml">+</mo><mn id="S2.E7.p1.3.3.m3.1.1.1.1.2.2.2.2" xref="S2.E7.p1.3.3.m3.1.1.1.1.2.2.2.2.cmml">1</mn></mrow><mo id="S2.E7.p1.3.3.m3.1.1.1.1.2.2.5" stretchy="false" xref="S2.E7.p1.3.3.m3.1.1.1.1.2.3.cmml">]</mo></mrow></mrow><mo id="S2.E7.p1.3.3.m3.1.1.1.2" lspace="0em" xref="S2.E7.p1.3.3.m3.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E7.p1.3.3.m3.1b"><apply id="S2.E7.p1.3.3.m3.1.1.1.1.cmml" xref="S2.E7.p1.3.3.m3.1.1.1"><in id="S2.E7.p1.3.3.m3.1.1.1.1.3.cmml" xref="S2.E7.p1.3.3.m3.1.1.1.1.3"></in><ci id="S2.E7.p1.3.3.m3.1.1.1.1.4.cmml" xref="S2.E7.p1.3.3.m3.1.1.1.1.4">𝑥</ci><interval closure="closed" id="S2.E7.p1.3.3.m3.1.1.1.1.2.3.cmml" xref="S2.E7.p1.3.3.m3.1.1.1.1.2.2"><apply id="S2.E7.p1.3.3.m3.1.1.1.1.1.1.1.cmml" xref="S2.E7.p1.3.3.m3.1.1.1.1.1.1.1"><minus id="S2.E7.p1.3.3.m3.1.1.1.1.1.1.1.1.cmml" xref="S2.E7.p1.3.3.m3.1.1.1.1.1.1.1"></minus><cn id="S2.E7.p1.3.3.m3.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S2.E7.p1.3.3.m3.1.1.1.1.1.1.1.2">1</cn></apply><apply id="S2.E7.p1.3.3.m3.1.1.1.1.2.2.2.cmml" xref="S2.E7.p1.3.3.m3.1.1.1.1.2.2.2"><plus id="S2.E7.p1.3.3.m3.1.1.1.1.2.2.2.1.cmml" xref="S2.E7.p1.3.3.m3.1.1.1.1.2.2.2"></plus><cn id="S2.E7.p1.3.3.m3.1.1.1.1.2.2.2.2.cmml" type="integer" xref="S2.E7.p1.3.3.m3.1.1.1.1.2.2.2.2">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E7.p1.3.3.m3.1c">x\in[-1,+1].</annotation><annotation encoding="application/x-llamapun" id="S2.E7.p1.3.3.m3.1d">italic_x ∈ [ - 1 , + 1 ] .</annotation></semantics></math></span></p> </div> </div> <div class="ltx_para" id="S2.SS2.p4"> <p class="ltx_p" id="S2.SS2.p4.5">Notably, if <math alttext="\mathsf{S}" class="ltx_Math" display="inline" id="S2.SS2.p4.1.m1.1"><semantics id="S2.SS2.p4.1.m1.1a"><mi id="S2.SS2.p4.1.m1.1.1" xref="S2.SS2.p4.1.m1.1.1.cmml">𝖲</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.1.m1.1b"><ci id="S2.SS2.p4.1.m1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1">𝖲</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.1.m1.1c">\mathsf{S}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.1.m1.1d">sansserif_S</annotation></semantics></math> is antisymmetric, then <math alttext="\mathsf{S}(0)=1/2" class="ltx_Math" display="inline" id="S2.SS2.p4.2.m2.1"><semantics id="S2.SS2.p4.2.m2.1a"><mrow id="S2.SS2.p4.2.m2.1.2" xref="S2.SS2.p4.2.m2.1.2.cmml"><mrow id="S2.SS2.p4.2.m2.1.2.2" xref="S2.SS2.p4.2.m2.1.2.2.cmml"><mi id="S2.SS2.p4.2.m2.1.2.2.2" xref="S2.SS2.p4.2.m2.1.2.2.2.cmml">𝖲</mi><mo id="S2.SS2.p4.2.m2.1.2.2.1" xref="S2.SS2.p4.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS2.p4.2.m2.1.2.2.3.2" xref="S2.SS2.p4.2.m2.1.2.2.cmml"><mo id="S2.SS2.p4.2.m2.1.2.2.3.2.1" stretchy="false" xref="S2.SS2.p4.2.m2.1.2.2.cmml">(</mo><mn id="S2.SS2.p4.2.m2.1.1" xref="S2.SS2.p4.2.m2.1.1.cmml">0</mn><mo id="S2.SS2.p4.2.m2.1.2.2.3.2.2" stretchy="false" xref="S2.SS2.p4.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p4.2.m2.1.2.1" xref="S2.SS2.p4.2.m2.1.2.1.cmml">=</mo><mrow id="S2.SS2.p4.2.m2.1.2.3" xref="S2.SS2.p4.2.m2.1.2.3.cmml"><mn id="S2.SS2.p4.2.m2.1.2.3.2" xref="S2.SS2.p4.2.m2.1.2.3.2.cmml">1</mn><mo id="S2.SS2.p4.2.m2.1.2.3.1" xref="S2.SS2.p4.2.m2.1.2.3.1.cmml">/</mo><mn id="S2.SS2.p4.2.m2.1.2.3.3" xref="S2.SS2.p4.2.m2.1.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.2.m2.1b"><apply id="S2.SS2.p4.2.m2.1.2.cmml" xref="S2.SS2.p4.2.m2.1.2"><eq id="S2.SS2.p4.2.m2.1.2.1.cmml" xref="S2.SS2.p4.2.m2.1.2.1"></eq><apply id="S2.SS2.p4.2.m2.1.2.2.cmml" xref="S2.SS2.p4.2.m2.1.2.2"><times id="S2.SS2.p4.2.m2.1.2.2.1.cmml" xref="S2.SS2.p4.2.m2.1.2.2.1"></times><ci id="S2.SS2.p4.2.m2.1.2.2.2.cmml" xref="S2.SS2.p4.2.m2.1.2.2.2">𝖲</ci><cn id="S2.SS2.p4.2.m2.1.1.cmml" type="integer" xref="S2.SS2.p4.2.m2.1.1">0</cn></apply><apply id="S2.SS2.p4.2.m2.1.2.3.cmml" xref="S2.SS2.p4.2.m2.1.2.3"><divide id="S2.SS2.p4.2.m2.1.2.3.1.cmml" xref="S2.SS2.p4.2.m2.1.2.3.1"></divide><cn id="S2.SS2.p4.2.m2.1.2.3.2.cmml" type="integer" xref="S2.SS2.p4.2.m2.1.2.3.2">1</cn><cn id="S2.SS2.p4.2.m2.1.2.3.3.cmml" type="integer" xref="S2.SS2.p4.2.m2.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.2.m2.1c">\mathsf{S}(0)=1/2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.2.m2.1d">sansserif_S ( 0 ) = 1 / 2</annotation></semantics></math>. Antisymmetry is a natural desideratum for selection functions for <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S2.SS2.p4.5.1">Max-DiCut</span>, since the operation of flipping all edges (i.e., switching to the “transpose” weight function <math alttext="w^{\top}(v_{1},v_{2})=w(v_{1},v_{2})" class="ltx_Math" display="inline" id="S2.SS2.p4.4.m4.4"><semantics id="S2.SS2.p4.4.m4.4a"><mrow id="S2.SS2.p4.4.m4.4.4" xref="S2.SS2.p4.4.m4.4.4.cmml"><mrow id="S2.SS2.p4.4.m4.2.2.2" xref="S2.SS2.p4.4.m4.2.2.2.cmml"><msup id="S2.SS2.p4.4.m4.2.2.2.4" xref="S2.SS2.p4.4.m4.2.2.2.4.cmml"><mi id="S2.SS2.p4.4.m4.2.2.2.4.2" xref="S2.SS2.p4.4.m4.2.2.2.4.2.cmml">w</mi><mo id="S2.SS2.p4.4.m4.2.2.2.4.3" xref="S2.SS2.p4.4.m4.2.2.2.4.3.cmml">⊤</mo></msup><mo id="S2.SS2.p4.4.m4.2.2.2.3" xref="S2.SS2.p4.4.m4.2.2.2.3.cmml">⁢</mo><mrow id="S2.SS2.p4.4.m4.2.2.2.2.2" xref="S2.SS2.p4.4.m4.2.2.2.2.3.cmml"><mo id="S2.SS2.p4.4.m4.2.2.2.2.2.3" stretchy="false" 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xref="S2.SS2.p4.4.m4.4.4.4.3.cmml">⁢</mo><mrow id="S2.SS2.p4.4.m4.4.4.4.2.2" xref="S2.SS2.p4.4.m4.4.4.4.2.3.cmml"><mo id="S2.SS2.p4.4.m4.4.4.4.2.2.3" stretchy="false" xref="S2.SS2.p4.4.m4.4.4.4.2.3.cmml">(</mo><msub id="S2.SS2.p4.4.m4.3.3.3.1.1.1" xref="S2.SS2.p4.4.m4.3.3.3.1.1.1.cmml"><mi id="S2.SS2.p4.4.m4.3.3.3.1.1.1.2" xref="S2.SS2.p4.4.m4.3.3.3.1.1.1.2.cmml">v</mi><mn id="S2.SS2.p4.4.m4.3.3.3.1.1.1.3" xref="S2.SS2.p4.4.m4.3.3.3.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS2.p4.4.m4.4.4.4.2.2.4" xref="S2.SS2.p4.4.m4.4.4.4.2.3.cmml">,</mo><msub id="S2.SS2.p4.4.m4.4.4.4.2.2.2" xref="S2.SS2.p4.4.m4.4.4.4.2.2.2.cmml"><mi id="S2.SS2.p4.4.m4.4.4.4.2.2.2.2" xref="S2.SS2.p4.4.m4.4.4.4.2.2.2.2.cmml">v</mi><mn id="S2.SS2.p4.4.m4.4.4.4.2.2.2.3" xref="S2.SS2.p4.4.m4.4.4.4.2.2.2.3.cmml">2</mn></msub><mo id="S2.SS2.p4.4.m4.4.4.4.2.2.5" stretchy="false" xref="S2.SS2.p4.4.m4.4.4.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.4.m4.4b"><apply 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end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_w ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>) preserves the <span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S2.SS2.p4.5.2">Max-DiCut</span> value; this operation also negates the bias of every vertex, and so antisymmetry implies that the output of the oblivious algorithm is also preserved.</p> </div> <div class="ltx_para" id="S2.SS2.p5"> <p class="ltx_p" id="S2.SS2.p5.1">Another class of selection functions we are interested in is “piecewise constant” functions:</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.E8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.E8.1.1.1">Definition 2.8</span></span><span class="ltx_text ltx_font_bold" id="S2.E8.2.2">.</span> </h6> <div class="ltx_para" id="S2.E8.p1"> 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end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\ell" class="ltx_Math" display="inline" id="S2.E8.p1.6.6.m5.1"><semantics id="S2.E8.p1.6.6.m5.1a"><mi id="S2.E8.p1.6.6.m5.1.1" mathvariant="normal" xref="S2.E8.p1.6.6.m5.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="S2.E8.p1.6.6.m5.1b"><ci id="S2.E8.p1.6.6.m5.1.1.cmml" xref="S2.E8.p1.6.6.m5.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.p1.6.6.m5.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="S2.E8.p1.6.6.m5.1d">roman_ℓ</annotation></semantics></math> values <math alttext="p_{1},\ldots,p_{\ell}" class="ltx_Math" display="inline" id="S2.E8.p1.7.7.m6.3"><semantics id="S2.E8.p1.7.7.m6.3a"><mrow id="S2.E8.p1.7.7.m6.3.3.2" xref="S2.E8.p1.7.7.m6.3.3.3.cmml"><msub id="S2.E8.p1.7.7.m6.2.2.1.1" xref="S2.E8.p1.7.7.m6.2.2.1.1.cmml"><mi id="S2.E8.p1.7.7.m6.2.2.1.1.2" xref="S2.E8.p1.7.7.m6.2.2.1.1.2.cmml">p</mi><mn id="S2.E8.p1.7.7.m6.2.2.1.1.3" xref="S2.E8.p1.7.7.m6.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.E8.p1.7.7.m6.3.3.2.3" xref="S2.E8.p1.7.7.m6.3.3.3.cmml">,</mo><mi id="S2.E8.p1.7.7.m6.1.1" mathvariant="normal" xref="S2.E8.p1.7.7.m6.1.1.cmml">…</mi><mo id="S2.E8.p1.7.7.m6.3.3.2.4" xref="S2.E8.p1.7.7.m6.3.3.3.cmml">,</mo><msub id="S2.E8.p1.7.7.m6.3.3.2.2" xref="S2.E8.p1.7.7.m6.3.3.2.2.cmml"><mi id="S2.E8.p1.7.7.m6.3.3.2.2.2" xref="S2.E8.p1.7.7.m6.3.3.2.2.2.cmml">p</mi><mi id="S2.E8.p1.7.7.m6.3.3.2.2.3" mathvariant="normal" xref="S2.E8.p1.7.7.m6.3.3.2.2.3.cmml">ℓ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.E8.p1.7.7.m6.3b"><list id="S2.E8.p1.7.7.m6.3.3.3.cmml" xref="S2.E8.p1.7.7.m6.3.3.2"><apply id="S2.E8.p1.7.7.m6.2.2.1.1.cmml" xref="S2.E8.p1.7.7.m6.2.2.1.1"><csymbol cd="ambiguous" id="S2.E8.p1.7.7.m6.2.2.1.1.1.cmml" xref="S2.E8.p1.7.7.m6.2.2.1.1">subscript</csymbol><ci id="S2.E8.p1.7.7.m6.2.2.1.1.2.cmml" xref="S2.E8.p1.7.7.m6.2.2.1.1.2">𝑝</ci><cn id="S2.E8.p1.7.7.m6.2.2.1.1.3.cmml" type="integer" xref="S2.E8.p1.7.7.m6.2.2.1.1.3">1</cn></apply><ci id="S2.E8.p1.7.7.m6.1.1.cmml" xref="S2.E8.p1.7.7.m6.1.1">…</ci><apply id="S2.E8.p1.7.7.m6.3.3.2.2.cmml" xref="S2.E8.p1.7.7.m6.3.3.2.2"><csymbol cd="ambiguous" id="S2.E8.p1.7.7.m6.3.3.2.2.1.cmml" xref="S2.E8.p1.7.7.m6.3.3.2.2">subscript</csymbol><ci id="S2.E8.p1.7.7.m6.3.3.2.2.2.cmml" xref="S2.E8.p1.7.7.m6.3.3.2.2.2">𝑝</ci><ci id="S2.E8.p1.7.7.m6.3.3.2.2.3.cmml" xref="S2.E8.p1.7.7.m6.3.3.2.2.3">ℓ</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.p1.7.7.m6.3c">p_{1},\ldots,p_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S2.E8.p1.7.7.m6.3d">italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>, such that <math alttext="\mathsf{S}(x)=p_{i}" class="ltx_Math" display="inline" id="S2.E8.p1.8.8.m7.1"><semantics id="S2.E8.p1.8.8.m7.1a"><mrow id="S2.E8.p1.8.8.m7.1.2" xref="S2.E8.p1.8.8.m7.1.2.cmml"><mrow id="S2.E8.p1.8.8.m7.1.2.2" xref="S2.E8.p1.8.8.m7.1.2.2.cmml"><mi id="S2.E8.p1.8.8.m7.1.2.2.2" xref="S2.E8.p1.8.8.m7.1.2.2.2.cmml">𝖲</mi><mo id="S2.E8.p1.8.8.m7.1.2.2.1" xref="S2.E8.p1.8.8.m7.1.2.2.1.cmml">⁢</mo><mrow id="S2.E8.p1.8.8.m7.1.2.2.3.2" xref="S2.E8.p1.8.8.m7.1.2.2.cmml"><mo id="S2.E8.p1.8.8.m7.1.2.2.3.2.1" stretchy="false" xref="S2.E8.p1.8.8.m7.1.2.2.cmml">(</mo><mi id="S2.E8.p1.8.8.m7.1.1" xref="S2.E8.p1.8.8.m7.1.1.cmml">x</mi><mo id="S2.E8.p1.8.8.m7.1.2.2.3.2.2" stretchy="false" xref="S2.E8.p1.8.8.m7.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.E8.p1.8.8.m7.1.2.1" xref="S2.E8.p1.8.8.m7.1.2.1.cmml">=</mo><msub id="S2.E8.p1.8.8.m7.1.2.3" xref="S2.E8.p1.8.8.m7.1.2.3.cmml"><mi id="S2.E8.p1.8.8.m7.1.2.3.2" xref="S2.E8.p1.8.8.m7.1.2.3.2.cmml">p</mi><mi id="S2.E8.p1.8.8.m7.1.2.3.3" xref="S2.E8.p1.8.8.m7.1.2.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.E8.p1.8.8.m7.1b"><apply id="S2.E8.p1.8.8.m7.1.2.cmml" xref="S2.E8.p1.8.8.m7.1.2"><eq id="S2.E8.p1.8.8.m7.1.2.1.cmml" xref="S2.E8.p1.8.8.m7.1.2.1"></eq><apply id="S2.E8.p1.8.8.m7.1.2.2.cmml" xref="S2.E8.p1.8.8.m7.1.2.2"><times id="S2.E8.p1.8.8.m7.1.2.2.1.cmml" xref="S2.E8.p1.8.8.m7.1.2.2.1"></times><ci id="S2.E8.p1.8.8.m7.1.2.2.2.cmml" xref="S2.E8.p1.8.8.m7.1.2.2.2">𝖲</ci><ci id="S2.E8.p1.8.8.m7.1.1.cmml" xref="S2.E8.p1.8.8.m7.1.1">𝑥</ci></apply><apply id="S2.E8.p1.8.8.m7.1.2.3.cmml" xref="S2.E8.p1.8.8.m7.1.2.3"><csymbol cd="ambiguous" id="S2.E8.p1.8.8.m7.1.2.3.1.cmml" xref="S2.E8.p1.8.8.m7.1.2.3">subscript</csymbol><ci id="S2.E8.p1.8.8.m7.1.2.3.2.cmml" xref="S2.E8.p1.8.8.m7.1.2.3.2">𝑝</ci><ci id="S2.E8.p1.8.8.m7.1.2.3.3.cmml" xref="S2.E8.p1.8.8.m7.1.2.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.p1.8.8.m7.1c">\mathsf{S}(x)=p_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.E8.p1.8.8.m7.1d">sansserif_S ( italic_x ) = italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="x\in I_{i}" class="ltx_Math" display="inline" id="S2.E8.p1.9.9.m8.1"><semantics id="S2.E8.p1.9.9.m8.1a"><mrow id="S2.E8.p1.9.9.m8.1.1" xref="S2.E8.p1.9.9.m8.1.1.cmml"><mi id="S2.E8.p1.9.9.m8.1.1.2" xref="S2.E8.p1.9.9.m8.1.1.2.cmml">x</mi><mo id="S2.E8.p1.9.9.m8.1.1.1" xref="S2.E8.p1.9.9.m8.1.1.1.cmml">∈</mo><msub id="S2.E8.p1.9.9.m8.1.1.3" xref="S2.E8.p1.9.9.m8.1.1.3.cmml"><mi id="S2.E8.p1.9.9.m8.1.1.3.2" xref="S2.E8.p1.9.9.m8.1.1.3.2.cmml">I</mi><mi id="S2.E8.p1.9.9.m8.1.1.3.3" xref="S2.E8.p1.9.9.m8.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.E8.p1.9.9.m8.1b"><apply id="S2.E8.p1.9.9.m8.1.1.cmml" xref="S2.E8.p1.9.9.m8.1.1"><in id="S2.E8.p1.9.9.m8.1.1.1.cmml" xref="S2.E8.p1.9.9.m8.1.1.1"></in><ci id="S2.E8.p1.9.9.m8.1.1.2.cmml" xref="S2.E8.p1.9.9.m8.1.1.2">𝑥</ci><apply id="S2.E8.p1.9.9.m8.1.1.3.cmml" xref="S2.E8.p1.9.9.m8.1.1.3"><csymbol cd="ambiguous" id="S2.E8.p1.9.9.m8.1.1.3.1.cmml" xref="S2.E8.p1.9.9.m8.1.1.3">subscript</csymbol><ci id="S2.E8.p1.9.9.m8.1.1.3.2.cmml" xref="S2.E8.p1.9.9.m8.1.1.3.2">𝐼</ci><ci id="S2.E8.p1.9.9.m8.1.1.3.3.cmml" xref="S2.E8.p1.9.9.m8.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.p1.9.9.m8.1c">x\in I_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.E8.p1.9.9.m8.1d">italic_x ∈ italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.3 </span>Linear program</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.2">For completeness, we include here the linear program introduced by <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">Sin23-kand</span>]</cite> for calculating the approximation ratio of an antisymmetric piecewise constant selection function. For <math alttext="\ell\in\mathbb{N}" class="ltx_Math" display="inline" id="S2.SS3.p1.1.m1.1"><semantics id="S2.SS3.p1.1.m1.1a"><mrow id="S2.SS3.p1.1.m1.1.1" xref="S2.SS3.p1.1.m1.1.1.cmml"><mi id="S2.SS3.p1.1.m1.1.1.2" mathvariant="normal" xref="S2.SS3.p1.1.m1.1.1.2.cmml">ℓ</mi><mo id="S2.SS3.p1.1.m1.1.1.1" xref="S2.SS3.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S2.SS3.p1.1.m1.1.1.3" xref="S2.SS3.p1.1.m1.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.1.m1.1b"><apply id="S2.SS3.p1.1.m1.1.1.cmml" xref="S2.SS3.p1.1.m1.1.1"><in id="S2.SS3.p1.1.m1.1.1.1.cmml" xref="S2.SS3.p1.1.m1.1.1.1"></in><ci id="S2.SS3.p1.1.m1.1.1.2.cmml" xref="S2.SS3.p1.1.m1.1.1.2">ℓ</ci><ci id="S2.SS3.p1.1.m1.1.1.3.cmml" xref="S2.SS3.p1.1.m1.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.1.m1.1c">\ell\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.1.m1.1d">roman_ℓ ∈ blackboard_N</annotation></semantics></math>, let <math alttext="[\pm\ell]=\{-\ell,\ldots,+\ell\}" class="ltx_Math" display="inline" id="S2.SS3.p1.2.m2.4"><semantics id="S2.SS3.p1.2.m2.4a"><mrow id="S2.SS3.p1.2.m2.4.4" xref="S2.SS3.p1.2.m2.4.4.cmml"><mrow id="S2.SS3.p1.2.m2.2.2.1.1" xref="S2.SS3.p1.2.m2.2.2.1.2.cmml"><mo id="S2.SS3.p1.2.m2.2.2.1.1.2" stretchy="false" xref="S2.SS3.p1.2.m2.2.2.1.2.1.cmml">[</mo><mrow id="S2.SS3.p1.2.m2.2.2.1.1.1" xref="S2.SS3.p1.2.m2.2.2.1.1.1.cmml"><mo id="S2.SS3.p1.2.m2.2.2.1.1.1a" xref="S2.SS3.p1.2.m2.2.2.1.1.1.cmml">±</mo><mi id="S2.SS3.p1.2.m2.2.2.1.1.1.2" mathvariant="normal" xref="S2.SS3.p1.2.m2.2.2.1.1.1.2.cmml">ℓ</mi></mrow><mo id="S2.SS3.p1.2.m2.2.2.1.1.3" stretchy="false" xref="S2.SS3.p1.2.m2.2.2.1.2.1.cmml">]</mo></mrow><mo id="S2.SS3.p1.2.m2.4.4.4" xref="S2.SS3.p1.2.m2.4.4.4.cmml">=</mo><mrow id="S2.SS3.p1.2.m2.4.4.3.2" xref="S2.SS3.p1.2.m2.4.4.3.3.cmml"><mo id="S2.SS3.p1.2.m2.4.4.3.2.3" stretchy="false" xref="S2.SS3.p1.2.m2.4.4.3.3.cmml">{</mo><mrow id="S2.SS3.p1.2.m2.3.3.2.1.1" xref="S2.SS3.p1.2.m2.3.3.2.1.1.cmml"><mo id="S2.SS3.p1.2.m2.3.3.2.1.1a" xref="S2.SS3.p1.2.m2.3.3.2.1.1.cmml">−</mo><mi id="S2.SS3.p1.2.m2.3.3.2.1.1.2" mathvariant="normal" xref="S2.SS3.p1.2.m2.3.3.2.1.1.2.cmml">ℓ</mi></mrow><mo id="S2.SS3.p1.2.m2.4.4.3.2.4" xref="S2.SS3.p1.2.m2.4.4.3.3.cmml">,</mo><mi id="S2.SS3.p1.2.m2.1.1" mathvariant="normal" xref="S2.SS3.p1.2.m2.1.1.cmml">…</mi><mo id="S2.SS3.p1.2.m2.4.4.3.2.5" xref="S2.SS3.p1.2.m2.4.4.3.3.cmml">,</mo><mrow id="S2.SS3.p1.2.m2.4.4.3.2.2" xref="S2.SS3.p1.2.m2.4.4.3.2.2.cmml"><mo id="S2.SS3.p1.2.m2.4.4.3.2.2a" xref="S2.SS3.p1.2.m2.4.4.3.2.2.cmml">+</mo><mi id="S2.SS3.p1.2.m2.4.4.3.2.2.2" mathvariant="normal" xref="S2.SS3.p1.2.m2.4.4.3.2.2.2.cmml">ℓ</mi></mrow><mo id="S2.SS3.p1.2.m2.4.4.3.2.6" stretchy="false" xref="S2.SS3.p1.2.m2.4.4.3.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.2.m2.4b"><apply id="S2.SS3.p1.2.m2.4.4.cmml" xref="S2.SS3.p1.2.m2.4.4"><eq id="S2.SS3.p1.2.m2.4.4.4.cmml" xref="S2.SS3.p1.2.m2.4.4.4"></eq><apply id="S2.SS3.p1.2.m2.2.2.1.2.cmml" xref="S2.SS3.p1.2.m2.2.2.1.1"><csymbol cd="latexml" id="S2.SS3.p1.2.m2.2.2.1.2.1.cmml" xref="S2.SS3.p1.2.m2.2.2.1.1.2">delimited-[]</csymbol><apply id="S2.SS3.p1.2.m2.2.2.1.1.1.cmml" xref="S2.SS3.p1.2.m2.2.2.1.1.1"><csymbol cd="latexml" id="S2.SS3.p1.2.m2.2.2.1.1.1.1.cmml" xref="S2.SS3.p1.2.m2.2.2.1.1.1">plus-or-minus</csymbol><ci id="S2.SS3.p1.2.m2.2.2.1.1.1.2.cmml" xref="S2.SS3.p1.2.m2.2.2.1.1.1.2">ℓ</ci></apply></apply><set id="S2.SS3.p1.2.m2.4.4.3.3.cmml" xref="S2.SS3.p1.2.m2.4.4.3.2"><apply id="S2.SS3.p1.2.m2.3.3.2.1.1.cmml" xref="S2.SS3.p1.2.m2.3.3.2.1.1"><minus id="S2.SS3.p1.2.m2.3.3.2.1.1.1.cmml" xref="S2.SS3.p1.2.m2.3.3.2.1.1"></minus><ci id="S2.SS3.p1.2.m2.3.3.2.1.1.2.cmml" xref="S2.SS3.p1.2.m2.3.3.2.1.1.2">ℓ</ci></apply><ci id="S2.SS3.p1.2.m2.1.1.cmml" xref="S2.SS3.p1.2.m2.1.1">…</ci><apply id="S2.SS3.p1.2.m2.4.4.3.2.2.cmml" xref="S2.SS3.p1.2.m2.4.4.3.2.2"><plus id="S2.SS3.p1.2.m2.4.4.3.2.2.1.cmml" xref="S2.SS3.p1.2.m2.4.4.3.2.2"></plus><ci id="S2.SS3.p1.2.m2.4.4.3.2.2.2.cmml" xref="S2.SS3.p1.2.m2.4.4.3.2.2.2">ℓ</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.2.m2.4c">[\pm\ell]=\{-\ell,\ldots,+\ell\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.2.m2.4d">[ ± roman_ℓ ] = { - roman_ℓ , … , + roman_ℓ }</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S2.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="S2.Thmtheorem1.1.1.1">Theorem 2.1</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.2.2"> </span>(LP for antisymmetric selection functions)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem1.p1"> <p class="ltx_p" id="S2.Thmtheorem1.p1.17"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem1.p1.17.17">Let <math alttext="\ell\in\mathbb{N}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.1.1.m1.1"><semantics id="S2.Thmtheorem1.p1.1.1.m1.1a"><mrow id="S2.Thmtheorem1.p1.1.1.m1.1.1" xref="S2.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem1.p1.1.1.m1.1.1.2" mathvariant="normal" xref="S2.Thmtheorem1.p1.1.1.m1.1.1.2.cmml">ℓ</mi><mo id="S2.Thmtheorem1.p1.1.1.m1.1.1.1" xref="S2.Thmtheorem1.p1.1.1.m1.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem1.p1.1.1.m1.1.1.3" xref="S2.Thmtheorem1.p1.1.1.m1.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.1.1.m1.1b"><apply id="S2.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p1.1.1.m1.1.1"><in id="S2.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.1.1.m1.1.1.1"></in><ci id="S2.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.1.1.m1.1.1.2">ℓ</ci><ci id="S2.Thmtheorem1.p1.1.1.m1.1.1.3.cmml" xref="S2.Thmtheorem1.p1.1.1.m1.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.1.1.m1.1c">\ell\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.1.1.m1.1d">roman_ℓ ∈ blackboard_N</annotation></semantics></math>. Let <math alttext="0\leq t_{0}\leq\cdots\leq t_{\ell}=1" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.2.2.m2.1"><semantics id="S2.Thmtheorem1.p1.2.2.m2.1a"><mrow id="S2.Thmtheorem1.p1.2.2.m2.1.1" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.cmml"><mn id="S2.Thmtheorem1.p1.2.2.m2.1.1.2" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.2.cmml">0</mn><mo id="S2.Thmtheorem1.p1.2.2.m2.1.1.3" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.3.cmml">≤</mo><msub id="S2.Thmtheorem1.p1.2.2.m2.1.1.4" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.4.cmml"><mi id="S2.Thmtheorem1.p1.2.2.m2.1.1.4.2" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.4.2.cmml">t</mi><mn id="S2.Thmtheorem1.p1.2.2.m2.1.1.4.3" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.4.3.cmml">0</mn></msub><mo id="S2.Thmtheorem1.p1.2.2.m2.1.1.5" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.5.cmml">≤</mo><mi id="S2.Thmtheorem1.p1.2.2.m2.1.1.6" mathvariant="normal" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.6.cmml">⋯</mi><mo id="S2.Thmtheorem1.p1.2.2.m2.1.1.7" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.7.cmml">≤</mo><msub id="S2.Thmtheorem1.p1.2.2.m2.1.1.8" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.8.cmml"><mi id="S2.Thmtheorem1.p1.2.2.m2.1.1.8.2" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.8.2.cmml">t</mi><mi id="S2.Thmtheorem1.p1.2.2.m2.1.1.8.3" mathvariant="normal" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.8.3.cmml">ℓ</mi></msub><mo id="S2.Thmtheorem1.p1.2.2.m2.1.1.9" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.9.cmml">=</mo><mn id="S2.Thmtheorem1.p1.2.2.m2.1.1.10" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.10.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.2.2.m2.1b"><apply id="S2.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1"><and id="S2.Thmtheorem1.p1.2.2.m2.1.1a.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1"></and><apply id="S2.Thmtheorem1.p1.2.2.m2.1.1b.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1"><leq id="S2.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.3"></leq><cn id="S2.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" type="integer" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.2">0</cn><apply id="S2.Thmtheorem1.p1.2.2.m2.1.1.4.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.4"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.2.2.m2.1.1.4.1.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.4">subscript</csymbol><ci id="S2.Thmtheorem1.p1.2.2.m2.1.1.4.2.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.4.2">𝑡</ci><cn id="S2.Thmtheorem1.p1.2.2.m2.1.1.4.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.4.3">0</cn></apply></apply><apply id="S2.Thmtheorem1.p1.2.2.m2.1.1c.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1"><leq id="S2.Thmtheorem1.p1.2.2.m2.1.1.5.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.5"></leq><share href="https://arxiv.org/html/2411.12976v1#S2.Thmtheorem1.p1.2.2.m2.1.1.4.cmml" id="S2.Thmtheorem1.p1.2.2.m2.1.1d.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1"></share><ci id="S2.Thmtheorem1.p1.2.2.m2.1.1.6.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.6">⋯</ci></apply><apply id="S2.Thmtheorem1.p1.2.2.m2.1.1e.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1"><leq id="S2.Thmtheorem1.p1.2.2.m2.1.1.7.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.7"></leq><share href="https://arxiv.org/html/2411.12976v1#S2.Thmtheorem1.p1.2.2.m2.1.1.6.cmml" id="S2.Thmtheorem1.p1.2.2.m2.1.1f.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1"></share><apply id="S2.Thmtheorem1.p1.2.2.m2.1.1.8.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.8"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.2.2.m2.1.1.8.1.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.8">subscript</csymbol><ci id="S2.Thmtheorem1.p1.2.2.m2.1.1.8.2.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.8.2">𝑡</ci><ci id="S2.Thmtheorem1.p1.2.2.m2.1.1.8.3.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.8.3">ℓ</ci></apply></apply><apply id="S2.Thmtheorem1.p1.2.2.m2.1.1g.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1"><eq id="S2.Thmtheorem1.p1.2.2.m2.1.1.9.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.9"></eq><share href="https://arxiv.org/html/2411.12976v1#S2.Thmtheorem1.p1.2.2.m2.1.1.8.cmml" id="S2.Thmtheorem1.p1.2.2.m2.1.1h.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1"></share><cn id="S2.Thmtheorem1.p1.2.2.m2.1.1.10.cmml" type="integer" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.10">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.2.2.m2.1c">0\leq t_{0}\leq\cdots\leq t_{\ell}=1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.2.2.m2.1d">0 ≤ italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≤ ⋯ ≤ italic_t start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT = 1</annotation></semantics></math>, and <math alttext="0\leq p_{1},\ldots,p_{\ell}\leq 1" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.3.3.m3.3"><semantics id="S2.Thmtheorem1.p1.3.3.m3.3a"><mrow id="S2.Thmtheorem1.p1.3.3.m3.3.3.2" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.3.cmml"><mrow id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.cmml"><mn id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.3" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.3.cmml">0</mn><mo id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.2" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.2.cmml">≤</mo><mrow id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.2.cmml"><msub id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.2" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.2.cmml">p</mi><mn id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.3" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.2" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.2.cmml">,</mo><mi id="S2.Thmtheorem1.p1.3.3.m3.1.1" mathvariant="normal" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.cmml">…</mi></mrow></mrow><mo id="S2.Thmtheorem1.p1.3.3.m3.3.3.2.3" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.3a.cmml">,</mo><mrow id="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.cmml"><msub id="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.2" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.2.cmml"><mi id="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.2.2" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.2.2.cmml">p</mi><mi id="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.2.3" mathvariant="normal" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.2.3.cmml">ℓ</mi></msub><mo id="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.1" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.cmml">≤</mo><mn id="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.3" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.3.3.m3.3b"><apply id="S2.Thmtheorem1.p1.3.3.m3.3.3.3.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.3.3.m3.3.3.3a.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.2.3">formulae-sequence</csymbol><apply id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1"><leq id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.2.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.2"></leq><cn id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.3">0</cn><list id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1"><apply id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1">subscript</csymbol><ci id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.2">𝑝</ci><cn id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.3">1</cn></apply><ci id="S2.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.1.1">…</ci></list></apply><apply id="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2"><leq id="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.1"></leq><apply id="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.2.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.2.1.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.2">subscript</csymbol><ci id="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.2.2.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.2.2">𝑝</ci><ci id="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.2.3.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.2.3">ℓ</ci></apply><cn id="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.3.3.m3.3.3.2.2.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.3.3.m3.3c">0\leq p_{1},\ldots,p_{\ell}\leq 1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.3.3.m3.3d">0 ≤ italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ≤ 1</annotation></semantics></math>. Define intervals <math alttext="I_{+i}:=(+t_{i-1},+t_{i}]" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.4.4.m4.2"><semantics id="S2.Thmtheorem1.p1.4.4.m4.2a"><mrow id="S2.Thmtheorem1.p1.4.4.m4.2.2" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.cmml"><msub id="S2.Thmtheorem1.p1.4.4.m4.2.2.4" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.4.cmml"><mi id="S2.Thmtheorem1.p1.4.4.m4.2.2.4.2" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.4.2.cmml">I</mi><mrow id="S2.Thmtheorem1.p1.4.4.m4.2.2.4.3" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.4.3.cmml"><mo id="S2.Thmtheorem1.p1.4.4.m4.2.2.4.3a" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.4.3.cmml">+</mo><mi id="S2.Thmtheorem1.p1.4.4.m4.2.2.4.3.2" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.4.3.2.cmml">i</mi></mrow></msub><mo id="S2.Thmtheorem1.p1.4.4.m4.2.2.3" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.3.cmml">:=</mo><mrow id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.3.cmml"><mo id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.3" stretchy="false" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.3.cmml">(</mo><mrow id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1a" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.cmml">+</mo><msub id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.cmml"><mi id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.cmml">t</mi><mrow id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3.cmml"><mi id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3.2" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3.2.cmml">i</mi><mo id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3.1" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3.1.cmml">−</mo><mn id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3.3" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3.3.cmml">1</mn></mrow></msub></mrow><mo id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.4" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.3.cmml">,</mo><mrow id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.cmml"><mo id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2a" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.cmml">+</mo><msub id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.cmml"><mi id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.2" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.2.cmml">t</mi><mi id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.3" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.3.cmml">i</mi></msub></mrow><mo id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.5" stretchy="false" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.3.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.4.4.m4.2b"><apply id="S2.Thmtheorem1.p1.4.4.m4.2.2.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.2.2"><csymbol cd="latexml" id="S2.Thmtheorem1.p1.4.4.m4.2.2.3.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.3">assign</csymbol><apply id="S2.Thmtheorem1.p1.4.4.m4.2.2.4.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.4"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.4.4.m4.2.2.4.1.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.4">subscript</csymbol><ci id="S2.Thmtheorem1.p1.4.4.m4.2.2.4.2.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.4.2">𝐼</ci><apply id="S2.Thmtheorem1.p1.4.4.m4.2.2.4.3.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.4.3"><plus id="S2.Thmtheorem1.p1.4.4.m4.2.2.4.3.1.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.4.3"></plus><ci id="S2.Thmtheorem1.p1.4.4.m4.2.2.4.3.2.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.4.3.2">𝑖</ci></apply></apply><interval closure="open-closed" id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.3.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2"><apply id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1"><plus id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1"></plus><apply id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.1.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.2">𝑡</ci><apply id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3"><minus id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3.1.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3.1"></minus><ci id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3.2.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3.2">𝑖</ci><cn id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.3.3">1</cn></apply></apply></apply><apply id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2"><plus id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.1.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2"></plus><apply id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.1.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2">subscript</csymbol><ci id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.2.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.2">𝑡</ci><ci id="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.3.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.3">𝑖</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.4.4.m4.2c">I_{+i}:=(+t_{i-1},+t_{i}]</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.4.4.m4.2d">italic_I start_POSTSUBSCRIPT + italic_i end_POSTSUBSCRIPT := ( + italic_t start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT , + italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ]</annotation></semantics></math> and <math alttext="I_{-i}:=[-t_{i},-t_{i-1})" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.5.5.m5.2"><semantics id="S2.Thmtheorem1.p1.5.5.m5.2a"><mrow id="S2.Thmtheorem1.p1.5.5.m5.2.2" 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xref="S2.Thmtheorem1.p1.5.5.m5.2.2.4.2">𝐼</ci><apply id="S2.Thmtheorem1.p1.5.5.m5.2.2.4.3.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.2.4.3"><minus id="S2.Thmtheorem1.p1.5.5.m5.2.2.4.3.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.2.4.3"></minus><ci id="S2.Thmtheorem1.p1.5.5.m5.2.2.4.3.2.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.2.4.3.2">𝑖</ci></apply></apply><interval closure="closed-open" id="S2.Thmtheorem1.p1.5.5.m5.2.2.2.3.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2"><apply id="S2.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.1.1.1"><minus id="S2.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.1.1.1"></minus><apply id="S2.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.2.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.2.2.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.2.2">𝑡</ci><ci id="S2.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.2.3.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.1.1.1.2.3">𝑖</ci></apply></apply><apply id="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2"><minus id="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2"></minus><apply id="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.2.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.2.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.2">subscript</csymbol><ci id="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.2.2.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.2.2">𝑡</ci><apply id="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.2.3.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.2.3"><minus id="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.2.3.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.2.3.1"></minus><ci id="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.2.3.2.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.2.3.2">𝑖</ci><cn id="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.2.3.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.5.5.m5.2.2.2.2.2.2.3.3">1</cn></apply></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.5.5.m5.2c">I_{-i}:=[-t_{i},-t_{i-1})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.5.5.m5.2d">italic_I start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT := [ - italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , - italic_t start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT )</annotation></semantics></math> for <math alttext="i\in[\ell]" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.6.6.m6.1"><semantics id="S2.Thmtheorem1.p1.6.6.m6.1a"><mrow id="S2.Thmtheorem1.p1.6.6.m6.1.2" xref="S2.Thmtheorem1.p1.6.6.m6.1.2.cmml"><mi id="S2.Thmtheorem1.p1.6.6.m6.1.2.2" xref="S2.Thmtheorem1.p1.6.6.m6.1.2.2.cmml">i</mi><mo id="S2.Thmtheorem1.p1.6.6.m6.1.2.1" xref="S2.Thmtheorem1.p1.6.6.m6.1.2.1.cmml">∈</mo><mrow id="S2.Thmtheorem1.p1.6.6.m6.1.2.3.2" xref="S2.Thmtheorem1.p1.6.6.m6.1.2.3.1.cmml"><mo id="S2.Thmtheorem1.p1.6.6.m6.1.2.3.2.1" stretchy="false" xref="S2.Thmtheorem1.p1.6.6.m6.1.2.3.1.1.cmml">[</mo><mi id="S2.Thmtheorem1.p1.6.6.m6.1.1" mathvariant="normal" xref="S2.Thmtheorem1.p1.6.6.m6.1.1.cmml">ℓ</mi><mo id="S2.Thmtheorem1.p1.6.6.m6.1.2.3.2.2" stretchy="false" xref="S2.Thmtheorem1.p1.6.6.m6.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.6.6.m6.1b"><apply id="S2.Thmtheorem1.p1.6.6.m6.1.2.cmml" xref="S2.Thmtheorem1.p1.6.6.m6.1.2"><in id="S2.Thmtheorem1.p1.6.6.m6.1.2.1.cmml" xref="S2.Thmtheorem1.p1.6.6.m6.1.2.1"></in><ci id="S2.Thmtheorem1.p1.6.6.m6.1.2.2.cmml" xref="S2.Thmtheorem1.p1.6.6.m6.1.2.2">𝑖</ci><apply id="S2.Thmtheorem1.p1.6.6.m6.1.2.3.1.cmml" xref="S2.Thmtheorem1.p1.6.6.m6.1.2.3.2"><csymbol cd="latexml" id="S2.Thmtheorem1.p1.6.6.m6.1.2.3.1.1.cmml" xref="S2.Thmtheorem1.p1.6.6.m6.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.Thmtheorem1.p1.6.6.m6.1.1.cmml" xref="S2.Thmtheorem1.p1.6.6.m6.1.1">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.6.6.m6.1c">i\in[\ell]</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.6.6.m6.1d">italic_i ∈ [ roman_ℓ ]</annotation></semantics></math> and <math alttext="I_{0}:=[-t_{0},+t_{0}]" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.7.7.m7.2"><semantics id="S2.Thmtheorem1.p1.7.7.m7.2a"><mrow id="S2.Thmtheorem1.p1.7.7.m7.2.2" xref="S2.Thmtheorem1.p1.7.7.m7.2.2.cmml"><msub id="S2.Thmtheorem1.p1.7.7.m7.2.2.4" xref="S2.Thmtheorem1.p1.7.7.m7.2.2.4.cmml"><mi id="S2.Thmtheorem1.p1.7.7.m7.2.2.4.2" xref="S2.Thmtheorem1.p1.7.7.m7.2.2.4.2.cmml">I</mi><mn id="S2.Thmtheorem1.p1.7.7.m7.2.2.4.3" xref="S2.Thmtheorem1.p1.7.7.m7.2.2.4.3.cmml">0</mn></msub><mo id="S2.Thmtheorem1.p1.7.7.m7.2.2.3" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem1.p1.7.7.m7.2.2.3.cmml">:=</mo><mrow id="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2" xref="S2.Thmtheorem1.p1.7.7.m7.2.2.2.3.cmml"><mo id="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.3" stretchy="false" xref="S2.Thmtheorem1.p1.7.7.m7.2.2.2.3.cmml">[</mo><mrow id="S2.Thmtheorem1.p1.7.7.m7.1.1.1.1.1" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem1.p1.7.7.m7.1.1.1.1.1a" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.cmml">−</mo><msub id="S2.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.2" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.2.cmml"><mi id="S2.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.2.2" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.2.2.cmml">t</mi><mn id="S2.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.2.3" xref="S2.Thmtheorem1.p1.7.7.m7.1.1.1.1.1.2.3.cmml">0</mn></msub></mrow><mo id="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.4" xref="S2.Thmtheorem1.p1.7.7.m7.2.2.2.3.cmml">,</mo><mrow id="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.2" xref="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.2.cmml"><mo id="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.2a" 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id="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.2.cmml" xref="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.2"><plus id="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.2.1.cmml" xref="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.2"></plus><apply id="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.2.2.cmml" xref="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.2.2.1.cmml" xref="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.2.2">subscript</csymbol><ci id="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.2.2.2.cmml" xref="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.2.2.2">𝑡</ci><cn id="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.2.2.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.7.7.m7.2.2.2.2.2.2.3">0</cn></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.7.7.m7.2c">I_{0}:=[-t_{0},+t_{0}]</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.7.7.m7.2d">italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT := [ - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , + italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ]</annotation></semantics></math>. Let <math alttext="\mathsf{S}:[-1,+1]\to[0,1]" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.8.8.m8.4"><semantics id="S2.Thmtheorem1.p1.8.8.m8.4a"><mrow id="S2.Thmtheorem1.p1.8.8.m8.4.4" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.cmml"><mi id="S2.Thmtheorem1.p1.8.8.m8.4.4.4" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.4.cmml">𝖲</mi><mo id="S2.Thmtheorem1.p1.8.8.m8.4.4.3" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.3.cmml">:</mo><mrow id="S2.Thmtheorem1.p1.8.8.m8.4.4.2" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.cmml"><mrow id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.3.cmml"><mo id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2.3" stretchy="false" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.3.cmml">[</mo><mrow id="S2.Thmtheorem1.p1.8.8.m8.3.3.1.1.1.1" xref="S2.Thmtheorem1.p1.8.8.m8.3.3.1.1.1.1.cmml"><mo id="S2.Thmtheorem1.p1.8.8.m8.3.3.1.1.1.1a" xref="S2.Thmtheorem1.p1.8.8.m8.3.3.1.1.1.1.cmml">−</mo><mn id="S2.Thmtheorem1.p1.8.8.m8.3.3.1.1.1.1.2" xref="S2.Thmtheorem1.p1.8.8.m8.3.3.1.1.1.1.2.cmml">1</mn></mrow><mo id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2.4" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.3.cmml">,</mo><mrow id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2.2" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2.2.cmml"><mo id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2.2a" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2.2.cmml">+</mo><mn id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2.2.2" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2.2.2.cmml">1</mn></mrow><mo id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2.5" stretchy="false" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.3.cmml">]</mo></mrow><mo id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.3" stretchy="false" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.3.cmml">→</mo><mrow id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.4.2" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.4.1.cmml"><mo id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.4.2.1" stretchy="false" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.4.1.cmml">[</mo><mn id="S2.Thmtheorem1.p1.8.8.m8.1.1" xref="S2.Thmtheorem1.p1.8.8.m8.1.1.cmml">0</mn><mo id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.4.2.2" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.4.1.cmml">,</mo><mn id="S2.Thmtheorem1.p1.8.8.m8.2.2" xref="S2.Thmtheorem1.p1.8.8.m8.2.2.cmml">1</mn><mo id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.4.2.3" stretchy="false" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.4.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.8.8.m8.4b"><apply id="S2.Thmtheorem1.p1.8.8.m8.4.4.cmml" xref="S2.Thmtheorem1.p1.8.8.m8.4.4"><ci id="S2.Thmtheorem1.p1.8.8.m8.4.4.3.cmml" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.3">:</ci><ci id="S2.Thmtheorem1.p1.8.8.m8.4.4.4.cmml" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.4">𝖲</ci><apply id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.cmml" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2"><ci id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.3.cmml" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.3">→</ci><interval closure="closed" id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.3.cmml" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2"><apply id="S2.Thmtheorem1.p1.8.8.m8.3.3.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.8.8.m8.3.3.1.1.1.1"><minus id="S2.Thmtheorem1.p1.8.8.m8.3.3.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.8.8.m8.3.3.1.1.1.1"></minus><cn id="S2.Thmtheorem1.p1.8.8.m8.3.3.1.1.1.1.2.cmml" type="integer" xref="S2.Thmtheorem1.p1.8.8.m8.3.3.1.1.1.1.2">1</cn></apply><apply id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2.2.cmml" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2.2"><plus id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2.2.1.cmml" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2.2"></plus><cn id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2.2.2.cmml" type="integer" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.2.2.2.2">1</cn></apply></interval><interval closure="closed" id="S2.Thmtheorem1.p1.8.8.m8.4.4.2.4.1.cmml" xref="S2.Thmtheorem1.p1.8.8.m8.4.4.2.4.2"><cn id="S2.Thmtheorem1.p1.8.8.m8.1.1.cmml" type="integer" xref="S2.Thmtheorem1.p1.8.8.m8.1.1">0</cn><cn id="S2.Thmtheorem1.p1.8.8.m8.2.2.cmml" type="integer" xref="S2.Thmtheorem1.p1.8.8.m8.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.8.8.m8.4c">\mathsf{S}:[-1,+1]\to[0,1]</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.8.8.m8.4d">sansserif_S : [ - 1 , + 1 ] → [ 0 , 1 ]</annotation></semantics></math> be the antisymmetric selection function which maps <math alttext="I_{+i}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.9.9.m9.1"><semantics id="S2.Thmtheorem1.p1.9.9.m9.1a"><msub id="S2.Thmtheorem1.p1.9.9.m9.1.1" xref="S2.Thmtheorem1.p1.9.9.m9.1.1.cmml"><mi id="S2.Thmtheorem1.p1.9.9.m9.1.1.2" xref="S2.Thmtheorem1.p1.9.9.m9.1.1.2.cmml">I</mi><mrow id="S2.Thmtheorem1.p1.9.9.m9.1.1.3" xref="S2.Thmtheorem1.p1.9.9.m9.1.1.3.cmml"><mo id="S2.Thmtheorem1.p1.9.9.m9.1.1.3a" xref="S2.Thmtheorem1.p1.9.9.m9.1.1.3.cmml">+</mo><mi id="S2.Thmtheorem1.p1.9.9.m9.1.1.3.2" xref="S2.Thmtheorem1.p1.9.9.m9.1.1.3.2.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.9.9.m9.1b"><apply id="S2.Thmtheorem1.p1.9.9.m9.1.1.cmml" xref="S2.Thmtheorem1.p1.9.9.m9.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.9.9.m9.1.1.1.cmml" xref="S2.Thmtheorem1.p1.9.9.m9.1.1">subscript</csymbol><ci id="S2.Thmtheorem1.p1.9.9.m9.1.1.2.cmml" xref="S2.Thmtheorem1.p1.9.9.m9.1.1.2">𝐼</ci><apply id="S2.Thmtheorem1.p1.9.9.m9.1.1.3.cmml" xref="S2.Thmtheorem1.p1.9.9.m9.1.1.3"><plus id="S2.Thmtheorem1.p1.9.9.m9.1.1.3.1.cmml" xref="S2.Thmtheorem1.p1.9.9.m9.1.1.3"></plus><ci id="S2.Thmtheorem1.p1.9.9.m9.1.1.3.2.cmml" xref="S2.Thmtheorem1.p1.9.9.m9.1.1.3.2">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.9.9.m9.1c">I_{+i}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.9.9.m9.1d">italic_I start_POSTSUBSCRIPT + italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="p_{i}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.10.10.m10.1"><semantics id="S2.Thmtheorem1.p1.10.10.m10.1a"><msub id="S2.Thmtheorem1.p1.10.10.m10.1.1" xref="S2.Thmtheorem1.p1.10.10.m10.1.1.cmml"><mi id="S2.Thmtheorem1.p1.10.10.m10.1.1.2" xref="S2.Thmtheorem1.p1.10.10.m10.1.1.2.cmml">p</mi><mi id="S2.Thmtheorem1.p1.10.10.m10.1.1.3" xref="S2.Thmtheorem1.p1.10.10.m10.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.10.10.m10.1b"><apply id="S2.Thmtheorem1.p1.10.10.m10.1.1.cmml" xref="S2.Thmtheorem1.p1.10.10.m10.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.10.10.m10.1.1.1.cmml" xref="S2.Thmtheorem1.p1.10.10.m10.1.1">subscript</csymbol><ci id="S2.Thmtheorem1.p1.10.10.m10.1.1.2.cmml" xref="S2.Thmtheorem1.p1.10.10.m10.1.1.2">𝑝</ci><ci id="S2.Thmtheorem1.p1.10.10.m10.1.1.3.cmml" xref="S2.Thmtheorem1.p1.10.10.m10.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.10.10.m10.1c">p_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.10.10.m10.1d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="I_{-i}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.11.11.m11.1"><semantics id="S2.Thmtheorem1.p1.11.11.m11.1a"><msub id="S2.Thmtheorem1.p1.11.11.m11.1.1" xref="S2.Thmtheorem1.p1.11.11.m11.1.1.cmml"><mi id="S2.Thmtheorem1.p1.11.11.m11.1.1.2" xref="S2.Thmtheorem1.p1.11.11.m11.1.1.2.cmml">I</mi><mrow id="S2.Thmtheorem1.p1.11.11.m11.1.1.3" xref="S2.Thmtheorem1.p1.11.11.m11.1.1.3.cmml"><mo id="S2.Thmtheorem1.p1.11.11.m11.1.1.3a" xref="S2.Thmtheorem1.p1.11.11.m11.1.1.3.cmml">−</mo><mi id="S2.Thmtheorem1.p1.11.11.m11.1.1.3.2" xref="S2.Thmtheorem1.p1.11.11.m11.1.1.3.2.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.11.11.m11.1b"><apply id="S2.Thmtheorem1.p1.11.11.m11.1.1.cmml" xref="S2.Thmtheorem1.p1.11.11.m11.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.11.11.m11.1.1.1.cmml" xref="S2.Thmtheorem1.p1.11.11.m11.1.1">subscript</csymbol><ci id="S2.Thmtheorem1.p1.11.11.m11.1.1.2.cmml" xref="S2.Thmtheorem1.p1.11.11.m11.1.1.2">𝐼</ci><apply id="S2.Thmtheorem1.p1.11.11.m11.1.1.3.cmml" xref="S2.Thmtheorem1.p1.11.11.m11.1.1.3"><minus id="S2.Thmtheorem1.p1.11.11.m11.1.1.3.1.cmml" xref="S2.Thmtheorem1.p1.11.11.m11.1.1.3"></minus><ci id="S2.Thmtheorem1.p1.11.11.m11.1.1.3.2.cmml" xref="S2.Thmtheorem1.p1.11.11.m11.1.1.3.2">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.11.11.m11.1c">I_{-i}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.11.11.m11.1d">italic_I start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="1-p_{i}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.12.12.m12.1"><semantics id="S2.Thmtheorem1.p1.12.12.m12.1a"><mrow id="S2.Thmtheorem1.p1.12.12.m12.1.1" xref="S2.Thmtheorem1.p1.12.12.m12.1.1.cmml"><mn id="S2.Thmtheorem1.p1.12.12.m12.1.1.2" xref="S2.Thmtheorem1.p1.12.12.m12.1.1.2.cmml">1</mn><mo id="S2.Thmtheorem1.p1.12.12.m12.1.1.1" xref="S2.Thmtheorem1.p1.12.12.m12.1.1.1.cmml">−</mo><msub id="S2.Thmtheorem1.p1.12.12.m12.1.1.3" xref="S2.Thmtheorem1.p1.12.12.m12.1.1.3.cmml"><mi id="S2.Thmtheorem1.p1.12.12.m12.1.1.3.2" xref="S2.Thmtheorem1.p1.12.12.m12.1.1.3.2.cmml">p</mi><mi id="S2.Thmtheorem1.p1.12.12.m12.1.1.3.3" xref="S2.Thmtheorem1.p1.12.12.m12.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.12.12.m12.1b"><apply id="S2.Thmtheorem1.p1.12.12.m12.1.1.cmml" xref="S2.Thmtheorem1.p1.12.12.m12.1.1"><minus id="S2.Thmtheorem1.p1.12.12.m12.1.1.1.cmml" xref="S2.Thmtheorem1.p1.12.12.m12.1.1.1"></minus><cn id="S2.Thmtheorem1.p1.12.12.m12.1.1.2.cmml" type="integer" xref="S2.Thmtheorem1.p1.12.12.m12.1.1.2">1</cn><apply id="S2.Thmtheorem1.p1.12.12.m12.1.1.3.cmml" xref="S2.Thmtheorem1.p1.12.12.m12.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.12.12.m12.1.1.3.1.cmml" xref="S2.Thmtheorem1.p1.12.12.m12.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem1.p1.12.12.m12.1.1.3.2.cmml" xref="S2.Thmtheorem1.p1.12.12.m12.1.1.3.2">𝑝</ci><ci id="S2.Thmtheorem1.p1.12.12.m12.1.1.3.3.cmml" xref="S2.Thmtheorem1.p1.12.12.m12.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.12.12.m12.1c">1-p_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.12.12.m12.1d">1 - italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="i\in[\ell]" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.13.13.m13.1"><semantics id="S2.Thmtheorem1.p1.13.13.m13.1a"><mrow id="S2.Thmtheorem1.p1.13.13.m13.1.2" xref="S2.Thmtheorem1.p1.13.13.m13.1.2.cmml"><mi id="S2.Thmtheorem1.p1.13.13.m13.1.2.2" xref="S2.Thmtheorem1.p1.13.13.m13.1.2.2.cmml">i</mi><mo id="S2.Thmtheorem1.p1.13.13.m13.1.2.1" xref="S2.Thmtheorem1.p1.13.13.m13.1.2.1.cmml">∈</mo><mrow id="S2.Thmtheorem1.p1.13.13.m13.1.2.3.2" xref="S2.Thmtheorem1.p1.13.13.m13.1.2.3.1.cmml"><mo id="S2.Thmtheorem1.p1.13.13.m13.1.2.3.2.1" stretchy="false" xref="S2.Thmtheorem1.p1.13.13.m13.1.2.3.1.1.cmml">[</mo><mi id="S2.Thmtheorem1.p1.13.13.m13.1.1" mathvariant="normal" xref="S2.Thmtheorem1.p1.13.13.m13.1.1.cmml">ℓ</mi><mo id="S2.Thmtheorem1.p1.13.13.m13.1.2.3.2.2" stretchy="false" xref="S2.Thmtheorem1.p1.13.13.m13.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.13.13.m13.1b"><apply id="S2.Thmtheorem1.p1.13.13.m13.1.2.cmml" xref="S2.Thmtheorem1.p1.13.13.m13.1.2"><in id="S2.Thmtheorem1.p1.13.13.m13.1.2.1.cmml" xref="S2.Thmtheorem1.p1.13.13.m13.1.2.1"></in><ci id="S2.Thmtheorem1.p1.13.13.m13.1.2.2.cmml" xref="S2.Thmtheorem1.p1.13.13.m13.1.2.2">𝑖</ci><apply id="S2.Thmtheorem1.p1.13.13.m13.1.2.3.1.cmml" xref="S2.Thmtheorem1.p1.13.13.m13.1.2.3.2"><csymbol cd="latexml" id="S2.Thmtheorem1.p1.13.13.m13.1.2.3.1.1.cmml" xref="S2.Thmtheorem1.p1.13.13.m13.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.Thmtheorem1.p1.13.13.m13.1.1.cmml" xref="S2.Thmtheorem1.p1.13.13.m13.1.1">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.13.13.m13.1c">i\in[\ell]</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.13.13.m13.1d">italic_i ∈ [ roman_ℓ ]</annotation></semantics></math> and <math alttext="I_{0}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.14.14.m14.1"><semantics id="S2.Thmtheorem1.p1.14.14.m14.1a"><msub id="S2.Thmtheorem1.p1.14.14.m14.1.1" xref="S2.Thmtheorem1.p1.14.14.m14.1.1.cmml"><mi id="S2.Thmtheorem1.p1.14.14.m14.1.1.2" xref="S2.Thmtheorem1.p1.14.14.m14.1.1.2.cmml">I</mi><mn id="S2.Thmtheorem1.p1.14.14.m14.1.1.3" xref="S2.Thmtheorem1.p1.14.14.m14.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.14.14.m14.1b"><apply id="S2.Thmtheorem1.p1.14.14.m14.1.1.cmml" xref="S2.Thmtheorem1.p1.14.14.m14.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.14.14.m14.1.1.1.cmml" xref="S2.Thmtheorem1.p1.14.14.m14.1.1">subscript</csymbol><ci id="S2.Thmtheorem1.p1.14.14.m14.1.1.2.cmml" xref="S2.Thmtheorem1.p1.14.14.m14.1.1.2">𝐼</ci><cn id="S2.Thmtheorem1.p1.14.14.m14.1.1.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.14.14.m14.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.14.14.m14.1c">I_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.14.14.m14.1d">italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="\frac{1}{2}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.15.15.m15.1"><semantics id="S2.Thmtheorem1.p1.15.15.m15.1a"><mfrac id="S2.Thmtheorem1.p1.15.15.m15.1.1" xref="S2.Thmtheorem1.p1.15.15.m15.1.1.cmml"><mn id="S2.Thmtheorem1.p1.15.15.m15.1.1.2" xref="S2.Thmtheorem1.p1.15.15.m15.1.1.2.cmml">1</mn><mn id="S2.Thmtheorem1.p1.15.15.m15.1.1.3" xref="S2.Thmtheorem1.p1.15.15.m15.1.1.3.cmml">2</mn></mfrac><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.15.15.m15.1b"><apply id="S2.Thmtheorem1.p1.15.15.m15.1.1.cmml" xref="S2.Thmtheorem1.p1.15.15.m15.1.1"><divide id="S2.Thmtheorem1.p1.15.15.m15.1.1.1.cmml" xref="S2.Thmtheorem1.p1.15.15.m15.1.1"></divide><cn id="S2.Thmtheorem1.p1.15.15.m15.1.1.2.cmml" type="integer" xref="S2.Thmtheorem1.p1.15.15.m15.1.1.2">1</cn><cn id="S2.Thmtheorem1.p1.15.15.m15.1.1.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.15.15.m15.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.15.15.m15.1c">\frac{1}{2}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.15.15.m15.1d">divide start_ARG 1 end_ARG start_ARG 2 end_ARG</annotation></semantics></math>. Then the approximation ratio <math alttext="\alpha(\mathcal{O}_{\mathsf{S}})" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.16.16.m16.1"><semantics id="S2.Thmtheorem1.p1.16.16.m16.1a"><mrow id="S2.Thmtheorem1.p1.16.16.m16.1.1" xref="S2.Thmtheorem1.p1.16.16.m16.1.1.cmml"><mi id="S2.Thmtheorem1.p1.16.16.m16.1.1.3" xref="S2.Thmtheorem1.p1.16.16.m16.1.1.3.cmml">α</mi><mo id="S2.Thmtheorem1.p1.16.16.m16.1.1.2" xref="S2.Thmtheorem1.p1.16.16.m16.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1" xref="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.1.cmml">(</mo><msub id="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.1" xref="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.1.2" xref="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.1.2.cmml">𝒪</mi><mi id="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.1.3" xref="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.1.3.cmml">𝖲</mi></msub><mo id="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.16.16.m16.1b"><apply id="S2.Thmtheorem1.p1.16.16.m16.1.1.cmml" xref="S2.Thmtheorem1.p1.16.16.m16.1.1"><times id="S2.Thmtheorem1.p1.16.16.m16.1.1.2.cmml" xref="S2.Thmtheorem1.p1.16.16.m16.1.1.2"></times><ci id="S2.Thmtheorem1.p1.16.16.m16.1.1.3.cmml" xref="S2.Thmtheorem1.p1.16.16.m16.1.1.3">𝛼</ci><apply id="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1">subscript</csymbol><ci id="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.1.2">𝒪</ci><ci id="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem1.p1.16.16.m16.1.1.1.1.1.3">𝖲</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.16.16.m16.1c">\alpha(\mathcal{O}_{\mathsf{S}})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.16.16.m16.1d">italic_α ( caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT )</annotation></semantics></math> achieved by <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.17.17.m17.1"><semantics id="S2.Thmtheorem1.p1.17.17.m17.1a"><msub id="S2.Thmtheorem1.p1.17.17.m17.1.1" xref="S2.Thmtheorem1.p1.17.17.m17.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem1.p1.17.17.m17.1.1.2" xref="S2.Thmtheorem1.p1.17.17.m17.1.1.2.cmml">𝒪</mi><mi id="S2.Thmtheorem1.p1.17.17.m17.1.1.3" xref="S2.Thmtheorem1.p1.17.17.m17.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.17.17.m17.1b"><apply id="S2.Thmtheorem1.p1.17.17.m17.1.1.cmml" xref="S2.Thmtheorem1.p1.17.17.m17.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.17.17.m17.1.1.1.cmml" xref="S2.Thmtheorem1.p1.17.17.m17.1.1">subscript</csymbol><ci id="S2.Thmtheorem1.p1.17.17.m17.1.1.2.cmml" xref="S2.Thmtheorem1.p1.17.17.m17.1.1.2">𝒪</ci><ci id="S2.Thmtheorem1.p1.17.17.m17.1.1.3.cmml" xref="S2.Thmtheorem1.p1.17.17.m17.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.17.17.m17.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.17.17.m17.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math> equals the value of the following linear program:</span></p> </div> <div class="ltx_para ltx_noindent" id="S2.Thmtheorem1.p2"> <svg class="ltx_picture" height="120.17" id="S2.Thmtheorem1.p2.pic1" overflow="visible" version="1.1" width="600"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,120.17) matrix(1 0 0 -1 0 0)"><g fill="#404040" fill-opacity="1.0"><path d="M 0 5.91 L 0 114.27 C 0 117.53 2.64 120.17 5.91 120.17 L 594.09 120.17 C 597.36 120.17 600 117.53 600 114.27 L 600 5.91 C 600 2.64 597.36 0 594.09 0 L 5.91 0 C 2.64 0 0 2.64 0 5.91 Z" style="stroke:none"></path></g><g fill="#F2F2F2" fill-opacity="1.0"><path d="M 1.97 5.91 L 1.97 114.27 C 1.97 116.44 3.73 118.21 5.91 118.21 L 594.09 118.21 C 596.27 118.21 598.03 116.44 598.03 114.27 L 598.03 5.91 C 598.03 3.73 596.27 1.97 594.09 1.97 L 5.91 1.97 C 3.73 1.97 1.97 3.73 1.97 5.91 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 21.65 13.78)"><foreignobject color="#000000" height="92.62" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="556.69"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="S2.Thmtheorem1.p2.pic1.1.1.1.1.1" style="width:402.3pt;"> <span class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx1"> <span id="S2.Ex8"><span class="ltx_equation ltx_eqn_row ltx_align_baseline"> <span class="ltx_eqn_cell ltx_eqn_center_padleft"></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\underset{\{w(\boldsymbol{c}):\boldsymbol{c}\in C\}}{\mathrm{% minimize}}\quad" class="ltx_Math" display="inline" id="S2.Ex8.m1.3"><semantics id="S2.Ex8.m1.3a"><mrow id="S2.Ex8.m1.3.4.2" xref="S2.Ex8.m1.3.3.cmml"><munder accentunder="true" id="S2.Ex8.m1.3.3" xref="S2.Ex8.m1.3.3.cmml"><mi id="S2.Ex8.m1.3.3.4" xref="S2.Ex8.m1.3.3.4.cmml">minimize</mi><mrow id="S2.Ex8.m1.3.3.3.3" xref="S2.Ex8.m1.3.3.3.4.cmml"><mo id="S2.Ex8.m1.3.3.3.3.3" stretchy="false" xref="S2.Ex8.m1.3.3.3.4.1.cmml">{</mo><mrow id="S2.Ex8.m1.2.2.2.2.1" xref="S2.Ex8.m1.2.2.2.2.1.cmml"><mi id="S2.Ex8.m1.2.2.2.2.1.2" xref="S2.Ex8.m1.2.2.2.2.1.2.cmml">w</mi><mo id="S2.Ex8.m1.2.2.2.2.1.1" xref="S2.Ex8.m1.2.2.2.2.1.1.cmml">⁢</mo><mrow id="S2.Ex8.m1.2.2.2.2.1.3.2" xref="S2.Ex8.m1.2.2.2.2.1.cmml"><mo id="S2.Ex8.m1.2.2.2.2.1.3.2.1" stretchy="false" xref="S2.Ex8.m1.2.2.2.2.1.cmml">(</mo><mi id="S2.Ex8.m1.1.1.1.1" xref="S2.Ex8.m1.1.1.1.1.cmml">𝒄</mi><mo id="S2.Ex8.m1.2.2.2.2.1.3.2.2" rspace="0.278em" stretchy="false" xref="S2.Ex8.m1.2.2.2.2.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex8.m1.3.3.3.3.4" rspace="0.278em" xref="S2.Ex8.m1.3.3.3.4.1.cmml">:</mo><mrow id="S2.Ex8.m1.3.3.3.3.2" xref="S2.Ex8.m1.3.3.3.3.2.cmml"><mi id="S2.Ex8.m1.3.3.3.3.2.2" xref="S2.Ex8.m1.3.3.3.3.2.2.cmml">𝒄</mi><mo id="S2.Ex8.m1.3.3.3.3.2.1" xref="S2.Ex8.m1.3.3.3.3.2.1.cmml">∈</mo><mi id="S2.Ex8.m1.3.3.3.3.2.3" xref="S2.Ex8.m1.3.3.3.3.2.3.cmml">C</mi></mrow><mo id="S2.Ex8.m1.3.3.3.3.5" stretchy="false" xref="S2.Ex8.m1.3.3.3.4.1.cmml">}</mo></mrow></munder><mspace id="S2.Ex8.m1.3.4.2.1" width="1em" xref="S2.Ex8.m1.3.3.cmml"></mspace></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex8.m1.3b"><apply id="S2.Ex8.m1.3.3.cmml" xref="S2.Ex8.m1.3.4.2"><apply id="S2.Ex8.m1.3.3.3.4.cmml" xref="S2.Ex8.m1.3.3.3.3"><csymbol cd="latexml" id="S2.Ex8.m1.3.3.3.4.1.cmml" xref="S2.Ex8.m1.3.3.3.3.3">conditional-set</csymbol><apply id="S2.Ex8.m1.2.2.2.2.1.cmml" xref="S2.Ex8.m1.2.2.2.2.1"><times id="S2.Ex8.m1.2.2.2.2.1.1.cmml" xref="S2.Ex8.m1.2.2.2.2.1.1"></times><ci id="S2.Ex8.m1.2.2.2.2.1.2.cmml" xref="S2.Ex8.m1.2.2.2.2.1.2">𝑤</ci><ci id="S2.Ex8.m1.1.1.1.1.cmml" xref="S2.Ex8.m1.1.1.1.1">𝒄</ci></apply><apply id="S2.Ex8.m1.3.3.3.3.2.cmml" xref="S2.Ex8.m1.3.3.3.3.2"><in id="S2.Ex8.m1.3.3.3.3.2.1.cmml" xref="S2.Ex8.m1.3.3.3.3.2.1"></in><ci id="S2.Ex8.m1.3.3.3.3.2.2.cmml" xref="S2.Ex8.m1.3.3.3.3.2.2">𝒄</ci><ci id="S2.Ex8.m1.3.3.3.3.2.3.cmml" xref="S2.Ex8.m1.3.3.3.3.2.3">𝐶</ci></apply></apply><ci id="S2.Ex8.m1.3.3.4.cmml" xref="S2.Ex8.m1.3.3.4">minimize</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex8.m1.3c">\displaystyle\underset{\{w(\boldsymbol{c}):\boldsymbol{c}\in C\}}{\mathrm{% minimize}}\quad</annotation><annotation encoding="application/x-llamapun" id="S2.Ex8.m1.3d">start_UNDERACCENT { italic_w ( bold_italic_c ) : bold_italic_c ∈ italic_C } end_UNDERACCENT start_ARG roman_minimize end_ARG</annotation></semantics></math></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{\boldsymbol{c}\in C}p(\boldsymbol{c})w(\boldsymbol{c})" class="ltx_Math" display="inline" id="S2.Ex8.m2.2"><semantics id="S2.Ex8.m2.2a"><mrow id="S2.Ex8.m2.2.3" xref="S2.Ex8.m2.2.3.cmml"><mstyle displaystyle="true" id="S2.Ex8.m2.2.3.1" xref="S2.Ex8.m2.2.3.1.cmml"><munder id="S2.Ex8.m2.2.3.1a" xref="S2.Ex8.m2.2.3.1.cmml"><mo id="S2.Ex8.m2.2.3.1.2" movablelimits="false" xref="S2.Ex8.m2.2.3.1.2.cmml">∑</mo><mrow id="S2.Ex8.m2.2.3.1.3" xref="S2.Ex8.m2.2.3.1.3.cmml"><mi id="S2.Ex8.m2.2.3.1.3.2" xref="S2.Ex8.m2.2.3.1.3.2.cmml">𝒄</mi><mo id="S2.Ex8.m2.2.3.1.3.1" xref="S2.Ex8.m2.2.3.1.3.1.cmml">∈</mo><mi id="S2.Ex8.m2.2.3.1.3.3" xref="S2.Ex8.m2.2.3.1.3.3.cmml">C</mi></mrow></munder></mstyle><mrow id="S2.Ex8.m2.2.3.2" xref="S2.Ex8.m2.2.3.2.cmml"><mi id="S2.Ex8.m2.2.3.2.2" xref="S2.Ex8.m2.2.3.2.2.cmml">p</mi><mo id="S2.Ex8.m2.2.3.2.1" xref="S2.Ex8.m2.2.3.2.1.cmml">⁢</mo><mrow id="S2.Ex8.m2.2.3.2.3.2" xref="S2.Ex8.m2.2.3.2.cmml"><mo id="S2.Ex8.m2.2.3.2.3.2.1" stretchy="false" xref="S2.Ex8.m2.2.3.2.cmml">(</mo><mi id="S2.Ex8.m2.1.1" xref="S2.Ex8.m2.1.1.cmml">𝒄</mi><mo id="S2.Ex8.m2.2.3.2.3.2.2" stretchy="false" xref="S2.Ex8.m2.2.3.2.cmml">)</mo></mrow><mo id="S2.Ex8.m2.2.3.2.1a" xref="S2.Ex8.m2.2.3.2.1.cmml">⁢</mo><mi id="S2.Ex8.m2.2.3.2.4" xref="S2.Ex8.m2.2.3.2.4.cmml">w</mi><mo id="S2.Ex8.m2.2.3.2.1b" xref="S2.Ex8.m2.2.3.2.1.cmml">⁢</mo><mrow id="S2.Ex8.m2.2.3.2.5.2" xref="S2.Ex8.m2.2.3.2.cmml"><mo id="S2.Ex8.m2.2.3.2.5.2.1" stretchy="false" xref="S2.Ex8.m2.2.3.2.cmml">(</mo><mi id="S2.Ex8.m2.2.2" xref="S2.Ex8.m2.2.2.cmml">𝒄</mi><mo id="S2.Ex8.m2.2.3.2.5.2.2" stretchy="false" xref="S2.Ex8.m2.2.3.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex8.m2.2b"><apply id="S2.Ex8.m2.2.3.cmml" xref="S2.Ex8.m2.2.3"><apply id="S2.Ex8.m2.2.3.1.cmml" xref="S2.Ex8.m2.2.3.1"><csymbol cd="ambiguous" id="S2.Ex8.m2.2.3.1.1.cmml" xref="S2.Ex8.m2.2.3.1">subscript</csymbol><sum id="S2.Ex8.m2.2.3.1.2.cmml" xref="S2.Ex8.m2.2.3.1.2"></sum><apply id="S2.Ex8.m2.2.3.1.3.cmml" xref="S2.Ex8.m2.2.3.1.3"><in id="S2.Ex8.m2.2.3.1.3.1.cmml" xref="S2.Ex8.m2.2.3.1.3.1"></in><ci id="S2.Ex8.m2.2.3.1.3.2.cmml" xref="S2.Ex8.m2.2.3.1.3.2">𝒄</ci><ci id="S2.Ex8.m2.2.3.1.3.3.cmml" xref="S2.Ex8.m2.2.3.1.3.3">𝐶</ci></apply></apply><apply id="S2.Ex8.m2.2.3.2.cmml" xref="S2.Ex8.m2.2.3.2"><times id="S2.Ex8.m2.2.3.2.1.cmml" xref="S2.Ex8.m2.2.3.2.1"></times><ci id="S2.Ex8.m2.2.3.2.2.cmml" xref="S2.Ex8.m2.2.3.2.2">𝑝</ci><ci id="S2.Ex8.m2.1.1.cmml" xref="S2.Ex8.m2.1.1">𝒄</ci><ci id="S2.Ex8.m2.2.3.2.4.cmml" xref="S2.Ex8.m2.2.3.2.4">𝑤</ci><ci id="S2.Ex8.m2.2.2.cmml" xref="S2.Ex8.m2.2.2">𝒄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex8.m2.2c">\displaystyle\sum_{\boldsymbol{c}\in C}p(\boldsymbol{c})w(\boldsymbol{c})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex8.m2.2d">∑ start_POSTSUBSCRIPT bold_italic_c ∈ italic_C end_POSTSUBSCRIPT italic_p ( bold_italic_c ) italic_w ( bold_italic_c )</annotation></semantics></math></span> <span class="ltx_eqn_cell ltx_eqn_center_padright" colspan="2"></span></span></span> <span id="S2.Ex9"><span class="ltx_equation ltx_eqn_row ltx_align_baseline"> <span class="ltx_eqn_cell ltx_eqn_center_padleft"></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mathrm{s.t.}" class="ltx_Math" display="inline" id="S2.Ex9.m1.3"><semantics id="S2.Ex9.m1.3a"><mrow id="S2.Ex9.m1.3.3.1"><mrow id="S2.Ex9.m1.3.3.1.1.2" xref="S2.Ex9.m1.3.3.1.1.1.cmml"><mi id="S2.Ex9.m1.1.1" mathvariant="normal" xref="S2.Ex9.m1.1.1.cmml">s</mi><mo id="S2.Ex9.m1.3.3.1.1.2.1" lspace="0em" rspace="0.167em" xref="S2.Ex9.m1.3.3.1.1.1a.cmml">.</mo><mi id="S2.Ex9.m1.2.2" mathvariant="normal" xref="S2.Ex9.m1.2.2.cmml">t</mi></mrow><mo id="S2.Ex9.m1.3.3.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex9.m1.3b"><apply id="S2.Ex9.m1.3.3.1.1.1.cmml" xref="S2.Ex9.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S2.Ex9.m1.3.3.1.1.1a.cmml" xref="S2.Ex9.m1.3.3.1.1.2.1">formulae-sequence</csymbol><ci id="S2.Ex9.m1.1.1.cmml" xref="S2.Ex9.m1.1.1">s</ci><ci id="S2.Ex9.m1.2.2.cmml" xref="S2.Ex9.m1.2.2">t</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex9.m1.3c">\displaystyle\mathrm{s.t.}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex9.m1.3d">roman_s . roman_t .</annotation></semantics></math></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle w(\boldsymbol{c})\geq 0" class="ltx_Math" display="inline" id="S2.Ex9.m2.1"><semantics id="S2.Ex9.m2.1a"><mrow id="S2.Ex9.m2.1.2" xref="S2.Ex9.m2.1.2.cmml"><mrow id="S2.Ex9.m2.1.2.2" xref="S2.Ex9.m2.1.2.2.cmml"><mi id="S2.Ex9.m2.1.2.2.2" xref="S2.Ex9.m2.1.2.2.2.cmml">w</mi><mo id="S2.Ex9.m2.1.2.2.1" xref="S2.Ex9.m2.1.2.2.1.cmml">⁢</mo><mrow id="S2.Ex9.m2.1.2.2.3.2" xref="S2.Ex9.m2.1.2.2.cmml"><mo id="S2.Ex9.m2.1.2.2.3.2.1" stretchy="false" xref="S2.Ex9.m2.1.2.2.cmml">(</mo><mi id="S2.Ex9.m2.1.1" xref="S2.Ex9.m2.1.1.cmml">𝒄</mi><mo id="S2.Ex9.m2.1.2.2.3.2.2" stretchy="false" xref="S2.Ex9.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Ex9.m2.1.2.1" xref="S2.Ex9.m2.1.2.1.cmml">≥</mo><mn id="S2.Ex9.m2.1.2.3" xref="S2.Ex9.m2.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex9.m2.1b"><apply id="S2.Ex9.m2.1.2.cmml" xref="S2.Ex9.m2.1.2"><geq id="S2.Ex9.m2.1.2.1.cmml" xref="S2.Ex9.m2.1.2.1"></geq><apply id="S2.Ex9.m2.1.2.2.cmml" xref="S2.Ex9.m2.1.2.2"><times id="S2.Ex9.m2.1.2.2.1.cmml" xref="S2.Ex9.m2.1.2.2.1"></times><ci id="S2.Ex9.m2.1.2.2.2.cmml" xref="S2.Ex9.m2.1.2.2.2">𝑤</ci><ci id="S2.Ex9.m2.1.1.cmml" xref="S2.Ex9.m2.1.1">𝒄</ci></apply><cn id="S2.Ex9.m2.1.2.3.cmml" type="integer" xref="S2.Ex9.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex9.m2.1c">\displaystyle w(\boldsymbol{c})\geq 0</annotation><annotation encoding="application/x-llamapun" id="S2.Ex9.m2.1d">italic_w ( bold_italic_c ) ≥ 0</annotation></semantics></math></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\forall\boldsymbol{c}\in C" class="ltx_Math" display="inline" id="S2.Ex9.m3.1"><semantics id="S2.Ex9.m3.1a"><mrow id="S2.Ex9.m3.1.1" xref="S2.Ex9.m3.1.1.cmml"><mrow id="S2.Ex9.m3.1.1.2" xref="S2.Ex9.m3.1.1.2.cmml"><mo id="S2.Ex9.m3.1.1.2.1" rspace="0.167em" xref="S2.Ex9.m3.1.1.2.1.cmml">∀</mo><mi id="S2.Ex9.m3.1.1.2.2" xref="S2.Ex9.m3.1.1.2.2.cmml">𝒄</mi></mrow><mo id="S2.Ex9.m3.1.1.1" xref="S2.Ex9.m3.1.1.1.cmml">∈</mo><mi id="S2.Ex9.m3.1.1.3" xref="S2.Ex9.m3.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex9.m3.1b"><apply id="S2.Ex9.m3.1.1.cmml" xref="S2.Ex9.m3.1.1"><in id="S2.Ex9.m3.1.1.1.cmml" xref="S2.Ex9.m3.1.1.1"></in><apply id="S2.Ex9.m3.1.1.2.cmml" xref="S2.Ex9.m3.1.1.2"><csymbol cd="latexml" id="S2.Ex9.m3.1.1.2.1.cmml" xref="S2.Ex9.m3.1.1.2.1">for-all</csymbol><ci id="S2.Ex9.m3.1.1.2.2.cmml" xref="S2.Ex9.m3.1.1.2.2">𝒄</ci></apply><ci id="S2.Ex9.m3.1.1.3.cmml" xref="S2.Ex9.m3.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex9.m3.1c">\displaystyle\forall\boldsymbol{c}\in C</annotation><annotation encoding="application/x-llamapun" id="S2.Ex9.m3.1d">∀ bold_italic_c ∈ italic_C</annotation></semantics></math></span> <span class="ltx_eqn_cell ltx_eqn_center_padright"></span></span></span> <span id="S2.Ex10"><span class="ltx_equation ltx_eqn_row ltx_align_baseline"> <span class="ltx_eqn_cell ltx_eqn_center_padleft"></span> <span class="ltx_td ltx_eqn_cell"></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{\boldsymbol{c}\in C^{+}}w(\boldsymbol{c})=1" class="ltx_Math" display="inline" id="S2.Ex10.m1.1"><semantics id="S2.Ex10.m1.1a"><mrow id="S2.Ex10.m1.1.2" xref="S2.Ex10.m1.1.2.cmml"><mrow id="S2.Ex10.m1.1.2.2" xref="S2.Ex10.m1.1.2.2.cmml"><mstyle displaystyle="true" id="S2.Ex10.m1.1.2.2.1" xref="S2.Ex10.m1.1.2.2.1.cmml"><munder id="S2.Ex10.m1.1.2.2.1a" xref="S2.Ex10.m1.1.2.2.1.cmml"><mo id="S2.Ex10.m1.1.2.2.1.2" movablelimits="false" xref="S2.Ex10.m1.1.2.2.1.2.cmml">∑</mo><mrow id="S2.Ex10.m1.1.2.2.1.3" xref="S2.Ex10.m1.1.2.2.1.3.cmml"><mi id="S2.Ex10.m1.1.2.2.1.3.2" xref="S2.Ex10.m1.1.2.2.1.3.2.cmml">𝒄</mi><mo id="S2.Ex10.m1.1.2.2.1.3.1" xref="S2.Ex10.m1.1.2.2.1.3.1.cmml">∈</mo><msup id="S2.Ex10.m1.1.2.2.1.3.3" xref="S2.Ex10.m1.1.2.2.1.3.3.cmml"><mi id="S2.Ex10.m1.1.2.2.1.3.3.2" xref="S2.Ex10.m1.1.2.2.1.3.3.2.cmml">C</mi><mo id="S2.Ex10.m1.1.2.2.1.3.3.3" xref="S2.Ex10.m1.1.2.2.1.3.3.3.cmml">+</mo></msup></mrow></munder></mstyle><mrow id="S2.Ex10.m1.1.2.2.2" xref="S2.Ex10.m1.1.2.2.2.cmml"><mi id="S2.Ex10.m1.1.2.2.2.2" xref="S2.Ex10.m1.1.2.2.2.2.cmml">w</mi><mo id="S2.Ex10.m1.1.2.2.2.1" xref="S2.Ex10.m1.1.2.2.2.1.cmml">⁢</mo><mrow id="S2.Ex10.m1.1.2.2.2.3.2" xref="S2.Ex10.m1.1.2.2.2.cmml"><mo id="S2.Ex10.m1.1.2.2.2.3.2.1" stretchy="false" xref="S2.Ex10.m1.1.2.2.2.cmml">(</mo><mi id="S2.Ex10.m1.1.1" xref="S2.Ex10.m1.1.1.cmml">𝒄</mi><mo id="S2.Ex10.m1.1.2.2.2.3.2.2" stretchy="false" xref="S2.Ex10.m1.1.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex10.m1.1.2.1" xref="S2.Ex10.m1.1.2.1.cmml">=</mo><mn id="S2.Ex10.m1.1.2.3" xref="S2.Ex10.m1.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex10.m1.1b"><apply id="S2.Ex10.m1.1.2.cmml" xref="S2.Ex10.m1.1.2"><eq id="S2.Ex10.m1.1.2.1.cmml" xref="S2.Ex10.m1.1.2.1"></eq><apply id="S2.Ex10.m1.1.2.2.cmml" xref="S2.Ex10.m1.1.2.2"><apply id="S2.Ex10.m1.1.2.2.1.cmml" xref="S2.Ex10.m1.1.2.2.1"><csymbol cd="ambiguous" id="S2.Ex10.m1.1.2.2.1.1.cmml" xref="S2.Ex10.m1.1.2.2.1">subscript</csymbol><sum id="S2.Ex10.m1.1.2.2.1.2.cmml" xref="S2.Ex10.m1.1.2.2.1.2"></sum><apply id="S2.Ex10.m1.1.2.2.1.3.cmml" xref="S2.Ex10.m1.1.2.2.1.3"><in id="S2.Ex10.m1.1.2.2.1.3.1.cmml" xref="S2.Ex10.m1.1.2.2.1.3.1"></in><ci id="S2.Ex10.m1.1.2.2.1.3.2.cmml" xref="S2.Ex10.m1.1.2.2.1.3.2">𝒄</ci><apply id="S2.Ex10.m1.1.2.2.1.3.3.cmml" xref="S2.Ex10.m1.1.2.2.1.3.3"><csymbol cd="ambiguous" id="S2.Ex10.m1.1.2.2.1.3.3.1.cmml" xref="S2.Ex10.m1.1.2.2.1.3.3">superscript</csymbol><ci id="S2.Ex10.m1.1.2.2.1.3.3.2.cmml" xref="S2.Ex10.m1.1.2.2.1.3.3.2">𝐶</ci><plus id="S2.Ex10.m1.1.2.2.1.3.3.3.cmml" xref="S2.Ex10.m1.1.2.2.1.3.3.3"></plus></apply></apply></apply><apply id="S2.Ex10.m1.1.2.2.2.cmml" xref="S2.Ex10.m1.1.2.2.2"><times id="S2.Ex10.m1.1.2.2.2.1.cmml" xref="S2.Ex10.m1.1.2.2.2.1"></times><ci id="S2.Ex10.m1.1.2.2.2.2.cmml" xref="S2.Ex10.m1.1.2.2.2.2">𝑤</ci><ci id="S2.Ex10.m1.1.1.cmml" xref="S2.Ex10.m1.1.1">𝒄</ci></apply></apply><cn id="S2.Ex10.m1.1.2.3.cmml" type="integer" xref="S2.Ex10.m1.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex10.m1.1c">\displaystyle\sum_{\boldsymbol{c}\in C^{+}}w(\boldsymbol{c})=1</annotation><annotation encoding="application/x-llamapun" id="S2.Ex10.m1.1d">∑ start_POSTSUBSCRIPT bold_italic_c ∈ italic_C start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_w ( bold_italic_c ) = 1</annotation></semantics></math></span> <span class="ltx_eqn_cell ltx_eqn_center_padright" colspan="2"></span></span></span> <span id="S2.Ex11"><span class="ltx_equation ltx_eqn_row ltx_align_baseline"> <span class="ltx_eqn_cell ltx_eqn_center_padleft"></span> <span class="ltx_td ltx_eqn_cell"></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle b_{i}(W^{+}(i)+W^{-}(i))\leq W^{+}(i)-W^{-}(i)\quad" class="ltx_Math" display="inline" id="S2.Ex11.m1.5"><semantics id="S2.Ex11.m1.5a"><mrow id="S2.Ex11.m1.5.5.1" xref="S2.Ex11.m1.5.5.1.1.cmml"><mrow id="S2.Ex11.m1.5.5.1.1" 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xref="S2.Ex12.m1.4.4">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex12.m1.5c">\displaystyle W^{+}(i)-W^{-}(i)\leq a_{i}(W^{+}(i)+W^{-}(i))\quad</annotation><annotation encoding="application/x-llamapun" id="S2.Ex12.m1.5d">italic_W start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_i ) - italic_W start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_i ) ≤ italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_W start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_i ) + italic_W start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_i ) )</annotation></semantics></math></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\forall i\in[\pm\ell]" class="ltx_Math" display="inline" id="S2.Ex12.m2.1"><semantics id="S2.Ex12.m2.1a"><mrow id="S2.Ex12.m2.1.1" xref="S2.Ex12.m2.1.1.cmml"><mrow id="S2.Ex12.m2.1.1.3" xref="S2.Ex12.m2.1.1.3.cmml"><mo id="S2.Ex12.m2.1.1.3.1" rspace="0.167em" 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id="S2.Ex12.m2.1.1.3.2.cmml" xref="S2.Ex12.m2.1.1.3.2">𝑖</ci></apply><apply id="S2.Ex12.m2.1.1.1.2.cmml" xref="S2.Ex12.m2.1.1.1.1"><csymbol cd="latexml" id="S2.Ex12.m2.1.1.1.2.1.cmml" xref="S2.Ex12.m2.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.Ex12.m2.1.1.1.1.1.cmml" xref="S2.Ex12.m2.1.1.1.1.1"><csymbol cd="latexml" id="S2.Ex12.m2.1.1.1.1.1.1.cmml" xref="S2.Ex12.m2.1.1.1.1.1">plus-or-minus</csymbol><ci id="S2.Ex12.m2.1.1.1.1.1.2.cmml" xref="S2.Ex12.m2.1.1.1.1.1.2">ℓ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex12.m2.1c">\displaystyle\forall i\in[\pm\ell]</annotation><annotation encoding="application/x-llamapun" id="S2.Ex12.m2.1d">∀ italic_i ∈ [ ± roman_ℓ ]</annotation></semantics></math></span> <span class="ltx_eqn_cell ltx_eqn_center_padright"></span></span></span> </span> </span></foreignobject></g></g></svg> </div> <div class="ltx_para" id="S2.Thmtheorem1.p3"> <p class="ltx_p" id="S2.Thmtheorem1.p3.1"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem1.p3.1.1">where we define the set of <math alttext="\mathbb{N}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p3.1.1.m1.1"><semantics id="S2.Thmtheorem1.p3.1.1.m1.1a"><mi id="S2.Thmtheorem1.p3.1.1.m1.1.1" xref="S2.Thmtheorem1.p3.1.1.m1.1.1.cmml">ℕ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p3.1.1.m1.1b"><ci id="S2.Thmtheorem1.p3.1.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p3.1.1.m1.1.1">ℕ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p3.1.1.m1.1c">\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p3.1.1.m1.1d">blackboard_N</annotation></semantics></math>-valued matrices:</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex13"> <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="C=\left\{\boldsymbol{c}\in\mathbb{N}^{\{\pm 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id="S2.Ex13.m1.7c">C=\left\{\boldsymbol{c}\in\mathbb{N}^{\{\pm 1\}\times[\pm\ell]}:\sum_{b\in\{% \pm 1\}}\sum_{i\in[\pm\ell]}c_{b,i}=2\right\};</annotation><annotation encoding="application/x-llamapun" id="S2.Ex13.m1.7d">italic_C = { bold_italic_c ∈ blackboard_N start_POSTSUPERSCRIPT { ± 1 } × [ ± roman_ℓ ] end_POSTSUPERSCRIPT : ∑ start_POSTSUBSCRIPT italic_b ∈ { ± 1 } end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i ∈ [ ± roman_ℓ ] end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_b , italic_i end_POSTSUBSCRIPT = 2 } ;</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem1.p3.3"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem1.p3.3.2">the subset of <math alttext="C" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p3.2.1.m1.1"><semantics id="S2.Thmtheorem1.p3.2.1.m1.1a"><mi id="S2.Thmtheorem1.p3.2.1.m1.1.1" xref="S2.Thmtheorem1.p3.2.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p3.2.1.m1.1b"><ci id="S2.Thmtheorem1.p3.2.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p3.2.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p3.2.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p3.2.1.m1.1d">italic_C</annotation></semantics></math> supported on the <math alttext="+1" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p3.3.2.m2.1"><semantics id="S2.Thmtheorem1.p3.3.2.m2.1a"><mrow id="S2.Thmtheorem1.p3.3.2.m2.1.1" xref="S2.Thmtheorem1.p3.3.2.m2.1.1.cmml"><mo id="S2.Thmtheorem1.p3.3.2.m2.1.1a" xref="S2.Thmtheorem1.p3.3.2.m2.1.1.cmml">+</mo><mn id="S2.Thmtheorem1.p3.3.2.m2.1.1.2" xref="S2.Thmtheorem1.p3.3.2.m2.1.1.2.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p3.3.2.m2.1b"><apply id="S2.Thmtheorem1.p3.3.2.m2.1.1.cmml" xref="S2.Thmtheorem1.p3.3.2.m2.1.1"><plus id="S2.Thmtheorem1.p3.3.2.m2.1.1.1.cmml" xref="S2.Thmtheorem1.p3.3.2.m2.1.1"></plus><cn id="S2.Thmtheorem1.p3.3.2.m2.1.1.2.cmml" type="integer" xref="S2.Thmtheorem1.p3.3.2.m2.1.1.2">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p3.3.2.m2.1c">+1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p3.3.2.m2.1d">+ 1</annotation></semantics></math> row:</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex14"> <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="C^{+}=\{\boldsymbol{c}\in C:c_{-1,i}=0\quad\forall i\in[\pm\ell]\};" class="ltx_Math" display="block" id="S2.Ex14.m1.3"><semantics id="S2.Ex14.m1.3a"><mrow id="S2.Ex14.m1.3.3.1" xref="S2.Ex14.m1.3.3.1.1.cmml"><mrow id="S2.Ex14.m1.3.3.1.1" xref="S2.Ex14.m1.3.3.1.1.cmml"><msup id="S2.Ex14.m1.3.3.1.1.4" xref="S2.Ex14.m1.3.3.1.1.4.cmml"><mi id="S2.Ex14.m1.3.3.1.1.4.2" xref="S2.Ex14.m1.3.3.1.1.4.2.cmml">C</mi><mo id="S2.Ex14.m1.3.3.1.1.4.3" xref="S2.Ex14.m1.3.3.1.1.4.3.cmml">+</mo></msup><mo id="S2.Ex14.m1.3.3.1.1.3" xref="S2.Ex14.m1.3.3.1.1.3.cmml">=</mo><mrow id="S2.Ex14.m1.3.3.1.1.2.2" xref="S2.Ex14.m1.3.3.1.1.2.3.cmml"><mo id="S2.Ex14.m1.3.3.1.1.2.2.3" stretchy="false" xref="S2.Ex14.m1.3.3.1.1.2.3.1.cmml">{</mo><mrow id="S2.Ex14.m1.3.3.1.1.1.1.1" xref="S2.Ex14.m1.3.3.1.1.1.1.1.cmml"><mi id="S2.Ex14.m1.3.3.1.1.1.1.1.2" xref="S2.Ex14.m1.3.3.1.1.1.1.1.2.cmml">𝒄</mi><mo id="S2.Ex14.m1.3.3.1.1.1.1.1.1" xref="S2.Ex14.m1.3.3.1.1.1.1.1.1.cmml">∈</mo><mi id="S2.Ex14.m1.3.3.1.1.1.1.1.3" xref="S2.Ex14.m1.3.3.1.1.1.1.1.3.cmml">C</mi></mrow><mo id="S2.Ex14.m1.3.3.1.1.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.Ex14.m1.3.3.1.1.2.3.1.cmml">:</mo><mrow id="S2.Ex14.m1.3.3.1.1.2.2.2.2" xref="S2.Ex14.m1.3.3.1.1.2.2.2.3.cmml"><mrow id="S2.Ex14.m1.3.3.1.1.2.2.2.1.1" xref="S2.Ex14.m1.3.3.1.1.2.2.2.1.1.cmml"><msub id="S2.Ex14.m1.3.3.1.1.2.2.2.1.1.2" xref="S2.Ex14.m1.3.3.1.1.2.2.2.1.1.2.cmml"><mi id="S2.Ex14.m1.3.3.1.1.2.2.2.1.1.2.2" xref="S2.Ex14.m1.3.3.1.1.2.2.2.1.1.2.2.cmml">c</mi><mrow id="S2.Ex14.m1.2.2.2.2" xref="S2.Ex14.m1.2.2.2.3.cmml"><mrow id="S2.Ex14.m1.2.2.2.2.1" xref="S2.Ex14.m1.2.2.2.2.1.cmml"><mo id="S2.Ex14.m1.2.2.2.2.1a" xref="S2.Ex14.m1.2.2.2.2.1.cmml">−</mo><mn id="S2.Ex14.m1.2.2.2.2.1.2" xref="S2.Ex14.m1.2.2.2.2.1.2.cmml">1</mn></mrow><mo id="S2.Ex14.m1.2.2.2.2.2" xref="S2.Ex14.m1.2.2.2.3.cmml">,</mo><mi id="S2.Ex14.m1.1.1.1.1" xref="S2.Ex14.m1.1.1.1.1.cmml">i</mi></mrow></msub><mo id="S2.Ex14.m1.3.3.1.1.2.2.2.1.1.1" xref="S2.Ex14.m1.3.3.1.1.2.2.2.1.1.1.cmml">=</mo><mn id="S2.Ex14.m1.3.3.1.1.2.2.2.1.1.3" xref="S2.Ex14.m1.3.3.1.1.2.2.2.1.1.3.cmml">0</mn></mrow><mspace id="S2.Ex14.m1.3.3.1.1.2.2.2.2.3" width="1.167em" xref="S2.Ex14.m1.3.3.1.1.2.2.2.3a.cmml"></mspace><mrow id="S2.Ex14.m1.3.3.1.1.2.2.2.2.2" xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.cmml"><mrow id="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.3" xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.3.cmml"><mo id="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.3.1" rspace="0.167em" xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.3.1.cmml">∀</mo><mi id="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.3.2" xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.3.2.cmml">i</mi></mrow><mo id="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.2" xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.2.cmml">∈</mo><mrow id="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1" xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.2.cmml"><mo id="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1.2" stretchy="false" xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.2.1.cmml">[</mo><mrow id="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1.1" xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1.1.cmml"><mo id="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1.1a" xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1.1.cmml">±</mo><mi id="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1.1.2" mathvariant="normal" xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1.1.2.cmml">ℓ</mi></mrow><mo id="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1.3" stretchy="false" xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.2.1.cmml">]</mo></mrow></mrow></mrow><mo id="S2.Ex14.m1.3.3.1.1.2.2.5" stretchy="false" xref="S2.Ex14.m1.3.3.1.1.2.3.1.cmml">}</mo></mrow></mrow><mo id="S2.Ex14.m1.3.3.1.2" xref="S2.Ex14.m1.3.3.1.1.cmml">;</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex14.m1.3b"><apply id="S2.Ex14.m1.3.3.1.1.cmml" xref="S2.Ex14.m1.3.3.1"><eq id="S2.Ex14.m1.3.3.1.1.3.cmml" xref="S2.Ex14.m1.3.3.1.1.3"></eq><apply id="S2.Ex14.m1.3.3.1.1.4.cmml" xref="S2.Ex14.m1.3.3.1.1.4"><csymbol cd="ambiguous" id="S2.Ex14.m1.3.3.1.1.4.1.cmml" xref="S2.Ex14.m1.3.3.1.1.4">superscript</csymbol><ci id="S2.Ex14.m1.3.3.1.1.4.2.cmml" xref="S2.Ex14.m1.3.3.1.1.4.2">𝐶</ci><plus id="S2.Ex14.m1.3.3.1.1.4.3.cmml" xref="S2.Ex14.m1.3.3.1.1.4.3"></plus></apply><apply id="S2.Ex14.m1.3.3.1.1.2.3.cmml" xref="S2.Ex14.m1.3.3.1.1.2.2"><csymbol cd="latexml" id="S2.Ex14.m1.3.3.1.1.2.3.1.cmml" xref="S2.Ex14.m1.3.3.1.1.2.2.3">conditional-set</csymbol><apply id="S2.Ex14.m1.3.3.1.1.1.1.1.cmml" 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xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1"><csymbol cd="latexml" id="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.2.1.cmml" xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1.2">delimited-[]</csymbol><apply id="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1.1.cmml" xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1.1"><csymbol cd="latexml" id="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1.1.1.cmml" xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1.1">plus-or-minus</csymbol><ci id="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1.1.2.cmml" xref="S2.Ex14.m1.3.3.1.1.2.2.2.2.2.1.1.1.2">ℓ</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex14.m1.3c">C^{+}=\{\boldsymbol{c}\in C:c_{-1,i}=0\quad\forall i\in[\pm\ell]\};</annotation><annotation encoding="application/x-llamapun" id="S2.Ex14.m1.3d">italic_C start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT = { bold_italic_c ∈ italic_C : italic_c start_POSTSUBSCRIPT - 1 , italic_i end_POSTSUBSCRIPT = 0 ∀ italic_i ∈ [ ± roman_ℓ ] } ;</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem1.p3.4"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem1.p3.4.1">a constant for each vector in <math alttext="C" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p3.4.1.m1.1"><semantics id="S2.Thmtheorem1.p3.4.1.m1.1a"><mi id="S2.Thmtheorem1.p3.4.1.m1.1.1" xref="S2.Thmtheorem1.p3.4.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p3.4.1.m1.1b"><ci id="S2.Thmtheorem1.p3.4.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p3.4.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p3.4.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p3.4.1.m1.1d">italic_C</annotation></semantics></math>:</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex15"> <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="p(\boldsymbol{c})=\left(\frac{1}{2}\right)^{c(+1,0)+c(-1,0)}\prod_{i\in[\ell]}% p_{i}^{c(+1,+i)+c(-1,-i)}(1-p_{i})^{c(+1,-i)+c(-i,+i)};" class="ltx_Math" display="block" id="S2.Ex15.m1.16"><semantics id="S2.Ex15.m1.16a"><mrow id="S2.Ex15.m1.16.16.1" xref="S2.Ex15.m1.16.16.1.1.cmml"><mrow id="S2.Ex15.m1.16.16.1.1" xref="S2.Ex15.m1.16.16.1.1.cmml"><mrow id="S2.Ex15.m1.16.16.1.1.3" xref="S2.Ex15.m1.16.16.1.1.3.cmml"><mi id="S2.Ex15.m1.16.16.1.1.3.2" xref="S2.Ex15.m1.16.16.1.1.3.2.cmml">p</mi><mo id="S2.Ex15.m1.16.16.1.1.3.1" xref="S2.Ex15.m1.16.16.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex15.m1.16.16.1.1.3.3.2" xref="S2.Ex15.m1.16.16.1.1.3.cmml"><mo id="S2.Ex15.m1.16.16.1.1.3.3.2.1" stretchy="false" xref="S2.Ex15.m1.16.16.1.1.3.cmml">(</mo><mi id="S2.Ex15.m1.14.14" xref="S2.Ex15.m1.14.14.cmml">𝒄</mi><mo id="S2.Ex15.m1.16.16.1.1.3.3.2.2" stretchy="false" xref="S2.Ex15.m1.16.16.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.Ex15.m1.16.16.1.1.2" xref="S2.Ex15.m1.16.16.1.1.2.cmml">=</mo><mrow id="S2.Ex15.m1.16.16.1.1.1" xref="S2.Ex15.m1.16.16.1.1.1.cmml"><msup id="S2.Ex15.m1.16.16.1.1.1.3" xref="S2.Ex15.m1.16.16.1.1.1.3.cmml"><mrow id="S2.Ex15.m1.16.16.1.1.1.3.2.2" xref="S2.Ex15.m1.15.15.cmml"><mo id="S2.Ex15.m1.16.16.1.1.1.3.2.2.1" xref="S2.Ex15.m1.15.15.cmml">(</mo><mfrac id="S2.Ex15.m1.15.15" xref="S2.Ex15.m1.15.15.cmml"><mn id="S2.Ex15.m1.15.15.2" xref="S2.Ex15.m1.15.15.2.cmml">1</mn><mn id="S2.Ex15.m1.15.15.3" xref="S2.Ex15.m1.15.15.3.cmml">2</mn></mfrac><mo id="S2.Ex15.m1.16.16.1.1.1.3.2.2.2" xref="S2.Ex15.m1.15.15.cmml">)</mo></mrow><mrow id="S2.Ex15.m1.4.4.4" xref="S2.Ex15.m1.4.4.4.cmml"><mrow id="S2.Ex15.m1.3.3.3.3" xref="S2.Ex15.m1.3.3.3.3.cmml"><mi id="S2.Ex15.m1.3.3.3.3.3" xref="S2.Ex15.m1.3.3.3.3.3.cmml">c</mi><mo id="S2.Ex15.m1.3.3.3.3.2" xref="S2.Ex15.m1.3.3.3.3.2.cmml">⁢</mo><mrow id="S2.Ex15.m1.3.3.3.3.1.1" xref="S2.Ex15.m1.3.3.3.3.1.2.cmml"><mo id="S2.Ex15.m1.3.3.3.3.1.1.2" stretchy="false" xref="S2.Ex15.m1.3.3.3.3.1.2.cmml">(</mo><mrow id="S2.Ex15.m1.3.3.3.3.1.1.1" xref="S2.Ex15.m1.3.3.3.3.1.1.1.cmml"><mo id="S2.Ex15.m1.3.3.3.3.1.1.1a" xref="S2.Ex15.m1.3.3.3.3.1.1.1.cmml">+</mo><mn id="S2.Ex15.m1.3.3.3.3.1.1.1.2" xref="S2.Ex15.m1.3.3.3.3.1.1.1.2.cmml">1</mn></mrow><mo id="S2.Ex15.m1.3.3.3.3.1.1.3" xref="S2.Ex15.m1.3.3.3.3.1.2.cmml">,</mo><mn id="S2.Ex15.m1.1.1.1.1" xref="S2.Ex15.m1.1.1.1.1.cmml">0</mn><mo id="S2.Ex15.m1.3.3.3.3.1.1.4" stretchy="false" xref="S2.Ex15.m1.3.3.3.3.1.2.cmml">)</mo></mrow></mrow><mo id="S2.Ex15.m1.4.4.4.5" xref="S2.Ex15.m1.4.4.4.5.cmml">+</mo><mrow id="S2.Ex15.m1.4.4.4.4" xref="S2.Ex15.m1.4.4.4.4.cmml"><mi id="S2.Ex15.m1.4.4.4.4.3" xref="S2.Ex15.m1.4.4.4.4.3.cmml">c</mi><mo id="S2.Ex15.m1.4.4.4.4.2" xref="S2.Ex15.m1.4.4.4.4.2.cmml">⁢</mo><mrow id="S2.Ex15.m1.4.4.4.4.1.1" xref="S2.Ex15.m1.4.4.4.4.1.2.cmml"><mo id="S2.Ex15.m1.4.4.4.4.1.1.2" stretchy="false" xref="S2.Ex15.m1.4.4.4.4.1.2.cmml">(</mo><mrow id="S2.Ex15.m1.4.4.4.4.1.1.1" xref="S2.Ex15.m1.4.4.4.4.1.1.1.cmml"><mo id="S2.Ex15.m1.4.4.4.4.1.1.1a" xref="S2.Ex15.m1.4.4.4.4.1.1.1.cmml">−</mo><mn id="S2.Ex15.m1.4.4.4.4.1.1.1.2" xref="S2.Ex15.m1.4.4.4.4.1.1.1.2.cmml">1</mn></mrow><mo id="S2.Ex15.m1.4.4.4.4.1.1.3" xref="S2.Ex15.m1.4.4.4.4.1.2.cmml">,</mo><mn id="S2.Ex15.m1.2.2.2.2" xref="S2.Ex15.m1.2.2.2.2.cmml">0</mn><mo id="S2.Ex15.m1.4.4.4.4.1.1.4" stretchy="false" xref="S2.Ex15.m1.4.4.4.4.1.2.cmml">)</mo></mrow></mrow></mrow></msup><mo id="S2.Ex15.m1.16.16.1.1.1.2" xref="S2.Ex15.m1.16.16.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex15.m1.16.16.1.1.1.1" xref="S2.Ex15.m1.16.16.1.1.1.1.cmml"><munder id="S2.Ex15.m1.16.16.1.1.1.1.2" xref="S2.Ex15.m1.16.16.1.1.1.1.2.cmml"><mo id="S2.Ex15.m1.16.16.1.1.1.1.2.2" movablelimits="false" xref="S2.Ex15.m1.16.16.1.1.1.1.2.2.cmml">∏</mo><mrow id="S2.Ex15.m1.5.5.1" xref="S2.Ex15.m1.5.5.1.cmml"><mi id="S2.Ex15.m1.5.5.1.3" xref="S2.Ex15.m1.5.5.1.3.cmml">i</mi><mo id="S2.Ex15.m1.5.5.1.2" xref="S2.Ex15.m1.5.5.1.2.cmml">∈</mo><mrow 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xref="S2.Ex15.m1.7.7.2.2.2.3.cmml"><mo id="S2.Ex15.m1.7.7.2.2.2.2.3" stretchy="false" xref="S2.Ex15.m1.7.7.2.2.2.3.cmml">(</mo><mrow id="S2.Ex15.m1.6.6.1.1.1.1.1" xref="S2.Ex15.m1.6.6.1.1.1.1.1.cmml"><mo id="S2.Ex15.m1.6.6.1.1.1.1.1a" xref="S2.Ex15.m1.6.6.1.1.1.1.1.cmml">+</mo><mn id="S2.Ex15.m1.6.6.1.1.1.1.1.2" xref="S2.Ex15.m1.6.6.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S2.Ex15.m1.7.7.2.2.2.2.4" xref="S2.Ex15.m1.7.7.2.2.2.3.cmml">,</mo><mrow id="S2.Ex15.m1.7.7.2.2.2.2.2" xref="S2.Ex15.m1.7.7.2.2.2.2.2.cmml"><mo id="S2.Ex15.m1.7.7.2.2.2.2.2a" xref="S2.Ex15.m1.7.7.2.2.2.2.2.cmml">+</mo><mi id="S2.Ex15.m1.7.7.2.2.2.2.2.2" xref="S2.Ex15.m1.7.7.2.2.2.2.2.2.cmml">i</mi></mrow><mo id="S2.Ex15.m1.7.7.2.2.2.2.5" stretchy="false" xref="S2.Ex15.m1.7.7.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.Ex15.m1.9.9.4.5" xref="S2.Ex15.m1.9.9.4.5.cmml">+</mo><mrow id="S2.Ex15.m1.9.9.4.4" xref="S2.Ex15.m1.9.9.4.4.cmml"><mi id="S2.Ex15.m1.9.9.4.4.4" xref="S2.Ex15.m1.9.9.4.4.4.cmml">c</mi><mo id="S2.Ex15.m1.9.9.4.4.3" xref="S2.Ex15.m1.9.9.4.4.3.cmml">⁢</mo><mrow id="S2.Ex15.m1.9.9.4.4.2.2" xref="S2.Ex15.m1.9.9.4.4.2.3.cmml"><mo id="S2.Ex15.m1.9.9.4.4.2.2.3" stretchy="false" xref="S2.Ex15.m1.9.9.4.4.2.3.cmml">(</mo><mrow id="S2.Ex15.m1.8.8.3.3.1.1.1" xref="S2.Ex15.m1.8.8.3.3.1.1.1.cmml"><mo id="S2.Ex15.m1.8.8.3.3.1.1.1a" xref="S2.Ex15.m1.8.8.3.3.1.1.1.cmml">−</mo><mn id="S2.Ex15.m1.8.8.3.3.1.1.1.2" xref="S2.Ex15.m1.8.8.3.3.1.1.1.2.cmml">1</mn></mrow><mo id="S2.Ex15.m1.9.9.4.4.2.2.4" xref="S2.Ex15.m1.9.9.4.4.2.3.cmml">,</mo><mrow id="S2.Ex15.m1.9.9.4.4.2.2.2" xref="S2.Ex15.m1.9.9.4.4.2.2.2.cmml"><mo id="S2.Ex15.m1.9.9.4.4.2.2.2a" xref="S2.Ex15.m1.9.9.4.4.2.2.2.cmml">−</mo><mi id="S2.Ex15.m1.9.9.4.4.2.2.2.2" xref="S2.Ex15.m1.9.9.4.4.2.2.2.2.cmml">i</mi></mrow><mo id="S2.Ex15.m1.9.9.4.4.2.2.5" stretchy="false" xref="S2.Ex15.m1.9.9.4.4.2.3.cmml">)</mo></mrow></mrow></mrow></msubsup><mo id="S2.Ex15.m1.16.16.1.1.1.1.1.2" xref="S2.Ex15.m1.16.16.1.1.1.1.1.2.cmml">⁢</mo><msup 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xref="S2.Ex15.m1.16.16.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.Ex15.m1.13.13.4" xref="S2.Ex15.m1.13.13.4.cmml"><mrow id="S2.Ex15.m1.11.11.2.2" xref="S2.Ex15.m1.11.11.2.2.cmml"><mi id="S2.Ex15.m1.11.11.2.2.4" xref="S2.Ex15.m1.11.11.2.2.4.cmml">c</mi><mo id="S2.Ex15.m1.11.11.2.2.3" xref="S2.Ex15.m1.11.11.2.2.3.cmml">⁢</mo><mrow id="S2.Ex15.m1.11.11.2.2.2.2" xref="S2.Ex15.m1.11.11.2.2.2.3.cmml"><mo id="S2.Ex15.m1.11.11.2.2.2.2.3" stretchy="false" xref="S2.Ex15.m1.11.11.2.2.2.3.cmml">(</mo><mrow id="S2.Ex15.m1.10.10.1.1.1.1.1" xref="S2.Ex15.m1.10.10.1.1.1.1.1.cmml"><mo id="S2.Ex15.m1.10.10.1.1.1.1.1a" xref="S2.Ex15.m1.10.10.1.1.1.1.1.cmml">+</mo><mn id="S2.Ex15.m1.10.10.1.1.1.1.1.2" xref="S2.Ex15.m1.10.10.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S2.Ex15.m1.11.11.2.2.2.2.4" xref="S2.Ex15.m1.11.11.2.2.2.3.cmml">,</mo><mrow id="S2.Ex15.m1.11.11.2.2.2.2.2" xref="S2.Ex15.m1.11.11.2.2.2.2.2.cmml"><mo id="S2.Ex15.m1.11.11.2.2.2.2.2a" xref="S2.Ex15.m1.11.11.2.2.2.2.2.cmml">−</mo><mi 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xref="S2.Ex15.m1.12.12.3.3.1.1.1"></minus><ci id="S2.Ex15.m1.12.12.3.3.1.1.1.2.cmml" xref="S2.Ex15.m1.12.12.3.3.1.1.1.2">𝑖</ci></apply><apply id="S2.Ex15.m1.13.13.4.4.2.2.2.cmml" xref="S2.Ex15.m1.13.13.4.4.2.2.2"><plus id="S2.Ex15.m1.13.13.4.4.2.2.2.1.cmml" xref="S2.Ex15.m1.13.13.4.4.2.2.2"></plus><ci id="S2.Ex15.m1.13.13.4.4.2.2.2.2.cmml" xref="S2.Ex15.m1.13.13.4.4.2.2.2.2">𝑖</ci></apply></interval></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex15.m1.16c">p(\boldsymbol{c})=\left(\frac{1}{2}\right)^{c(+1,0)+c(-1,0)}\prod_{i\in[\ell]}% p_{i}^{c(+1,+i)+c(-1,-i)}(1-p_{i})^{c(+1,-i)+c(-i,+i)};</annotation><annotation encoding="application/x-llamapun" id="S2.Ex15.m1.16d">italic_p ( bold_italic_c ) = ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_c ( + 1 , 0 ) + italic_c ( - 1 , 0 ) end_POSTSUPERSCRIPT ∏ start_POSTSUBSCRIPT italic_i ∈ [ roman_ℓ ] end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c ( + 1 , + italic_i ) + italic_c ( - 1 , - italic_i ) end_POSTSUPERSCRIPT ( 1 - italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_c ( + 1 , - italic_i ) + italic_c ( - italic_i , + italic_i ) end_POSTSUPERSCRIPT ;</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem1.p3.5"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem1.p3.5.1">constants for <math alttext="i\in[\pm\ell]" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p3.5.1.m1.1"><semantics id="S2.Thmtheorem1.p3.5.1.m1.1a"><mrow id="S2.Thmtheorem1.p3.5.1.m1.1.1" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.cmml"><mi id="S2.Thmtheorem1.p3.5.1.m1.1.1.3" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.3.cmml">i</mi><mo id="S2.Thmtheorem1.p3.5.1.m1.1.1.2" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.2.cmml">∈</mo><mrow id="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.1.2.cmml"><mo id="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.1.2.1.cmml">[</mo><mrow id="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1.1" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1.1a" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1.1.cmml">±</mo><mi id="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1.1.2" mathvariant="normal" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1.1.2.cmml">ℓ</mi></mrow><mo id="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p3.5.1.m1.1b"><apply id="S2.Thmtheorem1.p3.5.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p3.5.1.m1.1.1"><in id="S2.Thmtheorem1.p3.5.1.m1.1.1.2.cmml" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.2"></in><ci id="S2.Thmtheorem1.p3.5.1.m1.1.1.3.cmml" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.3">𝑖</ci><apply id="S2.Thmtheorem1.p3.5.1.m1.1.1.1.2.cmml" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1"><csymbol cd="latexml" id="S2.Thmtheorem1.p3.5.1.m1.1.1.1.2.1.cmml" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1.1"><csymbol cd="latexml" id="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1.1">plus-or-minus</csymbol><ci id="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem1.p3.5.1.m1.1.1.1.1.1.2">ℓ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p3.5.1.m1.1c">i\in[\pm\ell]</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p3.5.1.m1.1d">italic_i ∈ [ ± roman_ℓ ]</annotation></semantics></math>:</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex16"> <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="a_{i}=\sup I_{i}\text{ and }b_{i}\inf I_{i};" class="ltx_Math" display="block" id="S2.Ex16.m1.1"><semantics id="S2.Ex16.m1.1a"><mrow id="S2.Ex16.m1.1.1.1" xref="S2.Ex16.m1.1.1.1.1.cmml"><mrow id="S2.Ex16.m1.1.1.1.1" xref="S2.Ex16.m1.1.1.1.1.cmml"><msub id="S2.Ex16.m1.1.1.1.1.2" xref="S2.Ex16.m1.1.1.1.1.2.cmml"><mi id="S2.Ex16.m1.1.1.1.1.2.2" xref="S2.Ex16.m1.1.1.1.1.2.2.cmml">a</mi><mi id="S2.Ex16.m1.1.1.1.1.2.3" xref="S2.Ex16.m1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S2.Ex16.m1.1.1.1.1.1" rspace="0.1389em" xref="S2.Ex16.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.Ex16.m1.1.1.1.1.3" xref="S2.Ex16.m1.1.1.1.1.3.cmml"><mo id="S2.Ex16.m1.1.1.1.1.3.1" lspace="0.1389em" movablelimits="false" rspace="0.167em" xref="S2.Ex16.m1.1.1.1.1.3.1.cmml">sup</mo><mrow id="S2.Ex16.m1.1.1.1.1.3.2" xref="S2.Ex16.m1.1.1.1.1.3.2.cmml"><msub id="S2.Ex16.m1.1.1.1.1.3.2.2" xref="S2.Ex16.m1.1.1.1.1.3.2.2.cmml"><mi id="S2.Ex16.m1.1.1.1.1.3.2.2.2" xref="S2.Ex16.m1.1.1.1.1.3.2.2.2.cmml">I</mi><mi id="S2.Ex16.m1.1.1.1.1.3.2.2.3" xref="S2.Ex16.m1.1.1.1.1.3.2.2.3.cmml">i</mi></msub><mo id="S2.Ex16.m1.1.1.1.1.3.2.1" xref="S2.Ex16.m1.1.1.1.1.3.2.1.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S2.Ex16.m1.1.1.1.1.3.2.3" xref="S2.Ex16.m1.1.1.1.1.3.2.3a.cmml"> and </mtext><mo id="S2.Ex16.m1.1.1.1.1.3.2.1a" xref="S2.Ex16.m1.1.1.1.1.3.2.1.cmml">⁢</mo><msub id="S2.Ex16.m1.1.1.1.1.3.2.4" xref="S2.Ex16.m1.1.1.1.1.3.2.4.cmml"><mi id="S2.Ex16.m1.1.1.1.1.3.2.4.2" xref="S2.Ex16.m1.1.1.1.1.3.2.4.2.cmml">b</mi><mi id="S2.Ex16.m1.1.1.1.1.3.2.4.3" xref="S2.Ex16.m1.1.1.1.1.3.2.4.3.cmml">i</mi></msub><mo id="S2.Ex16.m1.1.1.1.1.3.2.1b" lspace="0.167em" xref="S2.Ex16.m1.1.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S2.Ex16.m1.1.1.1.1.3.2.5" xref="S2.Ex16.m1.1.1.1.1.3.2.5.cmml"><mo id="S2.Ex16.m1.1.1.1.1.3.2.5.1" movablelimits="false" rspace="0.167em" xref="S2.Ex16.m1.1.1.1.1.3.2.5.1.cmml">inf</mo><msub 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xref="S2.Ex16.m1.1.1.1.1.3.2.4.2">𝑏</ci><ci id="S2.Ex16.m1.1.1.1.1.3.2.4.3.cmml" xref="S2.Ex16.m1.1.1.1.1.3.2.4.3">𝑖</ci></apply><apply id="S2.Ex16.m1.1.1.1.1.3.2.5.cmml" xref="S2.Ex16.m1.1.1.1.1.3.2.5"><csymbol cd="latexml" id="S2.Ex16.m1.1.1.1.1.3.2.5.1.cmml" xref="S2.Ex16.m1.1.1.1.1.3.2.5.1">infimum</csymbol><apply id="S2.Ex16.m1.1.1.1.1.3.2.5.2.cmml" xref="S2.Ex16.m1.1.1.1.1.3.2.5.2"><csymbol cd="ambiguous" id="S2.Ex16.m1.1.1.1.1.3.2.5.2.1.cmml" xref="S2.Ex16.m1.1.1.1.1.3.2.5.2">subscript</csymbol><ci id="S2.Ex16.m1.1.1.1.1.3.2.5.2.2.cmml" xref="S2.Ex16.m1.1.1.1.1.3.2.5.2.2">𝐼</ci><ci id="S2.Ex16.m1.1.1.1.1.3.2.5.2.3.cmml" xref="S2.Ex16.m1.1.1.1.1.3.2.5.2.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex16.m1.1c">a_{i}=\sup I_{i}\text{ and }b_{i}\inf I_{i};</annotation><annotation encoding="application/x-llamapun" id="S2.Ex16.m1.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = roman_sup italic_I 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xref="S2.Thmtheorem1.p3.6.1.m1.2.2.1.1.1.cmml">⁢</mo><mrow id="S2.Thmtheorem1.p3.6.1.m1.2.2.1.1.3.2" xref="S2.Thmtheorem1.p3.6.1.m1.2.2.1.1.cmml"><mo id="S2.Thmtheorem1.p3.6.1.m1.2.2.1.1.3.2.1" stretchy="false" xref="S2.Thmtheorem1.p3.6.1.m1.2.2.1.1.cmml">(</mo><mi id="S2.Thmtheorem1.p3.6.1.m1.1.1" xref="S2.Thmtheorem1.p3.6.1.m1.1.1.cmml">𝐜</mi><mo id="S2.Thmtheorem1.p3.6.1.m1.2.2.1.1.3.2.2" stretchy="false" xref="S2.Thmtheorem1.p3.6.1.m1.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem1.p3.6.1.m1.2.2.1.3" stretchy="false" xref="S2.Thmtheorem1.p3.6.1.m1.2.2.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p3.6.1.m1.2b"><set id="S2.Thmtheorem1.p3.6.1.m1.2.2.2.cmml" xref="S2.Thmtheorem1.p3.6.1.m1.2.2.1"><apply id="S2.Thmtheorem1.p3.6.1.m1.2.2.1.1.cmml" xref="S2.Thmtheorem1.p3.6.1.m1.2.2.1.1"><times id="S2.Thmtheorem1.p3.6.1.m1.2.2.1.1.1.cmml" xref="S2.Thmtheorem1.p3.6.1.m1.2.2.1.1.1"></times><ci id="S2.Thmtheorem1.p3.6.1.m1.2.2.1.1.2.cmml" xref="S2.Thmtheorem1.p3.6.1.m1.2.2.1.1.2">𝑤</ci><ci id="S2.Thmtheorem1.p3.6.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p3.6.1.m1.1.1">𝐜</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p3.6.1.m1.2c">\{w(\boldsymbol{c})\}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p3.6.1.m1.2d">{ italic_w ( bold_italic_c ) }</annotation></semantics></math>, for <math alttext="i\in[\pm\ell]" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p3.7.2.m2.1"><semantics id="S2.Thmtheorem1.p3.7.2.m2.1a"><mrow id="S2.Thmtheorem1.p3.7.2.m2.1.1" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.cmml"><mi id="S2.Thmtheorem1.p3.7.2.m2.1.1.3" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.3.cmml">i</mi><mo id="S2.Thmtheorem1.p3.7.2.m2.1.1.2" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.2.cmml">∈</mo><mrow id="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.1.2.cmml"><mo id="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.1.2.1.cmml">[</mo><mrow id="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1.1" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1.1a" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1.1.cmml">±</mo><mi id="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1.1.2" mathvariant="normal" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1.1.2.cmml">ℓ</mi></mrow><mo id="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p3.7.2.m2.1b"><apply id="S2.Thmtheorem1.p3.7.2.m2.1.1.cmml" xref="S2.Thmtheorem1.p3.7.2.m2.1.1"><in id="S2.Thmtheorem1.p3.7.2.m2.1.1.2.cmml" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.2"></in><ci id="S2.Thmtheorem1.p3.7.2.m2.1.1.3.cmml" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.3">𝑖</ci><apply id="S2.Thmtheorem1.p3.7.2.m2.1.1.1.2.cmml" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1"><csymbol cd="latexml" id="S2.Thmtheorem1.p3.7.2.m2.1.1.1.2.1.cmml" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1.1"><csymbol cd="latexml" id="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1.1">plus-or-minus</csymbol><ci id="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem1.p3.7.2.m2.1.1.1.1.1.2">ℓ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p3.7.2.m2.1c">i\in[\pm\ell]</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p3.7.2.m2.1d">italic_i ∈ [ ± roman_ℓ ]</annotation></semantics></math>:</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex17"> <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="W^{+}(i)=\sum_{\boldsymbol{c}\in C}c_{i}^{+}w(\boldsymbol{c})\text{ and }W^{-}% (i)=\sum_{\boldsymbol{c}\in C}c_{i}^{-}w(\boldsymbol{c})." class="ltx_Math" display="block" id="S2.Ex17.m1.5"><semantics id="S2.Ex17.m1.5a"><mrow id="S2.Ex17.m1.5.5.1" xref="S2.Ex17.m1.5.5.1.1.cmml"><mrow id="S2.Ex17.m1.5.5.1.1" xref="S2.Ex17.m1.5.5.1.1.cmml"><mrow id="S2.Ex17.m1.5.5.1.1.2" xref="S2.Ex17.m1.5.5.1.1.2.cmml"><msup id="S2.Ex17.m1.5.5.1.1.2.2" xref="S2.Ex17.m1.5.5.1.1.2.2.cmml"><mi id="S2.Ex17.m1.5.5.1.1.2.2.2" xref="S2.Ex17.m1.5.5.1.1.2.2.2.cmml">W</mi><mo id="S2.Ex17.m1.5.5.1.1.2.2.3" xref="S2.Ex17.m1.5.5.1.1.2.2.3.cmml">+</mo></msup><mo id="S2.Ex17.m1.5.5.1.1.2.1" xref="S2.Ex17.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="S2.Ex17.m1.5.5.1.1.2.3.2" xref="S2.Ex17.m1.5.5.1.1.2.cmml"><mo id="S2.Ex17.m1.5.5.1.1.2.3.2.1" stretchy="false" xref="S2.Ex17.m1.5.5.1.1.2.cmml">(</mo><mi id="S2.Ex17.m1.1.1" xref="S2.Ex17.m1.1.1.cmml">i</mi><mo id="S2.Ex17.m1.5.5.1.1.2.3.2.2" stretchy="false" xref="S2.Ex17.m1.5.5.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.Ex17.m1.5.5.1.1.3" rspace="0.111em" xref="S2.Ex17.m1.5.5.1.1.3.cmml">=</mo><mrow id="S2.Ex17.m1.5.5.1.1.4" xref="S2.Ex17.m1.5.5.1.1.4.cmml"><munder id="S2.Ex17.m1.5.5.1.1.4.1" xref="S2.Ex17.m1.5.5.1.1.4.1.cmml"><mo id="S2.Ex17.m1.5.5.1.1.4.1.2" movablelimits="false" xref="S2.Ex17.m1.5.5.1.1.4.1.2.cmml">∑</mo><mrow id="S2.Ex17.m1.5.5.1.1.4.1.3" xref="S2.Ex17.m1.5.5.1.1.4.1.3.cmml"><mi id="S2.Ex17.m1.5.5.1.1.4.1.3.2" xref="S2.Ex17.m1.5.5.1.1.4.1.3.2.cmml">𝒄</mi><mo id="S2.Ex17.m1.5.5.1.1.4.1.3.1" xref="S2.Ex17.m1.5.5.1.1.4.1.3.1.cmml">∈</mo><mi id="S2.Ex17.m1.5.5.1.1.4.1.3.3" xref="S2.Ex17.m1.5.5.1.1.4.1.3.3.cmml">C</mi></mrow></munder><mrow id="S2.Ex17.m1.5.5.1.1.4.2" xref="S2.Ex17.m1.5.5.1.1.4.2.cmml"><msubsup id="S2.Ex17.m1.5.5.1.1.4.2.2" xref="S2.Ex17.m1.5.5.1.1.4.2.2.cmml"><mi id="S2.Ex17.m1.5.5.1.1.4.2.2.2.2" xref="S2.Ex17.m1.5.5.1.1.4.2.2.2.2.cmml">c</mi><mi id="S2.Ex17.m1.5.5.1.1.4.2.2.2.3" xref="S2.Ex17.m1.5.5.1.1.4.2.2.2.3.cmml">i</mi><mo id="S2.Ex17.m1.5.5.1.1.4.2.2.3" xref="S2.Ex17.m1.5.5.1.1.4.2.2.3.cmml">+</mo></msubsup><mo id="S2.Ex17.m1.5.5.1.1.4.2.1" xref="S2.Ex17.m1.5.5.1.1.4.2.1.cmml">⁢</mo><mi id="S2.Ex17.m1.5.5.1.1.4.2.3" xref="S2.Ex17.m1.5.5.1.1.4.2.3.cmml">w</mi><mo id="S2.Ex17.m1.5.5.1.1.4.2.1a" xref="S2.Ex17.m1.5.5.1.1.4.2.1.cmml">⁢</mo><mrow id="S2.Ex17.m1.5.5.1.1.4.2.4.2" xref="S2.Ex17.m1.5.5.1.1.4.2.cmml"><mo id="S2.Ex17.m1.5.5.1.1.4.2.4.2.1" stretchy="false" xref="S2.Ex17.m1.5.5.1.1.4.2.cmml">(</mo><mi id="S2.Ex17.m1.2.2" xref="S2.Ex17.m1.2.2.cmml">𝒄</mi><mo id="S2.Ex17.m1.5.5.1.1.4.2.4.2.2" stretchy="false" xref="S2.Ex17.m1.5.5.1.1.4.2.cmml">)</mo></mrow><mo id="S2.Ex17.m1.5.5.1.1.4.2.1b" xref="S2.Ex17.m1.5.5.1.1.4.2.1.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S2.Ex17.m1.5.5.1.1.4.2.5" xref="S2.Ex17.m1.5.5.1.1.4.2.5a.cmml"> and </mtext><mo id="S2.Ex17.m1.5.5.1.1.4.2.1c" xref="S2.Ex17.m1.5.5.1.1.4.2.1.cmml">⁢</mo><msup id="S2.Ex17.m1.5.5.1.1.4.2.6" xref="S2.Ex17.m1.5.5.1.1.4.2.6.cmml"><mi id="S2.Ex17.m1.5.5.1.1.4.2.6.2" xref="S2.Ex17.m1.5.5.1.1.4.2.6.2.cmml">W</mi><mo id="S2.Ex17.m1.5.5.1.1.4.2.6.3" xref="S2.Ex17.m1.5.5.1.1.4.2.6.3.cmml">−</mo></msup><mo id="S2.Ex17.m1.5.5.1.1.4.2.1d" xref="S2.Ex17.m1.5.5.1.1.4.2.1.cmml">⁢</mo><mrow id="S2.Ex17.m1.5.5.1.1.4.2.7.2" xref="S2.Ex17.m1.5.5.1.1.4.2.cmml"><mo id="S2.Ex17.m1.5.5.1.1.4.2.7.2.1" stretchy="false" xref="S2.Ex17.m1.5.5.1.1.4.2.cmml">(</mo><mi id="S2.Ex17.m1.3.3" xref="S2.Ex17.m1.3.3.cmml">i</mi><mo id="S2.Ex17.m1.5.5.1.1.4.2.7.2.2" stretchy="false" xref="S2.Ex17.m1.5.5.1.1.4.2.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex17.m1.5.5.1.1.5" rspace="0.111em" xref="S2.Ex17.m1.5.5.1.1.5.cmml">=</mo><mrow id="S2.Ex17.m1.5.5.1.1.6" xref="S2.Ex17.m1.5.5.1.1.6.cmml"><munder id="S2.Ex17.m1.5.5.1.1.6.1" xref="S2.Ex17.m1.5.5.1.1.6.1.cmml"><mo id="S2.Ex17.m1.5.5.1.1.6.1.2" movablelimits="false" xref="S2.Ex17.m1.5.5.1.1.6.1.2.cmml">∑</mo><mrow id="S2.Ex17.m1.5.5.1.1.6.1.3" xref="S2.Ex17.m1.5.5.1.1.6.1.3.cmml"><mi id="S2.Ex17.m1.5.5.1.1.6.1.3.2" xref="S2.Ex17.m1.5.5.1.1.6.1.3.2.cmml">𝒄</mi><mo id="S2.Ex17.m1.5.5.1.1.6.1.3.1" xref="S2.Ex17.m1.5.5.1.1.6.1.3.1.cmml">∈</mo><mi id="S2.Ex17.m1.5.5.1.1.6.1.3.3" xref="S2.Ex17.m1.5.5.1.1.6.1.3.3.cmml">C</mi></mrow></munder><mrow id="S2.Ex17.m1.5.5.1.1.6.2" xref="S2.Ex17.m1.5.5.1.1.6.2.cmml"><msubsup id="S2.Ex17.m1.5.5.1.1.6.2.2" xref="S2.Ex17.m1.5.5.1.1.6.2.2.cmml"><mi id="S2.Ex17.m1.5.5.1.1.6.2.2.2.2" xref="S2.Ex17.m1.5.5.1.1.6.2.2.2.2.cmml">c</mi><mi id="S2.Ex17.m1.5.5.1.1.6.2.2.2.3" xref="S2.Ex17.m1.5.5.1.1.6.2.2.2.3.cmml">i</mi><mo id="S2.Ex17.m1.5.5.1.1.6.2.2.3" xref="S2.Ex17.m1.5.5.1.1.6.2.2.3.cmml">−</mo></msubsup><mo id="S2.Ex17.m1.5.5.1.1.6.2.1" xref="S2.Ex17.m1.5.5.1.1.6.2.1.cmml">⁢</mo><mi id="S2.Ex17.m1.5.5.1.1.6.2.3" xref="S2.Ex17.m1.5.5.1.1.6.2.3.cmml">w</mi><mo id="S2.Ex17.m1.5.5.1.1.6.2.1a" xref="S2.Ex17.m1.5.5.1.1.6.2.1.cmml">⁢</mo><mrow id="S2.Ex17.m1.5.5.1.1.6.2.4.2" xref="S2.Ex17.m1.5.5.1.1.6.2.cmml"><mo id="S2.Ex17.m1.5.5.1.1.6.2.4.2.1" stretchy="false" xref="S2.Ex17.m1.5.5.1.1.6.2.cmml">(</mo><mi id="S2.Ex17.m1.4.4" xref="S2.Ex17.m1.4.4.cmml">𝒄</mi><mo id="S2.Ex17.m1.5.5.1.1.6.2.4.2.2" stretchy="false" xref="S2.Ex17.m1.5.5.1.1.6.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.Ex17.m1.5.5.1.2" lspace="0em" xref="S2.Ex17.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex17.m1.5b"><apply id="S2.Ex17.m1.5.5.1.1.cmml" xref="S2.Ex17.m1.5.5.1"><and id="S2.Ex17.m1.5.5.1.1a.cmml" xref="S2.Ex17.m1.5.5.1"></and><apply id="S2.Ex17.m1.5.5.1.1b.cmml" xref="S2.Ex17.m1.5.5.1"><eq id="S2.Ex17.m1.5.5.1.1.3.cmml" xref="S2.Ex17.m1.5.5.1.1.3"></eq><apply id="S2.Ex17.m1.5.5.1.1.2.cmml" xref="S2.Ex17.m1.5.5.1.1.2"><times id="S2.Ex17.m1.5.5.1.1.2.1.cmml" xref="S2.Ex17.m1.5.5.1.1.2.1"></times><apply id="S2.Ex17.m1.5.5.1.1.2.2.cmml" xref="S2.Ex17.m1.5.5.1.1.2.2"><csymbol cd="ambiguous" id="S2.Ex17.m1.5.5.1.1.2.2.1.cmml" xref="S2.Ex17.m1.5.5.1.1.2.2">superscript</csymbol><ci id="S2.Ex17.m1.5.5.1.1.2.2.2.cmml" 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end_POSTSUPERSCRIPT ( italic_i ) = ∑ start_POSTSUBSCRIPT bold_italic_c ∈ italic_C end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT italic_w ( bold_italic_c ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.1">We note that <span class="ltx_ERROR undefined" id="S2.SS3.p2.1.1">\textcite</span>FJ15 gave an alternative (but highly similar) LP which worked for arbitrary selection functions — i.e., not just antisymmetric ones — but required roughly twice as many variables. In this paper, we only run the LP for antisymmetric selection functions, so we use the LP from <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">Sin23-kand</span>]</cite> because of the cost savings.</p> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Improved oblivious algorithms (<a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem5" title="Theorem 1.5 (New upper bound). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.5</span></a>)</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">Our goal in this section is to prove <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem5" title="Theorem 1.5 (New upper bound). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.5</span></a>:</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.1">See <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem5" title="Theorem 1.5 (New upper bound). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">1.5</span></a></p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.1">To prove this theorem, we introduce a specific type of antisymmetric selection function we are interested in, namely, an “<em class="ltx_emph ltx_font_italic" id="S3.p3.1.1">S</em>-shaped” piecewise linear function:</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S3.E1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.E1.1.1.1">Definition 3.1</span></span><span class="ltx_text ltx_font_bold" id="S3.E1.2.2"> </span>(PL sigmoid functions)<span class="ltx_text ltx_font_bold" id="S3.E1.3.3">.</span> </h6> <div class="ltx_para" id="S3.E1.p1"> <p class="ltx_p" id="S3.E1.p1.1"><span class="ltx_text ltx_font_italic" id="S3.E1.p1.1.1">A <em class="ltx_emph ltx_font_upright" id="S3.E1.p1.1.1.1">piecewise linear <em class="ltx_emph ltx_font_italic" id="S3.E1.p1.1.1.1.1">(PL)</em> sigmoid function</em> is a selection function of the following form: For an <em class="ltx_emph ltx_font_upright" id="S3.E1.p1.1.1.2">intercept</em> parameter <math alttext="b\in[0,1]" class="ltx_Math" display="inline" id="S3.E1.p1.1.1.m1.2"><semantics id="S3.E1.p1.1.1.m1.2a"><mrow id="S3.E1.p1.1.1.m1.2.3" xref="S3.E1.p1.1.1.m1.2.3.cmml"><mi id="S3.E1.p1.1.1.m1.2.3.2" xref="S3.E1.p1.1.1.m1.2.3.2.cmml">b</mi><mo id="S3.E1.p1.1.1.m1.2.3.1" 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xref="S3.Ex1.m1.4.4.4.4.2.1.5"></leq><share href="https://arxiv.org/html/2411.12976v1#S3.Ex1.m1.4.4.4.4.2.1.4.cmml" id="S3.Ex1.m1.4.4.4.4.2.1d.cmml" xref="S3.Ex1.m1.4.4.4.4.2.1"></share><apply id="S3.Ex1.m1.4.4.4.4.2.1.6.cmml" xref="S3.Ex1.m1.4.4.4.4.2.1.6"><plus id="S3.Ex1.m1.4.4.4.4.2.1.6.1.cmml" xref="S3.Ex1.m1.4.4.4.4.2.1.6"></plus><ci id="S3.Ex1.m1.4.4.4.4.2.1.6.2.cmml" xref="S3.Ex1.m1.4.4.4.4.2.1.6.2">𝑏</ci></apply></apply></apply><cn id="S3.Ex1.m1.5.5.5.5.1.1.cmml" type="integer" xref="S3.Ex1.m1.5.5.5.5.1.1">1</cn><apply id="S3.Ex1.m1.6.6.6.6.2.1.1.1.cmml" xref="S3.Ex1.m1.6.6.6.6.2.1.1"><geq id="S3.Ex1.m1.6.6.6.6.2.1.1.1.1.cmml" xref="S3.Ex1.m1.6.6.6.6.2.1.1.1.1"></geq><ci id="S3.Ex1.m1.6.6.6.6.2.1.1.1.2.cmml" xref="S3.Ex1.m1.6.6.6.6.2.1.1.1.2">𝑥</ci><apply id="S3.Ex1.m1.6.6.6.6.2.1.1.1.3.cmml" xref="S3.Ex1.m1.6.6.6.6.2.1.1.1.3"><plus id="S3.Ex1.m1.6.6.6.6.2.1.1.1.3.1.cmml" xref="S3.Ex1.m1.6.6.6.6.2.1.1.1.3"></plus><ci id="S3.Ex1.m1.6.6.6.6.2.1.1.1.3.2.cmml" xref="S3.Ex1.m1.6.6.6.6.2.1.1.1.3.2">𝑏</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex1.m1.7c">\mathsf{PLSigmoid}_{b}(x)=\begin{cases}0&x\leq-b\\ 1/2+\frac{x}{2b}&-b\leq x\leq+b\\ 1&x\geq+b.\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m1.7d">sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_x ) = { start_ROW start_CELL 0 end_CELL start_CELL italic_x ≤ - italic_b end_CELL end_ROW start_ROW start_CELL 1 / 2 + divide start_ARG italic_x end_ARG start_ARG 2 italic_b end_ARG end_CELL start_CELL - italic_b ≤ italic_x ≤ + italic_b end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL italic_x ≥ + italic_b . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.4">Note that according to <math alttext="\mathsf{PLSigmoid}_{b}" class="ltx_Math" display="inline" id="S3.p4.1.m1.1"><semantics id="S3.p4.1.m1.1a"><msub id="S3.p4.1.m1.1.1" xref="S3.p4.1.m1.1.1.cmml"><mi id="S3.p4.1.m1.1.1.2" xref="S3.p4.1.m1.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S3.p4.1.m1.1.1.3" xref="S3.p4.1.m1.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.1b"><apply id="S3.p4.1.m1.1.1.cmml" xref="S3.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p4.1.m1.1.1.1.cmml" xref="S3.p4.1.m1.1.1">subscript</csymbol><ci id="S3.p4.1.m1.1.1.2.cmml" xref="S3.p4.1.m1.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S3.p4.1.m1.1.1.3.cmml" xref="S3.p4.1.m1.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.1c">\mathsf{PLSigmoid}_{b}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.1.m1.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>, vertices with bias exceeding <math alttext="b" class="ltx_Math" display="inline" id="S3.p4.2.m2.1"><semantics id="S3.p4.2.m2.1a"><mi id="S3.p4.2.m2.1.1" xref="S3.p4.2.m2.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S3.p4.2.m2.1b"><ci id="S3.p4.2.m2.1.1.cmml" xref="S3.p4.2.m2.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.2.m2.1c">b</annotation><annotation encoding="application/x-llamapun" id="S3.p4.2.m2.1d">italic_b</annotation></semantics></math> in magnitude are assigned deterministically (i.e., they are always assigned to <math alttext="+1" class="ltx_Math" display="inline" id="S3.p4.3.m3.1"><semantics id="S3.p4.3.m3.1a"><mrow id="S3.p4.3.m3.1.1" xref="S3.p4.3.m3.1.1.cmml"><mo id="S3.p4.3.m3.1.1a" xref="S3.p4.3.m3.1.1.cmml">+</mo><mn id="S3.p4.3.m3.1.1.2" xref="S3.p4.3.m3.1.1.2.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.3.m3.1b"><apply id="S3.p4.3.m3.1.1.cmml" xref="S3.p4.3.m3.1.1"><plus id="S3.p4.3.m3.1.1.1.cmml" xref="S3.p4.3.m3.1.1"></plus><cn 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1</annotation></semantics></math>, depending on their sign), and vertices with smaller bias interpolate linearly between these two extremes. 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40.62"></path></g></g></g></g></svg> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F1.10.5.1" style="font-size:90%;">Figure 1</span>: </span><span class="ltx_text" id="S3.F1.8.4" style="font-size:90%;">The step function <math alttext="\mathsf{PLSigmoid}_{1/2}" class="ltx_Math" display="inline" id="S3.F1.5.1.m1.1"><semantics id="S3.F1.5.1.m1.1b"><msub id="S3.F1.5.1.m1.1.1" xref="S3.F1.5.1.m1.1.1.cmml"><mi id="S3.F1.5.1.m1.1.1.2" xref="S3.F1.5.1.m1.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S3.F1.5.1.m1.1.1.3" xref="S3.F1.5.1.m1.1.1.3.cmml"><mn id="S3.F1.5.1.m1.1.1.3.2" xref="S3.F1.5.1.m1.1.1.3.2.cmml">1</mn><mo id="S3.F1.5.1.m1.1.1.3.1" xref="S3.F1.5.1.m1.1.1.3.1.cmml">/</mo><mn id="S3.F1.5.1.m1.1.1.3.3" xref="S3.F1.5.1.m1.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.F1.5.1.m1.1c"><apply id="S3.F1.5.1.m1.1.1.cmml" xref="S3.F1.5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.F1.5.1.m1.1.1.1.cmml" xref="S3.F1.5.1.m1.1.1">subscript</csymbol><ci id="S3.F1.5.1.m1.1.1.2.cmml" xref="S3.F1.5.1.m1.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S3.F1.5.1.m1.1.1.3.cmml" xref="S3.F1.5.1.m1.1.1.3"><divide id="S3.F1.5.1.m1.1.1.3.1.cmml" xref="S3.F1.5.1.m1.1.1.3.1"></divide><cn id="S3.F1.5.1.m1.1.1.3.2.cmml" type="integer" xref="S3.F1.5.1.m1.1.1.3.2">1</cn><cn id="S3.F1.5.1.m1.1.1.3.3.cmml" type="integer" xref="S3.F1.5.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F1.5.1.m1.1d">\mathsf{PLSigmoid}_{1/2}</annotation><annotation encoding="application/x-llamapun" id="S3.F1.5.1.m1.1e">sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math> and its discretization into <math alttext="\ell=5" class="ltx_Math" display="inline" id="S3.F1.6.2.m2.1"><semantics id="S3.F1.6.2.m2.1b"><mrow id="S3.F1.6.2.m2.1.1" xref="S3.F1.6.2.m2.1.1.cmml"><mi id="S3.F1.6.2.m2.1.1.2" mathvariant="normal" xref="S3.F1.6.2.m2.1.1.2.cmml">ℓ</mi><mo id="S3.F1.6.2.m2.1.1.1" xref="S3.F1.6.2.m2.1.1.1.cmml">=</mo><mn id="S3.F1.6.2.m2.1.1.3" xref="S3.F1.6.2.m2.1.1.3.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F1.6.2.m2.1c"><apply id="S3.F1.6.2.m2.1.1.cmml" xref="S3.F1.6.2.m2.1.1"><eq id="S3.F1.6.2.m2.1.1.1.cmml" xref="S3.F1.6.2.m2.1.1.1"></eq><ci id="S3.F1.6.2.m2.1.1.2.cmml" xref="S3.F1.6.2.m2.1.1.2">ℓ</ci><cn id="S3.F1.6.2.m2.1.1.3.cmml" type="integer" xref="S3.F1.6.2.m2.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F1.6.2.m2.1d">\ell=5</annotation><annotation encoding="application/x-llamapun" id="S3.F1.6.2.m2.1e">roman_ℓ = 5</annotation></semantics></math> positive bias classes. The discretization is the function in red. The jump discontinuities are notated using standard marks: Open circles are open interval ends and closed circles are closed interval ends. The continuous (non-discretized) function <math alttext="\mathsf{PLSigmoid}_{1/2}" class="ltx_Math" display="inline" id="S3.F1.7.3.m3.1"><semantics id="S3.F1.7.3.m3.1b"><msub id="S3.F1.7.3.m3.1.1" xref="S3.F1.7.3.m3.1.1.cmml"><mi id="S3.F1.7.3.m3.1.1.2" xref="S3.F1.7.3.m3.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S3.F1.7.3.m3.1.1.3" xref="S3.F1.7.3.m3.1.1.3.cmml"><mn id="S3.F1.7.3.m3.1.1.3.2" xref="S3.F1.7.3.m3.1.1.3.2.cmml">1</mn><mo id="S3.F1.7.3.m3.1.1.3.1" xref="S3.F1.7.3.m3.1.1.3.1.cmml">/</mo><mn id="S3.F1.7.3.m3.1.1.3.3" xref="S3.F1.7.3.m3.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.F1.7.3.m3.1c"><apply id="S3.F1.7.3.m3.1.1.cmml" xref="S3.F1.7.3.m3.1.1"><csymbol cd="ambiguous" id="S3.F1.7.3.m3.1.1.1.cmml" xref="S3.F1.7.3.m3.1.1">subscript</csymbol><ci id="S3.F1.7.3.m3.1.1.2.cmml" xref="S3.F1.7.3.m3.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S3.F1.7.3.m3.1.1.3.cmml" xref="S3.F1.7.3.m3.1.1.3"><divide id="S3.F1.7.3.m3.1.1.3.1.cmml" xref="S3.F1.7.3.m3.1.1.3.1"></divide><cn id="S3.F1.7.3.m3.1.1.3.2.cmml" type="integer" xref="S3.F1.7.3.m3.1.1.3.2">1</cn><cn id="S3.F1.7.3.m3.1.1.3.3.cmml" type="integer" xref="S3.F1.7.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F1.7.3.m3.1d">\mathsf{PLSigmoid}_{1/2}</annotation><annotation encoding="application/x-llamapun" id="S3.F1.7.3.m3.1e">sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math> disagrees with its discretization only within the interval <math alttext="[-1/2,+1/2]" class="ltx_Math" display="inline" id="S3.F1.8.4.m4.2"><semantics id="S3.F1.8.4.m4.2b"><mrow id="S3.F1.8.4.m4.2.2.2" xref="S3.F1.8.4.m4.2.2.3.cmml"><mo id="S3.F1.8.4.m4.2.2.2.3" stretchy="false" xref="S3.F1.8.4.m4.2.2.3.cmml">[</mo><mrow id="S3.F1.8.4.m4.1.1.1.1" xref="S3.F1.8.4.m4.1.1.1.1.cmml"><mo id="S3.F1.8.4.m4.1.1.1.1b" xref="S3.F1.8.4.m4.1.1.1.1.cmml">−</mo><mrow id="S3.F1.8.4.m4.1.1.1.1.2" xref="S3.F1.8.4.m4.1.1.1.1.2.cmml"><mn id="S3.F1.8.4.m4.1.1.1.1.2.2" xref="S3.F1.8.4.m4.1.1.1.1.2.2.cmml">1</mn><mo id="S3.F1.8.4.m4.1.1.1.1.2.1" xref="S3.F1.8.4.m4.1.1.1.1.2.1.cmml">/</mo><mn id="S3.F1.8.4.m4.1.1.1.1.2.3" xref="S3.F1.8.4.m4.1.1.1.1.2.3.cmml">2</mn></mrow></mrow><mo id="S3.F1.8.4.m4.2.2.2.4" xref="S3.F1.8.4.m4.2.2.3.cmml">,</mo><mrow id="S3.F1.8.4.m4.2.2.2.2" xref="S3.F1.8.4.m4.2.2.2.2.cmml"><mo id="S3.F1.8.4.m4.2.2.2.2b" xref="S3.F1.8.4.m4.2.2.2.2.cmml">+</mo><mrow id="S3.F1.8.4.m4.2.2.2.2.2" xref="S3.F1.8.4.m4.2.2.2.2.2.cmml"><mn id="S3.F1.8.4.m4.2.2.2.2.2.2" xref="S3.F1.8.4.m4.2.2.2.2.2.2.cmml">1</mn><mo id="S3.F1.8.4.m4.2.2.2.2.2.1" xref="S3.F1.8.4.m4.2.2.2.2.2.1.cmml">/</mo><mn id="S3.F1.8.4.m4.2.2.2.2.2.3" xref="S3.F1.8.4.m4.2.2.2.2.2.3.cmml">2</mn></mrow></mrow><mo id="S3.F1.8.4.m4.2.2.2.5" stretchy="false" xref="S3.F1.8.4.m4.2.2.3.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.F1.8.4.m4.2c"><interval closure="closed" id="S3.F1.8.4.m4.2.2.3.cmml" xref="S3.F1.8.4.m4.2.2.2"><apply id="S3.F1.8.4.m4.1.1.1.1.cmml" xref="S3.F1.8.4.m4.1.1.1.1"><minus id="S3.F1.8.4.m4.1.1.1.1.1.cmml" xref="S3.F1.8.4.m4.1.1.1.1"></minus><apply id="S3.F1.8.4.m4.1.1.1.1.2.cmml" xref="S3.F1.8.4.m4.1.1.1.1.2"><divide id="S3.F1.8.4.m4.1.1.1.1.2.1.cmml" xref="S3.F1.8.4.m4.1.1.1.1.2.1"></divide><cn id="S3.F1.8.4.m4.1.1.1.1.2.2.cmml" type="integer" xref="S3.F1.8.4.m4.1.1.1.1.2.2">1</cn><cn id="S3.F1.8.4.m4.1.1.1.1.2.3.cmml" type="integer" xref="S3.F1.8.4.m4.1.1.1.1.2.3">2</cn></apply></apply><apply id="S3.F1.8.4.m4.2.2.2.2.cmml" xref="S3.F1.8.4.m4.2.2.2.2"><plus id="S3.F1.8.4.m4.2.2.2.2.1.cmml" xref="S3.F1.8.4.m4.2.2.2.2"></plus><apply id="S3.F1.8.4.m4.2.2.2.2.2.cmml" xref="S3.F1.8.4.m4.2.2.2.2.2"><divide id="S3.F1.8.4.m4.2.2.2.2.2.1.cmml" xref="S3.F1.8.4.m4.2.2.2.2.2.1"></divide><cn id="S3.F1.8.4.m4.2.2.2.2.2.2.cmml" type="integer" xref="S3.F1.8.4.m4.2.2.2.2.2.2">1</cn><cn id="S3.F1.8.4.m4.2.2.2.2.2.3.cmml" type="integer" xref="S3.F1.8.4.m4.2.2.2.2.2.3">2</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.F1.8.4.m4.2d">[-1/2,+1/2]</annotation><annotation encoding="application/x-llamapun" id="S3.F1.8.4.m4.2e">[ - 1 / 2 , + 1 / 2 ]</annotation></semantics></math> (marked by the vertical dashed line segments). The continuous function is represented by the black line segment within this interval.</span></figcaption> </figure> <div class="ltx_para" id="S3.p5"> <p class="ltx_p" id="S3.p5.11">In the prior works <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">FJ15</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">Sin23-kand</span>]</cite>, the highest approximation ratios achieved by oblivious algorithms were found by using discretized versions of <math alttext="\mathsf{PLSigmoid}_{1/2}" class="ltx_Math" display="inline" id="S3.p5.1.m1.1"><semantics id="S3.p5.1.m1.1a"><msub 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">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S3.p5.1.m1.1.1.3" xref="S3.p5.1.m1.1.1.3.cmml"><mn id="S3.p5.1.m1.1.1.3.2" xref="S3.p5.1.m1.1.1.3.2.cmml">1</mn><mo id="S3.p5.1.m1.1.1.3.1" xref="S3.p5.1.m1.1.1.3.1.cmml">/</mo><mn id="S3.p5.1.m1.1.1.3.3" xref="S3.p5.1.m1.1.1.3.3.cmml">2</mn></mrow></msub><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">subscript</csymbol><ci id="S3.p5.1.m1.1.1.2.cmml" xref="S3.p5.1.m1.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S3.p5.1.m1.1.1.3.cmml" xref="S3.p5.1.m1.1.1.3"><divide id="S3.p5.1.m1.1.1.3.1.cmml" xref="S3.p5.1.m1.1.1.3.1"></divide><cn id="S3.p5.1.m1.1.1.3.2.cmml" type="integer" xref="S3.p5.1.m1.1.1.3.2">1</cn><cn id="S3.p5.1.m1.1.1.3.3.cmml" type="integer" xref="S3.p5.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.1.m1.1c">\mathsf{PLSigmoid}_{1/2}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.1.m1.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math>. These discretizations — in the antisymmetric case — are controlled by a parameter <math alttext="\ell" class="ltx_Math" display="inline" id="S3.p5.2.m2.1"><semantics id="S3.p5.2.m2.1a"><mi id="S3.p5.2.m2.1.1" mathvariant="normal" xref="S3.p5.2.m2.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="S3.p5.2.m2.1b"><ci id="S3.p5.2.m2.1.1.cmml" xref="S3.p5.2.m2.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.2.m2.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="S3.p5.2.m2.1d">roman_ℓ</annotation></semantics></math>, denoting the number of <em class="ltx_emph ltx_font_italic" id="S3.p5.11.1">positive</em> bias classes. (There are <math alttext="L=2\ell+1" class="ltx_Math" display="inline" id="S3.p5.3.m3.1"><semantics id="S3.p5.3.m3.1a"><mrow id="S3.p5.3.m3.1.1" xref="S3.p5.3.m3.1.1.cmml"><mi id="S3.p5.3.m3.1.1.2" xref="S3.p5.3.m3.1.1.2.cmml">L</mi><mo id="S3.p5.3.m3.1.1.1" xref="S3.p5.3.m3.1.1.1.cmml">=</mo><mrow id="S3.p5.3.m3.1.1.3" xref="S3.p5.3.m3.1.1.3.cmml"><mrow id="S3.p5.3.m3.1.1.3.2" xref="S3.p5.3.m3.1.1.3.2.cmml"><mn id="S3.p5.3.m3.1.1.3.2.2" xref="S3.p5.3.m3.1.1.3.2.2.cmml">2</mn><mo id="S3.p5.3.m3.1.1.3.2.1" xref="S3.p5.3.m3.1.1.3.2.1.cmml">⁢</mo><mi id="S3.p5.3.m3.1.1.3.2.3" mathvariant="normal" xref="S3.p5.3.m3.1.1.3.2.3.cmml">ℓ</mi></mrow><mo id="S3.p5.3.m3.1.1.3.1" xref="S3.p5.3.m3.1.1.3.1.cmml">+</mo><mn id="S3.p5.3.m3.1.1.3.3" xref="S3.p5.3.m3.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.3.m3.1b"><apply id="S3.p5.3.m3.1.1.cmml" xref="S3.p5.3.m3.1.1"><eq id="S3.p5.3.m3.1.1.1.cmml" xref="S3.p5.3.m3.1.1.1"></eq><ci id="S3.p5.3.m3.1.1.2.cmml" xref="S3.p5.3.m3.1.1.2">𝐿</ci><apply id="S3.p5.3.m3.1.1.3.cmml" xref="S3.p5.3.m3.1.1.3"><plus id="S3.p5.3.m3.1.1.3.1.cmml" xref="S3.p5.3.m3.1.1.3.1"></plus><apply id="S3.p5.3.m3.1.1.3.2.cmml" xref="S3.p5.3.m3.1.1.3.2"><times id="S3.p5.3.m3.1.1.3.2.1.cmml" xref="S3.p5.3.m3.1.1.3.2.1"></times><cn id="S3.p5.3.m3.1.1.3.2.2.cmml" type="integer" xref="S3.p5.3.m3.1.1.3.2.2">2</cn><ci id="S3.p5.3.m3.1.1.3.2.3.cmml" xref="S3.p5.3.m3.1.1.3.2.3">ℓ</ci></apply><cn id="S3.p5.3.m3.1.1.3.3.cmml" type="integer" xref="S3.p5.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.3.m3.1c">L=2\ell+1</annotation><annotation encoding="application/x-llamapun" id="S3.p5.3.m3.1d">italic_L = 2 roman_ℓ + 1</annotation></semantics></math> bias classes in general.) <span class="ltx_ERROR undefined" id="S3.p5.11.2">\textcite</span>Sin23-kand used <math alttext="\ell=200" class="ltx_Math" display="inline" id="S3.p5.4.m4.1"><semantics id="S3.p5.4.m4.1a"><mrow id="S3.p5.4.m4.1.1" xref="S3.p5.4.m4.1.1.cmml"><mi id="S3.p5.4.m4.1.1.2" mathvariant="normal" xref="S3.p5.4.m4.1.1.2.cmml">ℓ</mi><mo id="S3.p5.4.m4.1.1.1" xref="S3.p5.4.m4.1.1.1.cmml">=</mo><mn id="S3.p5.4.m4.1.1.3" xref="S3.p5.4.m4.1.1.3.cmml">200</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.4.m4.1b"><apply id="S3.p5.4.m4.1.1.cmml" xref="S3.p5.4.m4.1.1"><eq id="S3.p5.4.m4.1.1.1.cmml" xref="S3.p5.4.m4.1.1.1"></eq><ci id="S3.p5.4.m4.1.1.2.cmml" xref="S3.p5.4.m4.1.1.2">ℓ</ci><cn id="S3.p5.4.m4.1.1.3.cmml" type="integer" xref="S3.p5.4.m4.1.1.3">200</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.4.m4.1c">\ell=200</annotation><annotation encoding="application/x-llamapun" id="S3.p5.4.m4.1d">roman_ℓ = 200</annotation></semantics></math>, though it appears that his discretization was uniform and so it split the interval <math alttext="[1/2,1]" class="ltx_Math" display="inline" id="S3.p5.5.m5.2"><semantics id="S3.p5.5.m5.2a"><mrow id="S3.p5.5.m5.2.2.1" xref="S3.p5.5.m5.2.2.2.cmml"><mo id="S3.p5.5.m5.2.2.1.2" stretchy="false" xref="S3.p5.5.m5.2.2.2.cmml">[</mo><mrow id="S3.p5.5.m5.2.2.1.1" xref="S3.p5.5.m5.2.2.1.1.cmml"><mn id="S3.p5.5.m5.2.2.1.1.2" xref="S3.p5.5.m5.2.2.1.1.2.cmml">1</mn><mo id="S3.p5.5.m5.2.2.1.1.1" xref="S3.p5.5.m5.2.2.1.1.1.cmml">/</mo><mn id="S3.p5.5.m5.2.2.1.1.3" xref="S3.p5.5.m5.2.2.1.1.3.cmml">2</mn></mrow><mo id="S3.p5.5.m5.2.2.1.3" xref="S3.p5.5.m5.2.2.2.cmml">,</mo><mn id="S3.p5.5.m5.1.1" xref="S3.p5.5.m5.1.1.cmml">1</mn><mo id="S3.p5.5.m5.2.2.1.4" stretchy="false" xref="S3.p5.5.m5.2.2.2.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.5.m5.2b"><interval closure="closed" id="S3.p5.5.m5.2.2.2.cmml" xref="S3.p5.5.m5.2.2.1"><apply id="S3.p5.5.m5.2.2.1.1.cmml" xref="S3.p5.5.m5.2.2.1.1"><divide id="S3.p5.5.m5.2.2.1.1.1.cmml" xref="S3.p5.5.m5.2.2.1.1.1"></divide><cn id="S3.p5.5.m5.2.2.1.1.2.cmml" type="integer" xref="S3.p5.5.m5.2.2.1.1.2">1</cn><cn id="S3.p5.5.m5.2.2.1.1.3.cmml" type="integer" xref="S3.p5.5.m5.2.2.1.1.3">2</cn></apply><cn id="S3.p5.5.m5.1.1.cmml" type="integer" xref="S3.p5.5.m5.1.1">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.5.m5.2c">[1/2,1]</annotation><annotation encoding="application/x-llamapun" id="S3.p5.5.m5.2d">[ 1 / 2 , 1 ]</annotation></semantics></math> into <math alttext="\approx 100" class="ltx_Math" display="inline" id="S3.p5.6.m6.1"><semantics id="S3.p5.6.m6.1a"><mrow id="S3.p5.6.m6.1.1" xref="S3.p5.6.m6.1.1.cmml"><mi id="S3.p5.6.m6.1.1.2" xref="S3.p5.6.m6.1.1.2.cmml"></mi><mo id="S3.p5.6.m6.1.1.1" xref="S3.p5.6.m6.1.1.1.cmml">≈</mo><mn id="S3.p5.6.m6.1.1.3" xref="S3.p5.6.m6.1.1.3.cmml">100</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.6.m6.1b"><apply id="S3.p5.6.m6.1.1.cmml" xref="S3.p5.6.m6.1.1"><approx id="S3.p5.6.m6.1.1.1.cmml" xref="S3.p5.6.m6.1.1.1"></approx><csymbol cd="latexml" id="S3.p5.6.m6.1.1.2.cmml" xref="S3.p5.6.m6.1.1.2">absent</csymbol><cn id="S3.p5.6.m6.1.1.3.cmml" type="integer" xref="S3.p5.6.m6.1.1.3">100</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.6.m6.1c">\approx 100</annotation><annotation encoding="application/x-llamapun" id="S3.p5.6.m6.1d">≈ 100</annotation></semantics></math> bias classes, and we condense them into only one bias class. We also use a finer discretization with <math alttext="\ell=251" class="ltx_Math" display="inline" id="S3.p5.7.m7.1"><semantics id="S3.p5.7.m7.1a"><mrow id="S3.p5.7.m7.1.1" xref="S3.p5.7.m7.1.1.cmml"><mi id="S3.p5.7.m7.1.1.2" mathvariant="normal" xref="S3.p5.7.m7.1.1.2.cmml">ℓ</mi><mo id="S3.p5.7.m7.1.1.1" xref="S3.p5.7.m7.1.1.1.cmml">=</mo><mn id="S3.p5.7.m7.1.1.3" xref="S3.p5.7.m7.1.1.3.cmml">251</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.7.m7.1b"><apply id="S3.p5.7.m7.1.1.cmml" xref="S3.p5.7.m7.1.1"><eq id="S3.p5.7.m7.1.1.1.cmml" xref="S3.p5.7.m7.1.1.1"></eq><ci id="S3.p5.7.m7.1.1.2.cmml" xref="S3.p5.7.m7.1.1.2">ℓ</ci><cn id="S3.p5.7.m7.1.1.3.cmml" type="integer" xref="S3.p5.7.m7.1.1.3">251</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.7.m7.1c">\ell=251</annotation><annotation encoding="application/x-llamapun" id="S3.p5.7.m7.1d">roman_ℓ = 251</annotation></semantics></math>. Most importantly, we use a different intercept parameter <math alttext="b" class="ltx_Math" display="inline" id="S3.p5.8.m8.1"><semantics id="S3.p5.8.m8.1a"><mi id="S3.p5.8.m8.1.1" xref="S3.p5.8.m8.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S3.p5.8.m8.1b"><ci id="S3.p5.8.m8.1.1.cmml" xref="S3.p5.8.m8.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.8.m8.1c">b</annotation><annotation encoding="application/x-llamapun" id="S3.p5.8.m8.1d">italic_b</annotation></semantics></math>. To choose <math alttext="b" class="ltx_Math" display="inline" id="S3.p5.9.m9.1"><semantics id="S3.p5.9.m9.1a"><mi id="S3.p5.9.m9.1.1" xref="S3.p5.9.m9.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S3.p5.9.m9.1b"><ci id="S3.p5.9.m9.1.1.cmml" xref="S3.p5.9.m9.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.9.m9.1c">b</annotation><annotation encoding="application/x-llamapun" id="S3.p5.9.m9.1d">italic_b</annotation></semantics></math>, we performed a binary search using a discretization with <math alttext="\ell=51" class="ltx_Math" display="inline" id="S3.p5.10.m10.1"><semantics id="S3.p5.10.m10.1a"><mrow id="S3.p5.10.m10.1.1" xref="S3.p5.10.m10.1.1.cmml"><mi id="S3.p5.10.m10.1.1.2" mathvariant="normal" xref="S3.p5.10.m10.1.1.2.cmml">ℓ</mi><mo id="S3.p5.10.m10.1.1.1" xref="S3.p5.10.m10.1.1.1.cmml">=</mo><mn id="S3.p5.10.m10.1.1.3" xref="S3.p5.10.m10.1.1.3.cmml">51</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.10.m10.1b"><apply id="S3.p5.10.m10.1.1.cmml" xref="S3.p5.10.m10.1.1"><eq id="S3.p5.10.m10.1.1.1.cmml" xref="S3.p5.10.m10.1.1.1"></eq><ci id="S3.p5.10.m10.1.1.2.cmml" xref="S3.p5.10.m10.1.1.2">ℓ</ci><cn id="S3.p5.10.m10.1.1.3.cmml" type="integer" xref="S3.p5.10.m10.1.1.3">51</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.10.m10.1c">\ell=51</annotation><annotation encoding="application/x-llamapun" id="S3.p5.10.m10.1d">roman_ℓ = 51</annotation></semantics></math>. We found that an intercept at <math alttext="b=149/309" class="ltx_Math" display="inline" id="S3.p5.11.m11.1"><semantics id="S3.p5.11.m11.1a"><mrow id="S3.p5.11.m11.1.1" xref="S3.p5.11.m11.1.1.cmml"><mi id="S3.p5.11.m11.1.1.2" xref="S3.p5.11.m11.1.1.2.cmml">b</mi><mo id="S3.p5.11.m11.1.1.1" xref="S3.p5.11.m11.1.1.1.cmml">=</mo><mrow id="S3.p5.11.m11.1.1.3" xref="S3.p5.11.m11.1.1.3.cmml"><mn id="S3.p5.11.m11.1.1.3.2" xref="S3.p5.11.m11.1.1.3.2.cmml">149</mn><mo id="S3.p5.11.m11.1.1.3.1" xref="S3.p5.11.m11.1.1.3.1.cmml">/</mo><mn id="S3.p5.11.m11.1.1.3.3" xref="S3.p5.11.m11.1.1.3.3.cmml">309</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.11.m11.1b"><apply id="S3.p5.11.m11.1.1.cmml" xref="S3.p5.11.m11.1.1"><eq id="S3.p5.11.m11.1.1.1.cmml" xref="S3.p5.11.m11.1.1.1"></eq><ci id="S3.p5.11.m11.1.1.2.cmml" xref="S3.p5.11.m11.1.1.2">𝑏</ci><apply id="S3.p5.11.m11.1.1.3.cmml" xref="S3.p5.11.m11.1.1.3"><divide id="S3.p5.11.m11.1.1.3.1.cmml" xref="S3.p5.11.m11.1.1.3.1"></divide><cn id="S3.p5.11.m11.1.1.3.2.cmml" type="integer" xref="S3.p5.11.m11.1.1.3.2">149</cn><cn id="S3.p5.11.m11.1.1.3.3.cmml" type="integer" xref="S3.p5.11.m11.1.1.3.3">309</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.11.m11.1c">b=149/309</annotation><annotation encoding="application/x-llamapun" id="S3.p5.11.m11.1d">italic_b = 149 / 309</annotation></semantics></math> gives the best bound among the intercepts we inspected.</p> </div> <div class="ltx_proof" id="S3.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem5" title="Theorem 1.5 (New upper bound). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.5</span></a>.</h6> <div class="ltx_para" id="S3.1.p1"> <p class="ltx_p" id="S3.1.p1.10">We set <math alttext="b=149/309" class="ltx_Math" display="inline" id="S3.1.p1.1.m1.1"><semantics id="S3.1.p1.1.m1.1a"><mrow id="S3.1.p1.1.m1.1.1" xref="S3.1.p1.1.m1.1.1.cmml"><mi id="S3.1.p1.1.m1.1.1.2" xref="S3.1.p1.1.m1.1.1.2.cmml">b</mi><mo id="S3.1.p1.1.m1.1.1.1" xref="S3.1.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S3.1.p1.1.m1.1.1.3" xref="S3.1.p1.1.m1.1.1.3.cmml"><mn id="S3.1.p1.1.m1.1.1.3.2" xref="S3.1.p1.1.m1.1.1.3.2.cmml">149</mn><mo id="S3.1.p1.1.m1.1.1.3.1" xref="S3.1.p1.1.m1.1.1.3.1.cmml">/</mo><mn id="S3.1.p1.1.m1.1.1.3.3" xref="S3.1.p1.1.m1.1.1.3.3.cmml">309</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.1.m1.1b"><apply id="S3.1.p1.1.m1.1.1.cmml" xref="S3.1.p1.1.m1.1.1"><eq id="S3.1.p1.1.m1.1.1.1.cmml" xref="S3.1.p1.1.m1.1.1.1"></eq><ci id="S3.1.p1.1.m1.1.1.2.cmml" xref="S3.1.p1.1.m1.1.1.2">𝑏</ci><apply id="S3.1.p1.1.m1.1.1.3.cmml" xref="S3.1.p1.1.m1.1.1.3"><divide id="S3.1.p1.1.m1.1.1.3.1.cmml" xref="S3.1.p1.1.m1.1.1.3.1"></divide><cn id="S3.1.p1.1.m1.1.1.3.2.cmml" type="integer" xref="S3.1.p1.1.m1.1.1.3.2">149</cn><cn id="S3.1.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S3.1.p1.1.m1.1.1.3.3">309</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.1.m1.1c">b=149/309</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.1.m1.1d">italic_b = 149 / 309</annotation></semantics></math> and use a discretization of <math alttext="\mathsf{PLSigmoid}_{b}" class="ltx_Math" display="inline" id="S3.1.p1.2.m2.1"><semantics id="S3.1.p1.2.m2.1a"><msub id="S3.1.p1.2.m2.1.1" xref="S3.1.p1.2.m2.1.1.cmml"><mi id="S3.1.p1.2.m2.1.1.2" xref="S3.1.p1.2.m2.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S3.1.p1.2.m2.1.1.3" xref="S3.1.p1.2.m2.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="S3.1.p1.2.m2.1b"><apply id="S3.1.p1.2.m2.1.1.cmml" xref="S3.1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.1.p1.2.m2.1.1.1.cmml" xref="S3.1.p1.2.m2.1.1">subscript</csymbol><ci id="S3.1.p1.2.m2.1.1.2.cmml" xref="S3.1.p1.2.m2.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S3.1.p1.2.m2.1.1.3.cmml" xref="S3.1.p1.2.m2.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.2.m2.1c">\mathsf{PLSigmoid}_{b}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.2.m2.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\ell=251" class="ltx_Math" display="inline" id="S3.1.p1.3.m3.1"><semantics id="S3.1.p1.3.m3.1a"><mrow id="S3.1.p1.3.m3.1.1" xref="S3.1.p1.3.m3.1.1.cmml"><mi id="S3.1.p1.3.m3.1.1.2" mathvariant="normal" xref="S3.1.p1.3.m3.1.1.2.cmml">ℓ</mi><mo id="S3.1.p1.3.m3.1.1.1" xref="S3.1.p1.3.m3.1.1.1.cmml">=</mo><mn id="S3.1.p1.3.m3.1.1.3" xref="S3.1.p1.3.m3.1.1.3.cmml">251</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.3.m3.1b"><apply id="S3.1.p1.3.m3.1.1.cmml" xref="S3.1.p1.3.m3.1.1"><eq id="S3.1.p1.3.m3.1.1.1.cmml" xref="S3.1.p1.3.m3.1.1.1"></eq><ci id="S3.1.p1.3.m3.1.1.2.cmml" xref="S3.1.p1.3.m3.1.1.2">ℓ</ci><cn id="S3.1.p1.3.m3.1.1.3.cmml" type="integer" xref="S3.1.p1.3.m3.1.1.3">251</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.3.m3.1c">\ell=251</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.3.m3.1d">roman_ℓ = 251</annotation></semantics></math> classes. We plug this discretization into the linear program for calculating approximation ratios for antisymmetric functions (<a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S2.Thmtheorem1" title="Theorem 2.1 (LP for antisymmetric selection functions). ‣ 2.3 Linear program ‣ 2 Preliminaries ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2.1</span></a>). Ours is the natural discretization: We set <math alttext="t_{i}=\frac{i}{\ell-1}b" class="ltx_Math" display="inline" id="S3.1.p1.4.m4.1"><semantics id="S3.1.p1.4.m4.1a"><mrow id="S3.1.p1.4.m4.1.1" xref="S3.1.p1.4.m4.1.1.cmml"><msub id="S3.1.p1.4.m4.1.1.2" xref="S3.1.p1.4.m4.1.1.2.cmml"><mi id="S3.1.p1.4.m4.1.1.2.2" xref="S3.1.p1.4.m4.1.1.2.2.cmml">t</mi><mi id="S3.1.p1.4.m4.1.1.2.3" xref="S3.1.p1.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S3.1.p1.4.m4.1.1.1" xref="S3.1.p1.4.m4.1.1.1.cmml">=</mo><mrow id="S3.1.p1.4.m4.1.1.3" xref="S3.1.p1.4.m4.1.1.3.cmml"><mfrac id="S3.1.p1.4.m4.1.1.3.2" xref="S3.1.p1.4.m4.1.1.3.2.cmml"><mi id="S3.1.p1.4.m4.1.1.3.2.2" xref="S3.1.p1.4.m4.1.1.3.2.2.cmml">i</mi><mrow id="S3.1.p1.4.m4.1.1.3.2.3" xref="S3.1.p1.4.m4.1.1.3.2.3.cmml"><mi id="S3.1.p1.4.m4.1.1.3.2.3.2" mathvariant="normal" xref="S3.1.p1.4.m4.1.1.3.2.3.2.cmml">ℓ</mi><mo id="S3.1.p1.4.m4.1.1.3.2.3.1" xref="S3.1.p1.4.m4.1.1.3.2.3.1.cmml">−</mo><mn id="S3.1.p1.4.m4.1.1.3.2.3.3" xref="S3.1.p1.4.m4.1.1.3.2.3.3.cmml">1</mn></mrow></mfrac><mo id="S3.1.p1.4.m4.1.1.3.1" xref="S3.1.p1.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S3.1.p1.4.m4.1.1.3.3" xref="S3.1.p1.4.m4.1.1.3.3.cmml">b</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.4.m4.1b"><apply id="S3.1.p1.4.m4.1.1.cmml" xref="S3.1.p1.4.m4.1.1"><eq id="S3.1.p1.4.m4.1.1.1.cmml" xref="S3.1.p1.4.m4.1.1.1"></eq><apply id="S3.1.p1.4.m4.1.1.2.cmml" xref="S3.1.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.1.p1.4.m4.1.1.2.1.cmml" xref="S3.1.p1.4.m4.1.1.2">subscript</csymbol><ci id="S3.1.p1.4.m4.1.1.2.2.cmml" xref="S3.1.p1.4.m4.1.1.2.2">𝑡</ci><ci id="S3.1.p1.4.m4.1.1.2.3.cmml" xref="S3.1.p1.4.m4.1.1.2.3">𝑖</ci></apply><apply id="S3.1.p1.4.m4.1.1.3.cmml" xref="S3.1.p1.4.m4.1.1.3"><times id="S3.1.p1.4.m4.1.1.3.1.cmml" xref="S3.1.p1.4.m4.1.1.3.1"></times><apply id="S3.1.p1.4.m4.1.1.3.2.cmml" xref="S3.1.p1.4.m4.1.1.3.2"><divide id="S3.1.p1.4.m4.1.1.3.2.1.cmml" xref="S3.1.p1.4.m4.1.1.3.2"></divide><ci id="S3.1.p1.4.m4.1.1.3.2.2.cmml" xref="S3.1.p1.4.m4.1.1.3.2.2">𝑖</ci><apply id="S3.1.p1.4.m4.1.1.3.2.3.cmml" xref="S3.1.p1.4.m4.1.1.3.2.3"><minus id="S3.1.p1.4.m4.1.1.3.2.3.1.cmml" xref="S3.1.p1.4.m4.1.1.3.2.3.1"></minus><ci id="S3.1.p1.4.m4.1.1.3.2.3.2.cmml" xref="S3.1.p1.4.m4.1.1.3.2.3.2">ℓ</ci><cn id="S3.1.p1.4.m4.1.1.3.2.3.3.cmml" type="integer" xref="S3.1.p1.4.m4.1.1.3.2.3.3">1</cn></apply></apply><ci id="S3.1.p1.4.m4.1.1.3.3.cmml" xref="S3.1.p1.4.m4.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.4.m4.1c">t_{i}=\frac{i}{\ell-1}b</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.4.m4.1d">italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = divide start_ARG italic_i end_ARG start_ARG roman_ℓ - 1 end_ARG italic_b</annotation></semantics></math> for <math alttext="i\in\{0,\ldots,\ell-1\}" class="ltx_Math" display="inline" id="S3.1.p1.5.m5.3"><semantics id="S3.1.p1.5.m5.3a"><mrow id="S3.1.p1.5.m5.3.3" xref="S3.1.p1.5.m5.3.3.cmml"><mi id="S3.1.p1.5.m5.3.3.3" xref="S3.1.p1.5.m5.3.3.3.cmml">i</mi><mo id="S3.1.p1.5.m5.3.3.2" xref="S3.1.p1.5.m5.3.3.2.cmml">∈</mo><mrow id="S3.1.p1.5.m5.3.3.1.1" xref="S3.1.p1.5.m5.3.3.1.2.cmml"><mo id="S3.1.p1.5.m5.3.3.1.1.2" stretchy="false" xref="S3.1.p1.5.m5.3.3.1.2.cmml">{</mo><mn id="S3.1.p1.5.m5.1.1" xref="S3.1.p1.5.m5.1.1.cmml">0</mn><mo id="S3.1.p1.5.m5.3.3.1.1.3" xref="S3.1.p1.5.m5.3.3.1.2.cmml">,</mo><mi id="S3.1.p1.5.m5.2.2" mathvariant="normal" xref="S3.1.p1.5.m5.2.2.cmml">…</mi><mo id="S3.1.p1.5.m5.3.3.1.1.4" xref="S3.1.p1.5.m5.3.3.1.2.cmml">,</mo><mrow id="S3.1.p1.5.m5.3.3.1.1.1" xref="S3.1.p1.5.m5.3.3.1.1.1.cmml"><mi id="S3.1.p1.5.m5.3.3.1.1.1.2" mathvariant="normal" xref="S3.1.p1.5.m5.3.3.1.1.1.2.cmml">ℓ</mi><mo id="S3.1.p1.5.m5.3.3.1.1.1.1" xref="S3.1.p1.5.m5.3.3.1.1.1.1.cmml">−</mo><mn id="S3.1.p1.5.m5.3.3.1.1.1.3" xref="S3.1.p1.5.m5.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S3.1.p1.5.m5.3.3.1.1.5" stretchy="false" xref="S3.1.p1.5.m5.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.5.m5.3b"><apply id="S3.1.p1.5.m5.3.3.cmml" xref="S3.1.p1.5.m5.3.3"><in id="S3.1.p1.5.m5.3.3.2.cmml" xref="S3.1.p1.5.m5.3.3.2"></in><ci id="S3.1.p1.5.m5.3.3.3.cmml" xref="S3.1.p1.5.m5.3.3.3">𝑖</ci><set id="S3.1.p1.5.m5.3.3.1.2.cmml" xref="S3.1.p1.5.m5.3.3.1.1"><cn id="S3.1.p1.5.m5.1.1.cmml" type="integer" xref="S3.1.p1.5.m5.1.1">0</cn><ci id="S3.1.p1.5.m5.2.2.cmml" xref="S3.1.p1.5.m5.2.2">…</ci><apply id="S3.1.p1.5.m5.3.3.1.1.1.cmml" xref="S3.1.p1.5.m5.3.3.1.1.1"><minus id="S3.1.p1.5.m5.3.3.1.1.1.1.cmml" xref="S3.1.p1.5.m5.3.3.1.1.1.1"></minus><ci id="S3.1.p1.5.m5.3.3.1.1.1.2.cmml" xref="S3.1.p1.5.m5.3.3.1.1.1.2">ℓ</ci><cn id="S3.1.p1.5.m5.3.3.1.1.1.3.cmml" type="integer" xref="S3.1.p1.5.m5.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.5.m5.3c">i\in\{0,\ldots,\ell-1\}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.5.m5.3d">italic_i ∈ { 0 , … , roman_ℓ - 1 }</annotation></semantics></math>, and then <math alttext="t_{\ell}=1" class="ltx_Math" display="inline" id="S3.1.p1.6.m6.1"><semantics id="S3.1.p1.6.m6.1a"><mrow id="S3.1.p1.6.m6.1.1" xref="S3.1.p1.6.m6.1.1.cmml"><msub id="S3.1.p1.6.m6.1.1.2" xref="S3.1.p1.6.m6.1.1.2.cmml"><mi id="S3.1.p1.6.m6.1.1.2.2" xref="S3.1.p1.6.m6.1.1.2.2.cmml">t</mi><mi id="S3.1.p1.6.m6.1.1.2.3" mathvariant="normal" xref="S3.1.p1.6.m6.1.1.2.3.cmml">ℓ</mi></msub><mo id="S3.1.p1.6.m6.1.1.1" xref="S3.1.p1.6.m6.1.1.1.cmml">=</mo><mn id="S3.1.p1.6.m6.1.1.3" xref="S3.1.p1.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.6.m6.1b"><apply id="S3.1.p1.6.m6.1.1.cmml" xref="S3.1.p1.6.m6.1.1"><eq id="S3.1.p1.6.m6.1.1.1.cmml" xref="S3.1.p1.6.m6.1.1.1"></eq><apply id="S3.1.p1.6.m6.1.1.2.cmml" xref="S3.1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.1.p1.6.m6.1.1.2.1.cmml" xref="S3.1.p1.6.m6.1.1.2">subscript</csymbol><ci id="S3.1.p1.6.m6.1.1.2.2.cmml" xref="S3.1.p1.6.m6.1.1.2.2">𝑡</ci><ci id="S3.1.p1.6.m6.1.1.2.3.cmml" xref="S3.1.p1.6.m6.1.1.2.3">ℓ</ci></apply><cn id="S3.1.p1.6.m6.1.1.3.cmml" type="integer" xref="S3.1.p1.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.6.m6.1c">t_{\ell}=1</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.6.m6.1d">italic_t start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT = 1</annotation></semantics></math>.<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>À la <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S2.Thmtheorem1" title="Theorem 2.1 (LP for antisymmetric selection functions). ‣ 2.3 Linear program ‣ 2 Preliminaries ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2.1</span></a>, this creates <math alttext="L=2\ell+1=503" class="ltx_Math" display="inline" id="footnote4.m1.1"><semantics id="footnote4.m1.1b"><mrow id="footnote4.m1.1.1" xref="footnote4.m1.1.1.cmml"><mi id="footnote4.m1.1.1.2" xref="footnote4.m1.1.1.2.cmml">L</mi><mo id="footnote4.m1.1.1.3" xref="footnote4.m1.1.1.3.cmml">=</mo><mrow id="footnote4.m1.1.1.4" xref="footnote4.m1.1.1.4.cmml"><mrow id="footnote4.m1.1.1.4.2" xref="footnote4.m1.1.1.4.2.cmml"><mn id="footnote4.m1.1.1.4.2.2" xref="footnote4.m1.1.1.4.2.2.cmml">2</mn><mo id="footnote4.m1.1.1.4.2.1" xref="footnote4.m1.1.1.4.2.1.cmml">⁢</mo><mi id="footnote4.m1.1.1.4.2.3" mathvariant="normal" xref="footnote4.m1.1.1.4.2.3.cmml">ℓ</mi></mrow><mo id="footnote4.m1.1.1.4.1" xref="footnote4.m1.1.1.4.1.cmml">+</mo><mn id="footnote4.m1.1.1.4.3" xref="footnote4.m1.1.1.4.3.cmml">1</mn></mrow><mo id="footnote4.m1.1.1.5" xref="footnote4.m1.1.1.5.cmml">=</mo><mn id="footnote4.m1.1.1.6" xref="footnote4.m1.1.1.6.cmml">503</mn></mrow><annotation-xml encoding="MathML-Content" id="footnote4.m1.1c"><apply id="footnote4.m1.1.1.cmml" xref="footnote4.m1.1.1"><and id="footnote4.m1.1.1a.cmml" xref="footnote4.m1.1.1"></and><apply id="footnote4.m1.1.1b.cmml" xref="footnote4.m1.1.1"><eq id="footnote4.m1.1.1.3.cmml" xref="footnote4.m1.1.1.3"></eq><ci id="footnote4.m1.1.1.2.cmml" xref="footnote4.m1.1.1.2">𝐿</ci><apply id="footnote4.m1.1.1.4.cmml" xref="footnote4.m1.1.1.4"><plus id="footnote4.m1.1.1.4.1.cmml" xref="footnote4.m1.1.1.4.1"></plus><apply id="footnote4.m1.1.1.4.2.cmml" xref="footnote4.m1.1.1.4.2"><times id="footnote4.m1.1.1.4.2.1.cmml" xref="footnote4.m1.1.1.4.2.1"></times><cn id="footnote4.m1.1.1.4.2.2.cmml" type="integer" xref="footnote4.m1.1.1.4.2.2">2</cn><ci id="footnote4.m1.1.1.4.2.3.cmml" xref="footnote4.m1.1.1.4.2.3">ℓ</ci></apply><cn id="footnote4.m1.1.1.4.3.cmml" type="integer" xref="footnote4.m1.1.1.4.3">1</cn></apply></apply><apply id="footnote4.m1.1.1c.cmml" xref="footnote4.m1.1.1"><eq id="footnote4.m1.1.1.5.cmml" xref="footnote4.m1.1.1.5"></eq><share href="https://arxiv.org/html/2411.12976v1#footnote4.m1.1.1.4.cmml" id="footnote4.m1.1.1d.cmml" xref="footnote4.m1.1.1"></share><cn id="footnote4.m1.1.1.6.cmml" type="integer" xref="footnote4.m1.1.1.6">503</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote4.m1.1d">L=2\ell+1=503</annotation><annotation encoding="application/x-llamapun" id="footnote4.m1.1e">italic_L = 2 roman_ℓ + 1 = 503</annotation></semantics></math> bias classes, labeled <math alttext="I_{-\ell},\ldots,I_{+\ell}" class="ltx_Math" display="inline" id="footnote4.m2.3"><semantics id="footnote4.m2.3b"><mrow id="footnote4.m2.3.3.2" xref="footnote4.m2.3.3.3.cmml"><msub id="footnote4.m2.2.2.1.1" xref="footnote4.m2.2.2.1.1.cmml"><mi id="footnote4.m2.2.2.1.1.2" xref="footnote4.m2.2.2.1.1.2.cmml">I</mi><mrow id="footnote4.m2.2.2.1.1.3" xref="footnote4.m2.2.2.1.1.3.cmml"><mo id="footnote4.m2.2.2.1.1.3b" xref="footnote4.m2.2.2.1.1.3.cmml">−</mo><mi id="footnote4.m2.2.2.1.1.3.2" mathvariant="normal" xref="footnote4.m2.2.2.1.1.3.2.cmml">ℓ</mi></mrow></msub><mo id="footnote4.m2.3.3.2.3" xref="footnote4.m2.3.3.3.cmml">,</mo><mi id="footnote4.m2.1.1" mathvariant="normal" xref="footnote4.m2.1.1.cmml">…</mi><mo id="footnote4.m2.3.3.2.4" xref="footnote4.m2.3.3.3.cmml">,</mo><msub id="footnote4.m2.3.3.2.2" xref="footnote4.m2.3.3.2.2.cmml"><mi id="footnote4.m2.3.3.2.2.2" xref="footnote4.m2.3.3.2.2.2.cmml">I</mi><mrow id="footnote4.m2.3.3.2.2.3" xref="footnote4.m2.3.3.2.2.3.cmml"><mo id="footnote4.m2.3.3.2.2.3b" xref="footnote4.m2.3.3.2.2.3.cmml">+</mo><mi id="footnote4.m2.3.3.2.2.3.2" mathvariant="normal" xref="footnote4.m2.3.3.2.2.3.2.cmml">ℓ</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="footnote4.m2.3c"><list id="footnote4.m2.3.3.3.cmml" xref="footnote4.m2.3.3.2"><apply id="footnote4.m2.2.2.1.1.cmml" xref="footnote4.m2.2.2.1.1"><csymbol cd="ambiguous" id="footnote4.m2.2.2.1.1.1.cmml" xref="footnote4.m2.2.2.1.1">subscript</csymbol><ci id="footnote4.m2.2.2.1.1.2.cmml" xref="footnote4.m2.2.2.1.1.2">𝐼</ci><apply id="footnote4.m2.2.2.1.1.3.cmml" xref="footnote4.m2.2.2.1.1.3"><minus id="footnote4.m2.2.2.1.1.3.1.cmml" xref="footnote4.m2.2.2.1.1.3"></minus><ci id="footnote4.m2.2.2.1.1.3.2.cmml" xref="footnote4.m2.2.2.1.1.3.2">ℓ</ci></apply></apply><ci id="footnote4.m2.1.1.cmml" xref="footnote4.m2.1.1">…</ci><apply id="footnote4.m2.3.3.2.2.cmml" xref="footnote4.m2.3.3.2.2"><csymbol cd="ambiguous" id="footnote4.m2.3.3.2.2.1.cmml" xref="footnote4.m2.3.3.2.2">subscript</csymbol><ci id="footnote4.m2.3.3.2.2.2.cmml" xref="footnote4.m2.3.3.2.2.2">𝐼</ci><apply id="footnote4.m2.3.3.2.2.3.cmml" xref="footnote4.m2.3.3.2.2.3"><plus id="footnote4.m2.3.3.2.2.3.1.cmml" xref="footnote4.m2.3.3.2.2.3"></plus><ci id="footnote4.m2.3.3.2.2.3.2.cmml" xref="footnote4.m2.3.3.2.2.3.2">ℓ</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="footnote4.m2.3d">I_{-\ell},\ldots,I_{+\ell}</annotation><annotation encoding="application/x-llamapun" id="footnote4.m2.3e">italic_I start_POSTSUBSCRIPT - roman_ℓ end_POSTSUBSCRIPT , … , italic_I start_POSTSUBSCRIPT + roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>. In particular, <math alttext="I_{0}=[0,0]" class="ltx_Math" display="inline" id="footnote4.m3.2"><semantics id="footnote4.m3.2b"><mrow id="footnote4.m3.2.3" xref="footnote4.m3.2.3.cmml"><msub id="footnote4.m3.2.3.2" xref="footnote4.m3.2.3.2.cmml"><mi id="footnote4.m3.2.3.2.2" xref="footnote4.m3.2.3.2.2.cmml">I</mi><mn id="footnote4.m3.2.3.2.3" xref="footnote4.m3.2.3.2.3.cmml">0</mn></msub><mo id="footnote4.m3.2.3.1" xref="footnote4.m3.2.3.1.cmml">=</mo><mrow id="footnote4.m3.2.3.3.2" xref="footnote4.m3.2.3.3.1.cmml"><mo id="footnote4.m3.2.3.3.2.1" stretchy="false" xref="footnote4.m3.2.3.3.1.cmml">[</mo><mn id="footnote4.m3.1.1" xref="footnote4.m3.1.1.cmml">0</mn><mo id="footnote4.m3.2.3.3.2.2" xref="footnote4.m3.2.3.3.1.cmml">,</mo><mn id="footnote4.m3.2.2" xref="footnote4.m3.2.2.cmml">0</mn><mo id="footnote4.m3.2.3.3.2.3" stretchy="false" xref="footnote4.m3.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote4.m3.2c"><apply id="footnote4.m3.2.3.cmml" xref="footnote4.m3.2.3"><eq id="footnote4.m3.2.3.1.cmml" xref="footnote4.m3.2.3.1"></eq><apply id="footnote4.m3.2.3.2.cmml" xref="footnote4.m3.2.3.2"><csymbol cd="ambiguous" id="footnote4.m3.2.3.2.1.cmml" xref="footnote4.m3.2.3.2">subscript</csymbol><ci id="footnote4.m3.2.3.2.2.cmml" xref="footnote4.m3.2.3.2.2">𝐼</ci><cn id="footnote4.m3.2.3.2.3.cmml" type="integer" xref="footnote4.m3.2.3.2.3">0</cn></apply><interval closure="closed" id="footnote4.m3.2.3.3.1.cmml" xref="footnote4.m3.2.3.3.2"><cn id="footnote4.m3.1.1.cmml" type="integer" xref="footnote4.m3.1.1">0</cn><cn id="footnote4.m3.2.2.cmml" type="integer" xref="footnote4.m3.2.2">0</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote4.m3.2d">I_{0}=[0,0]</annotation><annotation encoding="application/x-llamapun" id="footnote4.m3.2e">italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = [ 0 , 0 ]</annotation></semantics></math>, <math alttext="I_{+i}=(\frac{i-1}{\ell-1}b,\frac{i}{\ell-1}b]" class="ltx_Math" display="inline" id="footnote4.m4.2"><semantics id="footnote4.m4.2b"><mrow id="footnote4.m4.2.2" xref="footnote4.m4.2.2.cmml"><msub id="footnote4.m4.2.2.4" xref="footnote4.m4.2.2.4.cmml"><mi id="footnote4.m4.2.2.4.2" xref="footnote4.m4.2.2.4.2.cmml">I</mi><mrow id="footnote4.m4.2.2.4.3" xref="footnote4.m4.2.2.4.3.cmml"><mo id="footnote4.m4.2.2.4.3b" xref="footnote4.m4.2.2.4.3.cmml">+</mo><mi id="footnote4.m4.2.2.4.3.2" xref="footnote4.m4.2.2.4.3.2.cmml">i</mi></mrow></msub><mo id="footnote4.m4.2.2.3" xref="footnote4.m4.2.2.3.cmml">=</mo><mrow id="footnote4.m4.2.2.2.2" xref="footnote4.m4.2.2.2.3.cmml"><mo id="footnote4.m4.2.2.2.2.3" stretchy="false" xref="footnote4.m4.2.2.2.3.cmml">(</mo><mrow id="footnote4.m4.1.1.1.1.1" xref="footnote4.m4.1.1.1.1.1.cmml"><mfrac id="footnote4.m4.1.1.1.1.1.2" xref="footnote4.m4.1.1.1.1.1.2.cmml"><mrow id="footnote4.m4.1.1.1.1.1.2.2" xref="footnote4.m4.1.1.1.1.1.2.2.cmml"><mi id="footnote4.m4.1.1.1.1.1.2.2.2" xref="footnote4.m4.1.1.1.1.1.2.2.2.cmml">i</mi><mo id="footnote4.m4.1.1.1.1.1.2.2.1" xref="footnote4.m4.1.1.1.1.1.2.2.1.cmml">−</mo><mn id="footnote4.m4.1.1.1.1.1.2.2.3" xref="footnote4.m4.1.1.1.1.1.2.2.3.cmml">1</mn></mrow><mrow id="footnote4.m4.1.1.1.1.1.2.3" xref="footnote4.m4.1.1.1.1.1.2.3.cmml"><mi id="footnote4.m4.1.1.1.1.1.2.3.2" mathvariant="normal" xref="footnote4.m4.1.1.1.1.1.2.3.2.cmml">ℓ</mi><mo id="footnote4.m4.1.1.1.1.1.2.3.1" xref="footnote4.m4.1.1.1.1.1.2.3.1.cmml">−</mo><mn id="footnote4.m4.1.1.1.1.1.2.3.3" xref="footnote4.m4.1.1.1.1.1.2.3.3.cmml">1</mn></mrow></mfrac><mo id="footnote4.m4.1.1.1.1.1.1" xref="footnote4.m4.1.1.1.1.1.1.cmml">⁢</mo><mi id="footnote4.m4.1.1.1.1.1.3" xref="footnote4.m4.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="footnote4.m4.2.2.2.2.4" xref="footnote4.m4.2.2.2.3.cmml">,</mo><mrow id="footnote4.m4.2.2.2.2.2" xref="footnote4.m4.2.2.2.2.2.cmml"><mfrac id="footnote4.m4.2.2.2.2.2.2" xref="footnote4.m4.2.2.2.2.2.2.cmml"><mi id="footnote4.m4.2.2.2.2.2.2.2" xref="footnote4.m4.2.2.2.2.2.2.2.cmml">i</mi><mrow id="footnote4.m4.2.2.2.2.2.2.3" xref="footnote4.m4.2.2.2.2.2.2.3.cmml"><mi id="footnote4.m4.2.2.2.2.2.2.3.2" mathvariant="normal" xref="footnote4.m4.2.2.2.2.2.2.3.2.cmml">ℓ</mi><mo id="footnote4.m4.2.2.2.2.2.2.3.1" xref="footnote4.m4.2.2.2.2.2.2.3.1.cmml">−</mo><mn id="footnote4.m4.2.2.2.2.2.2.3.3" xref="footnote4.m4.2.2.2.2.2.2.3.3.cmml">1</mn></mrow></mfrac><mo id="footnote4.m4.2.2.2.2.2.1" xref="footnote4.m4.2.2.2.2.2.1.cmml">⁢</mo><mi id="footnote4.m4.2.2.2.2.2.3" xref="footnote4.m4.2.2.2.2.2.3.cmml">b</mi></mrow><mo id="footnote4.m4.2.2.2.2.5" stretchy="false" xref="footnote4.m4.2.2.2.3.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote4.m4.2c"><apply id="footnote4.m4.2.2.cmml" xref="footnote4.m4.2.2"><eq id="footnote4.m4.2.2.3.cmml" xref="footnote4.m4.2.2.3"></eq><apply id="footnote4.m4.2.2.4.cmml" xref="footnote4.m4.2.2.4"><csymbol cd="ambiguous" id="footnote4.m4.2.2.4.1.cmml" xref="footnote4.m4.2.2.4">subscript</csymbol><ci id="footnote4.m4.2.2.4.2.cmml" xref="footnote4.m4.2.2.4.2">𝐼</ci><apply id="footnote4.m4.2.2.4.3.cmml" xref="footnote4.m4.2.2.4.3"><plus id="footnote4.m4.2.2.4.3.1.cmml" xref="footnote4.m4.2.2.4.3"></plus><ci id="footnote4.m4.2.2.4.3.2.cmml" xref="footnote4.m4.2.2.4.3.2">𝑖</ci></apply></apply><interval closure="open-closed" id="footnote4.m4.2.2.2.3.cmml" xref="footnote4.m4.2.2.2.2"><apply id="footnote4.m4.1.1.1.1.1.cmml" xref="footnote4.m4.1.1.1.1.1"><times id="footnote4.m4.1.1.1.1.1.1.cmml" xref="footnote4.m4.1.1.1.1.1.1"></times><apply id="footnote4.m4.1.1.1.1.1.2.cmml" xref="footnote4.m4.1.1.1.1.1.2"><divide id="footnote4.m4.1.1.1.1.1.2.1.cmml" xref="footnote4.m4.1.1.1.1.1.2"></divide><apply id="footnote4.m4.1.1.1.1.1.2.2.cmml" xref="footnote4.m4.1.1.1.1.1.2.2"><minus id="footnote4.m4.1.1.1.1.1.2.2.1.cmml" xref="footnote4.m4.1.1.1.1.1.2.2.1"></minus><ci id="footnote4.m4.1.1.1.1.1.2.2.2.cmml" xref="footnote4.m4.1.1.1.1.1.2.2.2">𝑖</ci><cn id="footnote4.m4.1.1.1.1.1.2.2.3.cmml" type="integer" xref="footnote4.m4.1.1.1.1.1.2.2.3">1</cn></apply><apply id="footnote4.m4.1.1.1.1.1.2.3.cmml" xref="footnote4.m4.1.1.1.1.1.2.3"><minus id="footnote4.m4.1.1.1.1.1.2.3.1.cmml" xref="footnote4.m4.1.1.1.1.1.2.3.1"></minus><ci id="footnote4.m4.1.1.1.1.1.2.3.2.cmml" xref="footnote4.m4.1.1.1.1.1.2.3.2">ℓ</ci><cn id="footnote4.m4.1.1.1.1.1.2.3.3.cmml" type="integer" xref="footnote4.m4.1.1.1.1.1.2.3.3">1</cn></apply></apply><ci id="footnote4.m4.1.1.1.1.1.3.cmml" xref="footnote4.m4.1.1.1.1.1.3">𝑏</ci></apply><apply id="footnote4.m4.2.2.2.2.2.cmml" xref="footnote4.m4.2.2.2.2.2"><times id="footnote4.m4.2.2.2.2.2.1.cmml" xref="footnote4.m4.2.2.2.2.2.1"></times><apply id="footnote4.m4.2.2.2.2.2.2.cmml" xref="footnote4.m4.2.2.2.2.2.2"><divide id="footnote4.m4.2.2.2.2.2.2.1.cmml" xref="footnote4.m4.2.2.2.2.2.2"></divide><ci id="footnote4.m4.2.2.2.2.2.2.2.cmml" xref="footnote4.m4.2.2.2.2.2.2.2">𝑖</ci><apply id="footnote4.m4.2.2.2.2.2.2.3.cmml" xref="footnote4.m4.2.2.2.2.2.2.3"><minus id="footnote4.m4.2.2.2.2.2.2.3.1.cmml" xref="footnote4.m4.2.2.2.2.2.2.3.1"></minus><ci id="footnote4.m4.2.2.2.2.2.2.3.2.cmml" xref="footnote4.m4.2.2.2.2.2.2.3.2">ℓ</ci><cn id="footnote4.m4.2.2.2.2.2.2.3.3.cmml" type="integer" xref="footnote4.m4.2.2.2.2.2.2.3.3">1</cn></apply></apply><ci id="footnote4.m4.2.2.2.2.2.3.cmml" xref="footnote4.m4.2.2.2.2.2.3">𝑏</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote4.m4.2d">I_{+i}=(\frac{i-1}{\ell-1}b,\frac{i}{\ell-1}b]</annotation><annotation encoding="application/x-llamapun" id="footnote4.m4.2e">italic_I start_POSTSUBSCRIPT + italic_i end_POSTSUBSCRIPT = ( divide start_ARG italic_i - 1 end_ARG start_ARG roman_ℓ - 1 end_ARG italic_b , divide start_ARG italic_i end_ARG start_ARG roman_ℓ - 1 end_ARG italic_b ]</annotation></semantics></math> for <math alttext="i\in[\ell-1]" class="ltx_Math" display="inline" id="footnote4.m5.1"><semantics id="footnote4.m5.1b"><mrow id="footnote4.m5.1.1" xref="footnote4.m5.1.1.cmml"><mi id="footnote4.m5.1.1.3" xref="footnote4.m5.1.1.3.cmml">i</mi><mo id="footnote4.m5.1.1.2" xref="footnote4.m5.1.1.2.cmml">∈</mo><mrow id="footnote4.m5.1.1.1.1" xref="footnote4.m5.1.1.1.2.cmml"><mo id="footnote4.m5.1.1.1.1.2" stretchy="false" xref="footnote4.m5.1.1.1.2.1.cmml">[</mo><mrow id="footnote4.m5.1.1.1.1.1" xref="footnote4.m5.1.1.1.1.1.cmml"><mi id="footnote4.m5.1.1.1.1.1.2" mathvariant="normal" xref="footnote4.m5.1.1.1.1.1.2.cmml">ℓ</mi><mo id="footnote4.m5.1.1.1.1.1.1" xref="footnote4.m5.1.1.1.1.1.1.cmml">−</mo><mn id="footnote4.m5.1.1.1.1.1.3" xref="footnote4.m5.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="footnote4.m5.1.1.1.1.3" stretchy="false" xref="footnote4.m5.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote4.m5.1c"><apply id="footnote4.m5.1.1.cmml" xref="footnote4.m5.1.1"><in id="footnote4.m5.1.1.2.cmml" xref="footnote4.m5.1.1.2"></in><ci id="footnote4.m5.1.1.3.cmml" xref="footnote4.m5.1.1.3">𝑖</ci><apply id="footnote4.m5.1.1.1.2.cmml" xref="footnote4.m5.1.1.1.1"><csymbol cd="latexml" id="footnote4.m5.1.1.1.2.1.cmml" xref="footnote4.m5.1.1.1.1.2">delimited-[]</csymbol><apply id="footnote4.m5.1.1.1.1.1.cmml" xref="footnote4.m5.1.1.1.1.1"><minus id="footnote4.m5.1.1.1.1.1.1.cmml" xref="footnote4.m5.1.1.1.1.1.1"></minus><ci id="footnote4.m5.1.1.1.1.1.2.cmml" xref="footnote4.m5.1.1.1.1.1.2">ℓ</ci><cn id="footnote4.m5.1.1.1.1.1.3.cmml" type="integer" xref="footnote4.m5.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote4.m5.1d">i\in[\ell-1]</annotation><annotation encoding="application/x-llamapun" id="footnote4.m5.1e">italic_i ∈ [ roman_ℓ - 1 ]</annotation></semantics></math>, and <math alttext="i_{+\ell}=(b,1]" class="ltx_Math" display="inline" id="footnote4.m6.2"><semantics id="footnote4.m6.2b"><mrow id="footnote4.m6.2.3" xref="footnote4.m6.2.3.cmml"><msub id="footnote4.m6.2.3.2" xref="footnote4.m6.2.3.2.cmml"><mi id="footnote4.m6.2.3.2.2" xref="footnote4.m6.2.3.2.2.cmml">i</mi><mrow id="footnote4.m6.2.3.2.3" xref="footnote4.m6.2.3.2.3.cmml"><mo id="footnote4.m6.2.3.2.3b" xref="footnote4.m6.2.3.2.3.cmml">+</mo><mi id="footnote4.m6.2.3.2.3.2" mathvariant="normal" xref="footnote4.m6.2.3.2.3.2.cmml">ℓ</mi></mrow></msub><mo id="footnote4.m6.2.3.1" xref="footnote4.m6.2.3.1.cmml">=</mo><mrow id="footnote4.m6.2.3.3.2" xref="footnote4.m6.2.3.3.1.cmml"><mo id="footnote4.m6.2.3.3.2.1" stretchy="false" xref="footnote4.m6.2.3.3.1.cmml">(</mo><mi id="footnote4.m6.1.1" xref="footnote4.m6.1.1.cmml">b</mi><mo id="footnote4.m6.2.3.3.2.2" xref="footnote4.m6.2.3.3.1.cmml">,</mo><mn id="footnote4.m6.2.2" xref="footnote4.m6.2.2.cmml">1</mn><mo id="footnote4.m6.2.3.3.2.3" stretchy="false" xref="footnote4.m6.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote4.m6.2c"><apply id="footnote4.m6.2.3.cmml" xref="footnote4.m6.2.3"><eq id="footnote4.m6.2.3.1.cmml" xref="footnote4.m6.2.3.1"></eq><apply id="footnote4.m6.2.3.2.cmml" xref="footnote4.m6.2.3.2"><csymbol cd="ambiguous" id="footnote4.m6.2.3.2.1.cmml" xref="footnote4.m6.2.3.2">subscript</csymbol><ci id="footnote4.m6.2.3.2.2.cmml" xref="footnote4.m6.2.3.2.2">𝑖</ci><apply id="footnote4.m6.2.3.2.3.cmml" xref="footnote4.m6.2.3.2.3"><plus id="footnote4.m6.2.3.2.3.1.cmml" xref="footnote4.m6.2.3.2.3"></plus><ci id="footnote4.m6.2.3.2.3.2.cmml" xref="footnote4.m6.2.3.2.3.2">ℓ</ci></apply></apply><interval closure="open-closed" id="footnote4.m6.2.3.3.1.cmml" xref="footnote4.m6.2.3.3.2"><ci id="footnote4.m6.1.1.cmml" xref="footnote4.m6.1.1">𝑏</ci><cn id="footnote4.m6.2.2.cmml" type="integer" xref="footnote4.m6.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote4.m6.2d">i_{+\ell}=(b,1]</annotation><annotation encoding="application/x-llamapun" id="footnote4.m6.2e">italic_i start_POSTSUBSCRIPT + roman_ℓ end_POSTSUBSCRIPT = ( italic_b , 1 ]</annotation></semantics></math>, and <math alttext="I_{-i}" class="ltx_Math" display="inline" id="footnote4.m7.1"><semantics id="footnote4.m7.1b"><msub id="footnote4.m7.1.1" xref="footnote4.m7.1.1.cmml"><mi id="footnote4.m7.1.1.2" xref="footnote4.m7.1.1.2.cmml">I</mi><mrow id="footnote4.m7.1.1.3" xref="footnote4.m7.1.1.3.cmml"><mo id="footnote4.m7.1.1.3b" xref="footnote4.m7.1.1.3.cmml">−</mo><mi id="footnote4.m7.1.1.3.2" xref="footnote4.m7.1.1.3.2.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="footnote4.m7.1c"><apply id="footnote4.m7.1.1.cmml" xref="footnote4.m7.1.1"><csymbol cd="ambiguous" id="footnote4.m7.1.1.1.cmml" xref="footnote4.m7.1.1">subscript</csymbol><ci id="footnote4.m7.1.1.2.cmml" xref="footnote4.m7.1.1.2">𝐼</ci><apply id="footnote4.m7.1.1.3.cmml" xref="footnote4.m7.1.1.3"><minus id="footnote4.m7.1.1.3.1.cmml" xref="footnote4.m7.1.1.3"></minus><ci id="footnote4.m7.1.1.3.2.cmml" xref="footnote4.m7.1.1.3.2">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote4.m7.1d">I_{-i}</annotation><annotation encoding="application/x-llamapun" id="footnote4.m7.1e">italic_I start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is defined symmetrically.</span></span></span> We set <math alttext="p_{i}=\frac{\mathsf{PLSigmoid}_{b}(\frac{i-1}{\ell-1})+\mathsf{PLSigmoid}_{b}(% \frac{i}{\ell-1})}{2}" class="ltx_Math" display="inline" id="S3.1.p1.7.m7.2"><semantics id="S3.1.p1.7.m7.2a"><mrow id="S3.1.p1.7.m7.2.3" xref="S3.1.p1.7.m7.2.3.cmml"><msub id="S3.1.p1.7.m7.2.3.2" xref="S3.1.p1.7.m7.2.3.2.cmml"><mi id="S3.1.p1.7.m7.2.3.2.2" xref="S3.1.p1.7.m7.2.3.2.2.cmml">p</mi><mi id="S3.1.p1.7.m7.2.3.2.3" xref="S3.1.p1.7.m7.2.3.2.3.cmml">i</mi></msub><mo id="S3.1.p1.7.m7.2.3.1" xref="S3.1.p1.7.m7.2.3.1.cmml">=</mo><mfrac id="S3.1.p1.7.m7.2.2" xref="S3.1.p1.7.m7.2.2.cmml"><mrow id="S3.1.p1.7.m7.2.2.2" xref="S3.1.p1.7.m7.2.2.2.cmml"><mrow id="S3.1.p1.7.m7.2.2.2.4" xref="S3.1.p1.7.m7.2.2.2.4.cmml"><msub id="S3.1.p1.7.m7.2.2.2.4.2" xref="S3.1.p1.7.m7.2.2.2.4.2.cmml"><mi id="S3.1.p1.7.m7.2.2.2.4.2.2" xref="S3.1.p1.7.m7.2.2.2.4.2.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S3.1.p1.7.m7.2.2.2.4.2.3" xref="S3.1.p1.7.m7.2.2.2.4.2.3.cmml">b</mi></msub><mo id="S3.1.p1.7.m7.2.2.2.4.1" xref="S3.1.p1.7.m7.2.2.2.4.1.cmml">⁢</mo><mrow id="S3.1.p1.7.m7.2.2.2.4.3.2" xref="S3.1.p1.7.m7.1.1.1.1.cmml"><mo id="S3.1.p1.7.m7.2.2.2.4.3.2.1" stretchy="false" xref="S3.1.p1.7.m7.1.1.1.1.cmml">(</mo><mfrac id="S3.1.p1.7.m7.1.1.1.1" xref="S3.1.p1.7.m7.1.1.1.1.cmml"><mrow id="S3.1.p1.7.m7.1.1.1.1.2" xref="S3.1.p1.7.m7.1.1.1.1.2.cmml"><mi id="S3.1.p1.7.m7.1.1.1.1.2.2" xref="S3.1.p1.7.m7.1.1.1.1.2.2.cmml">i</mi><mo id="S3.1.p1.7.m7.1.1.1.1.2.1" xref="S3.1.p1.7.m7.1.1.1.1.2.1.cmml">−</mo><mn id="S3.1.p1.7.m7.1.1.1.1.2.3" xref="S3.1.p1.7.m7.1.1.1.1.2.3.cmml">1</mn></mrow><mrow id="S3.1.p1.7.m7.1.1.1.1.3" xref="S3.1.p1.7.m7.1.1.1.1.3.cmml"><mi id="S3.1.p1.7.m7.1.1.1.1.3.2" mathvariant="normal" xref="S3.1.p1.7.m7.1.1.1.1.3.2.cmml">ℓ</mi><mo id="S3.1.p1.7.m7.1.1.1.1.3.1" xref="S3.1.p1.7.m7.1.1.1.1.3.1.cmml">−</mo><mn id="S3.1.p1.7.m7.1.1.1.1.3.3" xref="S3.1.p1.7.m7.1.1.1.1.3.3.cmml">1</mn></mrow></mfrac><mo id="S3.1.p1.7.m7.2.2.2.4.3.2.2" stretchy="false" xref="S3.1.p1.7.m7.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.1.p1.7.m7.2.2.2.3" xref="S3.1.p1.7.m7.2.2.2.3.cmml">+</mo><mrow id="S3.1.p1.7.m7.2.2.2.5" xref="S3.1.p1.7.m7.2.2.2.5.cmml"><msub id="S3.1.p1.7.m7.2.2.2.5.2" xref="S3.1.p1.7.m7.2.2.2.5.2.cmml"><mi id="S3.1.p1.7.m7.2.2.2.5.2.2" xref="S3.1.p1.7.m7.2.2.2.5.2.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S3.1.p1.7.m7.2.2.2.5.2.3" xref="S3.1.p1.7.m7.2.2.2.5.2.3.cmml">b</mi></msub><mo id="S3.1.p1.7.m7.2.2.2.5.1" xref="S3.1.p1.7.m7.2.2.2.5.1.cmml">⁢</mo><mrow id="S3.1.p1.7.m7.2.2.2.5.3.2" xref="S3.1.p1.7.m7.2.2.2.2.cmml"><mo id="S3.1.p1.7.m7.2.2.2.5.3.2.1" stretchy="false" xref="S3.1.p1.7.m7.2.2.2.2.cmml">(</mo><mfrac id="S3.1.p1.7.m7.2.2.2.2" xref="S3.1.p1.7.m7.2.2.2.2.cmml"><mi id="S3.1.p1.7.m7.2.2.2.2.2" xref="S3.1.p1.7.m7.2.2.2.2.2.cmml">i</mi><mrow id="S3.1.p1.7.m7.2.2.2.2.3" xref="S3.1.p1.7.m7.2.2.2.2.3.cmml"><mi id="S3.1.p1.7.m7.2.2.2.2.3.2" mathvariant="normal" xref="S3.1.p1.7.m7.2.2.2.2.3.2.cmml">ℓ</mi><mo id="S3.1.p1.7.m7.2.2.2.2.3.1" xref="S3.1.p1.7.m7.2.2.2.2.3.1.cmml">−</mo><mn id="S3.1.p1.7.m7.2.2.2.2.3.3" xref="S3.1.p1.7.m7.2.2.2.2.3.3.cmml">1</mn></mrow></mfrac><mo id="S3.1.p1.7.m7.2.2.2.5.3.2.2" stretchy="false" xref="S3.1.p1.7.m7.2.2.2.2.cmml">)</mo></mrow></mrow></mrow><mn id="S3.1.p1.7.m7.2.2.4" xref="S3.1.p1.7.m7.2.2.4.cmml">2</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.7.m7.2b"><apply id="S3.1.p1.7.m7.2.3.cmml" xref="S3.1.p1.7.m7.2.3"><eq id="S3.1.p1.7.m7.2.3.1.cmml" xref="S3.1.p1.7.m7.2.3.1"></eq><apply id="S3.1.p1.7.m7.2.3.2.cmml" xref="S3.1.p1.7.m7.2.3.2"><csymbol cd="ambiguous" id="S3.1.p1.7.m7.2.3.2.1.cmml" xref="S3.1.p1.7.m7.2.3.2">subscript</csymbol><ci id="S3.1.p1.7.m7.2.3.2.2.cmml" xref="S3.1.p1.7.m7.2.3.2.2">𝑝</ci><ci id="S3.1.p1.7.m7.2.3.2.3.cmml" xref="S3.1.p1.7.m7.2.3.2.3">𝑖</ci></apply><apply id="S3.1.p1.7.m7.2.2.cmml" xref="S3.1.p1.7.m7.2.2"><divide id="S3.1.p1.7.m7.2.2.3.cmml" xref="S3.1.p1.7.m7.2.2"></divide><apply id="S3.1.p1.7.m7.2.2.2.cmml" xref="S3.1.p1.7.m7.2.2.2"><plus id="S3.1.p1.7.m7.2.2.2.3.cmml" xref="S3.1.p1.7.m7.2.2.2.3"></plus><apply id="S3.1.p1.7.m7.2.2.2.4.cmml" xref="S3.1.p1.7.m7.2.2.2.4"><times id="S3.1.p1.7.m7.2.2.2.4.1.cmml" xref="S3.1.p1.7.m7.2.2.2.4.1"></times><apply id="S3.1.p1.7.m7.2.2.2.4.2.cmml" xref="S3.1.p1.7.m7.2.2.2.4.2"><csymbol cd="ambiguous" id="S3.1.p1.7.m7.2.2.2.4.2.1.cmml" xref="S3.1.p1.7.m7.2.2.2.4.2">subscript</csymbol><ci id="S3.1.p1.7.m7.2.2.2.4.2.2.cmml" xref="S3.1.p1.7.m7.2.2.2.4.2.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S3.1.p1.7.m7.2.2.2.4.2.3.cmml" xref="S3.1.p1.7.m7.2.2.2.4.2.3">𝑏</ci></apply><apply id="S3.1.p1.7.m7.1.1.1.1.cmml" xref="S3.1.p1.7.m7.2.2.2.4.3.2"><divide id="S3.1.p1.7.m7.1.1.1.1.1.cmml" xref="S3.1.p1.7.m7.2.2.2.4.3.2"></divide><apply id="S3.1.p1.7.m7.1.1.1.1.2.cmml" xref="S3.1.p1.7.m7.1.1.1.1.2"><minus id="S3.1.p1.7.m7.1.1.1.1.2.1.cmml" xref="S3.1.p1.7.m7.1.1.1.1.2.1"></minus><ci id="S3.1.p1.7.m7.1.1.1.1.2.2.cmml" xref="S3.1.p1.7.m7.1.1.1.1.2.2">𝑖</ci><cn id="S3.1.p1.7.m7.1.1.1.1.2.3.cmml" type="integer" xref="S3.1.p1.7.m7.1.1.1.1.2.3">1</cn></apply><apply id="S3.1.p1.7.m7.1.1.1.1.3.cmml" xref="S3.1.p1.7.m7.1.1.1.1.3"><minus id="S3.1.p1.7.m7.1.1.1.1.3.1.cmml" xref="S3.1.p1.7.m7.1.1.1.1.3.1"></minus><ci id="S3.1.p1.7.m7.1.1.1.1.3.2.cmml" xref="S3.1.p1.7.m7.1.1.1.1.3.2">ℓ</ci><cn id="S3.1.p1.7.m7.1.1.1.1.3.3.cmml" type="integer" xref="S3.1.p1.7.m7.1.1.1.1.3.3">1</cn></apply></apply></apply><apply id="S3.1.p1.7.m7.2.2.2.5.cmml" xref="S3.1.p1.7.m7.2.2.2.5"><times id="S3.1.p1.7.m7.2.2.2.5.1.cmml" xref="S3.1.p1.7.m7.2.2.2.5.1"></times><apply id="S3.1.p1.7.m7.2.2.2.5.2.cmml" xref="S3.1.p1.7.m7.2.2.2.5.2"><csymbol cd="ambiguous" id="S3.1.p1.7.m7.2.2.2.5.2.1.cmml" xref="S3.1.p1.7.m7.2.2.2.5.2">subscript</csymbol><ci id="S3.1.p1.7.m7.2.2.2.5.2.2.cmml" xref="S3.1.p1.7.m7.2.2.2.5.2.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S3.1.p1.7.m7.2.2.2.5.2.3.cmml" xref="S3.1.p1.7.m7.2.2.2.5.2.3">𝑏</ci></apply><apply id="S3.1.p1.7.m7.2.2.2.2.cmml" xref="S3.1.p1.7.m7.2.2.2.5.3.2"><divide id="S3.1.p1.7.m7.2.2.2.2.1.cmml" xref="S3.1.p1.7.m7.2.2.2.5.3.2"></divide><ci id="S3.1.p1.7.m7.2.2.2.2.2.cmml" xref="S3.1.p1.7.m7.2.2.2.2.2">𝑖</ci><apply id="S3.1.p1.7.m7.2.2.2.2.3.cmml" xref="S3.1.p1.7.m7.2.2.2.2.3"><minus id="S3.1.p1.7.m7.2.2.2.2.3.1.cmml" xref="S3.1.p1.7.m7.2.2.2.2.3.1"></minus><ci id="S3.1.p1.7.m7.2.2.2.2.3.2.cmml" xref="S3.1.p1.7.m7.2.2.2.2.3.2">ℓ</ci><cn id="S3.1.p1.7.m7.2.2.2.2.3.3.cmml" type="integer" xref="S3.1.p1.7.m7.2.2.2.2.3.3">1</cn></apply></apply></apply></apply><cn id="S3.1.p1.7.m7.2.2.4.cmml" type="integer" xref="S3.1.p1.7.m7.2.2.4">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.7.m7.2c">p_{i}=\frac{\mathsf{PLSigmoid}_{b}(\frac{i-1}{\ell-1})+\mathsf{PLSigmoid}_{b}(% \frac{i}{\ell-1})}{2}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.7.m7.2d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = divide start_ARG sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( divide start_ARG italic_i - 1 end_ARG start_ARG roman_ℓ - 1 end_ARG ) + sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( divide start_ARG italic_i end_ARG start_ARG roman_ℓ - 1 end_ARG ) end_ARG start_ARG 2 end_ARG</annotation></semantics></math> for <math alttext="i\in[\ell-1]" class="ltx_Math" display="inline" id="S3.1.p1.8.m8.1"><semantics id="S3.1.p1.8.m8.1a"><mrow id="S3.1.p1.8.m8.1.1" xref="S3.1.p1.8.m8.1.1.cmml"><mi id="S3.1.p1.8.m8.1.1.3" xref="S3.1.p1.8.m8.1.1.3.cmml">i</mi><mo id="S3.1.p1.8.m8.1.1.2" xref="S3.1.p1.8.m8.1.1.2.cmml">∈</mo><mrow id="S3.1.p1.8.m8.1.1.1.1" xref="S3.1.p1.8.m8.1.1.1.2.cmml"><mo id="S3.1.p1.8.m8.1.1.1.1.2" stretchy="false" xref="S3.1.p1.8.m8.1.1.1.2.1.cmml">[</mo><mrow id="S3.1.p1.8.m8.1.1.1.1.1" xref="S3.1.p1.8.m8.1.1.1.1.1.cmml"><mi id="S3.1.p1.8.m8.1.1.1.1.1.2" mathvariant="normal" xref="S3.1.p1.8.m8.1.1.1.1.1.2.cmml">ℓ</mi><mo id="S3.1.p1.8.m8.1.1.1.1.1.1" xref="S3.1.p1.8.m8.1.1.1.1.1.1.cmml">−</mo><mn id="S3.1.p1.8.m8.1.1.1.1.1.3" xref="S3.1.p1.8.m8.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.1.p1.8.m8.1.1.1.1.3" stretchy="false" xref="S3.1.p1.8.m8.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.8.m8.1b"><apply id="S3.1.p1.8.m8.1.1.cmml" xref="S3.1.p1.8.m8.1.1"><in id="S3.1.p1.8.m8.1.1.2.cmml" xref="S3.1.p1.8.m8.1.1.2"></in><ci id="S3.1.p1.8.m8.1.1.3.cmml" xref="S3.1.p1.8.m8.1.1.3">𝑖</ci><apply id="S3.1.p1.8.m8.1.1.1.2.cmml" xref="S3.1.p1.8.m8.1.1.1.1"><csymbol cd="latexml" id="S3.1.p1.8.m8.1.1.1.2.1.cmml" xref="S3.1.p1.8.m8.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.1.p1.8.m8.1.1.1.1.1.cmml" xref="S3.1.p1.8.m8.1.1.1.1.1"><minus id="S3.1.p1.8.m8.1.1.1.1.1.1.cmml" xref="S3.1.p1.8.m8.1.1.1.1.1.1"></minus><ci id="S3.1.p1.8.m8.1.1.1.1.1.2.cmml" xref="S3.1.p1.8.m8.1.1.1.1.1.2">ℓ</ci><cn id="S3.1.p1.8.m8.1.1.1.1.1.3.cmml" type="integer" xref="S3.1.p1.8.m8.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.8.m8.1c">i\in[\ell-1]</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.8.m8.1d">italic_i ∈ [ roman_ℓ - 1 ]</annotation></semantics></math>, and <math alttext="p_{\ell}=1" class="ltx_Math" display="inline" id="S3.1.p1.9.m9.1"><semantics id="S3.1.p1.9.m9.1a"><mrow id="S3.1.p1.9.m9.1.1" xref="S3.1.p1.9.m9.1.1.cmml"><msub id="S3.1.p1.9.m9.1.1.2" xref="S3.1.p1.9.m9.1.1.2.cmml"><mi id="S3.1.p1.9.m9.1.1.2.2" xref="S3.1.p1.9.m9.1.1.2.2.cmml">p</mi><mi id="S3.1.p1.9.m9.1.1.2.3" mathvariant="normal" xref="S3.1.p1.9.m9.1.1.2.3.cmml">ℓ</mi></msub><mo id="S3.1.p1.9.m9.1.1.1" xref="S3.1.p1.9.m9.1.1.1.cmml">=</mo><mn id="S3.1.p1.9.m9.1.1.3" xref="S3.1.p1.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.9.m9.1b"><apply id="S3.1.p1.9.m9.1.1.cmml" xref="S3.1.p1.9.m9.1.1"><eq id="S3.1.p1.9.m9.1.1.1.cmml" xref="S3.1.p1.9.m9.1.1.1"></eq><apply id="S3.1.p1.9.m9.1.1.2.cmml" xref="S3.1.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3.1.p1.9.m9.1.1.2.1.cmml" xref="S3.1.p1.9.m9.1.1.2">subscript</csymbol><ci id="S3.1.p1.9.m9.1.1.2.2.cmml" xref="S3.1.p1.9.m9.1.1.2.2">𝑝</ci><ci id="S3.1.p1.9.m9.1.1.2.3.cmml" xref="S3.1.p1.9.m9.1.1.2.3">ℓ</ci></apply><cn id="S3.1.p1.9.m9.1.1.3.cmml" type="integer" xref="S3.1.p1.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.9.m9.1c">p_{\ell}=1</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.9.m9.1d">italic_p start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT = 1</annotation></semantics></math>. Evaluating the linear program in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S2.Thmtheorem1" title="Theorem 2.1 (LP for antisymmetric selection functions). ‣ 2.3 Linear program ‣ 2 Preliminaries ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2.1</span></a> using the code in <a class="ltx_ref ltx_href ltx_font_typewriter" href="https://github.com/singerng/oblivious-csps/blob/main/figures/new_algorithm.py" title="">figures/new_algorithm.py</a> in the source repository, we deduce <math alttext="\alpha(\mathcal{O}_{\mathsf{PLSigmoid}_{b}})\geq 0.485275" class="ltx_Math" display="inline" id="S3.1.p1.10.m10.1"><semantics id="S3.1.p1.10.m10.1a"><mrow id="S3.1.p1.10.m10.1.1" xref="S3.1.p1.10.m10.1.1.cmml"><mrow id="S3.1.p1.10.m10.1.1.1" xref="S3.1.p1.10.m10.1.1.1.cmml"><mi id="S3.1.p1.10.m10.1.1.1.3" xref="S3.1.p1.10.m10.1.1.1.3.cmml">α</mi><mo id="S3.1.p1.10.m10.1.1.1.2" xref="S3.1.p1.10.m10.1.1.1.2.cmml">⁢</mo><mrow id="S3.1.p1.10.m10.1.1.1.1.1" xref="S3.1.p1.10.m10.1.1.1.1.1.1.cmml"><mo id="S3.1.p1.10.m10.1.1.1.1.1.2" stretchy="false" xref="S3.1.p1.10.m10.1.1.1.1.1.1.cmml">(</mo><msub id="S3.1.p1.10.m10.1.1.1.1.1.1" xref="S3.1.p1.10.m10.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.1.p1.10.m10.1.1.1.1.1.1.2" xref="S3.1.p1.10.m10.1.1.1.1.1.1.2.cmml">𝒪</mi><msub id="S3.1.p1.10.m10.1.1.1.1.1.1.3" xref="S3.1.p1.10.m10.1.1.1.1.1.1.3.cmml"><mi id="S3.1.p1.10.m10.1.1.1.1.1.1.3.2" xref="S3.1.p1.10.m10.1.1.1.1.1.1.3.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S3.1.p1.10.m10.1.1.1.1.1.1.3.3" xref="S3.1.p1.10.m10.1.1.1.1.1.1.3.3.cmml">b</mi></msub></msub><mo id="S3.1.p1.10.m10.1.1.1.1.1.3" stretchy="false" xref="S3.1.p1.10.m10.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.1.p1.10.m10.1.1.2" xref="S3.1.p1.10.m10.1.1.2.cmml">≥</mo><mn id="S3.1.p1.10.m10.1.1.3" xref="S3.1.p1.10.m10.1.1.3.cmml">0.485275</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.10.m10.1b"><apply id="S3.1.p1.10.m10.1.1.cmml" xref="S3.1.p1.10.m10.1.1"><geq id="S3.1.p1.10.m10.1.1.2.cmml" xref="S3.1.p1.10.m10.1.1.2"></geq><apply id="S3.1.p1.10.m10.1.1.1.cmml" xref="S3.1.p1.10.m10.1.1.1"><times id="S3.1.p1.10.m10.1.1.1.2.cmml" xref="S3.1.p1.10.m10.1.1.1.2"></times><ci id="S3.1.p1.10.m10.1.1.1.3.cmml" xref="S3.1.p1.10.m10.1.1.1.3">𝛼</ci><apply id="S3.1.p1.10.m10.1.1.1.1.1.1.cmml" xref="S3.1.p1.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.1.p1.10.m10.1.1.1.1.1.1.1.cmml" xref="S3.1.p1.10.m10.1.1.1.1.1">subscript</csymbol><ci id="S3.1.p1.10.m10.1.1.1.1.1.1.2.cmml" xref="S3.1.p1.10.m10.1.1.1.1.1.1.2">𝒪</ci><apply id="S3.1.p1.10.m10.1.1.1.1.1.1.3.cmml" xref="S3.1.p1.10.m10.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.1.p1.10.m10.1.1.1.1.1.1.3.1.cmml" xref="S3.1.p1.10.m10.1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.1.p1.10.m10.1.1.1.1.1.1.3.2.cmml" xref="S3.1.p1.10.m10.1.1.1.1.1.1.3.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S3.1.p1.10.m10.1.1.1.1.1.1.3.3.cmml" xref="S3.1.p1.10.m10.1.1.1.1.1.1.3.3">𝑏</ci></apply></apply></apply><cn id="S3.1.p1.10.m10.1.1.3.cmml" type="float" xref="S3.1.p1.10.m10.1.1.3">0.485275</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.10.m10.1c">\alpha(\mathcal{O}_{\mathsf{PLSigmoid}_{b}})\geq 0.485275</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.10.m10.1d">italic_α ( caligraphic_O start_POSTSUBSCRIPT sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ≥ 0.485275</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S3.p6"> <p class="ltx_p" id="S3.p6.1"><a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S3.F2" title="In 3 Improved oblivious algorithms (Theorem 1.5) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Fig.</span> <span class="ltx_text ltx_ref_tag">2</span></a> below depicts the effect of increasingly fine discretization on approximation ratio. We suspect that even further improvements are possible by running the LP for an even finer discretization, but we do not know how to calculate the limit of infinitely fine discretization (i.e., the value of the actual step function).</p> </div> <figure class="ltx_figure" id="S3.F2"><svg class="ltx_picture ltx_centering" height="327.47" id="S3.F2.pic1" overflow="visible" version="1.1" width="405.52"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,327.47) matrix(1 0 0 -1 0 0) translate(43.32,0) translate(0,18.42) matrix(1.0 0.0 0.0 1.0 -43.32 -18.42)"><g class="ltx_nestedsvg" transform="matrix(1 0 0 1 0 0) translate(43.32,0) translate(0,18.42)"><g color="#BFBFBF" fill="#BFBFBF" stroke="#BFBFBF" stroke-dasharray="3.0pt,3.0pt" stroke-dashoffset="0.0pt" stroke-width="0.4pt"><path d="M 0 0 L 357.73 0 M 0 60 L 357.73 60 M 0 119.99 L 357.73 119.99 M 0 179.99 L 357.73 179.99 M 0 239.99 L 357.73 239.99 M 0 299.99 L 357.73 299.99" style="fill:none"></path></g><g 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fineness of discretization affects the approximation ratio calculated by the linear program of <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S2.Thmtheorem1" title="Theorem 2.1 (LP for antisymmetric selection functions). ‣ 2.3 Linear program ‣ 2 Preliminaries ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2.1</span></a>, for two continuous selection functions: <math alttext="\mathsf{PLSigmoid}_{1/2}" class="ltx_Math" display="inline" id="S3.F2.3.1.m1.1"><semantics id="S3.F2.3.1.m1.1b"><msub id="S3.F2.3.1.m1.1.1" xref="S3.F2.3.1.m1.1.1.cmml"><mi id="S3.F2.3.1.m1.1.1.2" xref="S3.F2.3.1.m1.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S3.F2.3.1.m1.1.1.3" xref="S3.F2.3.1.m1.1.1.3.cmml"><mn id="S3.F2.3.1.m1.1.1.3.2" xref="S3.F2.3.1.m1.1.1.3.2.cmml">1</mn><mo id="S3.F2.3.1.m1.1.1.3.1" xref="S3.F2.3.1.m1.1.1.3.1.cmml">/</mo><mn id="S3.F2.3.1.m1.1.1.3.3" xref="S3.F2.3.1.m1.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.F2.3.1.m1.1c"><apply id="S3.F2.3.1.m1.1.1.cmml" xref="S3.F2.3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.F2.3.1.m1.1.1.1.cmml" xref="S3.F2.3.1.m1.1.1">subscript</csymbol><ci id="S3.F2.3.1.m1.1.1.2.cmml" xref="S3.F2.3.1.m1.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S3.F2.3.1.m1.1.1.3.cmml" xref="S3.F2.3.1.m1.1.1.3"><divide id="S3.F2.3.1.m1.1.1.3.1.cmml" xref="S3.F2.3.1.m1.1.1.3.1"></divide><cn id="S3.F2.3.1.m1.1.1.3.2.cmml" type="integer" xref="S3.F2.3.1.m1.1.1.3.2">1</cn><cn id="S3.F2.3.1.m1.1.1.3.3.cmml" type="integer" xref="S3.F2.3.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.3.1.m1.1d">\mathsf{PLSigmoid}_{1/2}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.3.1.m1.1e">sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathsf{PLSigmoid}_{149/309}" class="ltx_Math" display="inline" id="S3.F2.4.2.m2.1"><semantics id="S3.F2.4.2.m2.1b"><msub id="S3.F2.4.2.m2.1.1" xref="S3.F2.4.2.m2.1.1.cmml"><mi id="S3.F2.4.2.m2.1.1.2" xref="S3.F2.4.2.m2.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S3.F2.4.2.m2.1.1.3" xref="S3.F2.4.2.m2.1.1.3.cmml"><mn id="S3.F2.4.2.m2.1.1.3.2" xref="S3.F2.4.2.m2.1.1.3.2.cmml">149</mn><mo id="S3.F2.4.2.m2.1.1.3.1" xref="S3.F2.4.2.m2.1.1.3.1.cmml">/</mo><mn id="S3.F2.4.2.m2.1.1.3.3" xref="S3.F2.4.2.m2.1.1.3.3.cmml">309</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.F2.4.2.m2.1c"><apply id="S3.F2.4.2.m2.1.1.cmml" xref="S3.F2.4.2.m2.1.1"><csymbol cd="ambiguous" id="S3.F2.4.2.m2.1.1.1.cmml" xref="S3.F2.4.2.m2.1.1">subscript</csymbol><ci id="S3.F2.4.2.m2.1.1.2.cmml" xref="S3.F2.4.2.m2.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S3.F2.4.2.m2.1.1.3.cmml" xref="S3.F2.4.2.m2.1.1.3"><divide id="S3.F2.4.2.m2.1.1.3.1.cmml" xref="S3.F2.4.2.m2.1.1.3.1"></divide><cn id="S3.F2.4.2.m2.1.1.3.2.cmml" type="integer" xref="S3.F2.4.2.m2.1.1.3.2">149</cn><cn id="S3.F2.4.2.m2.1.1.3.3.cmml" type="integer" xref="S3.F2.4.2.m2.1.1.3.3">309</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.4.2.m2.1d">\mathsf{PLSigmoid}_{149/309}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.4.2.m2.1e">sansserif_PLSigmoid start_POSTSUBSCRIPT 149 / 309 end_POSTSUBSCRIPT</annotation></semantics></math>. Each point represents the approximation ratio of some oblivious algorithm, as calculated by the linear program in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S2.Thmtheorem1" title="Theorem 2.1 (LP for antisymmetric selection functions). ‣ 2.3 Linear program ‣ 2 Preliminaries ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2.1</span></a>. The horizontal axis records the number of bias classes (up to sign, i.e., as in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S2.Thmtheorem1" title="Theorem 2.1 (LP for antisymmetric selection functions). ‣ 2.3 Linear program ‣ 2 Preliminaries ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2.1</span></a>), and the vertical axis records the calculated approximation ratio. This plot was produced by <a class="ltx_ref ltx_href ltx_font_typewriter" href="https://github.com/singerng/oblivious-csps/blob/main/figures/discretization.py" title="">figures/discretization.py</a> in the source code.</span></figcaption> </figure> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Lower bounds</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">In this section, we present a number of lower bounds against oblivious algorithms. These bounds hold at varying levels of “granularity”: Some are only against individual selection functions (like <math alttext="\mathsf{PLSigmoid}_{1/2}" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><msub id="S4.p1.1.m1.1.1" xref="S4.p1.1.m1.1.1.cmml"><mi id="S4.p1.1.m1.1.1.2" xref="S4.p1.1.m1.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S4.p1.1.m1.1.1.3" xref="S4.p1.1.m1.1.1.3.cmml"><mn id="S4.p1.1.m1.1.1.3.2" xref="S4.p1.1.m1.1.1.3.2.cmml">1</mn><mo id="S4.p1.1.m1.1.1.3.1" xref="S4.p1.1.m1.1.1.3.1.cmml">/</mo><mn id="S4.p1.1.m1.1.1.3.3" xref="S4.p1.1.m1.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><apply id="S4.p1.1.m1.1.1.cmml" xref="S4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.p1.1.m1.1.1.1.cmml" xref="S4.p1.1.m1.1.1">subscript</csymbol><ci id="S4.p1.1.m1.1.1.2.cmml" xref="S4.p1.1.m1.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S4.p1.1.m1.1.1.3.cmml" xref="S4.p1.1.m1.1.1.3"><divide id="S4.p1.1.m1.1.1.3.1.cmml" xref="S4.p1.1.m1.1.1.3.1"></divide><cn id="S4.p1.1.m1.1.1.3.2.cmml" type="integer" xref="S4.p1.1.m1.1.1.3.2">1</cn><cn id="S4.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S4.p1.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">\mathsf{PLSigmoid}_{1/2}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math>, constructed in the last section), while others hold against all PL sigmoid functions, all antisymmetric functions, or even all functions. However, the proofs of these theorems are united by an underlying methodology for producing candidate lower bound graphs, which combines grid search and linear programming. We discuss this methodology next before turning to the proofs of specific bounds.</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>Methodology for finding hard instances</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">First, we describe our general strategy in this section for finding graphs that are hard for oblivious algorithms. This description is not strictly necessary for our proofs: One may simply take the explicit graphs we produce and verify that they have the desired lower bound properties. Also, some of the graphs are simple enough that one could have invented them by hand. However, we include this description to shed some light on where the more complex graphs come from. Also, when we can indeed prove such “uniform” lower bounds, the resulting quantitative bound will hold even against ensembles of oblivious algorithms.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.6">Ideally, given a set <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><ci id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">caligraphic_C</annotation></semantics></math> of oblivious algorithms, we would like to produce a graph <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><mi id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><ci id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">italic_G</annotation></semantics></math> and show that all algorithms in <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.1"><semantics id="S4.SS1.p2.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.1b"><ci id="S4.SS1.p2.3.m3.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.1c">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.1d">caligraphic_C</annotation></semantics></math> perform poorly on <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.1"><semantics id="S4.SS1.p2.4.m4.1a"><mi id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.1b"><ci id="S4.SS1.p2.4.m4.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.1d">italic_G</annotation></semantics></math>. This is, in some sense, “dual” to the problem of finding good oblivious algorithms, where we want a single oblivious algorithm that performs well on all graphs. In general, we do not have strong evidence as to whether or not this dual procedure is <em class="ltx_emph ltx_font_italic" id="S4.SS1.p2.6.1">tight</em>, i.e., whether we can certify optimal bounds on the performance of algorithms for the classes we’re interested in by coming up with a single hard graph <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m5.1"><semantics id="S4.SS1.p2.5.m5.1a"><mi id="S4.SS1.p2.5.m5.1.1" xref="S4.SS1.p2.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m5.1b"><ci id="S4.SS1.p2.5.m5.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m5.1d">italic_G</annotation></semantics></math>. Indeed, some of our proofs instead consider a small set of graphs and prove that any oblivious algorithm in <math alttext="\mathcal{C}" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m6.1"><semantics id="S4.SS1.p2.6.m6.1a"><mi class="ltx_font_mathcaligraphic" 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">\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m6.1d">caligraphic_C</annotation></semantics></math> must perform poorly on at least one of these algorithms; we do not know whether this potentially stronger proof method is tight, either.</p> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.12">Let <math alttext="L\in\mathbb{N}" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.1"><semantics id="S4.SS1.p3.1.m1.1a"><mrow id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml"><mi id="S4.SS1.p3.1.m1.1.1.2" xref="S4.SS1.p3.1.m1.1.1.2.cmml">L</mi><mo id="S4.SS1.p3.1.m1.1.1.1" xref="S4.SS1.p3.1.m1.1.1.1.cmml">∈</mo><mi id="S4.SS1.p3.1.m1.1.1.3" xref="S4.SS1.p3.1.m1.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.1b"><apply id="S4.SS1.p3.1.m1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1"><in id="S4.SS1.p3.1.m1.1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1.1"></in><ci id="S4.SS1.p3.1.m1.1.1.2.cmml" xref="S4.SS1.p3.1.m1.1.1.2">𝐿</ci><ci id="S4.SS1.p3.1.m1.1.1.3.cmml" xref="S4.SS1.p3.1.m1.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.1c">L\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.1d">italic_L ∈ blackboard_N</annotation></semantics></math>, <math alttext="-1\leq t_{1}<\cdots<t_{L}\leq+1" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.1"><semantics id="S4.SS1.p3.2.m2.1a"><mrow id="S4.SS1.p3.2.m2.1.1" xref="S4.SS1.p3.2.m2.1.1.cmml"><mrow id="S4.SS1.p3.2.m2.1.1.2" xref="S4.SS1.p3.2.m2.1.1.2.cmml"><mo id="S4.SS1.p3.2.m2.1.1.2a" xref="S4.SS1.p3.2.m2.1.1.2.cmml">−</mo><mn id="S4.SS1.p3.2.m2.1.1.2.2" xref="S4.SS1.p3.2.m2.1.1.2.2.cmml">1</mn></mrow><mo id="S4.SS1.p3.2.m2.1.1.3" xref="S4.SS1.p3.2.m2.1.1.3.cmml">≤</mo><msub id="S4.SS1.p3.2.m2.1.1.4" xref="S4.SS1.p3.2.m2.1.1.4.cmml"><mi id="S4.SS1.p3.2.m2.1.1.4.2" xref="S4.SS1.p3.2.m2.1.1.4.2.cmml">t</mi><mn id="S4.SS1.p3.2.m2.1.1.4.3" xref="S4.SS1.p3.2.m2.1.1.4.3.cmml">1</mn></msub><mo id="S4.SS1.p3.2.m2.1.1.5" xref="S4.SS1.p3.2.m2.1.1.5.cmml"><</mo><mi id="S4.SS1.p3.2.m2.1.1.6" mathvariant="normal" xref="S4.SS1.p3.2.m2.1.1.6.cmml">⋯</mi><mo id="S4.SS1.p3.2.m2.1.1.7" xref="S4.SS1.p3.2.m2.1.1.7.cmml"><</mo><msub id="S4.SS1.p3.2.m2.1.1.8" xref="S4.SS1.p3.2.m2.1.1.8.cmml"><mi id="S4.SS1.p3.2.m2.1.1.8.2" xref="S4.SS1.p3.2.m2.1.1.8.2.cmml">t</mi><mi id="S4.SS1.p3.2.m2.1.1.8.3" 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italic_t start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ≤ + 1</annotation></semantics></math>, and <math alttext="\mathcal{P}\subseteq[0,1]^{L}" class="ltx_Math" display="inline" id="S4.SS1.p3.3.m3.2"><semantics id="S4.SS1.p3.3.m3.2a"><mrow id="S4.SS1.p3.3.m3.2.3" xref="S4.SS1.p3.3.m3.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.3.m3.2.3.2" xref="S4.SS1.p3.3.m3.2.3.2.cmml">𝒫</mi><mo id="S4.SS1.p3.3.m3.2.3.1" xref="S4.SS1.p3.3.m3.2.3.1.cmml">⊆</mo><msup id="S4.SS1.p3.3.m3.2.3.3" xref="S4.SS1.p3.3.m3.2.3.3.cmml"><mrow id="S4.SS1.p3.3.m3.2.3.3.2.2" xref="S4.SS1.p3.3.m3.2.3.3.2.1.cmml"><mo id="S4.SS1.p3.3.m3.2.3.3.2.2.1" stretchy="false" xref="S4.SS1.p3.3.m3.2.3.3.2.1.cmml">[</mo><mn id="S4.SS1.p3.3.m3.1.1" xref="S4.SS1.p3.3.m3.1.1.cmml">0</mn><mo id="S4.SS1.p3.3.m3.2.3.3.2.2.2" xref="S4.SS1.p3.3.m3.2.3.3.2.1.cmml">,</mo><mn id="S4.SS1.p3.3.m3.2.2" xref="S4.SS1.p3.3.m3.2.2.cmml">1</mn><mo id="S4.SS1.p3.3.m3.2.3.3.2.2.3" stretchy="false" xref="S4.SS1.p3.3.m3.2.3.3.2.1.cmml">]</mo></mrow><mi id="S4.SS1.p3.3.m3.2.3.3.3" xref="S4.SS1.p3.3.m3.2.3.3.3.cmml">L</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.3.m3.2b"><apply id="S4.SS1.p3.3.m3.2.3.cmml" xref="S4.SS1.p3.3.m3.2.3"><subset id="S4.SS1.p3.3.m3.2.3.1.cmml" xref="S4.SS1.p3.3.m3.2.3.1"></subset><ci id="S4.SS1.p3.3.m3.2.3.2.cmml" xref="S4.SS1.p3.3.m3.2.3.2">𝒫</ci><apply id="S4.SS1.p3.3.m3.2.3.3.cmml" xref="S4.SS1.p3.3.m3.2.3.3"><csymbol cd="ambiguous" id="S4.SS1.p3.3.m3.2.3.3.1.cmml" xref="S4.SS1.p3.3.m3.2.3.3">superscript</csymbol><interval closure="closed" id="S4.SS1.p3.3.m3.2.3.3.2.1.cmml" xref="S4.SS1.p3.3.m3.2.3.3.2.2"><cn id="S4.SS1.p3.3.m3.1.1.cmml" type="integer" xref="S4.SS1.p3.3.m3.1.1">0</cn><cn id="S4.SS1.p3.3.m3.2.2.cmml" type="integer" xref="S4.SS1.p3.3.m3.2.2">1</cn></interval><ci id="S4.SS1.p3.3.m3.2.3.3.3.cmml" xref="S4.SS1.p3.3.m3.2.3.3.3">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.3.m3.2c">\mathcal{P}\subseteq[0,1]^{L}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.3.m3.2d">caligraphic_P ⊆ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT</annotation></semantics></math>. We will be interested in graphs that contain only vertices of biases <math alttext="t_{1},\ldots,t_{L}" class="ltx_Math" display="inline" id="S4.SS1.p3.4.m4.3"><semantics id="S4.SS1.p3.4.m4.3a"><mrow id="S4.SS1.p3.4.m4.3.3.2" xref="S4.SS1.p3.4.m4.3.3.3.cmml"><msub id="S4.SS1.p3.4.m4.2.2.1.1" xref="S4.SS1.p3.4.m4.2.2.1.1.cmml"><mi id="S4.SS1.p3.4.m4.2.2.1.1.2" xref="S4.SS1.p3.4.m4.2.2.1.1.2.cmml">t</mi><mn id="S4.SS1.p3.4.m4.2.2.1.1.3" xref="S4.SS1.p3.4.m4.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.SS1.p3.4.m4.3.3.2.3" xref="S4.SS1.p3.4.m4.3.3.3.cmml">,</mo><mi id="S4.SS1.p3.4.m4.1.1" mathvariant="normal" xref="S4.SS1.p3.4.m4.1.1.cmml">…</mi><mo id="S4.SS1.p3.4.m4.3.3.2.4" xref="S4.SS1.p3.4.m4.3.3.3.cmml">,</mo><msub id="S4.SS1.p3.4.m4.3.3.2.2" xref="S4.SS1.p3.4.m4.3.3.2.2.cmml"><mi id="S4.SS1.p3.4.m4.3.3.2.2.2" xref="S4.SS1.p3.4.m4.3.3.2.2.2.cmml">t</mi><mi id="S4.SS1.p3.4.m4.3.3.2.2.3" xref="S4.SS1.p3.4.m4.3.3.2.2.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.4.m4.3b"><list id="S4.SS1.p3.4.m4.3.3.3.cmml" xref="S4.SS1.p3.4.m4.3.3.2"><apply id="S4.SS1.p3.4.m4.2.2.1.1.cmml" xref="S4.SS1.p3.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.4.m4.2.2.1.1.1.cmml" xref="S4.SS1.p3.4.m4.2.2.1.1">subscript</csymbol><ci id="S4.SS1.p3.4.m4.2.2.1.1.2.cmml" xref="S4.SS1.p3.4.m4.2.2.1.1.2">𝑡</ci><cn id="S4.SS1.p3.4.m4.2.2.1.1.3.cmml" type="integer" xref="S4.SS1.p3.4.m4.2.2.1.1.3">1</cn></apply><ci id="S4.SS1.p3.4.m4.1.1.cmml" xref="S4.SS1.p3.4.m4.1.1">…</ci><apply id="S4.SS1.p3.4.m4.3.3.2.2.cmml" xref="S4.SS1.p3.4.m4.3.3.2.2"><csymbol cd="ambiguous" id="S4.SS1.p3.4.m4.3.3.2.2.1.cmml" xref="S4.SS1.p3.4.m4.3.3.2.2">subscript</csymbol><ci id="S4.SS1.p3.4.m4.3.3.2.2.2.cmml" xref="S4.SS1.p3.4.m4.3.3.2.2.2">𝑡</ci><ci id="S4.SS1.p3.4.m4.3.3.2.2.3.cmml" xref="S4.SS1.p3.4.m4.3.3.2.2.3">𝐿</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.4.m4.3c">t_{1},\ldots,t_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.4.m4.3d">italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. For any such graph <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.p3.5.m5.1"><semantics id="S4.SS1.p3.5.m5.1a"><mi id="S4.SS1.p3.5.m5.1.1" xref="S4.SS1.p3.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.5.m5.1b"><ci id="S4.SS1.p3.5.m5.1.1.cmml" xref="S4.SS1.p3.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.5.m5.1d">italic_G</annotation></semantics></math>, we can view a vector <math alttext="\boldsymbol{p}\in\mathcal{P}\subseteq[0,1]^{L}" class="ltx_Math" display="inline" id="S4.SS1.p3.6.m6.2"><semantics id="S4.SS1.p3.6.m6.2a"><mrow id="S4.SS1.p3.6.m6.2.3" xref="S4.SS1.p3.6.m6.2.3.cmml"><mi id="S4.SS1.p3.6.m6.2.3.2" xref="S4.SS1.p3.6.m6.2.3.2.cmml">𝒑</mi><mo id="S4.SS1.p3.6.m6.2.3.3" xref="S4.SS1.p3.6.m6.2.3.3.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.6.m6.2.3.4" xref="S4.SS1.p3.6.m6.2.3.4.cmml">𝒫</mi><mo id="S4.SS1.p3.6.m6.2.3.5" xref="S4.SS1.p3.6.m6.2.3.5.cmml">⊆</mo><msup id="S4.SS1.p3.6.m6.2.3.6" xref="S4.SS1.p3.6.m6.2.3.6.cmml"><mrow id="S4.SS1.p3.6.m6.2.3.6.2.2" xref="S4.SS1.p3.6.m6.2.3.6.2.1.cmml"><mo id="S4.SS1.p3.6.m6.2.3.6.2.2.1" stretchy="false" xref="S4.SS1.p3.6.m6.2.3.6.2.1.cmml">[</mo><mn id="S4.SS1.p3.6.m6.1.1" xref="S4.SS1.p3.6.m6.1.1.cmml">0</mn><mo id="S4.SS1.p3.6.m6.2.3.6.2.2.2" xref="S4.SS1.p3.6.m6.2.3.6.2.1.cmml">,</mo><mn id="S4.SS1.p3.6.m6.2.2" xref="S4.SS1.p3.6.m6.2.2.cmml">1</mn><mo id="S4.SS1.p3.6.m6.2.3.6.2.2.3" stretchy="false" xref="S4.SS1.p3.6.m6.2.3.6.2.1.cmml">]</mo></mrow><mi id="S4.SS1.p3.6.m6.2.3.6.3" xref="S4.SS1.p3.6.m6.2.3.6.3.cmml">L</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.6.m6.2b"><apply id="S4.SS1.p3.6.m6.2.3.cmml" xref="S4.SS1.p3.6.m6.2.3"><and id="S4.SS1.p3.6.m6.2.3a.cmml" xref="S4.SS1.p3.6.m6.2.3"></and><apply id="S4.SS1.p3.6.m6.2.3b.cmml" xref="S4.SS1.p3.6.m6.2.3"><in id="S4.SS1.p3.6.m6.2.3.3.cmml" xref="S4.SS1.p3.6.m6.2.3.3"></in><ci id="S4.SS1.p3.6.m6.2.3.2.cmml" xref="S4.SS1.p3.6.m6.2.3.2">𝒑</ci><ci id="S4.SS1.p3.6.m6.2.3.4.cmml" xref="S4.SS1.p3.6.m6.2.3.4">𝒫</ci></apply><apply id="S4.SS1.p3.6.m6.2.3c.cmml" xref="S4.SS1.p3.6.m6.2.3"><subset id="S4.SS1.p3.6.m6.2.3.5.cmml" xref="S4.SS1.p3.6.m6.2.3.5"></subset><share href="https://arxiv.org/html/2411.12976v1#S4.SS1.p3.6.m6.2.3.4.cmml" id="S4.SS1.p3.6.m6.2.3d.cmml" xref="S4.SS1.p3.6.m6.2.3"></share><apply id="S4.SS1.p3.6.m6.2.3.6.cmml" xref="S4.SS1.p3.6.m6.2.3.6"><csymbol cd="ambiguous" id="S4.SS1.p3.6.m6.2.3.6.1.cmml" xref="S4.SS1.p3.6.m6.2.3.6">superscript</csymbol><interval closure="closed" id="S4.SS1.p3.6.m6.2.3.6.2.1.cmml" xref="S4.SS1.p3.6.m6.2.3.6.2.2"><cn id="S4.SS1.p3.6.m6.1.1.cmml" type="integer" xref="S4.SS1.p3.6.m6.1.1">0</cn><cn id="S4.SS1.p3.6.m6.2.2.cmml" type="integer" xref="S4.SS1.p3.6.m6.2.2">1</cn></interval><ci id="S4.SS1.p3.6.m6.2.3.6.3.cmml" xref="S4.SS1.p3.6.m6.2.3.6.3">𝐿</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.6.m6.2c">\boldsymbol{p}\in\mathcal{P}\subseteq[0,1]^{L}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.6.m6.2d">bold_italic_p ∈ caligraphic_P ⊆ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT</annotation></semantics></math> as an oblivious algorithm: Each vertex of bias <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p3.7.m7.1"><semantics id="S4.SS1.p3.7.m7.1a"><mi id="S4.SS1.p3.7.m7.1.1" xref="S4.SS1.p3.7.m7.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.7.m7.1b"><ci id="S4.SS1.p3.7.m7.1.1.cmml" xref="S4.SS1.p3.7.m7.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.7.m7.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.7.m7.1d">italic_i</annotation></semantics></math> is assigned to <math alttext="1" class="ltx_Math" display="inline" id="S4.SS1.p3.8.m8.1"><semantics id="S4.SS1.p3.8.m8.1a"><mn id="S4.SS1.p3.8.m8.1.1" xref="S4.SS1.p3.8.m8.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.8.m8.1b"><cn id="S4.SS1.p3.8.m8.1.1.cmml" type="integer" xref="S4.SS1.p3.8.m8.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.8.m8.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.8.m8.1d">1</annotation></semantics></math> w.p. <math alttext="p(i)" class="ltx_Math" display="inline" id="S4.SS1.p3.9.m9.1"><semantics id="S4.SS1.p3.9.m9.1a"><mrow id="S4.SS1.p3.9.m9.1.2" xref="S4.SS1.p3.9.m9.1.2.cmml"><mi id="S4.SS1.p3.9.m9.1.2.2" xref="S4.SS1.p3.9.m9.1.2.2.cmml">p</mi><mo id="S4.SS1.p3.9.m9.1.2.1" xref="S4.SS1.p3.9.m9.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p3.9.m9.1.2.3.2" xref="S4.SS1.p3.9.m9.1.2.cmml"><mo id="S4.SS1.p3.9.m9.1.2.3.2.1" stretchy="false" xref="S4.SS1.p3.9.m9.1.2.cmml">(</mo><mi id="S4.SS1.p3.9.m9.1.1" xref="S4.SS1.p3.9.m9.1.1.cmml">i</mi><mo id="S4.SS1.p3.9.m9.1.2.3.2.2" stretchy="false" xref="S4.SS1.p3.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.9.m9.1b"><apply id="S4.SS1.p3.9.m9.1.2.cmml" xref="S4.SS1.p3.9.m9.1.2"><times id="S4.SS1.p3.9.m9.1.2.1.cmml" xref="S4.SS1.p3.9.m9.1.2.1"></times><ci id="S4.SS1.p3.9.m9.1.2.2.cmml" xref="S4.SS1.p3.9.m9.1.2.2">𝑝</ci><ci id="S4.SS1.p3.9.m9.1.1.cmml" xref="S4.SS1.p3.9.m9.1.1">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.9.m9.1c">p(i)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.9.m9.1d">italic_p ( italic_i )</annotation></semantics></math>. Our LP will output the graph which is “worst-case” for all algorithms in <math alttext="\mathcal{P}" class="ltx_Math" display="inline" id="S4.SS1.p3.10.m10.1"><semantics id="S4.SS1.p3.10.m10.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.10.m10.1.1" xref="S4.SS1.p3.10.m10.1.1.cmml">𝒫</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.10.m10.1b"><ci id="S4.SS1.p3.10.m10.1.1.cmml" xref="S4.SS1.p3.10.m10.1.1">𝒫</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.10.m10.1c">\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.10.m10.1d">caligraphic_P</annotation></semantics></math> (i.e., such that the performance of the best algorithm in <math alttext="\mathcal{P}" class="ltx_Math" display="inline" id="S4.SS1.p3.11.m11.1"><semantics id="S4.SS1.p3.11.m11.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.11.m11.1.1" xref="S4.SS1.p3.11.m11.1.1.cmml">𝒫</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.11.m11.1b"><ci id="S4.SS1.p3.11.m11.1.1.cmml" xref="S4.SS1.p3.11.m11.1.1">𝒫</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.11.m11.1c">\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.11.m11.1d">caligraphic_P</annotation></semantics></math> is minimized), among all graphs with vertices of bias <math alttext="t_{1},\ldots,t_{L}" class="ltx_Math" display="inline" id="S4.SS1.p3.12.m12.3"><semantics id="S4.SS1.p3.12.m12.3a"><mrow id="S4.SS1.p3.12.m12.3.3.2" xref="S4.SS1.p3.12.m12.3.3.3.cmml"><msub id="S4.SS1.p3.12.m12.2.2.1.1" xref="S4.SS1.p3.12.m12.2.2.1.1.cmml"><mi id="S4.SS1.p3.12.m12.2.2.1.1.2" xref="S4.SS1.p3.12.m12.2.2.1.1.2.cmml">t</mi><mn id="S4.SS1.p3.12.m12.2.2.1.1.3" xref="S4.SS1.p3.12.m12.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.SS1.p3.12.m12.3.3.2.3" xref="S4.SS1.p3.12.m12.3.3.3.cmml">,</mo><mi id="S4.SS1.p3.12.m12.1.1" mathvariant="normal" xref="S4.SS1.p3.12.m12.1.1.cmml">…</mi><mo id="S4.SS1.p3.12.m12.3.3.2.4" xref="S4.SS1.p3.12.m12.3.3.3.cmml">,</mo><msub id="S4.SS1.p3.12.m12.3.3.2.2" xref="S4.SS1.p3.12.m12.3.3.2.2.cmml"><mi id="S4.SS1.p3.12.m12.3.3.2.2.2" xref="S4.SS1.p3.12.m12.3.3.2.2.2.cmml">t</mi><mi id="S4.SS1.p3.12.m12.3.3.2.2.3" xref="S4.SS1.p3.12.m12.3.3.2.2.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.12.m12.3b"><list id="S4.SS1.p3.12.m12.3.3.3.cmml" xref="S4.SS1.p3.12.m12.3.3.2"><apply id="S4.SS1.p3.12.m12.2.2.1.1.cmml" xref="S4.SS1.p3.12.m12.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.12.m12.2.2.1.1.1.cmml" xref="S4.SS1.p3.12.m12.2.2.1.1">subscript</csymbol><ci id="S4.SS1.p3.12.m12.2.2.1.1.2.cmml" xref="S4.SS1.p3.12.m12.2.2.1.1.2">𝑡</ci><cn id="S4.SS1.p3.12.m12.2.2.1.1.3.cmml" type="integer" xref="S4.SS1.p3.12.m12.2.2.1.1.3">1</cn></apply><ci id="S4.SS1.p3.12.m12.1.1.cmml" xref="S4.SS1.p3.12.m12.1.1">…</ci><apply id="S4.SS1.p3.12.m12.3.3.2.2.cmml" xref="S4.SS1.p3.12.m12.3.3.2.2"><csymbol cd="ambiguous" 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id="S4.Ex1.m2.6.6.4.4.7.cmml" xref="S4.Ex1.m2.6.6.4.4.7">𝑤</ci><interval closure="open" id="S4.Ex1.m2.6.6.4.4.4.3.cmml" xref="S4.Ex1.m2.6.6.4.4.4.2"><apply id="S4.Ex1.m2.5.5.3.3.3.1.1.cmml" xref="S4.Ex1.m2.5.5.3.3.3.1.1"><csymbol cd="ambiguous" id="S4.Ex1.m2.5.5.3.3.3.1.1.1.cmml" xref="S4.Ex1.m2.5.5.3.3.3.1.1">subscript</csymbol><ci id="S4.Ex1.m2.5.5.3.3.3.1.1.2.cmml" xref="S4.Ex1.m2.5.5.3.3.3.1.1.2">𝑣</ci><cn id="S4.Ex1.m2.5.5.3.3.3.1.1.3.cmml" type="integer" xref="S4.Ex1.m2.5.5.3.3.3.1.1.3">1</cn></apply><apply id="S4.Ex1.m2.6.6.4.4.4.2.2.cmml" xref="S4.Ex1.m2.6.6.4.4.4.2.2"><csymbol cd="ambiguous" id="S4.Ex1.m2.6.6.4.4.4.2.2.1.cmml" xref="S4.Ex1.m2.6.6.4.4.4.2.2">subscript</csymbol><ci id="S4.Ex1.m2.6.6.4.4.4.2.2.2.cmml" xref="S4.Ex1.m2.6.6.4.4.4.2.2.2">𝑣</ci><cn id="S4.Ex1.m2.6.6.4.4.4.2.2.3.cmml" type="integer" xref="S4.Ex1.m2.6.6.4.4.4.2.2.3">2</cn></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex1.m2.6c">\displaystyle\max_{\boldsymbol{p}\in\mathcal{P}}\sum_{v_{1}=(b_{1},i_{1}),v_{2% }=(b_{2},i_{2})\in V}p(i_{1})(1-p(i_{2}))w(v_{1},v_{2})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex1.m2.6d">roman_max start_POSTSUBSCRIPT bold_italic_p ∈ caligraphic_P end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = ( italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∈ italic_V end_POSTSUBSCRIPT italic_p ( italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( 1 - italic_p ( italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) italic_w ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math></span> <span class="ltx_eqn_cell ltx_eqn_center_padright" colspan="2"></span></span></span> <span id="S4.Ex2"><span class="ltx_equation ltx_eqn_row ltx_align_baseline"> <span class="ltx_eqn_cell ltx_eqn_center_padleft"></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mathrm{s.t.}" class="ltx_Math" display="inline" id="S4.Ex2.m1.3"><semantics id="S4.Ex2.m1.3a"><mrow id="S4.Ex2.m1.3.3.1"><mrow id="S4.Ex2.m1.3.3.1.1.2" xref="S4.Ex2.m1.3.3.1.1.1.cmml"><mi id="S4.Ex2.m1.1.1" mathvariant="normal" xref="S4.Ex2.m1.1.1.cmml">s</mi><mo id="S4.Ex2.m1.3.3.1.1.2.1" lspace="0em" rspace="0.167em" xref="S4.Ex2.m1.3.3.1.1.1a.cmml">.</mo><mi id="S4.Ex2.m1.2.2" mathvariant="normal" xref="S4.Ex2.m1.2.2.cmml">t</mi></mrow><mo id="S4.Ex2.m1.3.3.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex2.m1.3b"><apply id="S4.Ex2.m1.3.3.1.1.1.cmml" xref="S4.Ex2.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S4.Ex2.m1.3.3.1.1.1a.cmml" xref="S4.Ex2.m1.3.3.1.1.2.1">formulae-sequence</csymbol><ci id="S4.Ex2.m1.1.1.cmml" xref="S4.Ex2.m1.1.1">s</ci><ci id="S4.Ex2.m1.2.2.cmml" xref="S4.Ex2.m1.2.2">t</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex2.m1.3c">\displaystyle\mathrm{s.t.}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex2.m1.3d">roman_s . roman_t .</annotation></semantics></math></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle w(v_{1},v_{2})\geq 0" class="ltx_Math" display="inline" id="S4.Ex2.m2.2"><semantics id="S4.Ex2.m2.2a"><mrow id="S4.Ex2.m2.2.2" xref="S4.Ex2.m2.2.2.cmml"><mrow id="S4.Ex2.m2.2.2.2" xref="S4.Ex2.m2.2.2.2.cmml"><mi id="S4.Ex2.m2.2.2.2.4" xref="S4.Ex2.m2.2.2.2.4.cmml">w</mi><mo id="S4.Ex2.m2.2.2.2.3" xref="S4.Ex2.m2.2.2.2.3.cmml">⁢</mo><mrow id="S4.Ex2.m2.2.2.2.2.2" xref="S4.Ex2.m2.2.2.2.2.3.cmml"><mo id="S4.Ex2.m2.2.2.2.2.2.3" stretchy="false" xref="S4.Ex2.m2.2.2.2.2.3.cmml">(</mo><msub id="S4.Ex2.m2.1.1.1.1.1.1" xref="S4.Ex2.m2.1.1.1.1.1.1.cmml"><mi id="S4.Ex2.m2.1.1.1.1.1.1.2" xref="S4.Ex2.m2.1.1.1.1.1.1.2.cmml">v</mi><mn id="S4.Ex2.m2.1.1.1.1.1.1.3" xref="S4.Ex2.m2.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Ex2.m2.2.2.2.2.2.4" xref="S4.Ex2.m2.2.2.2.2.3.cmml">,</mo><msub id="S4.Ex2.m2.2.2.2.2.2.2" xref="S4.Ex2.m2.2.2.2.2.2.2.cmml"><mi id="S4.Ex2.m2.2.2.2.2.2.2.2" xref="S4.Ex2.m2.2.2.2.2.2.2.2.cmml">v</mi><mn id="S4.Ex2.m2.2.2.2.2.2.2.3" xref="S4.Ex2.m2.2.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.Ex2.m2.2.2.2.2.2.5" stretchy="false" xref="S4.Ex2.m2.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Ex2.m2.2.2.3" xref="S4.Ex2.m2.2.2.3.cmml">≥</mo><mn id="S4.Ex2.m2.2.2.4" xref="S4.Ex2.m2.2.2.4.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex2.m2.2b"><apply id="S4.Ex2.m2.2.2.cmml" xref="S4.Ex2.m2.2.2"><geq id="S4.Ex2.m2.2.2.3.cmml" xref="S4.Ex2.m2.2.2.3"></geq><apply id="S4.Ex2.m2.2.2.2.cmml" xref="S4.Ex2.m2.2.2.2"><times id="S4.Ex2.m2.2.2.2.3.cmml" xref="S4.Ex2.m2.2.2.2.3"></times><ci id="S4.Ex2.m2.2.2.2.4.cmml" xref="S4.Ex2.m2.2.2.2.4">𝑤</ci><interval closure="open" id="S4.Ex2.m2.2.2.2.2.3.cmml" xref="S4.Ex2.m2.2.2.2.2.2"><apply id="S4.Ex2.m2.1.1.1.1.1.1.cmml" xref="S4.Ex2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex2.m2.1.1.1.1.1.1.1.cmml" xref="S4.Ex2.m2.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex2.m2.1.1.1.1.1.1.2.cmml" xref="S4.Ex2.m2.1.1.1.1.1.1.2">𝑣</ci><cn id="S4.Ex2.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Ex2.m2.1.1.1.1.1.1.3">1</cn></apply><apply id="S4.Ex2.m2.2.2.2.2.2.2.cmml" xref="S4.Ex2.m2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Ex2.m2.2.2.2.2.2.2.1.cmml" xref="S4.Ex2.m2.2.2.2.2.2.2">subscript</csymbol><ci id="S4.Ex2.m2.2.2.2.2.2.2.2.cmml" xref="S4.Ex2.m2.2.2.2.2.2.2.2">𝑣</ci><cn id="S4.Ex2.m2.2.2.2.2.2.2.3.cmml" type="integer" xref="S4.Ex2.m2.2.2.2.2.2.2.3">2</cn></apply></interval></apply><cn id="S4.Ex2.m2.2.2.4.cmml" type="integer" xref="S4.Ex2.m2.2.2.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex2.m2.2c">\displaystyle w(v_{1},v_{2})\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.Ex2.m2.2d">italic_w ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ≥ 0</annotation></semantics></math></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\forall v_{1},v_{2}\in V" class="ltx_Math" display="inline" id="S4.Ex2.m3.2"><semantics id="S4.Ex2.m3.2a"><mrow id="S4.Ex2.m3.2.2" xref="S4.Ex2.m3.2.2.cmml"><mrow id="S4.Ex2.m3.2.2.2.2" xref="S4.Ex2.m3.2.2.2.3.cmml"><mrow id="S4.Ex2.m3.1.1.1.1.1" xref="S4.Ex2.m3.1.1.1.1.1.cmml"><mo id="S4.Ex2.m3.1.1.1.1.1.1" rspace="0.167em" xref="S4.Ex2.m3.1.1.1.1.1.1.cmml">∀</mo><msub id="S4.Ex2.m3.1.1.1.1.1.2" xref="S4.Ex2.m3.1.1.1.1.1.2.cmml"><mi id="S4.Ex2.m3.1.1.1.1.1.2.2" xref="S4.Ex2.m3.1.1.1.1.1.2.2.cmml">v</mi><mn id="S4.Ex2.m3.1.1.1.1.1.2.3" xref="S4.Ex2.m3.1.1.1.1.1.2.3.cmml">1</mn></msub></mrow><mo id="S4.Ex2.m3.2.2.2.2.3" xref="S4.Ex2.m3.2.2.2.3.cmml">,</mo><msub id="S4.Ex2.m3.2.2.2.2.2" xref="S4.Ex2.m3.2.2.2.2.2.cmml"><mi id="S4.Ex2.m3.2.2.2.2.2.2" xref="S4.Ex2.m3.2.2.2.2.2.2.cmml">v</mi><mn id="S4.Ex2.m3.2.2.2.2.2.3" xref="S4.Ex2.m3.2.2.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S4.Ex2.m3.2.2.3" xref="S4.Ex2.m3.2.2.3.cmml">∈</mo><mi id="S4.Ex2.m3.2.2.4" xref="S4.Ex2.m3.2.2.4.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex2.m3.2b"><apply id="S4.Ex2.m3.2.2.cmml" xref="S4.Ex2.m3.2.2"><in id="S4.Ex2.m3.2.2.3.cmml" xref="S4.Ex2.m3.2.2.3"></in><list id="S4.Ex2.m3.2.2.2.3.cmml" xref="S4.Ex2.m3.2.2.2.2"><apply id="S4.Ex2.m3.1.1.1.1.1.cmml" xref="S4.Ex2.m3.1.1.1.1.1"><csymbol cd="latexml" id="S4.Ex2.m3.1.1.1.1.1.1.cmml" xref="S4.Ex2.m3.1.1.1.1.1.1">for-all</csymbol><apply id="S4.Ex2.m3.1.1.1.1.1.2.cmml" xref="S4.Ex2.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.Ex2.m3.1.1.1.1.1.2.1.cmml" xref="S4.Ex2.m3.1.1.1.1.1.2">subscript</csymbol><ci id="S4.Ex2.m3.1.1.1.1.1.2.2.cmml" xref="S4.Ex2.m3.1.1.1.1.1.2.2">𝑣</ci><cn id="S4.Ex2.m3.1.1.1.1.1.2.3.cmml" type="integer" xref="S4.Ex2.m3.1.1.1.1.1.2.3">1</cn></apply></apply><apply id="S4.Ex2.m3.2.2.2.2.2.cmml" xref="S4.Ex2.m3.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Ex2.m3.2.2.2.2.2.1.cmml" xref="S4.Ex2.m3.2.2.2.2.2">subscript</csymbol><ci id="S4.Ex2.m3.2.2.2.2.2.2.cmml" xref="S4.Ex2.m3.2.2.2.2.2.2">𝑣</ci><cn id="S4.Ex2.m3.2.2.2.2.2.3.cmml" type="integer" xref="S4.Ex2.m3.2.2.2.2.2.3">2</cn></apply></list><ci id="S4.Ex2.m3.2.2.4.cmml" xref="S4.Ex2.m3.2.2.4">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex2.m3.2c">\displaystyle\forall v_{1},v_{2}\in V</annotation><annotation encoding="application/x-llamapun" id="S4.Ex2.m3.2d">∀ italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_V</annotation></semantics></math></span> <span class="ltx_eqn_cell ltx_eqn_center_padright"></span></span></span> <span id="S4.Ex3"><span class="ltx_equation ltx_eqn_row ltx_align_baseline"> <span class="ltx_eqn_cell ltx_eqn_center_padleft"></span> <span class="ltx_td ltx_eqn_cell"></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{i_{1},i_{2}\in[L]}w((1,i_{1}),(0,i_{2}))=1" class="ltx_Math" display="inline" id="S4.Ex3.m1.7"><semantics id="S4.Ex3.m1.7a"><mrow id="S4.Ex3.m1.7.7" xref="S4.Ex3.m1.7.7.cmml"><mrow id="S4.Ex3.m1.7.7.2" xref="S4.Ex3.m1.7.7.2.cmml"><mstyle displaystyle="true" id="S4.Ex3.m1.7.7.2.3" xref="S4.Ex3.m1.7.7.2.3.cmml"><munder id="S4.Ex3.m1.7.7.2.3a" xref="S4.Ex3.m1.7.7.2.3.cmml"><mo id="S4.Ex3.m1.7.7.2.3.2" movablelimits="false" xref="S4.Ex3.m1.7.7.2.3.2.cmml">∑</mo><mrow id="S4.Ex3.m1.3.3.3" xref="S4.Ex3.m1.3.3.3.cmml"><mrow id="S4.Ex3.m1.3.3.3.3.2" xref="S4.Ex3.m1.3.3.3.3.3.cmml"><msub id="S4.Ex3.m1.2.2.2.2.1.1" xref="S4.Ex3.m1.2.2.2.2.1.1.cmml"><mi id="S4.Ex3.m1.2.2.2.2.1.1.2" xref="S4.Ex3.m1.2.2.2.2.1.1.2.cmml">i</mi><mn id="S4.Ex3.m1.2.2.2.2.1.1.3" xref="S4.Ex3.m1.2.2.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.Ex3.m1.3.3.3.3.2.3" xref="S4.Ex3.m1.3.3.3.3.3.cmml">,</mo><msub id="S4.Ex3.m1.3.3.3.3.2.2" xref="S4.Ex3.m1.3.3.3.3.2.2.cmml"><mi id="S4.Ex3.m1.3.3.3.3.2.2.2" xref="S4.Ex3.m1.3.3.3.3.2.2.2.cmml">i</mi><mn id="S4.Ex3.m1.3.3.3.3.2.2.3" xref="S4.Ex3.m1.3.3.3.3.2.2.3.cmml">2</mn></msub></mrow><mo id="S4.Ex3.m1.3.3.3.4" xref="S4.Ex3.m1.3.3.3.4.cmml">∈</mo><mrow id="S4.Ex3.m1.3.3.3.5.2" xref="S4.Ex3.m1.3.3.3.5.1.cmml"><mo id="S4.Ex3.m1.3.3.3.5.2.1" stretchy="false" xref="S4.Ex3.m1.3.3.3.5.1.1.cmml">[</mo><mi id="S4.Ex3.m1.1.1.1.1" xref="S4.Ex3.m1.1.1.1.1.cmml">L</mi><mo id="S4.Ex3.m1.3.3.3.5.2.2" stretchy="false" xref="S4.Ex3.m1.3.3.3.5.1.1.cmml">]</mo></mrow></mrow></munder></mstyle><mrow id="S4.Ex3.m1.7.7.2.2" xref="S4.Ex3.m1.7.7.2.2.cmml"><mi id="S4.Ex3.m1.7.7.2.2.4" xref="S4.Ex3.m1.7.7.2.2.4.cmml">w</mi><mo id="S4.Ex3.m1.7.7.2.2.3" xref="S4.Ex3.m1.7.7.2.2.3.cmml">⁢</mo><mrow id="S4.Ex3.m1.7.7.2.2.2.2" xref="S4.Ex3.m1.7.7.2.2.2.3.cmml"><mo id="S4.Ex3.m1.7.7.2.2.2.2.3" stretchy="false" xref="S4.Ex3.m1.7.7.2.2.2.3.cmml">(</mo><mrow id="S4.Ex3.m1.6.6.1.1.1.1.1.1" xref="S4.Ex3.m1.6.6.1.1.1.1.1.2.cmml"><mo id="S4.Ex3.m1.6.6.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex3.m1.6.6.1.1.1.1.1.2.cmml">(</mo><mn id="S4.Ex3.m1.4.4" xref="S4.Ex3.m1.4.4.cmml">1</mn><mo id="S4.Ex3.m1.6.6.1.1.1.1.1.1.3" xref="S4.Ex3.m1.6.6.1.1.1.1.1.2.cmml">,</mo><msub id="S4.Ex3.m1.6.6.1.1.1.1.1.1.1" xref="S4.Ex3.m1.6.6.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex3.m1.6.6.1.1.1.1.1.1.1.2" xref="S4.Ex3.m1.6.6.1.1.1.1.1.1.1.2.cmml">i</mi><mn id="S4.Ex3.m1.6.6.1.1.1.1.1.1.1.3" xref="S4.Ex3.m1.6.6.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Ex3.m1.6.6.1.1.1.1.1.1.4" stretchy="false" xref="S4.Ex3.m1.6.6.1.1.1.1.1.2.cmml">)</mo></mrow><mo id="S4.Ex3.m1.7.7.2.2.2.2.4" xref="S4.Ex3.m1.7.7.2.2.2.3.cmml">,</mo><mrow 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ltx_eqn_center_padright"></span></span></span> </span> </span></foreignobject></g></g></svg> </div> <div class="ltx_para" id="S4.SS1.p5"> <p class="ltx_p" id="S4.SS1.p5.4">where we define the linear functions</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="W^{+}(v_{1})=\sum_{v_{2}\in V}w(v_{1},v_{2})\text{ and }W^{-}(v_{1})=\sum_{v_{% 2}\in V}w(v_{2},v_{1})." class="ltx_Math" display="block" id="S4.Ex5.m1.1"><semantics id="S4.Ex5.m1.1a"><mrow id="S4.Ex5.m1.1.1.1" xref="S4.Ex5.m1.1.1.1.1.cmml"><mrow id="S4.Ex5.m1.1.1.1.1" xref="S4.Ex5.m1.1.1.1.1.cmml"><mrow id="S4.Ex5.m1.1.1.1.1.1" xref="S4.Ex5.m1.1.1.1.1.1.cmml"><msup id="S4.Ex5.m1.1.1.1.1.1.3" xref="S4.Ex5.m1.1.1.1.1.1.3.cmml"><mi id="S4.Ex5.m1.1.1.1.1.1.3.2" xref="S4.Ex5.m1.1.1.1.1.1.3.2.cmml">W</mi><mo id="S4.Ex5.m1.1.1.1.1.1.3.3" 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cd="ambiguous" id="S4.Ex5.m1.1.1.1.1.6.2.2.2.2.1.cmml" xref="S4.Ex5.m1.1.1.1.1.6.2.2.2.2">subscript</csymbol><ci id="S4.Ex5.m1.1.1.1.1.6.2.2.2.2.2.cmml" xref="S4.Ex5.m1.1.1.1.1.6.2.2.2.2.2">𝑣</ci><cn id="S4.Ex5.m1.1.1.1.1.6.2.2.2.2.3.cmml" type="integer" xref="S4.Ex5.m1.1.1.1.1.6.2.2.2.2.3">1</cn></apply></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex5.m1.1c">W^{+}(v_{1})=\sum_{v_{2}\in V}w(v_{1},v_{2})\text{ and }W^{-}(v_{1})=\sum_{v_{% 2}\in V}w(v_{2},v_{1}).</annotation><annotation encoding="application/x-llamapun" id="S4.Ex5.m1.1d">italic_W start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_V end_POSTSUBSCRIPT italic_w ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) and italic_W start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_V end_POSTSUBSCRIPT italic_w ( italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 1 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.SS1.p5.1">(While the objective, as written, is the maximum of a finite number of linear functions, this can be converted to a standard-form LP by using the standard trick which introduces one additional variable.) Note that every feasible solution to this linear program is a weighted, directed graph on <math alttext="V" class="ltx_Math" display="inline" id="S4.SS1.p5.1.m1.1"><semantics id="S4.SS1.p5.1.m1.1a"><mi id="S4.SS1.p5.1.m1.1.1" xref="S4.SS1.p5.1.m1.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.1.m1.1b"><ci id="S4.SS1.p5.1.m1.1.1.cmml" xref="S4.SS1.p5.1.m1.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.1.m1.1c">V</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.1.m1.1d">italic_V</annotation></semantics></math> where:</p> <ol class="ltx_enumerate" id="S4.I1"> <li class="ltx_item" id="S4.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S4.I1.i1.p1"> <p class="ltx_p" id="S4.I1.i1.p1.3">The cut from <math alttext="\{1\}\times[\pm\ell]" class="ltx_Math" display="inline" id="S4.I1.i1.p1.1.m1.2"><semantics id="S4.I1.i1.p1.1.m1.2a"><mrow id="S4.I1.i1.p1.1.m1.2.2" xref="S4.I1.i1.p1.1.m1.2.2.cmml"><mrow id="S4.I1.i1.p1.1.m1.2.2.3.2" xref="S4.I1.i1.p1.1.m1.2.2.3.1.cmml"><mo id="S4.I1.i1.p1.1.m1.2.2.3.2.1" stretchy="false" xref="S4.I1.i1.p1.1.m1.2.2.3.1.cmml">{</mo><mn id="S4.I1.i1.p1.1.m1.1.1" xref="S4.I1.i1.p1.1.m1.1.1.cmml">1</mn><mo id="S4.I1.i1.p1.1.m1.2.2.3.2.2" rspace="0.055em" stretchy="false" xref="S4.I1.i1.p1.1.m1.2.2.3.1.cmml">}</mo></mrow><mo id="S4.I1.i1.p1.1.m1.2.2.2" rspace="0.222em" xref="S4.I1.i1.p1.1.m1.2.2.2.cmml">×</mo><mrow id="S4.I1.i1.p1.1.m1.2.2.1.1" xref="S4.I1.i1.p1.1.m1.2.2.1.2.cmml"><mo id="S4.I1.i1.p1.1.m1.2.2.1.1.2" stretchy="false" xref="S4.I1.i1.p1.1.m1.2.2.1.2.1.cmml">[</mo><mrow id="S4.I1.i1.p1.1.m1.2.2.1.1.1" xref="S4.I1.i1.p1.1.m1.2.2.1.1.1.cmml"><mo id="S4.I1.i1.p1.1.m1.2.2.1.1.1a" xref="S4.I1.i1.p1.1.m1.2.2.1.1.1.cmml">±</mo><mi id="S4.I1.i1.p1.1.m1.2.2.1.1.1.2" mathvariant="normal" xref="S4.I1.i1.p1.1.m1.2.2.1.1.1.2.cmml">ℓ</mi></mrow><mo id="S4.I1.i1.p1.1.m1.2.2.1.1.3" stretchy="false" xref="S4.I1.i1.p1.1.m1.2.2.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.1.m1.2b"><apply id="S4.I1.i1.p1.1.m1.2.2.cmml" xref="S4.I1.i1.p1.1.m1.2.2"><times id="S4.I1.i1.p1.1.m1.2.2.2.cmml" xref="S4.I1.i1.p1.1.m1.2.2.2"></times><set id="S4.I1.i1.p1.1.m1.2.2.3.1.cmml" xref="S4.I1.i1.p1.1.m1.2.2.3.2"><cn id="S4.I1.i1.p1.1.m1.1.1.cmml" type="integer" xref="S4.I1.i1.p1.1.m1.1.1">1</cn></set><apply id="S4.I1.i1.p1.1.m1.2.2.1.2.cmml" xref="S4.I1.i1.p1.1.m1.2.2.1.1"><csymbol cd="latexml" id="S4.I1.i1.p1.1.m1.2.2.1.2.1.cmml" xref="S4.I1.i1.p1.1.m1.2.2.1.1.2">delimited-[]</csymbol><apply id="S4.I1.i1.p1.1.m1.2.2.1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.2.2.1.1.1"><csymbol cd="latexml" id="S4.I1.i1.p1.1.m1.2.2.1.1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.2.2.1.1.1">plus-or-minus</csymbol><ci id="S4.I1.i1.p1.1.m1.2.2.1.1.1.2.cmml" xref="S4.I1.i1.p1.1.m1.2.2.1.1.1.2">ℓ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.1.m1.2c">\{1\}\times[\pm\ell]</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.1.m1.2d">{ 1 } × [ ± roman_ℓ ]</annotation></semantics></math> to <math alttext="\{0\}\times[\pm\ell]" class="ltx_Math" display="inline" id="S4.I1.i1.p1.2.m2.2"><semantics id="S4.I1.i1.p1.2.m2.2a"><mrow id="S4.I1.i1.p1.2.m2.2.2" xref="S4.I1.i1.p1.2.m2.2.2.cmml"><mrow id="S4.I1.i1.p1.2.m2.2.2.3.2" xref="S4.I1.i1.p1.2.m2.2.2.3.1.cmml"><mo id="S4.I1.i1.p1.2.m2.2.2.3.2.1" stretchy="false" xref="S4.I1.i1.p1.2.m2.2.2.3.1.cmml">{</mo><mn id="S4.I1.i1.p1.2.m2.1.1" xref="S4.I1.i1.p1.2.m2.1.1.cmml">0</mn><mo id="S4.I1.i1.p1.2.m2.2.2.3.2.2" rspace="0.055em" stretchy="false" xref="S4.I1.i1.p1.2.m2.2.2.3.1.cmml">}</mo></mrow><mo id="S4.I1.i1.p1.2.m2.2.2.2" rspace="0.222em" xref="S4.I1.i1.p1.2.m2.2.2.2.cmml">×</mo><mrow id="S4.I1.i1.p1.2.m2.2.2.1.1" xref="S4.I1.i1.p1.2.m2.2.2.1.2.cmml"><mo id="S4.I1.i1.p1.2.m2.2.2.1.1.2" stretchy="false" xref="S4.I1.i1.p1.2.m2.2.2.1.2.1.cmml">[</mo><mrow id="S4.I1.i1.p1.2.m2.2.2.1.1.1" xref="S4.I1.i1.p1.2.m2.2.2.1.1.1.cmml"><mo id="S4.I1.i1.p1.2.m2.2.2.1.1.1a" xref="S4.I1.i1.p1.2.m2.2.2.1.1.1.cmml">±</mo><mi id="S4.I1.i1.p1.2.m2.2.2.1.1.1.2" mathvariant="normal" xref="S4.I1.i1.p1.2.m2.2.2.1.1.1.2.cmml">ℓ</mi></mrow><mo id="S4.I1.i1.p1.2.m2.2.2.1.1.3" stretchy="false" xref="S4.I1.i1.p1.2.m2.2.2.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.2.m2.2b"><apply id="S4.I1.i1.p1.2.m2.2.2.cmml" xref="S4.I1.i1.p1.2.m2.2.2"><times id="S4.I1.i1.p1.2.m2.2.2.2.cmml" xref="S4.I1.i1.p1.2.m2.2.2.2"></times><set id="S4.I1.i1.p1.2.m2.2.2.3.1.cmml" xref="S4.I1.i1.p1.2.m2.2.2.3.2"><cn id="S4.I1.i1.p1.2.m2.1.1.cmml" type="integer" xref="S4.I1.i1.p1.2.m2.1.1">0</cn></set><apply id="S4.I1.i1.p1.2.m2.2.2.1.2.cmml" xref="S4.I1.i1.p1.2.m2.2.2.1.1"><csymbol cd="latexml" id="S4.I1.i1.p1.2.m2.2.2.1.2.1.cmml" xref="S4.I1.i1.p1.2.m2.2.2.1.1.2">delimited-[]</csymbol><apply 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id="S4.I1.i2.p1.1.m1.2.3.1.cmml" xref="S4.I1.i2.p1.1.m1.2.3.2"><cn id="S4.I1.i2.p1.1.m1.1.1.cmml" type="integer" xref="S4.I1.i2.p1.1.m1.1.1">0</cn><ci id="S4.I1.i2.p1.1.m1.2.2.cmml" xref="S4.I1.i2.p1.1.m1.2.2">𝑖</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.1.m1.2c">(0,i)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.1.m1.2d">( 0 , italic_i )</annotation></semantics></math> and <math alttext="(1,i)" class="ltx_Math" display="inline" id="S4.I1.i2.p1.2.m2.2"><semantics id="S4.I1.i2.p1.2.m2.2a"><mrow id="S4.I1.i2.p1.2.m2.2.3.2" xref="S4.I1.i2.p1.2.m2.2.3.1.cmml"><mo id="S4.I1.i2.p1.2.m2.2.3.2.1" stretchy="false" xref="S4.I1.i2.p1.2.m2.2.3.1.cmml">(</mo><mn id="S4.I1.i2.p1.2.m2.1.1" xref="S4.I1.i2.p1.2.m2.1.1.cmml">1</mn><mo id="S4.I1.i2.p1.2.m2.2.3.2.2" xref="S4.I1.i2.p1.2.m2.2.3.1.cmml">,</mo><mi id="S4.I1.i2.p1.2.m2.2.2" xref="S4.I1.i2.p1.2.m2.2.2.cmml">i</mi><mo id="S4.I1.i2.p1.2.m2.2.3.2.3" stretchy="false" xref="S4.I1.i2.p1.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.2.m2.2b"><interval closure="open" id="S4.I1.i2.p1.2.m2.2.3.1.cmml" xref="S4.I1.i2.p1.2.m2.2.3.2"><cn id="S4.I1.i2.p1.2.m2.1.1.cmml" type="integer" xref="S4.I1.i2.p1.2.m2.1.1">1</cn><ci id="S4.I1.i2.p1.2.m2.2.2.cmml" xref="S4.I1.i2.p1.2.m2.2.2">𝑖</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.2.m2.2c">(1,i)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.2.m2.2d">( 1 , italic_i )</annotation></semantics></math> have bias <math alttext="t_{i}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.3.m3.1"><semantics id="S4.I1.i2.p1.3.m3.1a"><msub id="S4.I1.i2.p1.3.m3.1.1" xref="S4.I1.i2.p1.3.m3.1.1.cmml"><mi id="S4.I1.i2.p1.3.m3.1.1.2" xref="S4.I1.i2.p1.3.m3.1.1.2.cmml">t</mi><mi id="S4.I1.i2.p1.3.m3.1.1.3" xref="S4.I1.i2.p1.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.3.m3.1b"><apply id="S4.I1.i2.p1.3.m3.1.1.cmml" xref="S4.I1.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.I1.i2.p1.3.m3.1.1.1.cmml" xref="S4.I1.i2.p1.3.m3.1.1">subscript</csymbol><ci id="S4.I1.i2.p1.3.m3.1.1.2.cmml" xref="S4.I1.i2.p1.3.m3.1.1.2">𝑡</ci><ci id="S4.I1.i2.p1.3.m3.1.1.3.cmml" xref="S4.I1.i2.p1.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.3.m3.1c">t_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.3.m3.1d">italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (if they are nonisolated) for <math alttext="i\in[\pm\ell]" class="ltx_Math" display="inline" id="S4.I1.i2.p1.4.m4.1"><semantics id="S4.I1.i2.p1.4.m4.1a"><mrow id="S4.I1.i2.p1.4.m4.1.1" xref="S4.I1.i2.p1.4.m4.1.1.cmml"><mi id="S4.I1.i2.p1.4.m4.1.1.3" xref="S4.I1.i2.p1.4.m4.1.1.3.cmml">i</mi><mo id="S4.I1.i2.p1.4.m4.1.1.2" xref="S4.I1.i2.p1.4.m4.1.1.2.cmml">∈</mo><mrow id="S4.I1.i2.p1.4.m4.1.1.1.1" xref="S4.I1.i2.p1.4.m4.1.1.1.2.cmml"><mo id="S4.I1.i2.p1.4.m4.1.1.1.1.2" stretchy="false" xref="S4.I1.i2.p1.4.m4.1.1.1.2.1.cmml">[</mo><mrow id="S4.I1.i2.p1.4.m4.1.1.1.1.1" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1.cmml"><mo id="S4.I1.i2.p1.4.m4.1.1.1.1.1a" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1.cmml">±</mo><mi id="S4.I1.i2.p1.4.m4.1.1.1.1.1.2" mathvariant="normal" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1.2.cmml">ℓ</mi></mrow><mo id="S4.I1.i2.p1.4.m4.1.1.1.1.3" stretchy="false" xref="S4.I1.i2.p1.4.m4.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.4.m4.1b"><apply id="S4.I1.i2.p1.4.m4.1.1.cmml" xref="S4.I1.i2.p1.4.m4.1.1"><in id="S4.I1.i2.p1.4.m4.1.1.2.cmml" xref="S4.I1.i2.p1.4.m4.1.1.2"></in><ci id="S4.I1.i2.p1.4.m4.1.1.3.cmml" xref="S4.I1.i2.p1.4.m4.1.1.3">𝑖</ci><apply id="S4.I1.i2.p1.4.m4.1.1.1.2.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1.1"><csymbol cd="latexml" id="S4.I1.i2.p1.4.m4.1.1.1.2.1.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1.1.2">delimited-[]</csymbol><apply id="S4.I1.i2.p1.4.m4.1.1.1.1.1.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1"><csymbol cd="latexml" id="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1">plus-or-minus</csymbol><ci id="S4.I1.i2.p1.4.m4.1.1.1.1.1.2.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1.2">ℓ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.4.m4.1c">i\in[\pm\ell]</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.4.m4.1d">italic_i ∈ [ ± roman_ℓ ]</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S4.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S4.I1.i3.p1"> <p class="ltx_p" id="S4.I1.i3.p1.3">For <math alttext="\boldsymbol{p}\in[0,1]^{L}" class="ltx_Math" display="inline" id="S4.I1.i3.p1.1.m1.2"><semantics id="S4.I1.i3.p1.1.m1.2a"><mrow id="S4.I1.i3.p1.1.m1.2.3" xref="S4.I1.i3.p1.1.m1.2.3.cmml"><mi id="S4.I1.i3.p1.1.m1.2.3.2" xref="S4.I1.i3.p1.1.m1.2.3.2.cmml">𝒑</mi><mo id="S4.I1.i3.p1.1.m1.2.3.1" xref="S4.I1.i3.p1.1.m1.2.3.1.cmml">∈</mo><msup id="S4.I1.i3.p1.1.m1.2.3.3" xref="S4.I1.i3.p1.1.m1.2.3.3.cmml"><mrow id="S4.I1.i3.p1.1.m1.2.3.3.2.2" xref="S4.I1.i3.p1.1.m1.2.3.3.2.1.cmml"><mo id="S4.I1.i3.p1.1.m1.2.3.3.2.2.1" stretchy="false" xref="S4.I1.i3.p1.1.m1.2.3.3.2.1.cmml">[</mo><mn id="S4.I1.i3.p1.1.m1.1.1" xref="S4.I1.i3.p1.1.m1.1.1.cmml">0</mn><mo id="S4.I1.i3.p1.1.m1.2.3.3.2.2.2" xref="S4.I1.i3.p1.1.m1.2.3.3.2.1.cmml">,</mo><mn id="S4.I1.i3.p1.1.m1.2.2" xref="S4.I1.i3.p1.1.m1.2.2.cmml">1</mn><mo id="S4.I1.i3.p1.1.m1.2.3.3.2.2.3" stretchy="false" xref="S4.I1.i3.p1.1.m1.2.3.3.2.1.cmml">]</mo></mrow><mi id="S4.I1.i3.p1.1.m1.2.3.3.3" xref="S4.I1.i3.p1.1.m1.2.3.3.3.cmml">L</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i3.p1.1.m1.2b"><apply id="S4.I1.i3.p1.1.m1.2.3.cmml" xref="S4.I1.i3.p1.1.m1.2.3"><in id="S4.I1.i3.p1.1.m1.2.3.1.cmml" xref="S4.I1.i3.p1.1.m1.2.3.1"></in><ci id="S4.I1.i3.p1.1.m1.2.3.2.cmml" xref="S4.I1.i3.p1.1.m1.2.3.2">𝒑</ci><apply id="S4.I1.i3.p1.1.m1.2.3.3.cmml" xref="S4.I1.i3.p1.1.m1.2.3.3"><csymbol cd="ambiguous" id="S4.I1.i3.p1.1.m1.2.3.3.1.cmml" xref="S4.I1.i3.p1.1.m1.2.3.3">superscript</csymbol><interval closure="closed" id="S4.I1.i3.p1.1.m1.2.3.3.2.1.cmml" xref="S4.I1.i3.p1.1.m1.2.3.3.2.2"><cn id="S4.I1.i3.p1.1.m1.1.1.cmml" type="integer" xref="S4.I1.i3.p1.1.m1.1.1">0</cn><cn id="S4.I1.i3.p1.1.m1.2.2.cmml" type="integer" xref="S4.I1.i3.p1.1.m1.2.2">1</cn></interval><ci id="S4.I1.i3.p1.1.m1.2.3.3.3.cmml" xref="S4.I1.i3.p1.1.m1.2.3.3.3">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i3.p1.1.m1.2c">\boldsymbol{p}\in[0,1]^{L}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i3.p1.1.m1.2d">bold_italic_p ∈ [ 0 , 1 ] start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT</annotation></semantics></math>, the quantity <math alttext="\sum_{v_{1}=(b_{1},i_{1}),v_{2}=(b_{2},i_{2})\in V}p(i_{1})(1-p(i_{2}))w(v_{1}% ,v_{2})" class="ltx_Math" display="inline" id="S4.I1.i3.p1.2.m2.6"><semantics id="S4.I1.i3.p1.2.m2.6a"><mrow id="S4.I1.i3.p1.2.m2.6.6" xref="S4.I1.i3.p1.2.m2.6.6.cmml"><msub id="S4.I1.i3.p1.2.m2.6.6.5" xref="S4.I1.i3.p1.2.m2.6.6.5.cmml"><mo id="S4.I1.i3.p1.2.m2.6.6.5.2" xref="S4.I1.i3.p1.2.m2.6.6.5.2.cmml">∑</mo><mrow id="S4.I1.i3.p1.2.m2.2.2.2.2" xref="S4.I1.i3.p1.2.m2.2.2.2.3.cmml"><mrow id="S4.I1.i3.p1.2.m2.1.1.1.1.1" xref="S4.I1.i3.p1.2.m2.1.1.1.1.1.cmml"><msub id="S4.I1.i3.p1.2.m2.1.1.1.1.1.4" xref="S4.I1.i3.p1.2.m2.1.1.1.1.1.4.cmml"><mi id="S4.I1.i3.p1.2.m2.1.1.1.1.1.4.2" xref="S4.I1.i3.p1.2.m2.1.1.1.1.1.4.2.cmml">v</mi><mn id="S4.I1.i3.p1.2.m2.1.1.1.1.1.4.3" xref="S4.I1.i3.p1.2.m2.1.1.1.1.1.4.3.cmml">1</mn></msub><mo id="S4.I1.i3.p1.2.m2.1.1.1.1.1.3" xref="S4.I1.i3.p1.2.m2.1.1.1.1.1.3.cmml">=</mo><mrow id="S4.I1.i3.p1.2.m2.1.1.1.1.1.2.2" xref="S4.I1.i3.p1.2.m2.1.1.1.1.1.2.3.cmml"><mo id="S4.I1.i3.p1.2.m2.1.1.1.1.1.2.2.3" stretchy="false" xref="S4.I1.i3.p1.2.m2.1.1.1.1.1.2.3.cmml">(</mo><msub id="S4.I1.i3.p1.2.m2.1.1.1.1.1.1.1.1" xref="S4.I1.i3.p1.2.m2.1.1.1.1.1.1.1.1.cmml"><mi id="S4.I1.i3.p1.2.m2.1.1.1.1.1.1.1.1.2" xref="S4.I1.i3.p1.2.m2.1.1.1.1.1.1.1.1.2.cmml">b</mi><mn id="S4.I1.i3.p1.2.m2.1.1.1.1.1.1.1.1.3" xref="S4.I1.i3.p1.2.m2.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.I1.i3.p1.2.m2.1.1.1.1.1.2.2.4" xref="S4.I1.i3.p1.2.m2.1.1.1.1.1.2.3.cmml">,</mo><msub id="S4.I1.i3.p1.2.m2.1.1.1.1.1.2.2.2" xref="S4.I1.i3.p1.2.m2.1.1.1.1.1.2.2.2.cmml"><mi id="S4.I1.i3.p1.2.m2.1.1.1.1.1.2.2.2.2" xref="S4.I1.i3.p1.2.m2.1.1.1.1.1.2.2.2.2.cmml">i</mi><mn id="S4.I1.i3.p1.2.m2.1.1.1.1.1.2.2.2.3" xref="S4.I1.i3.p1.2.m2.1.1.1.1.1.2.2.2.3.cmml">1</mn></msub><mo id="S4.I1.i3.p1.2.m2.1.1.1.1.1.2.2.5" stretchy="false" xref="S4.I1.i3.p1.2.m2.1.1.1.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="S4.I1.i3.p1.2.m2.2.2.2.2.3" 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italic_p ( italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( 1 - italic_p ( italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) italic_w ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> calculates the expected weight cut by the oblivious algorithm <math alttext="\boldsymbol{p}" class="ltx_Math" display="inline" id="S4.I1.i3.p1.3.m3.1"><semantics id="S4.I1.i3.p1.3.m3.1a"><mi id="S4.I1.i3.p1.3.m3.1.1" xref="S4.I1.i3.p1.3.m3.1.1.cmml">𝒑</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i3.p1.3.m3.1b"><ci id="S4.I1.i3.p1.3.m3.1.1.cmml" xref="S4.I1.i3.p1.3.m3.1.1">𝒑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i3.p1.3.m3.1c">\boldsymbol{p}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i3.p1.3.m3.1d">bold_italic_p</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S4.SS1.p5.3">That this suffices to produce the worst-case graph is essentially a consequence of some reasoning due <span class="ltx_ERROR undefined" id="S4.SS1.p5.3.1">\textcite</span>FJ15 for a similar LP; the idea is to “condense” vertices into <math alttext="2L" class="ltx_Math" display="inline" id="S4.SS1.p5.2.m1.1"><semantics id="S4.SS1.p5.2.m1.1a"><mrow id="S4.SS1.p5.2.m1.1.1" xref="S4.SS1.p5.2.m1.1.1.cmml"><mn id="S4.SS1.p5.2.m1.1.1.2" xref="S4.SS1.p5.2.m1.1.1.2.cmml">2</mn><mo id="S4.SS1.p5.2.m1.1.1.1" xref="S4.SS1.p5.2.m1.1.1.1.cmml">⁢</mo><mi id="S4.SS1.p5.2.m1.1.1.3" xref="S4.SS1.p5.2.m1.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.2.m1.1b"><apply id="S4.SS1.p5.2.m1.1.1.cmml" xref="S4.SS1.p5.2.m1.1.1"><times id="S4.SS1.p5.2.m1.1.1.1.cmml" xref="S4.SS1.p5.2.m1.1.1.1"></times><cn id="S4.SS1.p5.2.m1.1.1.2.cmml" type="integer" xref="S4.SS1.p5.2.m1.1.1.2">2</cn><ci id="S4.SS1.p5.2.m1.1.1.3.cmml" xref="S4.SS1.p5.2.m1.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.2.m1.1c">2L</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.2.m1.1d">2 italic_L</annotation></semantics></math> equivalence classes based purely on their bias and their assignment in an optimal cut, and also to rescale so that the optimal cut has weight <math alttext="1" class="ltx_Math" display="inline" id="S4.SS1.p5.3.m2.1"><semantics id="S4.SS1.p5.3.m2.1a"><mn id="S4.SS1.p5.3.m2.1.1" xref="S4.SS1.p5.3.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.3.m2.1b"><cn id="S4.SS1.p5.3.m2.1.1.cmml" type="integer" xref="S4.SS1.p5.3.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.3.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.3.m2.1d">1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.p6"> <p class="ltx_p" id="S4.SS1.p6.11">However, to find lower bounds against concrete classes of selection functions, two issues remain. Firstly, we need to choose the biases <math alttext="b_{1},\ldots,b_{L}" class="ltx_Math" display="inline" id="S4.SS1.p6.1.m1.3"><semantics id="S4.SS1.p6.1.m1.3a"><mrow id="S4.SS1.p6.1.m1.3.3.2" xref="S4.SS1.p6.1.m1.3.3.3.cmml"><msub id="S4.SS1.p6.1.m1.2.2.1.1" xref="S4.SS1.p6.1.m1.2.2.1.1.cmml"><mi id="S4.SS1.p6.1.m1.2.2.1.1.2" xref="S4.SS1.p6.1.m1.2.2.1.1.2.cmml">b</mi><mn id="S4.SS1.p6.1.m1.2.2.1.1.3" xref="S4.SS1.p6.1.m1.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.SS1.p6.1.m1.3.3.2.3" xref="S4.SS1.p6.1.m1.3.3.3.cmml">,</mo><mi id="S4.SS1.p6.1.m1.1.1" mathvariant="normal" xref="S4.SS1.p6.1.m1.1.1.cmml">…</mi><mo id="S4.SS1.p6.1.m1.3.3.2.4" xref="S4.SS1.p6.1.m1.3.3.3.cmml">,</mo><msub id="S4.SS1.p6.1.m1.3.3.2.2" xref="S4.SS1.p6.1.m1.3.3.2.2.cmml"><mi id="S4.SS1.p6.1.m1.3.3.2.2.2" xref="S4.SS1.p6.1.m1.3.3.2.2.2.cmml">b</mi><mi id="S4.SS1.p6.1.m1.3.3.2.2.3" xref="S4.SS1.p6.1.m1.3.3.2.2.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.1.m1.3b"><list id="S4.SS1.p6.1.m1.3.3.3.cmml" xref="S4.SS1.p6.1.m1.3.3.2"><apply id="S4.SS1.p6.1.m1.2.2.1.1.cmml" xref="S4.SS1.p6.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.1.m1.2.2.1.1.1.cmml" xref="S4.SS1.p6.1.m1.2.2.1.1">subscript</csymbol><ci id="S4.SS1.p6.1.m1.2.2.1.1.2.cmml" xref="S4.SS1.p6.1.m1.2.2.1.1.2">𝑏</ci><cn id="S4.SS1.p6.1.m1.2.2.1.1.3.cmml" type="integer" xref="S4.SS1.p6.1.m1.2.2.1.1.3">1</cn></apply><ci id="S4.SS1.p6.1.m1.1.1.cmml" xref="S4.SS1.p6.1.m1.1.1">…</ci><apply id="S4.SS1.p6.1.m1.3.3.2.2.cmml" xref="S4.SS1.p6.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="S4.SS1.p6.1.m1.3.3.2.2.1.cmml" xref="S4.SS1.p6.1.m1.3.3.2.2">subscript</csymbol><ci id="S4.SS1.p6.1.m1.3.3.2.2.2.cmml" xref="S4.SS1.p6.1.m1.3.3.2.2.2">𝑏</ci><ci id="S4.SS1.p6.1.m1.3.3.2.2.3.cmml" xref="S4.SS1.p6.1.m1.3.3.2.2.3">𝐿</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.1.m1.3c">b_{1},\ldots,b_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.1.m1.3d">italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_b start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. We typically do this by either fixing a small value of <math alttext="L" class="ltx_Math" display="inline" id="S4.SS1.p6.2.m2.1"><semantics id="S4.SS1.p6.2.m2.1a"><mi id="S4.SS1.p6.2.m2.1.1" xref="S4.SS1.p6.2.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.2.m2.1b"><ci id="S4.SS1.p6.2.m2.1.1.cmml" xref="S4.SS1.p6.2.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.2.m2.1d">italic_L</annotation></semantics></math> and performing a grid search, or by picking uniformly spaced points. Secondly, in the case where we desire to prove a bound against a large or infinite set <math alttext="\mathcal{P}^{*}" class="ltx_Math" display="inline" id="S4.SS1.p6.3.m3.1"><semantics id="S4.SS1.p6.3.m3.1a"><msup id="S4.SS1.p6.3.m3.1.1" xref="S4.SS1.p6.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p6.3.m3.1.1.2" xref="S4.SS1.p6.3.m3.1.1.2.cmml">𝒫</mi><mo id="S4.SS1.p6.3.m3.1.1.3" xref="S4.SS1.p6.3.m3.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.3.m3.1b"><apply id="S4.SS1.p6.3.m3.1.1.cmml" xref="S4.SS1.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.3.m3.1.1.1.cmml" xref="S4.SS1.p6.3.m3.1.1">superscript</csymbol><ci id="S4.SS1.p6.3.m3.1.1.2.cmml" xref="S4.SS1.p6.3.m3.1.1.2">𝒫</ci><times id="S4.SS1.p6.3.m3.1.1.3.cmml" xref="S4.SS1.p6.3.m3.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.3.m3.1c">\mathcal{P}^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.3.m3.1d">caligraphic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> of selection functions — e.g., the set of all antisymmetric selection functions, or PL sigmoid selection functions — we need to reduce the set of selection functions to a tractable size. We typically do this by discretizing the space of all selection functions in some form to form some small subset <math alttext="\mathcal{P}" class="ltx_Math" display="inline" id="S4.SS1.p6.4.m4.1"><semantics id="S4.SS1.p6.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p6.4.m4.1.1" xref="S4.SS1.p6.4.m4.1.1.cmml">𝒫</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.4.m4.1b"><ci id="S4.SS1.p6.4.m4.1.1.cmml" xref="S4.SS1.p6.4.m4.1.1">𝒫</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.4.m4.1c">\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.4.m4.1d">caligraphic_P</annotation></semantics></math>. This does cause some additional loss in the approximation ratio, which will ideally be small. To carry out the proof, we first produce a candidate graph <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.p6.5.m5.1"><semantics id="S4.SS1.p6.5.m5.1a"><mi id="S4.SS1.p6.5.m5.1.1" xref="S4.SS1.p6.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.5.m5.1b"><ci id="S4.SS1.p6.5.m5.1.1.cmml" xref="S4.SS1.p6.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.5.m5.1d">italic_G</annotation></semantics></math> which holds against <math alttext="\mathcal{P}" class="ltx_Math" display="inline" id="S4.SS1.p6.6.m6.1"><semantics id="S4.SS1.p6.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p6.6.m6.1.1" xref="S4.SS1.p6.6.m6.1.1.cmml">𝒫</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.6.m6.1b"><ci id="S4.SS1.p6.6.m6.1.1.cmml" xref="S4.SS1.p6.6.m6.1.1">𝒫</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.6.m6.1c">\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.6.m6.1d">caligraphic_P</annotation></semantics></math>, and then (ideally) show that it holds almost as well against <math alttext="\mathcal{P}^{*}" class="ltx_Math" display="inline" id="S4.SS1.p6.7.m7.1"><semantics id="S4.SS1.p6.7.m7.1a"><msup id="S4.SS1.p6.7.m7.1.1" xref="S4.SS1.p6.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p6.7.m7.1.1.2" xref="S4.SS1.p6.7.m7.1.1.2.cmml">𝒫</mi><mo id="S4.SS1.p6.7.m7.1.1.3" xref="S4.SS1.p6.7.m7.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.7.m7.1b"><apply id="S4.SS1.p6.7.m7.1.1.cmml" xref="S4.SS1.p6.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.7.m7.1.1.1.cmml" xref="S4.SS1.p6.7.m7.1.1">superscript</csymbol><ci id="S4.SS1.p6.7.m7.1.1.2.cmml" xref="S4.SS1.p6.7.m7.1.1.2">𝒫</ci><times id="S4.SS1.p6.7.m7.1.1.3.cmml" xref="S4.SS1.p6.7.m7.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.7.m7.1c">\mathcal{P}^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.7.m7.1d">caligraphic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> by directly solving the maximization problem over <math alttext="\mathcal{P}^{*}" class="ltx_Math" display="inline" id="S4.SS1.p6.8.m8.1"><semantics id="S4.SS1.p6.8.m8.1a"><msup id="S4.SS1.p6.8.m8.1.1" xref="S4.SS1.p6.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p6.8.m8.1.1.2" xref="S4.SS1.p6.8.m8.1.1.2.cmml">𝒫</mi><mo id="S4.SS1.p6.8.m8.1.1.3" xref="S4.SS1.p6.8.m8.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.8.m8.1b"><apply id="S4.SS1.p6.8.m8.1.1.cmml" xref="S4.SS1.p6.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.8.m8.1.1.1.cmml" xref="S4.SS1.p6.8.m8.1.1">superscript</csymbol><ci id="S4.SS1.p6.8.m8.1.1.2.cmml" xref="S4.SS1.p6.8.m8.1.1.2">𝒫</ci><times id="S4.SS1.p6.8.m8.1.1.3.cmml" xref="S4.SS1.p6.8.m8.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.8.m8.1c">\mathcal{P}^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.8.m8.1d">caligraphic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. In the case where <math alttext="\mathcal{P}^{*}" class="ltx_Math" display="inline" id="S4.SS1.p6.9.m9.1"><semantics id="S4.SS1.p6.9.m9.1a"><msup id="S4.SS1.p6.9.m9.1.1" xref="S4.SS1.p6.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p6.9.m9.1.1.2" xref="S4.SS1.p6.9.m9.1.1.2.cmml">𝒫</mi><mo id="S4.SS1.p6.9.m9.1.1.3" xref="S4.SS1.p6.9.m9.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.9.m9.1b"><apply id="S4.SS1.p6.9.m9.1.1.cmml" xref="S4.SS1.p6.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.9.m9.1.1.1.cmml" xref="S4.SS1.p6.9.m9.1.1">superscript</csymbol><ci id="S4.SS1.p6.9.m9.1.1.2.cmml" xref="S4.SS1.p6.9.m9.1.1.2">𝒫</ci><times id="S4.SS1.p6.9.m9.1.1.3.cmml" xref="S4.SS1.p6.9.m9.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.9.m9.1c">\mathcal{P}^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.9.m9.1d">caligraphic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is the set of all antisymmetric functions (or all functions), maximizing the weight of the assignment over <math alttext="\mathcal{P}^{*}" class="ltx_Math" display="inline" id="S4.SS1.p6.10.m10.1"><semantics id="S4.SS1.p6.10.m10.1a"><msup id="S4.SS1.p6.10.m10.1.1" xref="S4.SS1.p6.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p6.10.m10.1.1.2" xref="S4.SS1.p6.10.m10.1.1.2.cmml">𝒫</mi><mo id="S4.SS1.p6.10.m10.1.1.3" xref="S4.SS1.p6.10.m10.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.10.m10.1b"><apply id="S4.SS1.p6.10.m10.1.1.cmml" xref="S4.SS1.p6.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.10.m10.1.1.1.cmml" xref="S4.SS1.p6.10.m10.1.1">superscript</csymbol><ci id="S4.SS1.p6.10.m10.1.1.2.cmml" xref="S4.SS1.p6.10.m10.1.1.2">𝒫</ci><times id="S4.SS1.p6.10.m10.1.1.3.cmml" xref="S4.SS1.p6.10.m10.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.10.m10.1c">\mathcal{P}^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.10.m10.1d">caligraphic_P start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> corresponds to solving an <math alttext="L" class="ltx_Math" display="inline" id="S4.SS1.p6.11.m11.1"><semantics id="S4.SS1.p6.11.m11.1a"><mi id="S4.SS1.p6.11.m11.1.1" xref="S4.SS1.p6.11.m11.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.11.m11.1b"><ci id="S4.SS1.p6.11.m11.1.1.cmml" xref="S4.SS1.p6.11.m11.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.11.m11.1c">L</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.11.m11.1d">italic_L</annotation></semantics></math>-variate quadratic optimization problem.</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>Bounds against <math alttext="\mathsf{PLSigmoid}_{1/2}" class="ltx_Math" display="inline" id="S4.SS2.1.m1.1"><semantics id="S4.SS2.1.m1.1b"><msub id="S4.SS2.1.m1.1.1" xref="S4.SS2.1.m1.1.1.cmml"><mi id="S4.SS2.1.m1.1.1.2" xref="S4.SS2.1.m1.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S4.SS2.1.m1.1.1.3" xref="S4.SS2.1.m1.1.1.3.cmml"><mn id="S4.SS2.1.m1.1.1.3.2" xref="S4.SS2.1.m1.1.1.3.2.cmml">1</mn><mo id="S4.SS2.1.m1.1.1.3.1" xref="S4.SS2.1.m1.1.1.3.1.cmml">/</mo><mn id="S4.SS2.1.m1.1.1.3.3" xref="S4.SS2.1.m1.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.1.m1.1c"><apply id="S4.SS2.1.m1.1.1.cmml" xref="S4.SS2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.1.m1.1.1.1.cmml" xref="S4.SS2.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.1.m1.1.1.2.cmml" xref="S4.SS2.1.m1.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S4.SS2.1.m1.1.1.3.cmml" xref="S4.SS2.1.m1.1.1.3"><divide id="S4.SS2.1.m1.1.1.3.1.cmml" xref="S4.SS2.1.m1.1.1.3.1"></divide><cn id="S4.SS2.1.m1.1.1.3.2.cmml" type="integer" xref="S4.SS2.1.m1.1.1.3.2">1</cn><cn id="S4.SS2.1.m1.1.1.3.3.cmml" type="integer" xref="S4.SS2.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.m1.1d">\mathsf{PLSigmoid}_{1/2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.m1.1e">sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math> (<a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem6" title="Theorem 1.6 (Lower bound for PL sigmoid selection with 𝑏=1/2 intercept). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.6</span></a>)</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">We have the following bound on <math alttext="\mathsf{PLSigmoid}_{1/2}" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><msub id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.p1.1.m1.1.1.2" xref="S4.SS2.p1.1.m1.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S4.SS2.p1.1.m1.1.1.3" xref="S4.SS2.p1.1.m1.1.1.3.cmml"><mn id="S4.SS2.p1.1.m1.1.1.3.2" xref="S4.SS2.p1.1.m1.1.1.3.2.cmml">1</mn><mo id="S4.SS2.p1.1.m1.1.1.3.1" xref="S4.SS2.p1.1.m1.1.1.3.1.cmml">/</mo><mn id="S4.SS2.p1.1.m1.1.1.3.3" xref="S4.SS2.p1.1.m1.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><apply id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S4.SS2.p1.1.m1.1.1.3.cmml" xref="S4.SS2.p1.1.m1.1.1.3"><divide id="S4.SS2.p1.1.m1.1.1.3.1.cmml" xref="S4.SS2.p1.1.m1.1.1.3.1"></divide><cn id="S4.SS2.p1.1.m1.1.1.3.2.cmml" type="integer" xref="S4.SS2.p1.1.m1.1.1.3.2">1</cn><cn id="S4.SS2.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S4.SS2.p1.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">\mathsf{PLSigmoid}_{1/2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math>:</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.1">See <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem6" title="Theorem 1.6 (Lower bound for PL sigmoid selection with 𝑏=1/2 intercept). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">1.6</span></a></p> </div> <figure class="ltx_figure" id="S4.F3"><svg class="ltx_picture ltx_centering" height="142.98" id="S4.F3.pic1" overflow="visible" version="1.1" width="50.35"><g fill="#000000" stroke="#000000" 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id="S4.F3.pic1.4.4.4.4.1.1">2</span></foreignobject></g></g><g stroke-width="0.85358pt"><path d="M -5.25 106.84 C -21.01 73.06 -21.01 45.05 -8.61 18.46" style="fill:none"></path><g stroke-dasharray="none" stroke-dashoffset="0.0pt" stroke-linejoin="miter" transform="matrix(0.42262 -0.90631 0.90631 0.42262 -8.61 18.46)"><path d="M 6.4 0 C 5.61 0.28 2.16 1.87 0 3.6 L 0 -3.6 C 2.16 -1.87 5.61 -0.28 6.4 0 Z"></path></g><g fill="#E6E6E6" stroke="#000000"><path d="M -24.37 51.76 h 14.61 v 14.59 h -14.61 Z"></path></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 -19.76 56.37)"><foreignobject height="5.36" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="5.39"><math alttext="c" class="ltx_Math" display="inline" id="S4.F3.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S4.F3.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S4.F3.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" mathsize="90%" xref="S4.F3.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.F3.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S4.F3.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S4.F3.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.F3.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_c</annotation></semantics></math></foreignobject></g></g><g stroke-width="0.85358pt"><path d="M 5.25 11.27 C 21.01 45.05 21.01 73.06 8.61 99.65" style="fill:none"></path><g stroke-dasharray="none" stroke-dashoffset="0.0pt" stroke-linejoin="miter" transform="matrix(-0.42262 0.90631 -0.90631 -0.42262 8.61 99.65)"><path d="M 6.4 0 C 5.61 0.28 2.16 1.87 0 3.6 L 0 -3.6 C 2.16 -1.87 5.61 -0.28 6.4 0 Z"></path></g><g fill="#E6E6E6" stroke="#000000"><path d="M 9.34 50.43 h 15.45 v 17.25 h -15.45 Z"></path></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 13.95 55.04)"><foreignobject height="8.03" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="6.23"><math alttext="1" class="ltx_Math" display="inline" id="S4.F3.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S4.F3.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mn id="S4.F3.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" mathsize="90%" xref="S4.F3.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.F3.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><cn id="S4.F3.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" type="integer" xref="S4.F3.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.F3.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">1</annotation></semantics></math></foreignobject></g></g></g></svg> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F3.23.10.1" style="font-size:90%;">Figure 3</span>: </span><em class="ltx_emph ltx_font_italic" id="S4.F3.24.11" style="font-size:90%;">Parameter:</em><span class="ltx_text" id="S4.F3.18.9" style="font-size:90%;"> <math alttext="c>1" class="ltx_Math" display="inline" id="S4.F3.10.1.m1.1"><semantics id="S4.F3.10.1.m1.1b"><mrow id="S4.F3.10.1.m1.1.1" xref="S4.F3.10.1.m1.1.1.cmml"><mi id="S4.F3.10.1.m1.1.1.2" xref="S4.F3.10.1.m1.1.1.2.cmml">c</mi><mo id="S4.F3.10.1.m1.1.1.1" xref="S4.F3.10.1.m1.1.1.1.cmml">></mo><mn id="S4.F3.10.1.m1.1.1.3" xref="S4.F3.10.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.10.1.m1.1c"><apply id="S4.F3.10.1.m1.1.1.cmml" xref="S4.F3.10.1.m1.1.1"><gt id="S4.F3.10.1.m1.1.1.1.cmml" xref="S4.F3.10.1.m1.1.1.1"></gt><ci id="S4.F3.10.1.m1.1.1.2.cmml" xref="S4.F3.10.1.m1.1.1.2">𝑐</ci><cn id="S4.F3.10.1.m1.1.1.3.cmml" type="integer" xref="S4.F3.10.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.10.1.m1.1d">c>1</annotation><annotation encoding="application/x-llamapun" id="S4.F3.10.1.m1.1e">italic_c > 1</annotation></semantics></math>. The <span class="ltx_text" id="S4.F3.18.9.1" style="background-color:#ABDEE6;">LIGHT BLUE</span> vertex (<math alttext="1" class="ltx_Math" display="inline" id="S4.F3.11.2.m2.1"><semantics id="S4.F3.11.2.m2.1b"><mn id="S4.F3.11.2.m2.1.1" xref="S4.F3.11.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.F3.11.2.m2.1c"><cn id="S4.F3.11.2.m2.1.1.cmml" type="integer" xref="S4.F3.11.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.11.2.m2.1d">1</annotation><annotation encoding="application/x-llamapun" id="S4.F3.11.2.m2.1e">1</annotation></semantics></math>) has bias <math alttext="+\frac{c-1}{c+1}" class="ltx_Math" display="inline" id="S4.F3.12.3.m3.1"><semantics id="S4.F3.12.3.m3.1b"><mrow id="S4.F3.12.3.m3.1.1" xref="S4.F3.12.3.m3.1.1.cmml"><mo id="S4.F3.12.3.m3.1.1b" xref="S4.F3.12.3.m3.1.1.cmml">+</mo><mfrac id="S4.F3.12.3.m3.1.1.2" xref="S4.F3.12.3.m3.1.1.2.cmml"><mrow id="S4.F3.12.3.m3.1.1.2.2" xref="S4.F3.12.3.m3.1.1.2.2.cmml"><mi id="S4.F3.12.3.m3.1.1.2.2.2" xref="S4.F3.12.3.m3.1.1.2.2.2.cmml">c</mi><mo id="S4.F3.12.3.m3.1.1.2.2.1" xref="S4.F3.12.3.m3.1.1.2.2.1.cmml">−</mo><mn id="S4.F3.12.3.m3.1.1.2.2.3" xref="S4.F3.12.3.m3.1.1.2.2.3.cmml">1</mn></mrow><mrow id="S4.F3.12.3.m3.1.1.2.3" xref="S4.F3.12.3.m3.1.1.2.3.cmml"><mi id="S4.F3.12.3.m3.1.1.2.3.2" xref="S4.F3.12.3.m3.1.1.2.3.2.cmml">c</mi><mo id="S4.F3.12.3.m3.1.1.2.3.1" xref="S4.F3.12.3.m3.1.1.2.3.1.cmml">+</mo><mn id="S4.F3.12.3.m3.1.1.2.3.3" xref="S4.F3.12.3.m3.1.1.2.3.3.cmml">1</mn></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.12.3.m3.1c"><apply id="S4.F3.12.3.m3.1.1.cmml" xref="S4.F3.12.3.m3.1.1"><plus id="S4.F3.12.3.m3.1.1.1.cmml" xref="S4.F3.12.3.m3.1.1"></plus><apply id="S4.F3.12.3.m3.1.1.2.cmml" xref="S4.F3.12.3.m3.1.1.2"><divide id="S4.F3.12.3.m3.1.1.2.1.cmml" xref="S4.F3.12.3.m3.1.1.2"></divide><apply id="S4.F3.12.3.m3.1.1.2.2.cmml" xref="S4.F3.12.3.m3.1.1.2.2"><minus id="S4.F3.12.3.m3.1.1.2.2.1.cmml" xref="S4.F3.12.3.m3.1.1.2.2.1"></minus><ci id="S4.F3.12.3.m3.1.1.2.2.2.cmml" xref="S4.F3.12.3.m3.1.1.2.2.2">𝑐</ci><cn id="S4.F3.12.3.m3.1.1.2.2.3.cmml" type="integer" xref="S4.F3.12.3.m3.1.1.2.2.3">1</cn></apply><apply id="S4.F3.12.3.m3.1.1.2.3.cmml" xref="S4.F3.12.3.m3.1.1.2.3"><plus id="S4.F3.12.3.m3.1.1.2.3.1.cmml" xref="S4.F3.12.3.m3.1.1.2.3.1"></plus><ci id="S4.F3.12.3.m3.1.1.2.3.2.cmml" xref="S4.F3.12.3.m3.1.1.2.3.2">𝑐</ci><cn id="S4.F3.12.3.m3.1.1.2.3.3.cmml" type="integer" xref="S4.F3.12.3.m3.1.1.2.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.12.3.m3.1d">+\frac{c-1}{c+1}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.12.3.m3.1e">+ divide start_ARG italic_c - 1 end_ARG start_ARG italic_c + 1 end_ARG</annotation></semantics></math>. The <span class="ltx_text" id="S4.F3.18.9.2" style="background-color:#FEE1E8;">PINK</span> vertex (<math alttext="2" class="ltx_Math" display="inline" id="S4.F3.13.4.m4.1"><semantics id="S4.F3.13.4.m4.1b"><mn id="S4.F3.13.4.m4.1.1" xref="S4.F3.13.4.m4.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S4.F3.13.4.m4.1c"><cn id="S4.F3.13.4.m4.1.1.cmml" type="integer" xref="S4.F3.13.4.m4.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.13.4.m4.1d">2</annotation><annotation encoding="application/x-llamapun" id="S4.F3.13.4.m4.1e">2</annotation></semantics></math>) has bias <math alttext="-\frac{c-1}{c+1}" class="ltx_Math" display="inline" id="S4.F3.14.5.m5.1"><semantics id="S4.F3.14.5.m5.1b"><mrow id="S4.F3.14.5.m5.1.1" xref="S4.F3.14.5.m5.1.1.cmml"><mo id="S4.F3.14.5.m5.1.1b" xref="S4.F3.14.5.m5.1.1.cmml">−</mo><mfrac id="S4.F3.14.5.m5.1.1.2" xref="S4.F3.14.5.m5.1.1.2.cmml"><mrow id="S4.F3.14.5.m5.1.1.2.2" xref="S4.F3.14.5.m5.1.1.2.2.cmml"><mi id="S4.F3.14.5.m5.1.1.2.2.2" xref="S4.F3.14.5.m5.1.1.2.2.2.cmml">c</mi><mo id="S4.F3.14.5.m5.1.1.2.2.1" xref="S4.F3.14.5.m5.1.1.2.2.1.cmml">−</mo><mn id="S4.F3.14.5.m5.1.1.2.2.3" xref="S4.F3.14.5.m5.1.1.2.2.3.cmml">1</mn></mrow><mrow id="S4.F3.14.5.m5.1.1.2.3" xref="S4.F3.14.5.m5.1.1.2.3.cmml"><mi id="S4.F3.14.5.m5.1.1.2.3.2" xref="S4.F3.14.5.m5.1.1.2.3.2.cmml">c</mi><mo id="S4.F3.14.5.m5.1.1.2.3.1" xref="S4.F3.14.5.m5.1.1.2.3.1.cmml">+</mo><mn id="S4.F3.14.5.m5.1.1.2.3.3" xref="S4.F3.14.5.m5.1.1.2.3.3.cmml">1</mn></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.14.5.m5.1c"><apply id="S4.F3.14.5.m5.1.1.cmml" xref="S4.F3.14.5.m5.1.1"><minus id="S4.F3.14.5.m5.1.1.1.cmml" xref="S4.F3.14.5.m5.1.1"></minus><apply id="S4.F3.14.5.m5.1.1.2.cmml" xref="S4.F3.14.5.m5.1.1.2"><divide id="S4.F3.14.5.m5.1.1.2.1.cmml" xref="S4.F3.14.5.m5.1.1.2"></divide><apply id="S4.F3.14.5.m5.1.1.2.2.cmml" xref="S4.F3.14.5.m5.1.1.2.2"><minus id="S4.F3.14.5.m5.1.1.2.2.1.cmml" xref="S4.F3.14.5.m5.1.1.2.2.1"></minus><ci id="S4.F3.14.5.m5.1.1.2.2.2.cmml" xref="S4.F3.14.5.m5.1.1.2.2.2">𝑐</ci><cn id="S4.F3.14.5.m5.1.1.2.2.3.cmml" type="integer" xref="S4.F3.14.5.m5.1.1.2.2.3">1</cn></apply><apply id="S4.F3.14.5.m5.1.1.2.3.cmml" xref="S4.F3.14.5.m5.1.1.2.3"><plus id="S4.F3.14.5.m5.1.1.2.3.1.cmml" xref="S4.F3.14.5.m5.1.1.2.3.1"></plus><ci id="S4.F3.14.5.m5.1.1.2.3.2.cmml" xref="S4.F3.14.5.m5.1.1.2.3.2">𝑐</ci><cn id="S4.F3.14.5.m5.1.1.2.3.3.cmml" type="integer" xref="S4.F3.14.5.m5.1.1.2.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.14.5.m5.1d">-\frac{c-1}{c+1}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.14.5.m5.1e">- divide start_ARG italic_c - 1 end_ARG start_ARG italic_c + 1 end_ARG</annotation></semantics></math>. The assignment <math alttext="1\to 1,2\to 0" class="ltx_Math" display="inline" id="S4.F3.15.6.m6.2"><semantics id="S4.F3.15.6.m6.2b"><mrow id="S4.F3.15.6.m6.2.2.2" xref="S4.F3.15.6.m6.2.2.3.cmml"><mrow id="S4.F3.15.6.m6.1.1.1.1" xref="S4.F3.15.6.m6.1.1.1.1.cmml"><mn id="S4.F3.15.6.m6.1.1.1.1.2" xref="S4.F3.15.6.m6.1.1.1.1.2.cmml">1</mn><mo id="S4.F3.15.6.m6.1.1.1.1.1" stretchy="false" xref="S4.F3.15.6.m6.1.1.1.1.1.cmml">→</mo><mn id="S4.F3.15.6.m6.1.1.1.1.3" xref="S4.F3.15.6.m6.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.F3.15.6.m6.2.2.2.3" xref="S4.F3.15.6.m6.2.2.3a.cmml">,</mo><mrow id="S4.F3.15.6.m6.2.2.2.2" xref="S4.F3.15.6.m6.2.2.2.2.cmml"><mn id="S4.F3.15.6.m6.2.2.2.2.2" xref="S4.F3.15.6.m6.2.2.2.2.2.cmml">2</mn><mo id="S4.F3.15.6.m6.2.2.2.2.1" stretchy="false" xref="S4.F3.15.6.m6.2.2.2.2.1.cmml">→</mo><mn id="S4.F3.15.6.m6.2.2.2.2.3" xref="S4.F3.15.6.m6.2.2.2.2.3.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.15.6.m6.2c"><apply id="S4.F3.15.6.m6.2.2.3.cmml" xref="S4.F3.15.6.m6.2.2.2"><csymbol cd="ambiguous" id="S4.F3.15.6.m6.2.2.3a.cmml" xref="S4.F3.15.6.m6.2.2.2.3">formulae-sequence</csymbol><apply id="S4.F3.15.6.m6.1.1.1.1.cmml" xref="S4.F3.15.6.m6.1.1.1.1"><ci id="S4.F3.15.6.m6.1.1.1.1.1.cmml" xref="S4.F3.15.6.m6.1.1.1.1.1">→</ci><cn id="S4.F3.15.6.m6.1.1.1.1.2.cmml" type="integer" xref="S4.F3.15.6.m6.1.1.1.1.2">1</cn><cn id="S4.F3.15.6.m6.1.1.1.1.3.cmml" type="integer" xref="S4.F3.15.6.m6.1.1.1.1.3">1</cn></apply><apply id="S4.F3.15.6.m6.2.2.2.2.cmml" xref="S4.F3.15.6.m6.2.2.2.2"><ci id="S4.F3.15.6.m6.2.2.2.2.1.cmml" xref="S4.F3.15.6.m6.2.2.2.2.1">→</ci><cn id="S4.F3.15.6.m6.2.2.2.2.2.cmml" type="integer" xref="S4.F3.15.6.m6.2.2.2.2.2">2</cn><cn id="S4.F3.15.6.m6.2.2.2.2.3.cmml" type="integer" xref="S4.F3.15.6.m6.2.2.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.15.6.m6.2d">1\to 1,2\to 0</annotation><annotation encoding="application/x-llamapun" id="S4.F3.15.6.m6.2e">1 → 1 , 2 → 0</annotation></semantics></math> satisfies weight <math alttext="c" class="ltx_Math" display="inline" id="S4.F3.16.7.m7.1"><semantics id="S4.F3.16.7.m7.1b"><mi id="S4.F3.16.7.m7.1.1" xref="S4.F3.16.7.m7.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.F3.16.7.m7.1c"><ci id="S4.F3.16.7.m7.1.1.cmml" xref="S4.F3.16.7.m7.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.16.7.m7.1d">c</annotation><annotation encoding="application/x-llamapun" id="S4.F3.16.7.m7.1e">italic_c</annotation></semantics></math>. An oblivious assignment <math alttext="1\to p,2\to q" class="ltx_Math" display="inline" id="S4.F3.17.8.m8.2"><semantics id="S4.F3.17.8.m8.2b"><mrow id="S4.F3.17.8.m8.2.2.2" xref="S4.F3.17.8.m8.2.2.3.cmml"><mrow id="S4.F3.17.8.m8.1.1.1.1" xref="S4.F3.17.8.m8.1.1.1.1.cmml"><mn id="S4.F3.17.8.m8.1.1.1.1.2" xref="S4.F3.17.8.m8.1.1.1.1.2.cmml">1</mn><mo id="S4.F3.17.8.m8.1.1.1.1.1" stretchy="false" xref="S4.F3.17.8.m8.1.1.1.1.1.cmml">→</mo><mi id="S4.F3.17.8.m8.1.1.1.1.3" xref="S4.F3.17.8.m8.1.1.1.1.3.cmml">p</mi></mrow><mo id="S4.F3.17.8.m8.2.2.2.3" xref="S4.F3.17.8.m8.2.2.3a.cmml">,</mo><mrow id="S4.F3.17.8.m8.2.2.2.2" xref="S4.F3.17.8.m8.2.2.2.2.cmml"><mn id="S4.F3.17.8.m8.2.2.2.2.2" xref="S4.F3.17.8.m8.2.2.2.2.2.cmml">2</mn><mo id="S4.F3.17.8.m8.2.2.2.2.1" stretchy="false" xref="S4.F3.17.8.m8.2.2.2.2.1.cmml">→</mo><mi id="S4.F3.17.8.m8.2.2.2.2.3" xref="S4.F3.17.8.m8.2.2.2.2.3.cmml">q</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.17.8.m8.2c"><apply id="S4.F3.17.8.m8.2.2.3.cmml" xref="S4.F3.17.8.m8.2.2.2"><csymbol cd="ambiguous" id="S4.F3.17.8.m8.2.2.3a.cmml" xref="S4.F3.17.8.m8.2.2.2.3">formulae-sequence</csymbol><apply id="S4.F3.17.8.m8.1.1.1.1.cmml" xref="S4.F3.17.8.m8.1.1.1.1"><ci id="S4.F3.17.8.m8.1.1.1.1.1.cmml" xref="S4.F3.17.8.m8.1.1.1.1.1">→</ci><cn id="S4.F3.17.8.m8.1.1.1.1.2.cmml" type="integer" xref="S4.F3.17.8.m8.1.1.1.1.2">1</cn><ci id="S4.F3.17.8.m8.1.1.1.1.3.cmml" xref="S4.F3.17.8.m8.1.1.1.1.3">𝑝</ci></apply><apply id="S4.F3.17.8.m8.2.2.2.2.cmml" xref="S4.F3.17.8.m8.2.2.2.2"><ci id="S4.F3.17.8.m8.2.2.2.2.1.cmml" xref="S4.F3.17.8.m8.2.2.2.2.1">→</ci><cn id="S4.F3.17.8.m8.2.2.2.2.2.cmml" type="integer" xref="S4.F3.17.8.m8.2.2.2.2.2">2</cn><ci id="S4.F3.17.8.m8.2.2.2.2.3.cmml" xref="S4.F3.17.8.m8.2.2.2.2.3">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.17.8.m8.2d">1\to p,2\to q</annotation><annotation encoding="application/x-llamapun" id="S4.F3.17.8.m8.2e">1 → italic_p , 2 → italic_q</annotation></semantics></math> satisfies weight <math alttext="p(1-q)c+q(1-p)" class="ltx_Math" display="inline" id="S4.F3.18.9.m9.2"><semantics id="S4.F3.18.9.m9.2b"><mrow id="S4.F3.18.9.m9.2.2" xref="S4.F3.18.9.m9.2.2.cmml"><mrow id="S4.F3.18.9.m9.1.1.1" xref="S4.F3.18.9.m9.1.1.1.cmml"><mi id="S4.F3.18.9.m9.1.1.1.3" xref="S4.F3.18.9.m9.1.1.1.3.cmml">p</mi><mo id="S4.F3.18.9.m9.1.1.1.2" xref="S4.F3.18.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S4.F3.18.9.m9.1.1.1.1.1" xref="S4.F3.18.9.m9.1.1.1.1.1.1.cmml"><mo id="S4.F3.18.9.m9.1.1.1.1.1.2" stretchy="false" xref="S4.F3.18.9.m9.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.F3.18.9.m9.1.1.1.1.1.1" xref="S4.F3.18.9.m9.1.1.1.1.1.1.cmml"><mn id="S4.F3.18.9.m9.1.1.1.1.1.1.2" xref="S4.F3.18.9.m9.1.1.1.1.1.1.2.cmml">1</mn><mo id="S4.F3.18.9.m9.1.1.1.1.1.1.1" xref="S4.F3.18.9.m9.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.F3.18.9.m9.1.1.1.1.1.1.3" xref="S4.F3.18.9.m9.1.1.1.1.1.1.3.cmml">q</mi></mrow><mo id="S4.F3.18.9.m9.1.1.1.1.1.3" stretchy="false" xref="S4.F3.18.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.F3.18.9.m9.1.1.1.2b" xref="S4.F3.18.9.m9.1.1.1.2.cmml">⁢</mo><mi id="S4.F3.18.9.m9.1.1.1.4" xref="S4.F3.18.9.m9.1.1.1.4.cmml">c</mi></mrow><mo id="S4.F3.18.9.m9.2.2.3" xref="S4.F3.18.9.m9.2.2.3.cmml">+</mo><mrow id="S4.F3.18.9.m9.2.2.2" xref="S4.F3.18.9.m9.2.2.2.cmml"><mi id="S4.F3.18.9.m9.2.2.2.3" xref="S4.F3.18.9.m9.2.2.2.3.cmml">q</mi><mo id="S4.F3.18.9.m9.2.2.2.2" xref="S4.F3.18.9.m9.2.2.2.2.cmml">⁢</mo><mrow id="S4.F3.18.9.m9.2.2.2.1.1" xref="S4.F3.18.9.m9.2.2.2.1.1.1.cmml"><mo id="S4.F3.18.9.m9.2.2.2.1.1.2" stretchy="false" xref="S4.F3.18.9.m9.2.2.2.1.1.1.cmml">(</mo><mrow id="S4.F3.18.9.m9.2.2.2.1.1.1" xref="S4.F3.18.9.m9.2.2.2.1.1.1.cmml"><mn id="S4.F3.18.9.m9.2.2.2.1.1.1.2" xref="S4.F3.18.9.m9.2.2.2.1.1.1.2.cmml">1</mn><mo id="S4.F3.18.9.m9.2.2.2.1.1.1.1" xref="S4.F3.18.9.m9.2.2.2.1.1.1.1.cmml">−</mo><mi id="S4.F3.18.9.m9.2.2.2.1.1.1.3" xref="S4.F3.18.9.m9.2.2.2.1.1.1.3.cmml">p</mi></mrow><mo id="S4.F3.18.9.m9.2.2.2.1.1.3" stretchy="false" xref="S4.F3.18.9.m9.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.18.9.m9.2c"><apply id="S4.F3.18.9.m9.2.2.cmml" xref="S4.F3.18.9.m9.2.2"><plus id="S4.F3.18.9.m9.2.2.3.cmml" xref="S4.F3.18.9.m9.2.2.3"></plus><apply id="S4.F3.18.9.m9.1.1.1.cmml" xref="S4.F3.18.9.m9.1.1.1"><times id="S4.F3.18.9.m9.1.1.1.2.cmml" xref="S4.F3.18.9.m9.1.1.1.2"></times><ci id="S4.F3.18.9.m9.1.1.1.3.cmml" xref="S4.F3.18.9.m9.1.1.1.3">𝑝</ci><apply id="S4.F3.18.9.m9.1.1.1.1.1.1.cmml" xref="S4.F3.18.9.m9.1.1.1.1.1"><minus id="S4.F3.18.9.m9.1.1.1.1.1.1.1.cmml" xref="S4.F3.18.9.m9.1.1.1.1.1.1.1"></minus><cn id="S4.F3.18.9.m9.1.1.1.1.1.1.2.cmml" type="integer" xref="S4.F3.18.9.m9.1.1.1.1.1.1.2">1</cn><ci id="S4.F3.18.9.m9.1.1.1.1.1.1.3.cmml" xref="S4.F3.18.9.m9.1.1.1.1.1.1.3">𝑞</ci></apply><ci id="S4.F3.18.9.m9.1.1.1.4.cmml" xref="S4.F3.18.9.m9.1.1.1.4">𝑐</ci></apply><apply id="S4.F3.18.9.m9.2.2.2.cmml" xref="S4.F3.18.9.m9.2.2.2"><times id="S4.F3.18.9.m9.2.2.2.2.cmml" xref="S4.F3.18.9.m9.2.2.2.2"></times><ci id="S4.F3.18.9.m9.2.2.2.3.cmml" xref="S4.F3.18.9.m9.2.2.2.3">𝑞</ci><apply id="S4.F3.18.9.m9.2.2.2.1.1.1.cmml" xref="S4.F3.18.9.m9.2.2.2.1.1"><minus id="S4.F3.18.9.m9.2.2.2.1.1.1.1.cmml" xref="S4.F3.18.9.m9.2.2.2.1.1.1.1"></minus><cn id="S4.F3.18.9.m9.2.2.2.1.1.1.2.cmml" type="integer" xref="S4.F3.18.9.m9.2.2.2.1.1.1.2">1</cn><ci id="S4.F3.18.9.m9.2.2.2.1.1.1.3.cmml" xref="S4.F3.18.9.m9.2.2.2.1.1.1.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.18.9.m9.2d">p(1-q)c+q(1-p)</annotation><annotation encoding="application/x-llamapun" id="S4.F3.18.9.m9.2e">italic_p ( 1 - italic_q ) italic_c + italic_q ( 1 - italic_p )</annotation></semantics></math>. (Every two-vertex graph (without self-loops and with two nontrivial edges) is isomorphic to this graph up to rescaling.)</span></figcaption> </figure> <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.p1.13">Let <math alttext="G" 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">G</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">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.1.m1.1d">italic_G</annotation></semantics></math> denote the graph in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S4.F3" title="In 4.2 Bounds against 𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽_{1/2} (Theorem 1.6) ‣ 4 Lower bounds ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Fig.</span> <span class="ltx_text ltx_ref_tag">3</span></a> with <math alttext="1<c<3" class="ltx_Math" display="inline" id="S4.SS2.2.p1.2.m2.1"><semantics id="S4.SS2.2.p1.2.m2.1a"><mrow id="S4.SS2.2.p1.2.m2.1.1" xref="S4.SS2.2.p1.2.m2.1.1.cmml"><mn id="S4.SS2.2.p1.2.m2.1.1.2" xref="S4.SS2.2.p1.2.m2.1.1.2.cmml">1</mn><mo id="S4.SS2.2.p1.2.m2.1.1.3" xref="S4.SS2.2.p1.2.m2.1.1.3.cmml"><</mo><mi id="S4.SS2.2.p1.2.m2.1.1.4" xref="S4.SS2.2.p1.2.m2.1.1.4.cmml">c</mi><mo id="S4.SS2.2.p1.2.m2.1.1.5" xref="S4.SS2.2.p1.2.m2.1.1.5.cmml"><</mo><mn id="S4.SS2.2.p1.2.m2.1.1.6" xref="S4.SS2.2.p1.2.m2.1.1.6.cmml">3</mn></mrow><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"><and id="S4.SS2.2.p1.2.m2.1.1a.cmml" xref="S4.SS2.2.p1.2.m2.1.1"></and><apply id="S4.SS2.2.p1.2.m2.1.1b.cmml" xref="S4.SS2.2.p1.2.m2.1.1"><lt id="S4.SS2.2.p1.2.m2.1.1.3.cmml" xref="S4.SS2.2.p1.2.m2.1.1.3"></lt><cn id="S4.SS2.2.p1.2.m2.1.1.2.cmml" type="integer" xref="S4.SS2.2.p1.2.m2.1.1.2">1</cn><ci id="S4.SS2.2.p1.2.m2.1.1.4.cmml" xref="S4.SS2.2.p1.2.m2.1.1.4">𝑐</ci></apply><apply id="S4.SS2.2.p1.2.m2.1.1c.cmml" xref="S4.SS2.2.p1.2.m2.1.1"><lt id="S4.SS2.2.p1.2.m2.1.1.5.cmml" xref="S4.SS2.2.p1.2.m2.1.1.5"></lt><share href="https://arxiv.org/html/2411.12976v1#S4.SS2.2.p1.2.m2.1.1.4.cmml" id="S4.SS2.2.p1.2.m2.1.1d.cmml" xref="S4.SS2.2.p1.2.m2.1.1"></share><cn id="S4.SS2.2.p1.2.m2.1.1.6.cmml" type="integer" xref="S4.SS2.2.p1.2.m2.1.1.6">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.2.m2.1c">1<c<3</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.2.m2.1d">1 < italic_c < 3</annotation></semantics></math> TBD. Vertex <math alttext="1" class="ltx_Math" display="inline" id="S4.SS2.2.p1.3.m3.1"><semantics id="S4.SS2.2.p1.3.m3.1a"><mn id="S4.SS2.2.p1.3.m3.1.1" xref="S4.SS2.2.p1.3.m3.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.3.m3.1b"><cn id="S4.SS2.2.p1.3.m3.1.1.cmml" type="integer" xref="S4.SS2.2.p1.3.m3.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.3.m3.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.3.m3.1d">1</annotation></semantics></math> has bias <math alttext="+\frac{c-1}{c+1}" class="ltx_Math" display="inline" id="S4.SS2.2.p1.4.m4.1"><semantics id="S4.SS2.2.p1.4.m4.1a"><mrow id="S4.SS2.2.p1.4.m4.1.1" xref="S4.SS2.2.p1.4.m4.1.1.cmml"><mo id="S4.SS2.2.p1.4.m4.1.1a" xref="S4.SS2.2.p1.4.m4.1.1.cmml">+</mo><mfrac id="S4.SS2.2.p1.4.m4.1.1.2" xref="S4.SS2.2.p1.4.m4.1.1.2.cmml"><mrow id="S4.SS2.2.p1.4.m4.1.1.2.2" xref="S4.SS2.2.p1.4.m4.1.1.2.2.cmml"><mi id="S4.SS2.2.p1.4.m4.1.1.2.2.2" xref="S4.SS2.2.p1.4.m4.1.1.2.2.2.cmml">c</mi><mo id="S4.SS2.2.p1.4.m4.1.1.2.2.1" xref="S4.SS2.2.p1.4.m4.1.1.2.2.1.cmml">−</mo><mn id="S4.SS2.2.p1.4.m4.1.1.2.2.3" xref="S4.SS2.2.p1.4.m4.1.1.2.2.3.cmml">1</mn></mrow><mrow id="S4.SS2.2.p1.4.m4.1.1.2.3" xref="S4.SS2.2.p1.4.m4.1.1.2.3.cmml"><mi id="S4.SS2.2.p1.4.m4.1.1.2.3.2" xref="S4.SS2.2.p1.4.m4.1.1.2.3.2.cmml">c</mi><mo id="S4.SS2.2.p1.4.m4.1.1.2.3.1" xref="S4.SS2.2.p1.4.m4.1.1.2.3.1.cmml">+</mo><mn id="S4.SS2.2.p1.4.m4.1.1.2.3.3" xref="S4.SS2.2.p1.4.m4.1.1.2.3.3.cmml">1</mn></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.4.m4.1b"><apply id="S4.SS2.2.p1.4.m4.1.1.cmml" xref="S4.SS2.2.p1.4.m4.1.1"><plus id="S4.SS2.2.p1.4.m4.1.1.1.cmml" xref="S4.SS2.2.p1.4.m4.1.1"></plus><apply id="S4.SS2.2.p1.4.m4.1.1.2.cmml" xref="S4.SS2.2.p1.4.m4.1.1.2"><divide id="S4.SS2.2.p1.4.m4.1.1.2.1.cmml" xref="S4.SS2.2.p1.4.m4.1.1.2"></divide><apply id="S4.SS2.2.p1.4.m4.1.1.2.2.cmml" xref="S4.SS2.2.p1.4.m4.1.1.2.2"><minus id="S4.SS2.2.p1.4.m4.1.1.2.2.1.cmml" xref="S4.SS2.2.p1.4.m4.1.1.2.2.1"></minus><ci id="S4.SS2.2.p1.4.m4.1.1.2.2.2.cmml" xref="S4.SS2.2.p1.4.m4.1.1.2.2.2">𝑐</ci><cn id="S4.SS2.2.p1.4.m4.1.1.2.2.3.cmml" type="integer" xref="S4.SS2.2.p1.4.m4.1.1.2.2.3">1</cn></apply><apply id="S4.SS2.2.p1.4.m4.1.1.2.3.cmml" xref="S4.SS2.2.p1.4.m4.1.1.2.3"><plus id="S4.SS2.2.p1.4.m4.1.1.2.3.1.cmml" xref="S4.SS2.2.p1.4.m4.1.1.2.3.1"></plus><ci id="S4.SS2.2.p1.4.m4.1.1.2.3.2.cmml" xref="S4.SS2.2.p1.4.m4.1.1.2.3.2">𝑐</ci><cn id="S4.SS2.2.p1.4.m4.1.1.2.3.3.cmml" type="integer" xref="S4.SS2.2.p1.4.m4.1.1.2.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.4.m4.1c">+\frac{c-1}{c+1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.4.m4.1d">+ divide start_ARG italic_c - 1 end_ARG start_ARG italic_c + 1 end_ARG</annotation></semantics></math>. Since <math alttext="c<3" class="ltx_Math" display="inline" id="S4.SS2.2.p1.5.m5.1"><semantics id="S4.SS2.2.p1.5.m5.1a"><mrow id="S4.SS2.2.p1.5.m5.1.1" xref="S4.SS2.2.p1.5.m5.1.1.cmml"><mi id="S4.SS2.2.p1.5.m5.1.1.2" xref="S4.SS2.2.p1.5.m5.1.1.2.cmml">c</mi><mo id="S4.SS2.2.p1.5.m5.1.1.1" xref="S4.SS2.2.p1.5.m5.1.1.1.cmml"><</mo><mn id="S4.SS2.2.p1.5.m5.1.1.3" xref="S4.SS2.2.p1.5.m5.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.5.m5.1b"><apply id="S4.SS2.2.p1.5.m5.1.1.cmml" xref="S4.SS2.2.p1.5.m5.1.1"><lt id="S4.SS2.2.p1.5.m5.1.1.1.cmml" xref="S4.SS2.2.p1.5.m5.1.1.1"></lt><ci id="S4.SS2.2.p1.5.m5.1.1.2.cmml" xref="S4.SS2.2.p1.5.m5.1.1.2">𝑐</ci><cn id="S4.SS2.2.p1.5.m5.1.1.3.cmml" type="integer" xref="S4.SS2.2.p1.5.m5.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.5.m5.1c">c<3</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.5.m5.1d">italic_c < 3</annotation></semantics></math>, <math alttext="\frac{c-1}{c+1}<\frac{1}{2}" class="ltx_Math" display="inline" id="S4.SS2.2.p1.6.m6.1"><semantics id="S4.SS2.2.p1.6.m6.1a"><mrow id="S4.SS2.2.p1.6.m6.1.1" xref="S4.SS2.2.p1.6.m6.1.1.cmml"><mfrac id="S4.SS2.2.p1.6.m6.1.1.2" xref="S4.SS2.2.p1.6.m6.1.1.2.cmml"><mrow id="S4.SS2.2.p1.6.m6.1.1.2.2" xref="S4.SS2.2.p1.6.m6.1.1.2.2.cmml"><mi id="S4.SS2.2.p1.6.m6.1.1.2.2.2" xref="S4.SS2.2.p1.6.m6.1.1.2.2.2.cmml">c</mi><mo id="S4.SS2.2.p1.6.m6.1.1.2.2.1" xref="S4.SS2.2.p1.6.m6.1.1.2.2.1.cmml">−</mo><mn id="S4.SS2.2.p1.6.m6.1.1.2.2.3" xref="S4.SS2.2.p1.6.m6.1.1.2.2.3.cmml">1</mn></mrow><mrow id="S4.SS2.2.p1.6.m6.1.1.2.3" xref="S4.SS2.2.p1.6.m6.1.1.2.3.cmml"><mi id="S4.SS2.2.p1.6.m6.1.1.2.3.2" xref="S4.SS2.2.p1.6.m6.1.1.2.3.2.cmml">c</mi><mo id="S4.SS2.2.p1.6.m6.1.1.2.3.1" xref="S4.SS2.2.p1.6.m6.1.1.2.3.1.cmml">+</mo><mn id="S4.SS2.2.p1.6.m6.1.1.2.3.3" xref="S4.SS2.2.p1.6.m6.1.1.2.3.3.cmml">1</mn></mrow></mfrac><mo id="S4.SS2.2.p1.6.m6.1.1.1" xref="S4.SS2.2.p1.6.m6.1.1.1.cmml"><</mo><mfrac id="S4.SS2.2.p1.6.m6.1.1.3" xref="S4.SS2.2.p1.6.m6.1.1.3.cmml"><mn id="S4.SS2.2.p1.6.m6.1.1.3.2" xref="S4.SS2.2.p1.6.m6.1.1.3.2.cmml">1</mn><mn id="S4.SS2.2.p1.6.m6.1.1.3.3" xref="S4.SS2.2.p1.6.m6.1.1.3.3.cmml">2</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.6.m6.1b"><apply id="S4.SS2.2.p1.6.m6.1.1.cmml" xref="S4.SS2.2.p1.6.m6.1.1"><lt id="S4.SS2.2.p1.6.m6.1.1.1.cmml" xref="S4.SS2.2.p1.6.m6.1.1.1"></lt><apply id="S4.SS2.2.p1.6.m6.1.1.2.cmml" xref="S4.SS2.2.p1.6.m6.1.1.2"><divide id="S4.SS2.2.p1.6.m6.1.1.2.1.cmml" xref="S4.SS2.2.p1.6.m6.1.1.2"></divide><apply id="S4.SS2.2.p1.6.m6.1.1.2.2.cmml" xref="S4.SS2.2.p1.6.m6.1.1.2.2"><minus id="S4.SS2.2.p1.6.m6.1.1.2.2.1.cmml" xref="S4.SS2.2.p1.6.m6.1.1.2.2.1"></minus><ci id="S4.SS2.2.p1.6.m6.1.1.2.2.2.cmml" xref="S4.SS2.2.p1.6.m6.1.1.2.2.2">𝑐</ci><cn id="S4.SS2.2.p1.6.m6.1.1.2.2.3.cmml" type="integer" xref="S4.SS2.2.p1.6.m6.1.1.2.2.3">1</cn></apply><apply id="S4.SS2.2.p1.6.m6.1.1.2.3.cmml" xref="S4.SS2.2.p1.6.m6.1.1.2.3"><plus id="S4.SS2.2.p1.6.m6.1.1.2.3.1.cmml" xref="S4.SS2.2.p1.6.m6.1.1.2.3.1"></plus><ci id="S4.SS2.2.p1.6.m6.1.1.2.3.2.cmml" xref="S4.SS2.2.p1.6.m6.1.1.2.3.2">𝑐</ci><cn id="S4.SS2.2.p1.6.m6.1.1.2.3.3.cmml" type="integer" xref="S4.SS2.2.p1.6.m6.1.1.2.3.3">1</cn></apply></apply><apply id="S4.SS2.2.p1.6.m6.1.1.3.cmml" xref="S4.SS2.2.p1.6.m6.1.1.3"><divide id="S4.SS2.2.p1.6.m6.1.1.3.1.cmml" xref="S4.SS2.2.p1.6.m6.1.1.3"></divide><cn id="S4.SS2.2.p1.6.m6.1.1.3.2.cmml" type="integer" xref="S4.SS2.2.p1.6.m6.1.1.3.2">1</cn><cn id="S4.SS2.2.p1.6.m6.1.1.3.3.cmml" type="integer" xref="S4.SS2.2.p1.6.m6.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.6.m6.1c">\frac{c-1}{c+1}<\frac{1}{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.6.m6.1d">divide start_ARG italic_c - 1 end_ARG start_ARG italic_c + 1 end_ARG < divide start_ARG 1 end_ARG start_ARG 2 end_ARG</annotation></semantics></math>. Thus, <math alttext="\mathsf{PLSigmoid}_{1/2}" class="ltx_Math" display="inline" id="S4.SS2.2.p1.7.m7.1"><semantics id="S4.SS2.2.p1.7.m7.1a"><msub id="S4.SS2.2.p1.7.m7.1.1" xref="S4.SS2.2.p1.7.m7.1.1.cmml"><mi id="S4.SS2.2.p1.7.m7.1.1.2" xref="S4.SS2.2.p1.7.m7.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S4.SS2.2.p1.7.m7.1.1.3" xref="S4.SS2.2.p1.7.m7.1.1.3.cmml"><mn id="S4.SS2.2.p1.7.m7.1.1.3.2" xref="S4.SS2.2.p1.7.m7.1.1.3.2.cmml">1</mn><mo id="S4.SS2.2.p1.7.m7.1.1.3.1" xref="S4.SS2.2.p1.7.m7.1.1.3.1.cmml">/</mo><mn id="S4.SS2.2.p1.7.m7.1.1.3.3" xref="S4.SS2.2.p1.7.m7.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.7.m7.1b"><apply id="S4.SS2.2.p1.7.m7.1.1.cmml" xref="S4.SS2.2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p1.7.m7.1.1.1.cmml" xref="S4.SS2.2.p1.7.m7.1.1">subscript</csymbol><ci id="S4.SS2.2.p1.7.m7.1.1.2.cmml" xref="S4.SS2.2.p1.7.m7.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S4.SS2.2.p1.7.m7.1.1.3.cmml" xref="S4.SS2.2.p1.7.m7.1.1.3"><divide id="S4.SS2.2.p1.7.m7.1.1.3.1.cmml" xref="S4.SS2.2.p1.7.m7.1.1.3.1"></divide><cn id="S4.SS2.2.p1.7.m7.1.1.3.2.cmml" type="integer" xref="S4.SS2.2.p1.7.m7.1.1.3.2">1</cn><cn id="S4.SS2.2.p1.7.m7.1.1.3.3.cmml" type="integer" xref="S4.SS2.2.p1.7.m7.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.7.m7.1c">\mathsf{PLSigmoid}_{1/2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.7.m7.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math> assigns vertex <math alttext="1" class="ltx_Math" display="inline" id="S4.SS2.2.p1.8.m8.1"><semantics id="S4.SS2.2.p1.8.m8.1a"><mn id="S4.SS2.2.p1.8.m8.1.1" xref="S4.SS2.2.p1.8.m8.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.8.m8.1b"><cn id="S4.SS2.2.p1.8.m8.1.1.cmml" type="integer" xref="S4.SS2.2.p1.8.m8.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.8.m8.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.8.m8.1d">1</annotation></semantics></math> to <math alttext="1" class="ltx_Math" display="inline" id="S4.SS2.2.p1.9.m9.1"><semantics id="S4.SS2.2.p1.9.m9.1a"><mn id="S4.SS2.2.p1.9.m9.1.1" xref="S4.SS2.2.p1.9.m9.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.9.m9.1b"><cn id="S4.SS2.2.p1.9.m9.1.1.cmml" type="integer" xref="S4.SS2.2.p1.9.m9.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.9.m9.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.9.m9.1d">1</annotation></semantics></math> w.p. <math alttext="p:=\frac{1}{2}+\frac{c-1}{c+1}" class="ltx_Math" display="inline" id="S4.SS2.2.p1.10.m10.1"><semantics id="S4.SS2.2.p1.10.m10.1a"><mrow id="S4.SS2.2.p1.10.m10.1.1" xref="S4.SS2.2.p1.10.m10.1.1.cmml"><mi id="S4.SS2.2.p1.10.m10.1.1.2" xref="S4.SS2.2.p1.10.m10.1.1.2.cmml">p</mi><mo id="S4.SS2.2.p1.10.m10.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS2.2.p1.10.m10.1.1.1.cmml">:=</mo><mrow id="S4.SS2.2.p1.10.m10.1.1.3" xref="S4.SS2.2.p1.10.m10.1.1.3.cmml"><mfrac id="S4.SS2.2.p1.10.m10.1.1.3.2" xref="S4.SS2.2.p1.10.m10.1.1.3.2.cmml"><mn id="S4.SS2.2.p1.10.m10.1.1.3.2.2" xref="S4.SS2.2.p1.10.m10.1.1.3.2.2.cmml">1</mn><mn id="S4.SS2.2.p1.10.m10.1.1.3.2.3" xref="S4.SS2.2.p1.10.m10.1.1.3.2.3.cmml">2</mn></mfrac><mo id="S4.SS2.2.p1.10.m10.1.1.3.1" xref="S4.SS2.2.p1.10.m10.1.1.3.1.cmml">+</mo><mfrac id="S4.SS2.2.p1.10.m10.1.1.3.3" xref="S4.SS2.2.p1.10.m10.1.1.3.3.cmml"><mrow id="S4.SS2.2.p1.10.m10.1.1.3.3.2" xref="S4.SS2.2.p1.10.m10.1.1.3.3.2.cmml"><mi id="S4.SS2.2.p1.10.m10.1.1.3.3.2.2" xref="S4.SS2.2.p1.10.m10.1.1.3.3.2.2.cmml">c</mi><mo id="S4.SS2.2.p1.10.m10.1.1.3.3.2.1" xref="S4.SS2.2.p1.10.m10.1.1.3.3.2.1.cmml">−</mo><mn id="S4.SS2.2.p1.10.m10.1.1.3.3.2.3" xref="S4.SS2.2.p1.10.m10.1.1.3.3.2.3.cmml">1</mn></mrow><mrow id="S4.SS2.2.p1.10.m10.1.1.3.3.3" xref="S4.SS2.2.p1.10.m10.1.1.3.3.3.cmml"><mi id="S4.SS2.2.p1.10.m10.1.1.3.3.3.2" xref="S4.SS2.2.p1.10.m10.1.1.3.3.3.2.cmml">c</mi><mo id="S4.SS2.2.p1.10.m10.1.1.3.3.3.1" xref="S4.SS2.2.p1.10.m10.1.1.3.3.3.1.cmml">+</mo><mn id="S4.SS2.2.p1.10.m10.1.1.3.3.3.3" xref="S4.SS2.2.p1.10.m10.1.1.3.3.3.3.cmml">1</mn></mrow></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.10.m10.1b"><apply id="S4.SS2.2.p1.10.m10.1.1.cmml" xref="S4.SS2.2.p1.10.m10.1.1"><csymbol cd="latexml" id="S4.SS2.2.p1.10.m10.1.1.1.cmml" xref="S4.SS2.2.p1.10.m10.1.1.1">assign</csymbol><ci id="S4.SS2.2.p1.10.m10.1.1.2.cmml" xref="S4.SS2.2.p1.10.m10.1.1.2">𝑝</ci><apply id="S4.SS2.2.p1.10.m10.1.1.3.cmml" xref="S4.SS2.2.p1.10.m10.1.1.3"><plus id="S4.SS2.2.p1.10.m10.1.1.3.1.cmml" xref="S4.SS2.2.p1.10.m10.1.1.3.1"></plus><apply id="S4.SS2.2.p1.10.m10.1.1.3.2.cmml" xref="S4.SS2.2.p1.10.m10.1.1.3.2"><divide id="S4.SS2.2.p1.10.m10.1.1.3.2.1.cmml" xref="S4.SS2.2.p1.10.m10.1.1.3.2"></divide><cn id="S4.SS2.2.p1.10.m10.1.1.3.2.2.cmml" type="integer" xref="S4.SS2.2.p1.10.m10.1.1.3.2.2">1</cn><cn id="S4.SS2.2.p1.10.m10.1.1.3.2.3.cmml" type="integer" xref="S4.SS2.2.p1.10.m10.1.1.3.2.3">2</cn></apply><apply id="S4.SS2.2.p1.10.m10.1.1.3.3.cmml" xref="S4.SS2.2.p1.10.m10.1.1.3.3"><divide id="S4.SS2.2.p1.10.m10.1.1.3.3.1.cmml" xref="S4.SS2.2.p1.10.m10.1.1.3.3"></divide><apply id="S4.SS2.2.p1.10.m10.1.1.3.3.2.cmml" xref="S4.SS2.2.p1.10.m10.1.1.3.3.2"><minus id="S4.SS2.2.p1.10.m10.1.1.3.3.2.1.cmml" xref="S4.SS2.2.p1.10.m10.1.1.3.3.2.1"></minus><ci id="S4.SS2.2.p1.10.m10.1.1.3.3.2.2.cmml" xref="S4.SS2.2.p1.10.m10.1.1.3.3.2.2">𝑐</ci><cn id="S4.SS2.2.p1.10.m10.1.1.3.3.2.3.cmml" type="integer" xref="S4.SS2.2.p1.10.m10.1.1.3.3.2.3">1</cn></apply><apply id="S4.SS2.2.p1.10.m10.1.1.3.3.3.cmml" xref="S4.SS2.2.p1.10.m10.1.1.3.3.3"><plus id="S4.SS2.2.p1.10.m10.1.1.3.3.3.1.cmml" xref="S4.SS2.2.p1.10.m10.1.1.3.3.3.1"></plus><ci id="S4.SS2.2.p1.10.m10.1.1.3.3.3.2.cmml" xref="S4.SS2.2.p1.10.m10.1.1.3.3.3.2">𝑐</ci><cn id="S4.SS2.2.p1.10.m10.1.1.3.3.3.3.cmml" type="integer" xref="S4.SS2.2.p1.10.m10.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.10.m10.1c">p:=\frac{1}{2}+\frac{c-1}{c+1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.10.m10.1d">italic_p := divide start_ARG 1 end_ARG start_ARG 2 end_ARG + divide start_ARG italic_c - 1 end_ARG start_ARG italic_c + 1 end_ARG</annotation></semantics></math>, and <math alttext="2" class="ltx_Math" display="inline" id="S4.SS2.2.p1.11.m11.1"><semantics id="S4.SS2.2.p1.11.m11.1a"><mn id="S4.SS2.2.p1.11.m11.1.1" xref="S4.SS2.2.p1.11.m11.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.11.m11.1b"><cn id="S4.SS2.2.p1.11.m11.1.1.cmml" type="integer" xref="S4.SS2.2.p1.11.m11.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.11.m11.1c">2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.11.m11.1d">2</annotation></semantics></math> to <math alttext="1" class="ltx_Math" display="inline" id="S4.SS2.2.p1.12.m12.1"><semantics id="S4.SS2.2.p1.12.m12.1a"><mn id="S4.SS2.2.p1.12.m12.1.1" xref="S4.SS2.2.p1.12.m12.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.12.m12.1b"><cn id="S4.SS2.2.p1.12.m12.1.1.cmml" type="integer" xref="S4.SS2.2.p1.12.m12.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.12.m12.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.12.m12.1d">1</annotation></semantics></math> w.p. <math alttext="1-p" class="ltx_Math" display="inline" id="S4.SS2.2.p1.13.m13.1"><semantics id="S4.SS2.2.p1.13.m13.1a"><mrow id="S4.SS2.2.p1.13.m13.1.1" xref="S4.SS2.2.p1.13.m13.1.1.cmml"><mn id="S4.SS2.2.p1.13.m13.1.1.2" xref="S4.SS2.2.p1.13.m13.1.1.2.cmml">1</mn><mo id="S4.SS2.2.p1.13.m13.1.1.1" xref="S4.SS2.2.p1.13.m13.1.1.1.cmml">−</mo><mi id="S4.SS2.2.p1.13.m13.1.1.3" xref="S4.SS2.2.p1.13.m13.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.13.m13.1b"><apply id="S4.SS2.2.p1.13.m13.1.1.cmml" xref="S4.SS2.2.p1.13.m13.1.1"><minus id="S4.SS2.2.p1.13.m13.1.1.1.cmml" xref="S4.SS2.2.p1.13.m13.1.1.1"></minus><cn id="S4.SS2.2.p1.13.m13.1.1.2.cmml" type="integer" xref="S4.SS2.2.p1.13.m13.1.1.2">1</cn><ci id="S4.SS2.2.p1.13.m13.1.1.3.cmml" xref="S4.SS2.2.p1.13.m13.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.13.m13.1c">1-p</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.13.m13.1d">1 - italic_p</annotation></semantics></math>, satisfying weight</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="p^{2}c+q^{2}=\left(\frac{1}{2}+\frac{c-1}{c+1}\right)^{2}c+\left(\frac{1}{2}-% \frac{c-1}{c+1}\right)^{2}." class="ltx_Math" display="block" id="S4.Ex6.m1.1"><semantics id="S4.Ex6.m1.1a"><mrow id="S4.Ex6.m1.1.1.1" xref="S4.Ex6.m1.1.1.1.1.cmml"><mrow id="S4.Ex6.m1.1.1.1.1" xref="S4.Ex6.m1.1.1.1.1.cmml"><mrow id="S4.Ex6.m1.1.1.1.1.4" xref="S4.Ex6.m1.1.1.1.1.4.cmml"><mrow id="S4.Ex6.m1.1.1.1.1.4.2" xref="S4.Ex6.m1.1.1.1.1.4.2.cmml"><msup id="S4.Ex6.m1.1.1.1.1.4.2.2" xref="S4.Ex6.m1.1.1.1.1.4.2.2.cmml"><mi id="S4.Ex6.m1.1.1.1.1.4.2.2.2" xref="S4.Ex6.m1.1.1.1.1.4.2.2.2.cmml">p</mi><mn id="S4.Ex6.m1.1.1.1.1.4.2.2.3" xref="S4.Ex6.m1.1.1.1.1.4.2.2.3.cmml">2</mn></msup><mo id="S4.Ex6.m1.1.1.1.1.4.2.1" xref="S4.Ex6.m1.1.1.1.1.4.2.1.cmml">⁢</mo><mi id="S4.Ex6.m1.1.1.1.1.4.2.3" xref="S4.Ex6.m1.1.1.1.1.4.2.3.cmml">c</mi></mrow><mo id="S4.Ex6.m1.1.1.1.1.4.1" xref="S4.Ex6.m1.1.1.1.1.4.1.cmml">+</mo><msup id="S4.Ex6.m1.1.1.1.1.4.3" xref="S4.Ex6.m1.1.1.1.1.4.3.cmml"><mi id="S4.Ex6.m1.1.1.1.1.4.3.2" xref="S4.Ex6.m1.1.1.1.1.4.3.2.cmml">q</mi><mn id="S4.Ex6.m1.1.1.1.1.4.3.3" xref="S4.Ex6.m1.1.1.1.1.4.3.3.cmml">2</mn></msup></mrow><mo id="S4.Ex6.m1.1.1.1.1.3" xref="S4.Ex6.m1.1.1.1.1.3.cmml">=</mo><mrow id="S4.Ex6.m1.1.1.1.1.2" xref="S4.Ex6.m1.1.1.1.1.2.cmml"><mrow id="S4.Ex6.m1.1.1.1.1.1.1" xref="S4.Ex6.m1.1.1.1.1.1.1.cmml"><msup id="S4.Ex6.m1.1.1.1.1.1.1.1" xref="S4.Ex6.m1.1.1.1.1.1.1.1.cmml"><mrow id="S4.Ex6.m1.1.1.1.1.1.1.1.1.1" xref="S4.Ex6.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex6.m1.1.1.1.1.1.1.1.1.1.2" xref="S4.Ex6.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex6.m1.1.1.1.1.1.1.1.1.1.1" xref="S4.Ex6.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mfrac id="S4.Ex6.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S4.Ex6.m1.1.1.1.1.1.1.1.1.1.1.2.cmml"><mn id="S4.Ex6.m1.1.1.1.1.1.1.1.1.1.1.2.2" xref="S4.Ex6.m1.1.1.1.1.1.1.1.1.1.1.2.2.cmml">1</mn><mn id="S4.Ex6.m1.1.1.1.1.1.1.1.1.1.1.2.3" xref="S4.Ex6.m1.1.1.1.1.1.1.1.1.1.1.2.3.cmml">2</mn></mfrac><mo 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encoding="application/x-llamapun" id="S4.Ex6.m1.1d">italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_c + italic_q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG + divide start_ARG italic_c - 1 end_ARG start_ARG italic_c + 1 end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_c + ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG - divide start_ARG italic_c - 1 end_ARG start_ARG italic_c + 1 end_ARG ) start_POSTSUPERSCRIPT 2 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.p1.14">Thus, <math alttext="\mathsf{PLSigmoid}_{1/2}" class="ltx_Math" display="inline" id="S4.SS2.2.p1.14.m1.1"><semantics id="S4.SS2.2.p1.14.m1.1a"><msub id="S4.SS2.2.p1.14.m1.1.1" xref="S4.SS2.2.p1.14.m1.1.1.cmml"><mi id="S4.SS2.2.p1.14.m1.1.1.2" xref="S4.SS2.2.p1.14.m1.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S4.SS2.2.p1.14.m1.1.1.3" xref="S4.SS2.2.p1.14.m1.1.1.3.cmml"><mn id="S4.SS2.2.p1.14.m1.1.1.3.2" xref="S4.SS2.2.p1.14.m1.1.1.3.2.cmml">1</mn><mo id="S4.SS2.2.p1.14.m1.1.1.3.1" xref="S4.SS2.2.p1.14.m1.1.1.3.1.cmml">/</mo><mn id="S4.SS2.2.p1.14.m1.1.1.3.3" xref="S4.SS2.2.p1.14.m1.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.14.m1.1b"><apply id="S4.SS2.2.p1.14.m1.1.1.cmml" xref="S4.SS2.2.p1.14.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p1.14.m1.1.1.1.cmml" xref="S4.SS2.2.p1.14.m1.1.1">subscript</csymbol><ci id="S4.SS2.2.p1.14.m1.1.1.2.cmml" xref="S4.SS2.2.p1.14.m1.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S4.SS2.2.p1.14.m1.1.1.3.cmml" xref="S4.SS2.2.p1.14.m1.1.1.3"><divide id="S4.SS2.2.p1.14.m1.1.1.3.1.cmml" xref="S4.SS2.2.p1.14.m1.1.1.3.1"></divide><cn id="S4.SS2.2.p1.14.m1.1.1.3.2.cmml" type="integer" xref="S4.SS2.2.p1.14.m1.1.1.3.2">1</cn><cn id="S4.SS2.2.p1.14.m1.1.1.3.3.cmml" type="integer" xref="S4.SS2.2.p1.14.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.14.m1.1c">\mathsf{PLSigmoid}_{1/2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.14.m1.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math> achieves ratio at most</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex7"> <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="p^{2}+q^{2}c^{-1}=\left(\frac{1}{2}+\frac{c-1}{c+1}\right)^{2}+\left(\frac{1}{% 2}-\frac{c-1}{c+1}\right)^{2}c^{-1}" class="ltx_Math" display="block" id="S4.Ex7.m1.2"><semantics id="S4.Ex7.m1.2a"><mrow id="S4.Ex7.m1.2.2" xref="S4.Ex7.m1.2.2.cmml"><mrow id="S4.Ex7.m1.2.2.4" xref="S4.Ex7.m1.2.2.4.cmml"><msup id="S4.Ex7.m1.2.2.4.2" xref="S4.Ex7.m1.2.2.4.2.cmml"><mi id="S4.Ex7.m1.2.2.4.2.2" 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type="integer" xref="S4.Ex7.m1.2.2.2.2.1.1.1.1.2.3">2</cn></apply><apply id="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.cmml" xref="S4.Ex7.m1.2.2.2.2.1.1.1.1.3"><divide id="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.1.cmml" xref="S4.Ex7.m1.2.2.2.2.1.1.1.1.3"></divide><apply id="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.2.cmml" xref="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.2"><minus id="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.2.1.cmml" xref="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.2.1"></minus><ci id="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.2.2.cmml" xref="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.2.2">𝑐</ci><cn id="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.2.3.cmml" type="integer" xref="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.2.3">1</cn></apply><apply id="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.3.cmml" xref="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.3"><plus id="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.3.1.cmml" xref="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.3.1"></plus><ci id="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.3.2.cmml" xref="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.3.2">𝑐</ci><cn id="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.3.3.cmml" type="integer" xref="S4.Ex7.m1.2.2.2.2.1.1.1.1.3.3.3">1</cn></apply></apply></apply><cn id="S4.Ex7.m1.2.2.2.2.1.3.cmml" type="integer" xref="S4.Ex7.m1.2.2.2.2.1.3">2</cn></apply><apply id="S4.Ex7.m1.2.2.2.2.3.cmml" xref="S4.Ex7.m1.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.Ex7.m1.2.2.2.2.3.1.cmml" xref="S4.Ex7.m1.2.2.2.2.3">superscript</csymbol><ci id="S4.Ex7.m1.2.2.2.2.3.2.cmml" xref="S4.Ex7.m1.2.2.2.2.3.2">𝑐</ci><apply id="S4.Ex7.m1.2.2.2.2.3.3.cmml" xref="S4.Ex7.m1.2.2.2.2.3.3"><minus id="S4.Ex7.m1.2.2.2.2.3.3.1.cmml" xref="S4.Ex7.m1.2.2.2.2.3.3"></minus><cn id="S4.Ex7.m1.2.2.2.2.3.3.2.cmml" type="integer" xref="S4.Ex7.m1.2.2.2.2.3.3.2">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex7.m1.2c">p^{2}+q^{2}c^{-1}=\left(\frac{1}{2}+\frac{c-1}{c+1}\right)^{2}+\left(\frac{1}{% 2}-\frac{c-1}{c+1}\right)^{2}c^{-1}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex7.m1.2d">italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_c start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG + divide start_ARG italic_c - 1 end_ARG start_ARG italic_c + 1 end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG - divide start_ARG italic_c - 1 end_ARG start_ARG italic_c + 1 end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_c start_POSTSUPERSCRIPT - 1 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.p1.17">for all <math alttext="1<c<3" class="ltx_Math" display="inline" id="S4.SS2.2.p1.15.m1.1"><semantics id="S4.SS2.2.p1.15.m1.1a"><mrow id="S4.SS2.2.p1.15.m1.1.1" xref="S4.SS2.2.p1.15.m1.1.1.cmml"><mn id="S4.SS2.2.p1.15.m1.1.1.2" xref="S4.SS2.2.p1.15.m1.1.1.2.cmml">1</mn><mo id="S4.SS2.2.p1.15.m1.1.1.3" xref="S4.SS2.2.p1.15.m1.1.1.3.cmml"><</mo><mi id="S4.SS2.2.p1.15.m1.1.1.4" xref="S4.SS2.2.p1.15.m1.1.1.4.cmml">c</mi><mo id="S4.SS2.2.p1.15.m1.1.1.5" xref="S4.SS2.2.p1.15.m1.1.1.5.cmml"><</mo><mn id="S4.SS2.2.p1.15.m1.1.1.6" xref="S4.SS2.2.p1.15.m1.1.1.6.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.15.m1.1b"><apply id="S4.SS2.2.p1.15.m1.1.1.cmml" xref="S4.SS2.2.p1.15.m1.1.1"><and id="S4.SS2.2.p1.15.m1.1.1a.cmml" xref="S4.SS2.2.p1.15.m1.1.1"></and><apply id="S4.SS2.2.p1.15.m1.1.1b.cmml" xref="S4.SS2.2.p1.15.m1.1.1"><lt id="S4.SS2.2.p1.15.m1.1.1.3.cmml" xref="S4.SS2.2.p1.15.m1.1.1.3"></lt><cn id="S4.SS2.2.p1.15.m1.1.1.2.cmml" type="integer" xref="S4.SS2.2.p1.15.m1.1.1.2">1</cn><ci id="S4.SS2.2.p1.15.m1.1.1.4.cmml" xref="S4.SS2.2.p1.15.m1.1.1.4">𝑐</ci></apply><apply id="S4.SS2.2.p1.15.m1.1.1c.cmml" xref="S4.SS2.2.p1.15.m1.1.1"><lt id="S4.SS2.2.p1.15.m1.1.1.5.cmml" xref="S4.SS2.2.p1.15.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.12976v1#S4.SS2.2.p1.15.m1.1.1.4.cmml" id="S4.SS2.2.p1.15.m1.1.1d.cmml" xref="S4.SS2.2.p1.15.m1.1.1"></share><cn id="S4.SS2.2.p1.15.m1.1.1.6.cmml" type="integer" xref="S4.SS2.2.p1.15.m1.1.1.6">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.15.m1.1c">1<c<3</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.15.m1.1d">1 < italic_c < 3</annotation></semantics></math>. Setting<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>This value of <math alttext="c" 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">c</mi><annotation-xml encoding="MathML-Content" id="footnote5.m1.1c"><ci id="footnote5.m1.1.1.cmml" xref="footnote5.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote5.m1.1d">c</annotation><annotation encoding="application/x-llamapun" id="footnote5.m1.1e">italic_c</annotation></semantics></math> gives the minimum possible bound (though this fact is not necessary).</span></span></span> <math alttext="c=(9+12\sqrt{2})/23" class="ltx_Math" display="inline" id="S4.SS2.2.p1.16.m2.1"><semantics id="S4.SS2.2.p1.16.m2.1a"><mrow id="S4.SS2.2.p1.16.m2.1.1" xref="S4.SS2.2.p1.16.m2.1.1.cmml"><mi id="S4.SS2.2.p1.16.m2.1.1.3" xref="S4.SS2.2.p1.16.m2.1.1.3.cmml">c</mi><mo id="S4.SS2.2.p1.16.m2.1.1.2" xref="S4.SS2.2.p1.16.m2.1.1.2.cmml">=</mo><mrow id="S4.SS2.2.p1.16.m2.1.1.1" xref="S4.SS2.2.p1.16.m2.1.1.1.cmml"><mrow id="S4.SS2.2.p1.16.m2.1.1.1.1.1" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.cmml"><mo id="S4.SS2.2.p1.16.m2.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.cmml"><mn id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.2" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.2.cmml">9</mn><mo id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.1" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.cmml"><mn id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.2" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.2.cmml">12</mn><mo id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.1" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.1.cmml">⁢</mo><msqrt id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.3" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.3.cmml"><mn id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.3.2" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.3.2.cmml">2</mn></msqrt></mrow></mrow><mo id="S4.SS2.2.p1.16.m2.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.2.p1.16.m2.1.1.1.2" xref="S4.SS2.2.p1.16.m2.1.1.1.2.cmml">/</mo><mn id="S4.SS2.2.p1.16.m2.1.1.1.3" xref="S4.SS2.2.p1.16.m2.1.1.1.3.cmml">23</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.16.m2.1b"><apply id="S4.SS2.2.p1.16.m2.1.1.cmml" xref="S4.SS2.2.p1.16.m2.1.1"><eq id="S4.SS2.2.p1.16.m2.1.1.2.cmml" xref="S4.SS2.2.p1.16.m2.1.1.2"></eq><ci id="S4.SS2.2.p1.16.m2.1.1.3.cmml" xref="S4.SS2.2.p1.16.m2.1.1.3">𝑐</ci><apply id="S4.SS2.2.p1.16.m2.1.1.1.cmml" xref="S4.SS2.2.p1.16.m2.1.1.1"><divide id="S4.SS2.2.p1.16.m2.1.1.1.2.cmml" xref="S4.SS2.2.p1.16.m2.1.1.1.2"></divide><apply id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.cmml" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1"><plus id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.1.cmml" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.1"></plus><cn id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.2">9</cn><apply id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.cmml" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3"><times id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.1.cmml" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.1"></times><cn id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.2">12</cn><apply id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.3.cmml" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.3"><root id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.3a.cmml" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.3"></root><cn id="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S4.SS2.2.p1.16.m2.1.1.1.1.1.1.3.3.2">2</cn></apply></apply></apply><cn id="S4.SS2.2.p1.16.m2.1.1.1.3.cmml" type="integer" xref="S4.SS2.2.p1.16.m2.1.1.1.3">23</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.16.m2.1c">c=(9+12\sqrt{2})/23</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.16.m2.1d">italic_c = ( 9 + 12 square-root start_ARG 2 end_ARG ) / 23</annotation></semantics></math>, the expression is equal to <math alttext="6\sqrt{2}-8\approx 0.485282" class="ltx_Math" display="inline" id="S4.SS2.2.p1.17.m3.1"><semantics id="S4.SS2.2.p1.17.m3.1a"><mrow id="S4.SS2.2.p1.17.m3.1.1" xref="S4.SS2.2.p1.17.m3.1.1.cmml"><mrow id="S4.SS2.2.p1.17.m3.1.1.2" xref="S4.SS2.2.p1.17.m3.1.1.2.cmml"><mrow id="S4.SS2.2.p1.17.m3.1.1.2.2" xref="S4.SS2.2.p1.17.m3.1.1.2.2.cmml"><mn id="S4.SS2.2.p1.17.m3.1.1.2.2.2" xref="S4.SS2.2.p1.17.m3.1.1.2.2.2.cmml">6</mn><mo id="S4.SS2.2.p1.17.m3.1.1.2.2.1" xref="S4.SS2.2.p1.17.m3.1.1.2.2.1.cmml">⁢</mo><msqrt id="S4.SS2.2.p1.17.m3.1.1.2.2.3" xref="S4.SS2.2.p1.17.m3.1.1.2.2.3.cmml"><mn id="S4.SS2.2.p1.17.m3.1.1.2.2.3.2" xref="S4.SS2.2.p1.17.m3.1.1.2.2.3.2.cmml">2</mn></msqrt></mrow><mo id="S4.SS2.2.p1.17.m3.1.1.2.1" xref="S4.SS2.2.p1.17.m3.1.1.2.1.cmml">−</mo><mn id="S4.SS2.2.p1.17.m3.1.1.2.3" xref="S4.SS2.2.p1.17.m3.1.1.2.3.cmml">8</mn></mrow><mo id="S4.SS2.2.p1.17.m3.1.1.1" xref="S4.SS2.2.p1.17.m3.1.1.1.cmml">≈</mo><mn id="S4.SS2.2.p1.17.m3.1.1.3" xref="S4.SS2.2.p1.17.m3.1.1.3.cmml">0.485282</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p1.17.m3.1b"><apply id="S4.SS2.2.p1.17.m3.1.1.cmml" xref="S4.SS2.2.p1.17.m3.1.1"><approx id="S4.SS2.2.p1.17.m3.1.1.1.cmml" xref="S4.SS2.2.p1.17.m3.1.1.1"></approx><apply id="S4.SS2.2.p1.17.m3.1.1.2.cmml" xref="S4.SS2.2.p1.17.m3.1.1.2"><minus id="S4.SS2.2.p1.17.m3.1.1.2.1.cmml" xref="S4.SS2.2.p1.17.m3.1.1.2.1"></minus><apply id="S4.SS2.2.p1.17.m3.1.1.2.2.cmml" xref="S4.SS2.2.p1.17.m3.1.1.2.2"><times id="S4.SS2.2.p1.17.m3.1.1.2.2.1.cmml" xref="S4.SS2.2.p1.17.m3.1.1.2.2.1"></times><cn id="S4.SS2.2.p1.17.m3.1.1.2.2.2.cmml" type="integer" xref="S4.SS2.2.p1.17.m3.1.1.2.2.2">6</cn><apply id="S4.SS2.2.p1.17.m3.1.1.2.2.3.cmml" xref="S4.SS2.2.p1.17.m3.1.1.2.2.3"><root id="S4.SS2.2.p1.17.m3.1.1.2.2.3a.cmml" xref="S4.SS2.2.p1.17.m3.1.1.2.2.3"></root><cn id="S4.SS2.2.p1.17.m3.1.1.2.2.3.2.cmml" type="integer" xref="S4.SS2.2.p1.17.m3.1.1.2.2.3.2">2</cn></apply></apply><cn id="S4.SS2.2.p1.17.m3.1.1.2.3.cmml" type="integer" xref="S4.SS2.2.p1.17.m3.1.1.2.3">8</cn></apply><cn id="S4.SS2.2.p1.17.m3.1.1.3.cmml" type="float" xref="S4.SS2.2.p1.17.m3.1.1.3">0.485282</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p1.17.m3.1c">6\sqrt{2}-8\approx 0.485282</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p1.17.m3.1d">6 square-root start_ARG 2 end_ARG - 8 ≈ 0.485282</annotation></semantics></math>, as desired. ∎</p> </div> </div> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.3 </span>Lower bound for PL sigmoid functions (<a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem7" title="Theorem 1.7 (Lower bound for PL sigmoid selection with arbitrary intercept). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.7</span></a>)</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.1">See <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem7" title="Theorem 1.7 (Lower bound for PL sigmoid selection with arbitrary intercept). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">1.7</span></a></p> </div> <div class="ltx_proof" id="S4.SS3.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem7" title="Theorem 1.7 (Lower bound for PL sigmoid selection with arbitrary intercept). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.7</span></a>.</h6> <div class="ltx_para" id="S4.SS3.1.p1"> <p class="ltx_p" id="S4.SS3.1.p1.2">We prove this theorem by considering three separate cases based on the value of the intercept <math alttext="b" class="ltx_Math" display="inline" id="S4.SS3.1.p1.1.m1.1"><semantics id="S4.SS3.1.p1.1.m1.1a"><mi id="S4.SS3.1.p1.1.m1.1.1" xref="S4.SS3.1.p1.1.m1.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.1.m1.1b"><ci id="S4.SS3.1.p1.1.m1.1.1.cmml" xref="S4.SS3.1.p1.1.m1.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.1.m1.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.1.m1.1d">italic_b</annotation></semantics></math> of <math alttext="\mathsf{PLSigmoid}_{b}" class="ltx_Math" display="inline" id="S4.SS3.1.p1.2.m2.1"><semantics id="S4.SS3.1.p1.2.m2.1a"><msub id="S4.SS3.1.p1.2.m2.1.1" xref="S4.SS3.1.p1.2.m2.1.1.cmml"><mi id="S4.SS3.1.p1.2.m2.1.1.2" xref="S4.SS3.1.p1.2.m2.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S4.SS3.1.p1.2.m2.1.1.3" xref="S4.SS3.1.p1.2.m2.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.1.p1.2.m2.1b"><apply id="S4.SS3.1.p1.2.m2.1.1.cmml" xref="S4.SS3.1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.1.p1.2.m2.1.1.1.cmml" xref="S4.SS3.1.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS3.1.p1.2.m2.1.1.2.cmml" xref="S4.SS3.1.p1.2.m2.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S4.SS3.1.p1.2.m2.1.1.3.cmml" xref="S4.SS3.1.p1.2.m2.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.1.p1.2.m2.1c">\mathsf{PLSigmoid}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.1.p1.2.m2.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>:</p> </div> </div> <section class="ltx_paragraph" id="S4.SS3.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Case 1: <math alttext="1/2\leq b\leq 1" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.1.m1.1"><semantics id="S4.SS3.SSS0.Px1.1.m1.1b"><mrow id="S4.SS3.SSS0.Px1.1.m1.1.1" xref="S4.SS3.SSS0.Px1.1.m1.1.1.cmml"><mrow id="S4.SS3.SSS0.Px1.1.m1.1.1.2" xref="S4.SS3.SSS0.Px1.1.m1.1.1.2.cmml"><mn id="S4.SS3.SSS0.Px1.1.m1.1.1.2.2" xref="S4.SS3.SSS0.Px1.1.m1.1.1.2.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px1.1.m1.1.1.2.1" xref="S4.SS3.SSS0.Px1.1.m1.1.1.2.1.cmml">/</mo><mn id="S4.SS3.SSS0.Px1.1.m1.1.1.2.3" xref="S4.SS3.SSS0.Px1.1.m1.1.1.2.3.cmml">2</mn></mrow><mo id="S4.SS3.SSS0.Px1.1.m1.1.1.3" xref="S4.SS3.SSS0.Px1.1.m1.1.1.3.cmml">≤</mo><mi id="S4.SS3.SSS0.Px1.1.m1.1.1.4" xref="S4.SS3.SSS0.Px1.1.m1.1.1.4.cmml">b</mi><mo id="S4.SS3.SSS0.Px1.1.m1.1.1.5" xref="S4.SS3.SSS0.Px1.1.m1.1.1.5.cmml">≤</mo><mn id="S4.SS3.SSS0.Px1.1.m1.1.1.6" xref="S4.SS3.SSS0.Px1.1.m1.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.1.m1.1c"><apply id="S4.SS3.SSS0.Px1.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px1.1.m1.1.1"><and id="S4.SS3.SSS0.Px1.1.m1.1.1a.cmml" xref="S4.SS3.SSS0.Px1.1.m1.1.1"></and><apply id="S4.SS3.SSS0.Px1.1.m1.1.1b.cmml" xref="S4.SS3.SSS0.Px1.1.m1.1.1"><leq id="S4.SS3.SSS0.Px1.1.m1.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.1.m1.1.1.3"></leq><apply id="S4.SS3.SSS0.Px1.1.m1.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.1.m1.1.1.2"><divide id="S4.SS3.SSS0.Px1.1.m1.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px1.1.m1.1.1.2.1"></divide><cn id="S4.SS3.SSS0.Px1.1.m1.1.1.2.2.cmml" type="integer" xref="S4.SS3.SSS0.Px1.1.m1.1.1.2.2">1</cn><cn id="S4.SS3.SSS0.Px1.1.m1.1.1.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px1.1.m1.1.1.2.3">2</cn></apply><ci id="S4.SS3.SSS0.Px1.1.m1.1.1.4.cmml" xref="S4.SS3.SSS0.Px1.1.m1.1.1.4">𝑏</ci></apply><apply id="S4.SS3.SSS0.Px1.1.m1.1.1c.cmml" xref="S4.SS3.SSS0.Px1.1.m1.1.1"><leq id="S4.SS3.SSS0.Px1.1.m1.1.1.5.cmml" xref="S4.SS3.SSS0.Px1.1.m1.1.1.5"></leq><share href="https://arxiv.org/html/2411.12976v1#S4.SS3.SSS0.Px1.1.m1.1.1.4.cmml" id="S4.SS3.SSS0.Px1.1.m1.1.1d.cmml" xref="S4.SS3.SSS0.Px1.1.m1.1.1"></share><cn id="S4.SS3.SSS0.Px1.1.m1.1.1.6.cmml" type="integer" xref="S4.SS3.SSS0.Px1.1.m1.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.1.m1.1d">1/2\leq b\leq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.1.m1.1e">1 / 2 ≤ italic_b ≤ 1</annotation></semantics></math>.</h4> <div class="ltx_para" id="S4.SS3.SSS0.Px1.p1"> <p class="ltx_p" id="S4.SS3.SSS0.Px1.p1.1">For this case, we consider the graph from <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem6" title="Theorem 1.6 (Lower bound for PL sigmoid selection with 𝑏=1/2 intercept). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.6</span></a> with <math alttext="c=1.12916" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p1.1.m1.1"><semantics id="S4.SS3.SSS0.Px1.p1.1.m1.1a"><mrow id="S4.SS3.SSS0.Px1.p1.1.m1.1.1" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1.cmml"><mi id="S4.SS3.SSS0.Px1.p1.1.m1.1.1.2" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1.2.cmml">c</mi><mo id="S4.SS3.SSS0.Px1.p1.1.m1.1.1.1" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1.1.cmml">=</mo><mn id="S4.SS3.SSS0.Px1.p1.1.m1.1.1.3" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1.3.cmml">1.12916</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p1.1.m1.1b"><apply id="S4.SS3.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1"><eq id="S4.SS3.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1.1"></eq><ci id="S4.SS3.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1.2">𝑐</ci><cn id="S4.SS3.SSS0.Px1.p1.1.m1.1.1.3.cmml" type="float" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1.3">1.12916</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p1.1.m1.1c">c=1.12916</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p1.1.m1.1d">italic_c = 1.12916</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS3.SSS0.Px1.p2"> <p class="ltx_p" id="S4.SS3.SSS0.Px1.p2.3">It follows from the proof of <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem6" title="Theorem 1.6 (Lower bound for PL sigmoid selection with 𝑏=1/2 intercept). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.6</span></a> that <math alttext="\mathsf{PLSigmoid}_{1/2}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.1.m1.1"><semantics id="S4.SS3.SSS0.Px1.p2.1.m1.1a"><msub id="S4.SS3.SSS0.Px1.p2.1.m1.1.1" xref="S4.SS3.SSS0.Px1.p2.1.m1.1.1.cmml"><mi id="S4.SS3.SSS0.Px1.p2.1.m1.1.1.2" xref="S4.SS3.SSS0.Px1.p2.1.m1.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow id="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3" xref="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3.cmml"><mn id="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3.2" xref="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3.1" xref="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3.1.cmml">/</mo><mn id="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3.3" xref="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.1.m1.1b"><apply id="S4.SS3.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p2.1.m1.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.1.m1.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px1.p2.1.m1.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.1.m1.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><apply id="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3"><divide id="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3.1"></divide><cn id="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3.2.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3.2">1</cn><cn id="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3.3.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.1.m1.1c">\mathsf{PLSigmoid}_{1/2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.1.m1.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT</annotation></semantics></math> achieves an approximation ratio of at most <math alttext="0.485282" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.2.m2.1"><semantics id="S4.SS3.SSS0.Px1.p2.2.m2.1a"><mn id="S4.SS3.SSS0.Px1.p2.2.m2.1.1" xref="S4.SS3.SSS0.Px1.p2.2.m2.1.1.cmml">0.485282</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.2.m2.1b"><cn id="S4.SS3.SSS0.Px1.p2.2.m2.1.1.cmml" type="float" xref="S4.SS3.SSS0.Px1.p2.2.m2.1.1">0.485282</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.2.m2.1c">0.485282</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.2.m2.1d">0.485282</annotation></semantics></math> on this graph. The approximation ratio of <math alttext="\mathsf{PLSigmoid}_{b}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.3.m3.1"><semantics id="S4.SS3.SSS0.Px1.p2.3.m3.1a"><msub id="S4.SS3.SSS0.Px1.p2.3.m3.1.1" xref="S4.SS3.SSS0.Px1.p2.3.m3.1.1.cmml"><mi id="S4.SS3.SSS0.Px1.p2.3.m3.1.1.2" xref="S4.SS3.SSS0.Px1.p2.3.m3.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S4.SS3.SSS0.Px1.p2.3.m3.1.1.3" xref="S4.SS3.SSS0.Px1.p2.3.m3.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.3.m3.1b"><apply id="S4.SS3.SSS0.Px1.p2.3.m3.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p2.3.m3.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.3.m3.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px1.p2.3.m3.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.3.m3.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S4.SS3.SSS0.Px1.p2.3.m3.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.3.m3.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.3.m3.1c">\mathsf{PLSigmoid}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.3.m3.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math> on this graph is given by</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex8"> <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{1.12916p^{2}+(1-p)^{2}}{1.12916}\,," class="ltx_Math" display="block" id="S4.Ex8.m1.1"><semantics id="S4.Ex8.m1.1a"><mrow id="S4.Ex8.m1.1.2.2" xref="S4.Ex8.m1.1.1.cmml"><mfrac id="S4.Ex8.m1.1.1" xref="S4.Ex8.m1.1.1.cmml"><mrow id="S4.Ex8.m1.1.1.1" xref="S4.Ex8.m1.1.1.1.cmml"><mrow id="S4.Ex8.m1.1.1.1.3" xref="S4.Ex8.m1.1.1.1.3.cmml"><mn id="S4.Ex8.m1.1.1.1.3.2" xref="S4.Ex8.m1.1.1.1.3.2.cmml">1.12916</mn><mo id="S4.Ex8.m1.1.1.1.3.1" xref="S4.Ex8.m1.1.1.1.3.1.cmml">⁢</mo><msup id="S4.Ex8.m1.1.1.1.3.3" xref="S4.Ex8.m1.1.1.1.3.3.cmml"><mi id="S4.Ex8.m1.1.1.1.3.3.2" xref="S4.Ex8.m1.1.1.1.3.3.2.cmml">p</mi><mn id="S4.Ex8.m1.1.1.1.3.3.3" xref="S4.Ex8.m1.1.1.1.3.3.3.cmml">2</mn></msup></mrow><mo id="S4.Ex8.m1.1.1.1.2" xref="S4.Ex8.m1.1.1.1.2.cmml">+</mo><msup id="S4.Ex8.m1.1.1.1.1" xref="S4.Ex8.m1.1.1.1.1.cmml"><mrow id="S4.Ex8.m1.1.1.1.1.1.1" xref="S4.Ex8.m1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex8.m1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex8.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex8.m1.1.1.1.1.1.1.1" xref="S4.Ex8.m1.1.1.1.1.1.1.1.cmml"><mn id="S4.Ex8.m1.1.1.1.1.1.1.1.2" xref="S4.Ex8.m1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S4.Ex8.m1.1.1.1.1.1.1.1.1" xref="S4.Ex8.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.Ex8.m1.1.1.1.1.1.1.1.3" xref="S4.Ex8.m1.1.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="S4.Ex8.m1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex8.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S4.Ex8.m1.1.1.1.1.3" xref="S4.Ex8.m1.1.1.1.1.3.cmml">2</mn></msup></mrow><mn id="S4.Ex8.m1.1.1.3" xref="S4.Ex8.m1.1.1.3.cmml">1.12916</mn></mfrac><mo id="S4.Ex8.m1.1.2.2.1" lspace="0.170em" xref="S4.Ex8.m1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex8.m1.1b"><apply id="S4.Ex8.m1.1.1.cmml" xref="S4.Ex8.m1.1.2.2"><divide id="S4.Ex8.m1.1.1.2.cmml" xref="S4.Ex8.m1.1.2.2"></divide><apply id="S4.Ex8.m1.1.1.1.cmml" xref="S4.Ex8.m1.1.1.1"><plus id="S4.Ex8.m1.1.1.1.2.cmml" xref="S4.Ex8.m1.1.1.1.2"></plus><apply id="S4.Ex8.m1.1.1.1.3.cmml" xref="S4.Ex8.m1.1.1.1.3"><times id="S4.Ex8.m1.1.1.1.3.1.cmml" xref="S4.Ex8.m1.1.1.1.3.1"></times><cn id="S4.Ex8.m1.1.1.1.3.2.cmml" type="float" xref="S4.Ex8.m1.1.1.1.3.2">1.12916</cn><apply id="S4.Ex8.m1.1.1.1.3.3.cmml" xref="S4.Ex8.m1.1.1.1.3.3"><csymbol cd="ambiguous" id="S4.Ex8.m1.1.1.1.3.3.1.cmml" xref="S4.Ex8.m1.1.1.1.3.3">superscript</csymbol><ci id="S4.Ex8.m1.1.1.1.3.3.2.cmml" xref="S4.Ex8.m1.1.1.1.3.3.2">𝑝</ci><cn id="S4.Ex8.m1.1.1.1.3.3.3.cmml" type="integer" xref="S4.Ex8.m1.1.1.1.3.3.3">2</cn></apply></apply><apply id="S4.Ex8.m1.1.1.1.1.cmml" xref="S4.Ex8.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex8.m1.1.1.1.1.2.cmml" xref="S4.Ex8.m1.1.1.1.1">superscript</csymbol><apply id="S4.Ex8.m1.1.1.1.1.1.1.1.cmml" xref="S4.Ex8.m1.1.1.1.1.1.1"><minus id="S4.Ex8.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex8.m1.1.1.1.1.1.1.1.1"></minus><cn id="S4.Ex8.m1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S4.Ex8.m1.1.1.1.1.1.1.1.2">1</cn><ci id="S4.Ex8.m1.1.1.1.1.1.1.1.3.cmml" xref="S4.Ex8.m1.1.1.1.1.1.1.1.3">𝑝</ci></apply><cn id="S4.Ex8.m1.1.1.1.1.3.cmml" type="integer" xref="S4.Ex8.m1.1.1.1.1.3">2</cn></apply></apply><cn id="S4.Ex8.m1.1.1.3.cmml" type="float" xref="S4.Ex8.m1.1.1.3">1.12916</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex8.m1.1c">\frac{1.12916p^{2}+(1-p)^{2}}{1.12916}\,,</annotation><annotation encoding="application/x-llamapun" id="S4.Ex8.m1.1d">divide start_ARG 1.12916 italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 1 - italic_p ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 1.12916 end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS3.SSS0.Px1.p2.20">where <math alttext="p=\mathsf{S}(b_{1})" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.4.m1.1"><semantics id="S4.SS3.SSS0.Px1.p2.4.m1.1a"><mrow id="S4.SS3.SSS0.Px1.p2.4.m1.1.1" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.cmml"><mi id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.3" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.3.cmml">p</mi><mo id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.2" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.2.cmml">=</mo><mrow id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.cmml"><mi id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.3" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.3.cmml">𝖲</mi><mo id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.2" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.1.cmml"><mo id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.2" stretchy="false" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.1" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.1.cmml"><mi id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.1.2" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.1.2.cmml">b</mi><mn id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.1.3" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.3" stretchy="false" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.4.m1.1b"><apply id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1"><eq id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.2"></eq><ci id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.3">𝑝</ci><apply id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1"><times id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.2"></times><ci id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.3">𝖲</ci><apply id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.1.2">𝑏</ci><cn id="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.4.m1.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.4.m1.1c">p=\mathsf{S}(b_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.4.m1.1d">italic_p = sansserif_S ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="b_{1}=0.12916/2.12916" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.5.m2.1"><semantics id="S4.SS3.SSS0.Px1.p2.5.m2.1a"><mrow id="S4.SS3.SSS0.Px1.p2.5.m2.1.1" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.cmml"><msub id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.2" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.2.cmml"><mi id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.2.2" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.2.2.cmml">b</mi><mn id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.2.3" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.1" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.1.cmml">=</mo><mrow id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3.cmml"><mn id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3.2" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3.2.cmml">0.12916</mn><mo id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3.1" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3.1.cmml">/</mo><mn id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3.3" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3.3.cmml">2.12916</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.5.m2.1b"><apply id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1"><eq id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.1"></eq><apply id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.2">subscript</csymbol><ci id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.2.2.cmml" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.2.2">𝑏</ci><cn id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.2.3">1</cn></apply><apply id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3"><divide id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3.1"></divide><cn id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3.2.cmml" type="float" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3.2">0.12916</cn><cn id="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3.3.cmml" type="float" xref="S4.SS3.SSS0.Px1.p2.5.m2.1.1.3.3">2.12916</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.5.m2.1c">b_{1}=0.12916/2.12916</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.5.m2.1d">italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.12916 / 2.12916</annotation></semantics></math>. We now argue that <math alttext="1/2\leq b\leq 1" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.6.m3.1"><semantics id="S4.SS3.SSS0.Px1.p2.6.m3.1a"><mrow id="S4.SS3.SSS0.Px1.p2.6.m3.1.1" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.cmml"><mrow id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2.cmml"><mn id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2.2" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2.1" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2.1.cmml">/</mo><mn id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2.3" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2.3.cmml">2</mn></mrow><mo id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.3" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.3.cmml">≤</mo><mi id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.4" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.4.cmml">b</mi><mo id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.5" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.5.cmml">≤</mo><mn id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.6" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.6.m3.1b"><apply id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1"><and id="S4.SS3.SSS0.Px1.p2.6.m3.1.1a.cmml" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1"></and><apply id="S4.SS3.SSS0.Px1.p2.6.m3.1.1b.cmml" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1"><leq id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.3"></leq><apply id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2"><divide id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2.1"></divide><cn id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2.2.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2.2">1</cn><cn id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.2.3">2</cn></apply><ci id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.4.cmml" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.4">𝑏</ci></apply><apply id="S4.SS3.SSS0.Px1.p2.6.m3.1.1c.cmml" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1"><leq id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.5.cmml" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.5"></leq><share href="https://arxiv.org/html/2411.12976v1#S4.SS3.SSS0.Px1.p2.6.m3.1.1.4.cmml" id="S4.SS3.SSS0.Px1.p2.6.m3.1.1d.cmml" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1"></share><cn id="S4.SS3.SSS0.Px1.p2.6.m3.1.1.6.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.6.m3.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.6.m3.1c">1/2\leq b\leq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.6.m3.1d">1 / 2 ≤ italic_b ≤ 1</annotation></semantics></math> implies that <math alttext="\mathsf{PLSigmoid}_{b}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.7.m4.1"><semantics id="S4.SS3.SSS0.Px1.p2.7.m4.1a"><msub id="S4.SS3.SSS0.Px1.p2.7.m4.1.1" xref="S4.SS3.SSS0.Px1.p2.7.m4.1.1.cmml"><mi id="S4.SS3.SSS0.Px1.p2.7.m4.1.1.2" xref="S4.SS3.SSS0.Px1.p2.7.m4.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S4.SS3.SSS0.Px1.p2.7.m4.1.1.3" xref="S4.SS3.SSS0.Px1.p2.7.m4.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.7.m4.1b"><apply id="S4.SS3.SSS0.Px1.p2.7.m4.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.7.m4.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p2.7.m4.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.7.m4.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px1.p2.7.m4.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.7.m4.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S4.SS3.SSS0.Px1.p2.7.m4.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.7.m4.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.7.m4.1c">\mathsf{PLSigmoid}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.7.m4.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math> gets an approximation ratio of at most <math alttext="0.485282" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.8.m5.1"><semantics id="S4.SS3.SSS0.Px1.p2.8.m5.1a"><mn id="S4.SS3.SSS0.Px1.p2.8.m5.1.1" xref="S4.SS3.SSS0.Px1.p2.8.m5.1.1.cmml">0.485282</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.8.m5.1b"><cn id="S4.SS3.SSS0.Px1.p2.8.m5.1.1.cmml" type="float" xref="S4.SS3.SSS0.Px1.p2.8.m5.1.1">0.485282</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.8.m5.1c">0.485282</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.8.m5.1d">0.485282</annotation></semantics></math> on this graph. Observe that the derivative of <math alttext="1.12916p^{2}+(1-p)^{2}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.9.m6.1"><semantics id="S4.SS3.SSS0.Px1.p2.9.m6.1a"><mrow id="S4.SS3.SSS0.Px1.p2.9.m6.1.1" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.cmml"><mrow id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.cmml"><mn id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.2" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.2.cmml">1.12916</mn><mo id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.1" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.1.cmml">⁢</mo><msup id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.3" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.3.cmml"><mi id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.3.2" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.3.2.cmml">p</mi><mn id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.3.3" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.3.3.cmml">2</mn></msup></mrow><mo id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.2" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.2.cmml">+</mo><msup id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.cmml"><mrow id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.cmml"><mo id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.2" stretchy="false" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.cmml"><mn id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.2" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.1" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.3" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.3" stretchy="false" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.3" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.9.m6.1b"><apply id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1"><plus id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.2"></plus><apply id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3"><times id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.1"></times><cn id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.2.cmml" type="float" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.2">1.12916</cn><apply id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.3.cmml" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.3.1.cmml" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.3">superscript</csymbol><ci id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.3.2.cmml" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.3.2">𝑝</ci><cn id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.3.3.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.3.3.3">2</cn></apply></apply><apply id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1">superscript</csymbol><apply id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1"><minus id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.1"></minus><cn id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.2">1</cn><ci id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.1.1.1.3">𝑝</ci></apply><cn id="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.3.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.9.m6.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.9.m6.1c">1.12916p^{2}+(1-p)^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.9.m6.1d">1.12916 italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 1 - italic_p ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> is <math alttext="0" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.10.m7.1"><semantics id="S4.SS3.SSS0.Px1.p2.10.m7.1a"><mn id="S4.SS3.SSS0.Px1.p2.10.m7.1.1" xref="S4.SS3.SSS0.Px1.p2.10.m7.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.10.m7.1b"><cn id="S4.SS3.SSS0.Px1.p2.10.m7.1.1.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.10.m7.1.1">0</cn></annotation-xml></semantics></math> at <math alttext="1/2.12916\approx 0.46967" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.11.m8.1"><semantics id="S4.SS3.SSS0.Px1.p2.11.m8.1a"><mrow id="S4.SS3.SSS0.Px1.p2.11.m8.1.1" xref="S4.SS3.SSS0.Px1.p2.11.m8.1.1.cmml"><mrow id="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2" xref="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2.cmml"><mn id="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2.2" xref="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2.1" xref="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2.1.cmml">/</mo><mn id="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2.3" xref="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2.3.cmml">2.12916</mn></mrow><mo id="S4.SS3.SSS0.Px1.p2.11.m8.1.1.1" xref="S4.SS3.SSS0.Px1.p2.11.m8.1.1.1.cmml">≈</mo><mn id="S4.SS3.SSS0.Px1.p2.11.m8.1.1.3" xref="S4.SS3.SSS0.Px1.p2.11.m8.1.1.3.cmml">0.46967</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.11.m8.1b"><apply id="S4.SS3.SSS0.Px1.p2.11.m8.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.11.m8.1.1"><approx id="S4.SS3.SSS0.Px1.p2.11.m8.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.11.m8.1.1.1"></approx><apply id="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2"><divide id="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2.1"></divide><cn id="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2.2.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2.2">1</cn><cn id="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2.3.cmml" type="float" xref="S4.SS3.SSS0.Px1.p2.11.m8.1.1.2.3">2.12916</cn></apply><cn id="S4.SS3.SSS0.Px1.p2.11.m8.1.1.3.cmml" type="float" xref="S4.SS3.SSS0.Px1.p2.11.m8.1.1.3">0.46967</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.11.m8.1c">1/2.12916\approx 0.46967</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.11.m8.1d">1 / 2.12916 ≈ 0.46967</annotation></semantics></math>. Hence in the range <math alttext="0.46967\leq p\leq 1" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.12.m9.1"><semantics id="S4.SS3.SSS0.Px1.p2.12.m9.1a"><mrow id="S4.SS3.SSS0.Px1.p2.12.m9.1.1" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1.cmml"><mn id="S4.SS3.SSS0.Px1.p2.12.m9.1.1.2" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1.2.cmml">0.46967</mn><mo id="S4.SS3.SSS0.Px1.p2.12.m9.1.1.3" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1.3.cmml">≤</mo><mi id="S4.SS3.SSS0.Px1.p2.12.m9.1.1.4" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1.4.cmml">p</mi><mo id="S4.SS3.SSS0.Px1.p2.12.m9.1.1.5" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1.5.cmml">≤</mo><mn id="S4.SS3.SSS0.Px1.p2.12.m9.1.1.6" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.12.m9.1b"><apply id="S4.SS3.SSS0.Px1.p2.12.m9.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1"><and id="S4.SS3.SSS0.Px1.p2.12.m9.1.1a.cmml" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1"></and><apply id="S4.SS3.SSS0.Px1.p2.12.m9.1.1b.cmml" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1"><leq id="S4.SS3.SSS0.Px1.p2.12.m9.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1.3"></leq><cn id="S4.SS3.SSS0.Px1.p2.12.m9.1.1.2.cmml" type="float" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1.2">0.46967</cn><ci id="S4.SS3.SSS0.Px1.p2.12.m9.1.1.4.cmml" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1.4">𝑝</ci></apply><apply id="S4.SS3.SSS0.Px1.p2.12.m9.1.1c.cmml" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1"><leq id="S4.SS3.SSS0.Px1.p2.12.m9.1.1.5.cmml" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1.5"></leq><share href="https://arxiv.org/html/2411.12976v1#S4.SS3.SSS0.Px1.p2.12.m9.1.1.4.cmml" id="S4.SS3.SSS0.Px1.p2.12.m9.1.1d.cmml" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1"></share><cn id="S4.SS3.SSS0.Px1.p2.12.m9.1.1.6.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.12.m9.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.12.m9.1c">0.46967\leq p\leq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.12.m9.1d">0.46967 ≤ italic_p ≤ 1</annotation></semantics></math>, <math alttext="1.12916p^{2}+(1-p)^{2}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.13.m10.1"><semantics id="S4.SS3.SSS0.Px1.p2.13.m10.1a"><mrow id="S4.SS3.SSS0.Px1.p2.13.m10.1.1" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.cmml"><mrow id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.cmml"><mn id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.2" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.2.cmml">1.12916</mn><mo id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.1" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.1.cmml">⁢</mo><msup id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.3" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.3.cmml"><mi id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.3.2" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.3.2.cmml">p</mi><mn id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.3.3" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.3.3.cmml">2</mn></msup></mrow><mo id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.2" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.2.cmml">+</mo><msup id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.cmml"><mrow id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.cmml"><mo id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.2" stretchy="false" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.cmml"><mn id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.2" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.1" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.3" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.3" stretchy="false" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.3" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.13.m10.1b"><apply id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1"><plus id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.2"></plus><apply id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3"><times id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.1"></times><cn id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.2.cmml" type="float" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.2">1.12916</cn><apply id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.3.cmml" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.3.1.cmml" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.3">superscript</csymbol><ci id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.3.2.cmml" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.3.2">𝑝</ci><cn id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.3.3.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.3.3.3">2</cn></apply></apply><apply id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1">superscript</csymbol><apply id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1"><minus id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.1"></minus><cn id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.2">1</cn><ci id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.1.1.1.3">𝑝</ci></apply><cn id="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.3.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.13.m10.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.13.m10.1c">1.12916p^{2}+(1-p)^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.13.m10.1d">1.12916 italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 1 - italic_p ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> is an increasing function of <math alttext="p" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.14.m11.1"><semantics id="S4.SS3.SSS0.Px1.p2.14.m11.1a"><mi id="S4.SS3.SSS0.Px1.p2.14.m11.1.1" xref="S4.SS3.SSS0.Px1.p2.14.m11.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.14.m11.1b"><ci id="S4.SS3.SSS0.Px1.p2.14.m11.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.14.m11.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.14.m11.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.14.m11.1d">italic_p</annotation></semantics></math>. By the definition of <math alttext="\mathsf{PLSigmoid}_{b}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.15.m12.1"><semantics id="S4.SS3.SSS0.Px1.p2.15.m12.1a"><msub id="S4.SS3.SSS0.Px1.p2.15.m12.1.1" xref="S4.SS3.SSS0.Px1.p2.15.m12.1.1.cmml"><mi id="S4.SS3.SSS0.Px1.p2.15.m12.1.1.2" xref="S4.SS3.SSS0.Px1.p2.15.m12.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S4.SS3.SSS0.Px1.p2.15.m12.1.1.3" xref="S4.SS3.SSS0.Px1.p2.15.m12.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.15.m12.1b"><apply id="S4.SS3.SSS0.Px1.p2.15.m12.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.15.m12.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p2.15.m12.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.15.m12.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px1.p2.15.m12.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.15.m12.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S4.SS3.SSS0.Px1.p2.15.m12.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.15.m12.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.15.m12.1c">\mathsf{PLSigmoid}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.15.m12.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>, we have that <math alttext="p=1/2+b_{1}/2b" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.16.m13.1"><semantics id="S4.SS3.SSS0.Px1.p2.16.m13.1a"><mrow id="S4.SS3.SSS0.Px1.p2.16.m13.1.1" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.cmml"><mi id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.2" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.2.cmml">p</mi><mo id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.1" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.1.cmml">=</mo><mrow id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.cmml"><mrow id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2.cmml"><mn id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2.2" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2.1" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2.1.cmml">/</mo><mn id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2.3" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2.3.cmml">2</mn></mrow><mo id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.1" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.1.cmml">+</mo><mrow id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.cmml"><mrow id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.cmml"><msub id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.2" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.2.cmml"><mi id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.2.2" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.2.2.cmml">b</mi><mn id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.2.3" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.2.3.cmml">1</mn></msub><mo id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.1" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.1.cmml">/</mo><mn id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.3" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.3.cmml">2</mn></mrow><mo id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.1" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.1.cmml">⁢</mo><mi id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.3" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.3.cmml">b</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.16.m13.1b"><apply id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1"><eq id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.1"></eq><ci id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.2">𝑝</ci><apply id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3"><plus id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.1"></plus><apply id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2.cmml" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2"><divide id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2.1.cmml" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2.1"></divide><cn id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2.2.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2.2">1</cn><cn id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.2.3">2</cn></apply><apply id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.cmml" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3"><times id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.1.cmml" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.1"></times><apply id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.cmml" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2"><divide id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.1.cmml" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.1"></divide><apply id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.2.cmml" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.2.1.cmml" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.2">subscript</csymbol><ci id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.2.2.cmml" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.2.2">𝑏</ci><cn id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.2.3">1</cn></apply><cn id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.2.3">2</cn></apply><ci id="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.3.cmml" xref="S4.SS3.SSS0.Px1.p2.16.m13.1.1.3.3.3">𝑏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.16.m13.1c">p=1/2+b_{1}/2b</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.16.m13.1d">italic_p = 1 / 2 + italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / 2 italic_b</annotation></semantics></math> when <math alttext="b\geq 1/2" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.17.m14.1"><semantics id="S4.SS3.SSS0.Px1.p2.17.m14.1a"><mrow id="S4.SS3.SSS0.Px1.p2.17.m14.1.1" xref="S4.SS3.SSS0.Px1.p2.17.m14.1.1.cmml"><mi id="S4.SS3.SSS0.Px1.p2.17.m14.1.1.2" xref="S4.SS3.SSS0.Px1.p2.17.m14.1.1.2.cmml">b</mi><mo id="S4.SS3.SSS0.Px1.p2.17.m14.1.1.1" xref="S4.SS3.SSS0.Px1.p2.17.m14.1.1.1.cmml">≥</mo><mrow id="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3" xref="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3.cmml"><mn id="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3.2" xref="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3.1" xref="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3.1.cmml">/</mo><mn id="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3.3" xref="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.17.m14.1b"><apply id="S4.SS3.SSS0.Px1.p2.17.m14.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.17.m14.1.1"><geq id="S4.SS3.SSS0.Px1.p2.17.m14.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.17.m14.1.1.1"></geq><ci id="S4.SS3.SSS0.Px1.p2.17.m14.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.17.m14.1.1.2">𝑏</ci><apply id="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3"><divide id="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3.1"></divide><cn id="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3.2.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3.2">1</cn><cn id="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3.3.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.17.m14.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.17.m14.1c">b\geq 1/2</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.17.m14.1d">italic_b ≥ 1 / 2</annotation></semantics></math>. Hence, for <math alttext="1/2\leq b\leq 1" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.18.m15.1"><semantics id="S4.SS3.SSS0.Px1.p2.18.m15.1a"><mrow id="S4.SS3.SSS0.Px1.p2.18.m15.1.1" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.cmml"><mrow id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2.cmml"><mn id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2.2" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2.1" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2.1.cmml">/</mo><mn id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2.3" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2.3.cmml">2</mn></mrow><mo id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.3" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.3.cmml">≤</mo><mi id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.4" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.4.cmml">b</mi><mo id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.5" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.5.cmml">≤</mo><mn id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.6" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.18.m15.1b"><apply id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1"><and id="S4.SS3.SSS0.Px1.p2.18.m15.1.1a.cmml" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1"></and><apply id="S4.SS3.SSS0.Px1.p2.18.m15.1.1b.cmml" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1"><leq id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.3"></leq><apply id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2"><divide id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2.1"></divide><cn id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2.2.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2.2">1</cn><cn id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.2.3">2</cn></apply><ci id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.4.cmml" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.4">𝑏</ci></apply><apply id="S4.SS3.SSS0.Px1.p2.18.m15.1.1c.cmml" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1"><leq id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.5.cmml" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.5"></leq><share href="https://arxiv.org/html/2411.12976v1#S4.SS3.SSS0.Px1.p2.18.m15.1.1.4.cmml" id="S4.SS3.SSS0.Px1.p2.18.m15.1.1d.cmml" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1"></share><cn id="S4.SS3.SSS0.Px1.p2.18.m15.1.1.6.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.18.m15.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.18.m15.1c">1/2\leq b\leq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.18.m15.1d">1 / 2 ≤ italic_b ≤ 1</annotation></semantics></math>, the largest possible approximation is achieved at <math alttext="b=1/2" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.19.m16.1"><semantics id="S4.SS3.SSS0.Px1.p2.19.m16.1a"><mrow id="S4.SS3.SSS0.Px1.p2.19.m16.1.1" xref="S4.SS3.SSS0.Px1.p2.19.m16.1.1.cmml"><mi id="S4.SS3.SSS0.Px1.p2.19.m16.1.1.2" xref="S4.SS3.SSS0.Px1.p2.19.m16.1.1.2.cmml">b</mi><mo id="S4.SS3.SSS0.Px1.p2.19.m16.1.1.1" xref="S4.SS3.SSS0.Px1.p2.19.m16.1.1.1.cmml">=</mo><mrow id="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3" xref="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3.cmml"><mn id="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3.2" xref="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3.1" xref="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3.1.cmml">/</mo><mn id="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3.3" xref="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.19.m16.1b"><apply id="S4.SS3.SSS0.Px1.p2.19.m16.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.19.m16.1.1"><eq id="S4.SS3.SSS0.Px1.p2.19.m16.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p2.19.m16.1.1.1"></eq><ci id="S4.SS3.SSS0.Px1.p2.19.m16.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p2.19.m16.1.1.2">𝑏</ci><apply id="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3"><divide id="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3.1"></divide><cn id="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3.2.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3.2">1</cn><cn id="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3.3.cmml" type="integer" xref="S4.SS3.SSS0.Px1.p2.19.m16.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.19.m16.1c">b=1/2</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.19.m16.1d">italic_b = 1 / 2</annotation></semantics></math> and we already established that this is at most <math alttext="0.485282" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p2.20.m17.1"><semantics id="S4.SS3.SSS0.Px1.p2.20.m17.1a"><mn id="S4.SS3.SSS0.Px1.p2.20.m17.1.1" xref="S4.SS3.SSS0.Px1.p2.20.m17.1.1.cmml">0.485282</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p2.20.m17.1b"><cn id="S4.SS3.SSS0.Px1.p2.20.m17.1.1.cmml" type="float" xref="S4.SS3.SSS0.Px1.p2.20.m17.1.1">0.485282</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p2.20.m17.1c">0.485282</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p2.20.m17.1d">0.485282</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS3.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Case 2: <math alttext="0.225\leq b\leq 1/2" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.1.m1.1"><semantics id="S4.SS3.SSS0.Px2.1.m1.1b"><mrow id="S4.SS3.SSS0.Px2.1.m1.1.1" xref="S4.SS3.SSS0.Px2.1.m1.1.1.cmml"><mn id="S4.SS3.SSS0.Px2.1.m1.1.1.2" xref="S4.SS3.SSS0.Px2.1.m1.1.1.2.cmml">0.225</mn><mo id="S4.SS3.SSS0.Px2.1.m1.1.1.3" xref="S4.SS3.SSS0.Px2.1.m1.1.1.3.cmml">≤</mo><mi id="S4.SS3.SSS0.Px2.1.m1.1.1.4" xref="S4.SS3.SSS0.Px2.1.m1.1.1.4.cmml">b</mi><mo id="S4.SS3.SSS0.Px2.1.m1.1.1.5" xref="S4.SS3.SSS0.Px2.1.m1.1.1.5.cmml">≤</mo><mrow id="S4.SS3.SSS0.Px2.1.m1.1.1.6" xref="S4.SS3.SSS0.Px2.1.m1.1.1.6.cmml"><mn id="S4.SS3.SSS0.Px2.1.m1.1.1.6.2" xref="S4.SS3.SSS0.Px2.1.m1.1.1.6.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px2.1.m1.1.1.6.1" xref="S4.SS3.SSS0.Px2.1.m1.1.1.6.1.cmml">/</mo><mn id="S4.SS3.SSS0.Px2.1.m1.1.1.6.3" xref="S4.SS3.SSS0.Px2.1.m1.1.1.6.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.1.m1.1c"><apply id="S4.SS3.SSS0.Px2.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1"><and id="S4.SS3.SSS0.Px2.1.m1.1.1a.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1"></and><apply id="S4.SS3.SSS0.Px2.1.m1.1.1b.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1"><leq id="S4.SS3.SSS0.Px2.1.m1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.3"></leq><cn id="S4.SS3.SSS0.Px2.1.m1.1.1.2.cmml" type="float" xref="S4.SS3.SSS0.Px2.1.m1.1.1.2">0.225</cn><ci id="S4.SS3.SSS0.Px2.1.m1.1.1.4.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.4">𝑏</ci></apply><apply id="S4.SS3.SSS0.Px2.1.m1.1.1c.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1"><leq id="S4.SS3.SSS0.Px2.1.m1.1.1.5.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.5"></leq><share href="https://arxiv.org/html/2411.12976v1#S4.SS3.SSS0.Px2.1.m1.1.1.4.cmml" id="S4.SS3.SSS0.Px2.1.m1.1.1d.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1"></share><apply id="S4.SS3.SSS0.Px2.1.m1.1.1.6.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.6"><divide id="S4.SS3.SSS0.Px2.1.m1.1.1.6.1.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.6.1"></divide><cn id="S4.SS3.SSS0.Px2.1.m1.1.1.6.2.cmml" type="integer" xref="S4.SS3.SSS0.Px2.1.m1.1.1.6.2">1</cn><cn id="S4.SS3.SSS0.Px2.1.m1.1.1.6.3.cmml" type="integer" xref="S4.SS3.SSS0.Px2.1.m1.1.1.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.1.m1.1d">0.225\leq b\leq 1/2</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.1.m1.1e">0.225 ≤ italic_b ≤ 1 / 2</annotation></semantics></math>.</h4> <div class="ltx_para" id="S4.SS3.SSS0.Px2.p1"> <p class="ltx_p" id="S4.SS3.SSS0.Px2.p1.2">Let <math alttext="b_{i}=-0.475+0.05(i-1)" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p1.1.m1.1"><semantics id="S4.SS3.SSS0.Px2.p1.1.m1.1a"><mrow id="S4.SS3.SSS0.Px2.p1.1.m1.1.1" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.cmml"><msub id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.3" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.3.cmml"><mi id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.3.2" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.3.2.cmml">b</mi><mi id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.3.3" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.3.3.cmml">i</mi></msub><mo id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.2" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.2.cmml">=</mo><mrow id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.cmml"><mrow id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.3" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.3.cmml"><mo id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.3a" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.3.cmml">−</mo><mn id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.3.2" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.3.2.cmml">0.475</mn></mrow><mo id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.2" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.2.cmml">+</mo><mrow id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.cmml"><mn id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.3" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.3.cmml">0.05</mn><mo id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.2" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.2" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.2.cmml">i</mi><mo id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.1" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.3" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p1.1.m1.1b"><apply id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1"><eq id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.2"></eq><apply id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.3.2.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.3.2">𝑏</ci><ci id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.3.3.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.3.3">𝑖</ci></apply><apply id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1"><plus id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.2"></plus><apply id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.3"><minus id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.3"></minus><cn id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.3.2.cmml" type="float" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.3.2">0.475</cn></apply><apply id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1"><times id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.2"></times><cn id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.3.cmml" type="float" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.3">0.05</cn><apply id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1"><minus id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.1"></minus><ci id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.2">𝑖</ci><cn id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p1.1.m1.1c">b_{i}=-0.475+0.05(i-1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p1.1.m1.1d">italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 0.475 + 0.05 ( italic_i - 1 )</annotation></semantics></math> for all <math alttext="i\in\{1,\dots,20\}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p1.2.m2.3"><semantics id="S4.SS3.SSS0.Px2.p1.2.m2.3a"><mrow id="S4.SS3.SSS0.Px2.p1.2.m2.3.4" xref="S4.SS3.SSS0.Px2.p1.2.m2.3.4.cmml"><mi id="S4.SS3.SSS0.Px2.p1.2.m2.3.4.2" xref="S4.SS3.SSS0.Px2.p1.2.m2.3.4.2.cmml">i</mi><mo id="S4.SS3.SSS0.Px2.p1.2.m2.3.4.1" xref="S4.SS3.SSS0.Px2.p1.2.m2.3.4.1.cmml">∈</mo><mrow id="S4.SS3.SSS0.Px2.p1.2.m2.3.4.3.2" xref="S4.SS3.SSS0.Px2.p1.2.m2.3.4.3.1.cmml"><mo id="S4.SS3.SSS0.Px2.p1.2.m2.3.4.3.2.1" stretchy="false" xref="S4.SS3.SSS0.Px2.p1.2.m2.3.4.3.1.cmml">{</mo><mn id="S4.SS3.SSS0.Px2.p1.2.m2.1.1" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.cmml">1</mn><mo id="S4.SS3.SSS0.Px2.p1.2.m2.3.4.3.2.2" xref="S4.SS3.SSS0.Px2.p1.2.m2.3.4.3.1.cmml">,</mo><mi id="S4.SS3.SSS0.Px2.p1.2.m2.2.2" mathvariant="normal" xref="S4.SS3.SSS0.Px2.p1.2.m2.2.2.cmml">…</mi><mo id="S4.SS3.SSS0.Px2.p1.2.m2.3.4.3.2.3" xref="S4.SS3.SSS0.Px2.p1.2.m2.3.4.3.1.cmml">,</mo><mn id="S4.SS3.SSS0.Px2.p1.2.m2.3.3" xref="S4.SS3.SSS0.Px2.p1.2.m2.3.3.cmml">20</mn><mo id="S4.SS3.SSS0.Px2.p1.2.m2.3.4.3.2.4" stretchy="false" xref="S4.SS3.SSS0.Px2.p1.2.m2.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p1.2.m2.3b"><apply id="S4.SS3.SSS0.Px2.p1.2.m2.3.4.cmml" xref="S4.SS3.SSS0.Px2.p1.2.m2.3.4"><in id="S4.SS3.SSS0.Px2.p1.2.m2.3.4.1.cmml" xref="S4.SS3.SSS0.Px2.p1.2.m2.3.4.1"></in><ci id="S4.SS3.SSS0.Px2.p1.2.m2.3.4.2.cmml" xref="S4.SS3.SSS0.Px2.p1.2.m2.3.4.2">𝑖</ci><set id="S4.SS3.SSS0.Px2.p1.2.m2.3.4.3.1.cmml" xref="S4.SS3.SSS0.Px2.p1.2.m2.3.4.3.2"><cn id="S4.SS3.SSS0.Px2.p1.2.m2.1.1.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1">1</cn><ci id="S4.SS3.SSS0.Px2.p1.2.m2.2.2.cmml" xref="S4.SS3.SSS0.Px2.p1.2.m2.2.2">…</ci><cn id="S4.SS3.SSS0.Px2.p1.2.m2.3.3.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p1.2.m2.3.3">20</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p1.2.m2.3c">i\in\{1,\dots,20\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p1.2.m2.3d">italic_i ∈ { 1 , … , 20 }</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS3.SSS0.Px2.p2"> <p class="ltx_p" id="S4.SS3.SSS0.Px2.p2.14">We describe a concrete graph <math alttext="G_{\mathrm{LP}}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p2.1.m1.1"><semantics id="S4.SS3.SSS0.Px2.p2.1.m1.1a"><msub id="S4.SS3.SSS0.Px2.p2.1.m1.1.1" xref="S4.SS3.SSS0.Px2.p2.1.m1.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p2.1.m1.1.1.2" xref="S4.SS3.SSS0.Px2.p2.1.m1.1.1.2.cmml">G</mi><mi id="S4.SS3.SSS0.Px2.p2.1.m1.1.1.3" xref="S4.SS3.SSS0.Px2.p2.1.m1.1.1.3.cmml">LP</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p2.1.m1.1b"><apply id="S4.SS3.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p2.1.m1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.1.m1.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p2.1.m1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p2.1.m1.1.1.2">𝐺</ci><ci id="S4.SS3.SSS0.Px2.p2.1.m1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p2.1.m1.1.1.3">LP</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p2.1.m1.1c">G_{\mathrm{LP}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p2.1.m1.1d">italic_G start_POSTSUBSCRIPT roman_LP end_POSTSUBSCRIPT</annotation></semantics></math>, which we visualize in the next figure.<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>This graph was originally calculated by the LP paradigm described in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S4.SS1" title="4.1 Methodology for finding hard instances ‣ 4 Lower bounds ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.1</span></a>. <math alttext="\mathcal{P}" class="ltx_Math" display="inline" id="footnote6.m1.1"><semantics id="footnote6.m1.1b"><mi class="ltx_font_mathcaligraphic" id="footnote6.m1.1.1" xref="footnote6.m1.1.1.cmml">𝒫</mi><annotation-xml encoding="MathML-Content" id="footnote6.m1.1c"><ci id="footnote6.m1.1.1.cmml" xref="footnote6.m1.1.1">𝒫</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m1.1d">\mathcal{P}</annotation><annotation encoding="application/x-llamapun" id="footnote6.m1.1e">caligraphic_P</annotation></semantics></math> consisted of the single rounding function <math alttext="\mathsf{PLSigmoid}_{149/309}" class="ltx_Math" display="inline" id="footnote6.m2.1"><semantics id="footnote6.m2.1b"><msub 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">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mrow 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">149</mn><mo id="footnote6.m2.1.1.3.1" xref="footnote6.m2.1.1.3.1.cmml">/</mo><mn id="footnote6.m2.1.1.3.3" xref="footnote6.m2.1.1.3.3.cmml">309</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="footnote6.m2.1c"><apply id="footnote6.m2.1.1.cmml" xref="footnote6.m2.1.1"><csymbol cd="ambiguous" id="footnote6.m2.1.1.1.cmml" xref="footnote6.m2.1.1">subscript</csymbol><ci id="footnote6.m2.1.1.2.cmml" xref="footnote6.m2.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</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.m2.1.1.3.1"></divide><cn id="footnote6.m2.1.1.3.2.cmml" type="integer" xref="footnote6.m2.1.1.3.2">149</cn><cn id="footnote6.m2.1.1.3.3.cmml" type="integer" xref="footnote6.m2.1.1.3.3">309</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote6.m2.1d">\mathsf{PLSigmoid}_{149/309}</annotation><annotation encoding="application/x-llamapun" id="footnote6.m2.1e">sansserif_PLSigmoid start_POSTSUBSCRIPT 149 / 309 end_POSTSUBSCRIPT</annotation></semantics></math>. The particularly simple structure of the found solution let us describe it analytically, as we do here.</span></span></span> <math alttext="G_{\mathrm{LP}}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p2.2.m2.1"><semantics id="S4.SS3.SSS0.Px2.p2.2.m2.1a"><msub id="S4.SS3.SSS0.Px2.p2.2.m2.1.1" xref="S4.SS3.SSS0.Px2.p2.2.m2.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p2.2.m2.1.1.2" xref="S4.SS3.SSS0.Px2.p2.2.m2.1.1.2.cmml">G</mi><mi id="S4.SS3.SSS0.Px2.p2.2.m2.1.1.3" xref="S4.SS3.SSS0.Px2.p2.2.m2.1.1.3.cmml">LP</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p2.2.m2.1b"><apply id="S4.SS3.SSS0.Px2.p2.2.m2.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p2.2.m2.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.2.m2.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p2.2.m2.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p2.2.m2.1.1.2">𝐺</ci><ci id="S4.SS3.SSS0.Px2.p2.2.m2.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p2.2.m2.1.1.3">LP</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p2.2.m2.1c">G_{\mathrm{LP}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p2.2.m2.1d">italic_G start_POSTSUBSCRIPT roman_LP end_POSTSUBSCRIPT</annotation></semantics></math> has <math alttext="36" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p2.3.m3.1"><semantics id="S4.SS3.SSS0.Px2.p2.3.m3.1a"><mn id="S4.SS3.SSS0.Px2.p2.3.m3.1.1" xref="S4.SS3.SSS0.Px2.p2.3.m3.1.1.cmml">36</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p2.3.m3.1b"><cn id="S4.SS3.SSS0.Px2.p2.3.m3.1.1.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p2.3.m3.1.1">36</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p2.3.m3.1c">36</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p2.3.m3.1d">36</annotation></semantics></math> vertices, which we label <math alttext="\{1,\ldots,18\}" 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xref="S4.SS3.SSS0.Px2.p2.4.m4.3.4.2"><cn id="S4.SS3.SSS0.Px2.p2.4.m4.1.1.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p2.4.m4.1.1">1</cn><ci id="S4.SS3.SSS0.Px2.p2.4.m4.2.2.cmml" xref="S4.SS3.SSS0.Px2.p2.4.m4.2.2">…</ci><cn id="S4.SS3.SSS0.Px2.p2.4.m4.3.3.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p2.4.m4.3.3">18</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p2.4.m4.3c">\{1,\ldots,18\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p2.4.m4.3d">{ 1 , … , 18 }</annotation></semantics></math> and <math alttext="\{3^{\prime},\ldots,20^{\prime}\}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p2.5.m5.3"><semantics id="S4.SS3.SSS0.Px2.p2.5.m5.3a"><mrow id="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2" xref="S4.SS3.SSS0.Px2.p2.5.m5.3.3.3.cmml"><mo id="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.3" stretchy="false" xref="S4.SS3.SSS0.Px2.p2.5.m5.3.3.3.cmml">{</mo><msup id="S4.SS3.SSS0.Px2.p2.5.m5.2.2.1.1" xref="S4.SS3.SSS0.Px2.p2.5.m5.2.2.1.1.cmml"><mn id="S4.SS3.SSS0.Px2.p2.5.m5.2.2.1.1.2" xref="S4.SS3.SSS0.Px2.p2.5.m5.2.2.1.1.2.cmml">3</mn><mo id="S4.SS3.SSS0.Px2.p2.5.m5.2.2.1.1.3" xref="S4.SS3.SSS0.Px2.p2.5.m5.2.2.1.1.3.cmml">′</mo></msup><mo id="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.4" xref="S4.SS3.SSS0.Px2.p2.5.m5.3.3.3.cmml">,</mo><mi id="S4.SS3.SSS0.Px2.p2.5.m5.1.1" mathvariant="normal" xref="S4.SS3.SSS0.Px2.p2.5.m5.1.1.cmml">…</mi><mo id="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.5" xref="S4.SS3.SSS0.Px2.p2.5.m5.3.3.3.cmml">,</mo><msup id="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.2" xref="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.2.cmml"><mn id="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.2.2" xref="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.2.2.cmml">20</mn><mo id="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.2.3" xref="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.2.3.cmml">′</mo></msup><mo id="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.6" stretchy="false" xref="S4.SS3.SSS0.Px2.p2.5.m5.3.3.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p2.5.m5.3b"><set id="S4.SS3.SSS0.Px2.p2.5.m5.3.3.3.cmml" xref="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2"><apply id="S4.SS3.SSS0.Px2.p2.5.m5.2.2.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.5.m5.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p2.5.m5.2.2.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.5.m5.2.2.1.1">superscript</csymbol><cn id="S4.SS3.SSS0.Px2.p2.5.m5.2.2.1.1.2.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p2.5.m5.2.2.1.1.2">3</cn><ci id="S4.SS3.SSS0.Px2.p2.5.m5.2.2.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p2.5.m5.2.2.1.1.3">′</ci></apply><ci id="S4.SS3.SSS0.Px2.p2.5.m5.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.5.m5.1.1">…</ci><apply id="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.2.cmml" xref="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.2.1.cmml" xref="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.2">superscript</csymbol><cn id="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.2.2.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.2.2">20</cn><ci id="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.2.3.cmml" xref="S4.SS3.SSS0.Px2.p2.5.m5.3.3.2.2.3">′</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p2.5.m5.3c">\{3^{\prime},\ldots,20^{\prime}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p2.5.m5.3d">{ 3 start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , … , 20 start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT }</annotation></semantics></math>. The graph has the property that for each <math alttext="i\in\{1,\ldots,20\}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p2.6.m6.3"><semantics id="S4.SS3.SSS0.Px2.p2.6.m6.3a"><mrow id="S4.SS3.SSS0.Px2.p2.6.m6.3.4" xref="S4.SS3.SSS0.Px2.p2.6.m6.3.4.cmml"><mi id="S4.SS3.SSS0.Px2.p2.6.m6.3.4.2" xref="S4.SS3.SSS0.Px2.p2.6.m6.3.4.2.cmml">i</mi><mo id="S4.SS3.SSS0.Px2.p2.6.m6.3.4.1" xref="S4.SS3.SSS0.Px2.p2.6.m6.3.4.1.cmml">∈</mo><mrow id="S4.SS3.SSS0.Px2.p2.6.m6.3.4.3.2" xref="S4.SS3.SSS0.Px2.p2.6.m6.3.4.3.1.cmml"><mo id="S4.SS3.SSS0.Px2.p2.6.m6.3.4.3.2.1" stretchy="false" xref="S4.SS3.SSS0.Px2.p2.6.m6.3.4.3.1.cmml">{</mo><mn id="S4.SS3.SSS0.Px2.p2.6.m6.1.1" xref="S4.SS3.SSS0.Px2.p2.6.m6.1.1.cmml">1</mn><mo id="S4.SS3.SSS0.Px2.p2.6.m6.3.4.3.2.2" xref="S4.SS3.SSS0.Px2.p2.6.m6.3.4.3.1.cmml">,</mo><mi id="S4.SS3.SSS0.Px2.p2.6.m6.2.2" mathvariant="normal" xref="S4.SS3.SSS0.Px2.p2.6.m6.2.2.cmml">…</mi><mo id="S4.SS3.SSS0.Px2.p2.6.m6.3.4.3.2.3" xref="S4.SS3.SSS0.Px2.p2.6.m6.3.4.3.1.cmml">,</mo><mn id="S4.SS3.SSS0.Px2.p2.6.m6.3.3" xref="S4.SS3.SSS0.Px2.p2.6.m6.3.3.cmml">20</mn><mo id="S4.SS3.SSS0.Px2.p2.6.m6.3.4.3.2.4" stretchy="false" xref="S4.SS3.SSS0.Px2.p2.6.m6.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p2.6.m6.3b"><apply id="S4.SS3.SSS0.Px2.p2.6.m6.3.4.cmml" xref="S4.SS3.SSS0.Px2.p2.6.m6.3.4"><in id="S4.SS3.SSS0.Px2.p2.6.m6.3.4.1.cmml" xref="S4.SS3.SSS0.Px2.p2.6.m6.3.4.1"></in><ci id="S4.SS3.SSS0.Px2.p2.6.m6.3.4.2.cmml" xref="S4.SS3.SSS0.Px2.p2.6.m6.3.4.2">𝑖</ci><set id="S4.SS3.SSS0.Px2.p2.6.m6.3.4.3.1.cmml" xref="S4.SS3.SSS0.Px2.p2.6.m6.3.4.3.2"><cn id="S4.SS3.SSS0.Px2.p2.6.m6.1.1.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p2.6.m6.1.1">1</cn><ci id="S4.SS3.SSS0.Px2.p2.6.m6.2.2.cmml" xref="S4.SS3.SSS0.Px2.p2.6.m6.2.2">…</ci><cn id="S4.SS3.SSS0.Px2.p2.6.m6.3.3.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p2.6.m6.3.3">20</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p2.6.m6.3c">i\in\{1,\ldots,20\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p2.6.m6.3d">italic_i ∈ { 1 , … , 20 }</annotation></semantics></math>, the vertices <math alttext="i" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p2.7.m7.1"><semantics id="S4.SS3.SSS0.Px2.p2.7.m7.1a"><mi id="S4.SS3.SSS0.Px2.p2.7.m7.1.1" xref="S4.SS3.SSS0.Px2.p2.7.m7.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p2.7.m7.1b"><ci id="S4.SS3.SSS0.Px2.p2.7.m7.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.7.m7.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p2.7.m7.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p2.7.m7.1d">italic_i</annotation></semantics></math> and <math alttext="i^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p2.8.m8.1"><semantics id="S4.SS3.SSS0.Px2.p2.8.m8.1a"><msup id="S4.SS3.SSS0.Px2.p2.8.m8.1.1" xref="S4.SS3.SSS0.Px2.p2.8.m8.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p2.8.m8.1.1.2" xref="S4.SS3.SSS0.Px2.p2.8.m8.1.1.2.cmml">i</mi><mo id="S4.SS3.SSS0.Px2.p2.8.m8.1.1.3" xref="S4.SS3.SSS0.Px2.p2.8.m8.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p2.8.m8.1b"><apply id="S4.SS3.SSS0.Px2.p2.8.m8.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p2.8.m8.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.8.m8.1.1">superscript</csymbol><ci id="S4.SS3.SSS0.Px2.p2.8.m8.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p2.8.m8.1.1.2">𝑖</ci><ci id="S4.SS3.SSS0.Px2.p2.8.m8.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p2.8.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p2.8.m8.1c">i^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p2.8.m8.1d">italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> have bias <math alttext="b_{i}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p2.9.m9.1"><semantics id="S4.SS3.SSS0.Px2.p2.9.m9.1a"><msub id="S4.SS3.SSS0.Px2.p2.9.m9.1.1" xref="S4.SS3.SSS0.Px2.p2.9.m9.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p2.9.m9.1.1.2" xref="S4.SS3.SSS0.Px2.p2.9.m9.1.1.2.cmml">b</mi><mi id="S4.SS3.SSS0.Px2.p2.9.m9.1.1.3" xref="S4.SS3.SSS0.Px2.p2.9.m9.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p2.9.m9.1b"><apply id="S4.SS3.SSS0.Px2.p2.9.m9.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p2.9.m9.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.9.m9.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p2.9.m9.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p2.9.m9.1.1.2">𝑏</ci><ci id="S4.SS3.SSS0.Px2.p2.9.m9.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p2.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p2.9.m9.1c">b_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p2.9.m9.1d">italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. (We treat the biases of the nonexistent vertices <math alttext="1^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p2.10.m10.1"><semantics id="S4.SS3.SSS0.Px2.p2.10.m10.1a"><msup id="S4.SS3.SSS0.Px2.p2.10.m10.1.1" xref="S4.SS3.SSS0.Px2.p2.10.m10.1.1.cmml"><mn id="S4.SS3.SSS0.Px2.p2.10.m10.1.1.2" xref="S4.SS3.SSS0.Px2.p2.10.m10.1.1.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px2.p2.10.m10.1.1.3" xref="S4.SS3.SSS0.Px2.p2.10.m10.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p2.10.m10.1b"><apply id="S4.SS3.SSS0.Px2.p2.10.m10.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p2.10.m10.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.10.m10.1.1">superscript</csymbol><cn id="S4.SS3.SSS0.Px2.p2.10.m10.1.1.2.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p2.10.m10.1.1.2">1</cn><ci id="S4.SS3.SSS0.Px2.p2.10.m10.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p2.10.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p2.10.m10.1c">1^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p2.10.m10.1d">1 start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="2^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p2.11.m11.1"><semantics id="S4.SS3.SSS0.Px2.p2.11.m11.1a"><msup id="S4.SS3.SSS0.Px2.p2.11.m11.1.1" xref="S4.SS3.SSS0.Px2.p2.11.m11.1.1.cmml"><mn id="S4.SS3.SSS0.Px2.p2.11.m11.1.1.2" xref="S4.SS3.SSS0.Px2.p2.11.m11.1.1.2.cmml">2</mn><mo id="S4.SS3.SSS0.Px2.p2.11.m11.1.1.3" xref="S4.SS3.SSS0.Px2.p2.11.m11.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p2.11.m11.1b"><apply id="S4.SS3.SSS0.Px2.p2.11.m11.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p2.11.m11.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.11.m11.1.1">superscript</csymbol><cn id="S4.SS3.SSS0.Px2.p2.11.m11.1.1.2.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p2.11.m11.1.1.2">2</cn><ci id="S4.SS3.SSS0.Px2.p2.11.m11.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p2.11.m11.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p2.11.m11.1c">2^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p2.11.m11.1d">2 start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="19" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p2.12.m12.1"><semantics id="S4.SS3.SSS0.Px2.p2.12.m12.1a"><mn id="S4.SS3.SSS0.Px2.p2.12.m12.1.1" xref="S4.SS3.SSS0.Px2.p2.12.m12.1.1.cmml">19</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p2.12.m12.1b"><cn id="S4.SS3.SSS0.Px2.p2.12.m12.1.1.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p2.12.m12.1.1">19</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p2.12.m12.1c">19</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p2.12.m12.1d">19</annotation></semantics></math>, and <math alttext="20" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p2.13.m13.1"><semantics id="S4.SS3.SSS0.Px2.p2.13.m13.1a"><mn id="S4.SS3.SSS0.Px2.p2.13.m13.1.1" xref="S4.SS3.SSS0.Px2.p2.13.m13.1.1.cmml">20</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p2.13.m13.1b"><cn id="S4.SS3.SSS0.Px2.p2.13.m13.1.1.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p2.13.m13.1.1">20</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p2.13.m13.1c">20</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p2.13.m13.1d">20</annotation></semantics></math> as vacuous.) Further, <math alttext="G_{\mathrm{LP}}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p2.14.m14.1"><semantics id="S4.SS3.SSS0.Px2.p2.14.m14.1a"><msub id="S4.SS3.SSS0.Px2.p2.14.m14.1.1" xref="S4.SS3.SSS0.Px2.p2.14.m14.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p2.14.m14.1.1.2" xref="S4.SS3.SSS0.Px2.p2.14.m14.1.1.2.cmml">G</mi><mi id="S4.SS3.SSS0.Px2.p2.14.m14.1.1.3" xref="S4.SS3.SSS0.Px2.p2.14.m14.1.1.3.cmml">LP</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p2.14.m14.1b"><apply id="S4.SS3.SSS0.Px2.p2.14.m14.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p2.14.m14.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p2.14.m14.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p2.14.m14.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p2.14.m14.1.1.2">𝐺</ci><ci id="S4.SS3.SSS0.Px2.p2.14.m14.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p2.14.m14.1.1.3">LP</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p2.14.m14.1c">G_{\mathrm{LP}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p2.14.m14.1d">italic_G start_POSTSUBSCRIPT roman_LP end_POSTSUBSCRIPT</annotation></semantics></math> has the following very simple unweighted edge structure:</p> <ol class="ltx_enumerate" id="S4.I2"> <li class="ltx_item" id="S4.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S4.I2.i1.p1"> <p class="ltx_p" id="S4.I2.i1.p1.2">For each <math alttext="i\in\{1,\ldots,18\}" class="ltx_Math" display="inline" id="S4.I2.i1.p1.1.m1.3"><semantics id="S4.I2.i1.p1.1.m1.3a"><mrow id="S4.I2.i1.p1.1.m1.3.4" xref="S4.I2.i1.p1.1.m1.3.4.cmml"><mi id="S4.I2.i1.p1.1.m1.3.4.2" xref="S4.I2.i1.p1.1.m1.3.4.2.cmml">i</mi><mo id="S4.I2.i1.p1.1.m1.3.4.1" xref="S4.I2.i1.p1.1.m1.3.4.1.cmml">∈</mo><mrow id="S4.I2.i1.p1.1.m1.3.4.3.2" xref="S4.I2.i1.p1.1.m1.3.4.3.1.cmml"><mo id="S4.I2.i1.p1.1.m1.3.4.3.2.1" stretchy="false" xref="S4.I2.i1.p1.1.m1.3.4.3.1.cmml">{</mo><mn id="S4.I2.i1.p1.1.m1.1.1" xref="S4.I2.i1.p1.1.m1.1.1.cmml">1</mn><mo id="S4.I2.i1.p1.1.m1.3.4.3.2.2" xref="S4.I2.i1.p1.1.m1.3.4.3.1.cmml">,</mo><mi id="S4.I2.i1.p1.1.m1.2.2" mathvariant="normal" xref="S4.I2.i1.p1.1.m1.2.2.cmml">…</mi><mo id="S4.I2.i1.p1.1.m1.3.4.3.2.3" xref="S4.I2.i1.p1.1.m1.3.4.3.1.cmml">,</mo><mn id="S4.I2.i1.p1.1.m1.3.3" xref="S4.I2.i1.p1.1.m1.3.3.cmml">18</mn><mo id="S4.I2.i1.p1.1.m1.3.4.3.2.4" stretchy="false" xref="S4.I2.i1.p1.1.m1.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i1.p1.1.m1.3b"><apply id="S4.I2.i1.p1.1.m1.3.4.cmml" xref="S4.I2.i1.p1.1.m1.3.4"><in id="S4.I2.i1.p1.1.m1.3.4.1.cmml" xref="S4.I2.i1.p1.1.m1.3.4.1"></in><ci id="S4.I2.i1.p1.1.m1.3.4.2.cmml" xref="S4.I2.i1.p1.1.m1.3.4.2">𝑖</ci><set id="S4.I2.i1.p1.1.m1.3.4.3.1.cmml" xref="S4.I2.i1.p1.1.m1.3.4.3.2"><cn id="S4.I2.i1.p1.1.m1.1.1.cmml" type="integer" xref="S4.I2.i1.p1.1.m1.1.1">1</cn><ci id="S4.I2.i1.p1.1.m1.2.2.cmml" xref="S4.I2.i1.p1.1.m1.2.2">…</ci><cn id="S4.I2.i1.p1.1.m1.3.3.cmml" type="integer" xref="S4.I2.i1.p1.1.m1.3.3">18</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i1.p1.1.m1.3c">i\in\{1,\ldots,18\}</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i1.p1.1.m1.3d">italic_i ∈ { 1 , … , 18 }</annotation></semantics></math>, there is an edge <math alttext="i\to(i+2)^{\prime}" class="ltx_Math" display="inline" id="S4.I2.i1.p1.2.m2.1"><semantics id="S4.I2.i1.p1.2.m2.1a"><mrow id="S4.I2.i1.p1.2.m2.1.1" xref="S4.I2.i1.p1.2.m2.1.1.cmml"><mi id="S4.I2.i1.p1.2.m2.1.1.3" xref="S4.I2.i1.p1.2.m2.1.1.3.cmml">i</mi><mo id="S4.I2.i1.p1.2.m2.1.1.2" stretchy="false" xref="S4.I2.i1.p1.2.m2.1.1.2.cmml">→</mo><msup id="S4.I2.i1.p1.2.m2.1.1.1" xref="S4.I2.i1.p1.2.m2.1.1.1.cmml"><mrow id="S4.I2.i1.p1.2.m2.1.1.1.1.1" xref="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="S4.I2.i1.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.I2.i1.p1.2.m2.1.1.1.1.1.1" xref="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.2" xref="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.2.cmml">i</mi><mo id="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.1" xref="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.1.cmml">+</mo><mn id="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.3" xref="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.3.cmml">2</mn></mrow><mo id="S4.I2.i1.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.I2.i1.p1.2.m2.1.1.1.3" xref="S4.I2.i1.p1.2.m2.1.1.1.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i1.p1.2.m2.1b"><apply id="S4.I2.i1.p1.2.m2.1.1.cmml" xref="S4.I2.i1.p1.2.m2.1.1"><ci id="S4.I2.i1.p1.2.m2.1.1.2.cmml" xref="S4.I2.i1.p1.2.m2.1.1.2">→</ci><ci id="S4.I2.i1.p1.2.m2.1.1.3.cmml" xref="S4.I2.i1.p1.2.m2.1.1.3">𝑖</ci><apply id="S4.I2.i1.p1.2.m2.1.1.1.cmml" xref="S4.I2.i1.p1.2.m2.1.1.1"><csymbol cd="ambiguous" id="S4.I2.i1.p1.2.m2.1.1.1.2.cmml" xref="S4.I2.i1.p1.2.m2.1.1.1">superscript</csymbol><apply id="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S4.I2.i1.p1.2.m2.1.1.1.1.1"><plus id="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.1"></plus><ci id="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.2">𝑖</ci><cn id="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.I2.i1.p1.2.m2.1.1.1.1.1.1.3">2</cn></apply><ci id="S4.I2.i1.p1.2.m2.1.1.1.3.cmml" xref="S4.I2.i1.p1.2.m2.1.1.1.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i1.p1.2.m2.1c">i\to(i+2)^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i1.p1.2.m2.1d">italic_i → ( italic_i + 2 ) start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S4.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S4.I2.i2.p1"> <p class="ltx_p" id="S4.I2.i2.p1.2">For each <math alttext="i\in\{1,\ldots,17\}" class="ltx_Math" display="inline" id="S4.I2.i2.p1.1.m1.3"><semantics id="S4.I2.i2.p1.1.m1.3a"><mrow id="S4.I2.i2.p1.1.m1.3.4" xref="S4.I2.i2.p1.1.m1.3.4.cmml"><mi id="S4.I2.i2.p1.1.m1.3.4.2" xref="S4.I2.i2.p1.1.m1.3.4.2.cmml">i</mi><mo id="S4.I2.i2.p1.1.m1.3.4.1" xref="S4.I2.i2.p1.1.m1.3.4.1.cmml">∈</mo><mrow id="S4.I2.i2.p1.1.m1.3.4.3.2" xref="S4.I2.i2.p1.1.m1.3.4.3.1.cmml"><mo id="S4.I2.i2.p1.1.m1.3.4.3.2.1" stretchy="false" xref="S4.I2.i2.p1.1.m1.3.4.3.1.cmml">{</mo><mn id="S4.I2.i2.p1.1.m1.1.1" xref="S4.I2.i2.p1.1.m1.1.1.cmml">1</mn><mo id="S4.I2.i2.p1.1.m1.3.4.3.2.2" xref="S4.I2.i2.p1.1.m1.3.4.3.1.cmml">,</mo><mi id="S4.I2.i2.p1.1.m1.2.2" mathvariant="normal" xref="S4.I2.i2.p1.1.m1.2.2.cmml">…</mi><mo id="S4.I2.i2.p1.1.m1.3.4.3.2.3" xref="S4.I2.i2.p1.1.m1.3.4.3.1.cmml">,</mo><mn id="S4.I2.i2.p1.1.m1.3.3" xref="S4.I2.i2.p1.1.m1.3.3.cmml">17</mn><mo id="S4.I2.i2.p1.1.m1.3.4.3.2.4" stretchy="false" xref="S4.I2.i2.p1.1.m1.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i2.p1.1.m1.3b"><apply id="S4.I2.i2.p1.1.m1.3.4.cmml" xref="S4.I2.i2.p1.1.m1.3.4"><in id="S4.I2.i2.p1.1.m1.3.4.1.cmml" xref="S4.I2.i2.p1.1.m1.3.4.1"></in><ci id="S4.I2.i2.p1.1.m1.3.4.2.cmml" xref="S4.I2.i2.p1.1.m1.3.4.2">𝑖</ci><set id="S4.I2.i2.p1.1.m1.3.4.3.1.cmml" xref="S4.I2.i2.p1.1.m1.3.4.3.2"><cn id="S4.I2.i2.p1.1.m1.1.1.cmml" type="integer" xref="S4.I2.i2.p1.1.m1.1.1">1</cn><ci id="S4.I2.i2.p1.1.m1.2.2.cmml" xref="S4.I2.i2.p1.1.m1.2.2">…</ci><cn id="S4.I2.i2.p1.1.m1.3.3.cmml" type="integer" xref="S4.I2.i2.p1.1.m1.3.3">17</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i2.p1.1.m1.3c">i\in\{1,\ldots,17\}</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.p1.1.m1.3d">italic_i ∈ { 1 , … , 17 }</annotation></semantics></math>, there is an edge <math alttext="(i+3)^{\prime}\to i" class="ltx_Math" display="inline" id="S4.I2.i2.p1.2.m2.1"><semantics id="S4.I2.i2.p1.2.m2.1a"><mrow id="S4.I2.i2.p1.2.m2.1.1" xref="S4.I2.i2.p1.2.m2.1.1.cmml"><msup id="S4.I2.i2.p1.2.m2.1.1.1" xref="S4.I2.i2.p1.2.m2.1.1.1.cmml"><mrow id="S4.I2.i2.p1.2.m2.1.1.1.1.1" xref="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="S4.I2.i2.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.I2.i2.p1.2.m2.1.1.1.1.1.1" xref="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.2" xref="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.2.cmml">i</mi><mo id="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.1" xref="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.1.cmml">+</mo><mn id="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.3" xref="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.3.cmml">3</mn></mrow><mo id="S4.I2.i2.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.I2.i2.p1.2.m2.1.1.1.3" xref="S4.I2.i2.p1.2.m2.1.1.1.3.cmml">′</mo></msup><mo id="S4.I2.i2.p1.2.m2.1.1.2" stretchy="false" xref="S4.I2.i2.p1.2.m2.1.1.2.cmml">→</mo><mi id="S4.I2.i2.p1.2.m2.1.1.3" xref="S4.I2.i2.p1.2.m2.1.1.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i2.p1.2.m2.1b"><apply id="S4.I2.i2.p1.2.m2.1.1.cmml" xref="S4.I2.i2.p1.2.m2.1.1"><ci id="S4.I2.i2.p1.2.m2.1.1.2.cmml" xref="S4.I2.i2.p1.2.m2.1.1.2">→</ci><apply id="S4.I2.i2.p1.2.m2.1.1.1.cmml" xref="S4.I2.i2.p1.2.m2.1.1.1"><csymbol cd="ambiguous" id="S4.I2.i2.p1.2.m2.1.1.1.2.cmml" xref="S4.I2.i2.p1.2.m2.1.1.1">superscript</csymbol><apply id="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S4.I2.i2.p1.2.m2.1.1.1.1.1"><plus id="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.1"></plus><ci id="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.2">𝑖</ci><cn id="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.I2.i2.p1.2.m2.1.1.1.1.1.1.3">3</cn></apply><ci id="S4.I2.i2.p1.2.m2.1.1.1.3.cmml" xref="S4.I2.i2.p1.2.m2.1.1.1.3">′</ci></apply><ci id="S4.I2.i2.p1.2.m2.1.1.3.cmml" xref="S4.I2.i2.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i2.p1.2.m2.1c">(i+3)^{\prime}\to i</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.p1.2.m2.1d">( italic_i + 3 ) start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT → italic_i</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S4.I2.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S4.I2.i3.p1"> <p class="ltx_p" id="S4.I2.i3.p1.5">There are edges <math alttext="3^{\prime}\to 1" class="ltx_Math" display="inline" id="S4.I2.i3.p1.1.m1.1"><semantics id="S4.I2.i3.p1.1.m1.1a"><mrow id="S4.I2.i3.p1.1.m1.1.1" xref="S4.I2.i3.p1.1.m1.1.1.cmml"><msup id="S4.I2.i3.p1.1.m1.1.1.2" xref="S4.I2.i3.p1.1.m1.1.1.2.cmml"><mn id="S4.I2.i3.p1.1.m1.1.1.2.2" xref="S4.I2.i3.p1.1.m1.1.1.2.2.cmml">3</mn><mo id="S4.I2.i3.p1.1.m1.1.1.2.3" xref="S4.I2.i3.p1.1.m1.1.1.2.3.cmml">′</mo></msup><mo id="S4.I2.i3.p1.1.m1.1.1.1" stretchy="false" xref="S4.I2.i3.p1.1.m1.1.1.1.cmml">→</mo><mn id="S4.I2.i3.p1.1.m1.1.1.3" xref="S4.I2.i3.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i3.p1.1.m1.1b"><apply id="S4.I2.i3.p1.1.m1.1.1.cmml" xref="S4.I2.i3.p1.1.m1.1.1"><ci id="S4.I2.i3.p1.1.m1.1.1.1.cmml" xref="S4.I2.i3.p1.1.m1.1.1.1">→</ci><apply id="S4.I2.i3.p1.1.m1.1.1.2.cmml" xref="S4.I2.i3.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.I2.i3.p1.1.m1.1.1.2.1.cmml" xref="S4.I2.i3.p1.1.m1.1.1.2">superscript</csymbol><cn id="S4.I2.i3.p1.1.m1.1.1.2.2.cmml" type="integer" xref="S4.I2.i3.p1.1.m1.1.1.2.2">3</cn><ci id="S4.I2.i3.p1.1.m1.1.1.2.3.cmml" xref="S4.I2.i3.p1.1.m1.1.1.2.3">′</ci></apply><cn id="S4.I2.i3.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.I2.i3.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i3.p1.1.m1.1c">3^{\prime}\to 1</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i3.p1.1.m1.1d">3 start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT → 1</annotation></semantics></math> and <math alttext="20^{\prime}" class="ltx_Math" display="inline" id="S4.I2.i3.p1.2.m2.1"><semantics id="S4.I2.i3.p1.2.m2.1a"><msup id="S4.I2.i3.p1.2.m2.1.1" xref="S4.I2.i3.p1.2.m2.1.1.cmml"><mn id="S4.I2.i3.p1.2.m2.1.1.2" xref="S4.I2.i3.p1.2.m2.1.1.2.cmml">20</mn><mo id="S4.I2.i3.p1.2.m2.1.1.3" xref="S4.I2.i3.p1.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I2.i3.p1.2.m2.1b"><apply id="S4.I2.i3.p1.2.m2.1.1.cmml" xref="S4.I2.i3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.I2.i3.p1.2.m2.1.1.1.cmml" xref="S4.I2.i3.p1.2.m2.1.1">superscript</csymbol><cn id="S4.I2.i3.p1.2.m2.1.1.2.cmml" type="integer" xref="S4.I2.i3.p1.2.m2.1.1.2">20</cn><ci id="S4.I2.i3.p1.2.m2.1.1.3.cmml" xref="S4.I2.i3.p1.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i3.p1.2.m2.1c">20^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i3.p1.2.m2.1d">20 start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="18" class="ltx_Math" display="inline" id="S4.I2.i3.p1.3.m3.1"><semantics id="S4.I2.i3.p1.3.m3.1a"><mn id="S4.I2.i3.p1.3.m3.1.1" xref="S4.I2.i3.p1.3.m3.1.1.cmml">18</mn><annotation-xml encoding="MathML-Content" id="S4.I2.i3.p1.3.m3.1b"><cn id="S4.I2.i3.p1.3.m3.1.1.cmml" type="integer" xref="S4.I2.i3.p1.3.m3.1.1">18</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i3.p1.3.m3.1c">18</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i3.p1.3.m3.1d">18</annotation></semantics></math>. (Note that <math alttext="3^{\prime}" class="ltx_Math" display="inline" id="S4.I2.i3.p1.4.m4.1"><semantics id="S4.I2.i3.p1.4.m4.1a"><msup id="S4.I2.i3.p1.4.m4.1.1" xref="S4.I2.i3.p1.4.m4.1.1.cmml"><mn id="S4.I2.i3.p1.4.m4.1.1.2" xref="S4.I2.i3.p1.4.m4.1.1.2.cmml">3</mn><mo id="S4.I2.i3.p1.4.m4.1.1.3" xref="S4.I2.i3.p1.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I2.i3.p1.4.m4.1b"><apply id="S4.I2.i3.p1.4.m4.1.1.cmml" xref="S4.I2.i3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.I2.i3.p1.4.m4.1.1.1.cmml" xref="S4.I2.i3.p1.4.m4.1.1">superscript</csymbol><cn id="S4.I2.i3.p1.4.m4.1.1.2.cmml" type="integer" xref="S4.I2.i3.p1.4.m4.1.1.2">3</cn><ci id="S4.I2.i3.p1.4.m4.1.1.3.cmml" xref="S4.I2.i3.p1.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i3.p1.4.m4.1c">3^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i3.p1.4.m4.1d">3 start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="20^{\prime}" class="ltx_Math" display="inline" id="S4.I2.i3.p1.5.m5.1"><semantics id="S4.I2.i3.p1.5.m5.1a"><msup id="S4.I2.i3.p1.5.m5.1.1" xref="S4.I2.i3.p1.5.m5.1.1.cmml"><mn id="S4.I2.i3.p1.5.m5.1.1.2" xref="S4.I2.i3.p1.5.m5.1.1.2.cmml">20</mn><mo id="S4.I2.i3.p1.5.m5.1.1.3" xref="S4.I2.i3.p1.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I2.i3.p1.5.m5.1b"><apply id="S4.I2.i3.p1.5.m5.1.1.cmml" xref="S4.I2.i3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.I2.i3.p1.5.m5.1.1.1.cmml" xref="S4.I2.i3.p1.5.m5.1.1">superscript</csymbol><cn id="S4.I2.i3.p1.5.m5.1.1.2.cmml" type="integer" xref="S4.I2.i3.p1.5.m5.1.1.2">20</cn><ci id="S4.I2.i3.p1.5.m5.1.1.3.cmml" xref="S4.I2.i3.p1.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i3.p1.5.m5.1c">20^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i3.p1.5.m5.1d">20 start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> have no edges of type (2).)</p> </div> </li> </ol> </div> <div class="ltx_para" id="S4.SS3.SSS0.Px2.p3"> <p class="ltx_p" id="S4.SS3.SSS0.Px2.p3.4">One may verify that the bias constraints and the edge structure together determine every edge weight up to vertex rescaling: E.g., if a vertex <math alttext="v" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p3.1.m1.1"><semantics id="S4.SS3.SSS0.Px2.p3.1.m1.1a"><mi id="S4.SS3.SSS0.Px2.p3.1.m1.1.1" xref="S4.SS3.SSS0.Px2.p3.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p3.1.m1.1b"><ci id="S4.SS3.SSS0.Px2.p3.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p3.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p3.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p3.1.m1.1d">italic_v</annotation></semantics></math> has bias <math alttext="b" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p3.2.m2.1"><semantics id="S4.SS3.SSS0.Px2.p3.2.m2.1a"><mi id="S4.SS3.SSS0.Px2.p3.2.m2.1.1" xref="S4.SS3.SSS0.Px2.p3.2.m2.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p3.2.m2.1b"><ci id="S4.SS3.SSS0.Px2.p3.2.m2.1.1.cmml" xref="S4.SS3.SSS0.Px2.p3.2.m2.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p3.2.m2.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p3.2.m2.1d">italic_b</annotation></semantics></math> and indegree <math alttext="w" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p3.3.m3.1"><semantics id="S4.SS3.SSS0.Px2.p3.3.m3.1a"><mi id="S4.SS3.SSS0.Px2.p3.3.m3.1.1" xref="S4.SS3.SSS0.Px2.p3.3.m3.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p3.3.m3.1b"><ci id="S4.SS3.SSS0.Px2.p3.3.m3.1.1.cmml" xref="S4.SS3.SSS0.Px2.p3.3.m3.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p3.3.m3.1c">w</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p3.3.m3.1d">italic_w</annotation></semantics></math>, then it must have outdegree <math alttext="w\frac{1+b}{1-b}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p3.4.m4.1"><semantics id="S4.SS3.SSS0.Px2.p3.4.m4.1a"><mrow id="S4.SS3.SSS0.Px2.p3.4.m4.1.1" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.2" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.2.cmml">w</mi><mo id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.1" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.1.cmml">⁢</mo><mfrac id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.cmml"><mrow id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2.cmml"><mn id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2.2" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2.1" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2.1.cmml">+</mo><mi id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2.3" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2.3.cmml">b</mi></mrow><mrow id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3.cmml"><mn id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3.2" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3.1" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3.1.cmml">−</mo><mi id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3.3" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3.3.cmml">b</mi></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p3.4.m4.1b"><apply id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.cmml" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1"><times id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.1"></times><ci id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.2">𝑤</ci><apply id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3"><divide id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3"></divide><apply id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2.cmml" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2"><plus id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2.1.cmml" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2.1"></plus><cn id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2.2.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2.2">1</cn><ci id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2.3.cmml" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.2.3">𝑏</ci></apply><apply id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3.cmml" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3"><minus id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3.1.cmml" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3.1"></minus><cn id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3.2.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3.2">1</cn><ci id="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3.3.cmml" xref="S4.SS3.SSS0.Px2.p3.4.m4.1.1.3.3.3">𝑏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p3.4.m4.1c">w\frac{1+b}{1-b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p3.4.m4.1d">italic_w divide start_ARG 1 + italic_b end_ARG start_ARG 1 - italic_b end_ARG</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S4.F4"><svg class="ltx_picture ltx_centering" height="122.72" id="S4.F4.pic1" overflow="visible" version="1.1" width="626.66"><g transform="translate(0,122.72) matrix(1 0 0 -1 0 0) translate(-17.38,0) translate(0,14.11)"><g fill="#000000" stroke="#000000" stroke-width="0.4pt"><path d="M 45.33 0 C 45.33 7.64 39.14 13.84 31.5 13.84 C 23.85 13.84 17.66 7.64 17.66 0 C 17.66 -7.64 23.85 -13.84 31.5 -13.84 C 39.14 -13.84 45.33 -7.64 45.33 0 Z M 31.5 0" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 28.04 -4.46)"><foreignobject height="8.92" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="6.92"><math alttext="1" class="ltx_Math" display="inline" id="S4.F4.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics 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style="font-size:90%;">Figure 4</span>: </span><span class="ltx_text" id="S4.F4.3.2" style="font-size:90%;">The (unweighted version) of the graph used in Case 2 of the proof of <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem7" title="Theorem 1.7 (Lower bound for PL sigmoid selection with arbitrary intercept). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.7</span></a>.</span></figcaption> </figure> <div class="ltx_para" id="S4.SS3.SSS0.Px2.p4"> <p class="ltx_p" id="S4.SS3.SSS0.Px2.p4.12">Since <math alttext="G_{\mathrm{LP}}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.1.m1.1"><semantics id="S4.SS3.SSS0.Px2.p4.1.m1.1a"><msub id="S4.SS3.SSS0.Px2.p4.1.m1.1.1" xref="S4.SS3.SSS0.Px2.p4.1.m1.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p4.1.m1.1.1.2" xref="S4.SS3.SSS0.Px2.p4.1.m1.1.1.2.cmml">G</mi><mi 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sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT )</annotation></semantics></math> is a quadratic function of the selection probabilities <math alttext="\{\mathsf{PLSigmoid}_{b}(v)_{v\in V(G)}\}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.4.m4.3"><semantics id="S4.SS3.SSS0.Px2.p4.4.m4.3a"><mrow id="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1" xref="S4.SS3.SSS0.Px2.p4.4.m4.3.3.2.cmml"><mo id="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.2" stretchy="false" xref="S4.SS3.SSS0.Px2.p4.4.m4.3.3.2.cmml">{</mo><mrow id="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1" xref="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.cmml"><msub id="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.2" xref="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.2.cmml"><mi id="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.2.2" xref="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.2.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.2.3" xref="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.2.3.cmml">b</mi></msub><mo id="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.1" 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xref="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.1"></times><apply id="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.2">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.2.2.cmml" xref="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.2.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.2.3.cmml" xref="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.2.3">𝑏</ci></apply><apply id="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px2.p4.4.m4.3.3.1.1.3">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.4.m4.2.2.cmml" xref="S4.SS3.SSS0.Px2.p4.4.m4.2.2">𝑣</ci><apply id="S4.SS3.SSS0.Px2.p4.4.m4.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.4.m4.1.1.1"><in id="S4.SS3.SSS0.Px2.p4.4.m4.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.4.m4.1.1.1.2"></in><ci id="S4.SS3.SSS0.Px2.p4.4.m4.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.4.m4.1.1.1.3">𝑣</ci><apply id="S4.SS3.SSS0.Px2.p4.4.m4.1.1.1.4.cmml" xref="S4.SS3.SSS0.Px2.p4.4.m4.1.1.1.4"><times id="S4.SS3.SSS0.Px2.p4.4.m4.1.1.1.4.1.cmml" xref="S4.SS3.SSS0.Px2.p4.4.m4.1.1.1.4.1"></times><ci id="S4.SS3.SSS0.Px2.p4.4.m4.1.1.1.4.2.cmml" xref="S4.SS3.SSS0.Px2.p4.4.m4.1.1.1.4.2">𝑉</ci><ci id="S4.SS3.SSS0.Px2.p4.4.m4.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.4.m4.1.1.1.1">𝐺</ci></apply></apply></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.4.m4.3c">\{\mathsf{PLSigmoid}_{b}(v)_{v\in V(G)}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.4.m4.3d">{ sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_v ) start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_G ) end_POSTSUBSCRIPT }</annotation></semantics></math>. Further, recall that vertices in <math alttext="G_{\mathrm{LP}}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.5.m5.1"><semantics id="S4.SS3.SSS0.Px2.p4.5.m5.1a"><msub id="S4.SS3.SSS0.Px2.p4.5.m5.1.1" xref="S4.SS3.SSS0.Px2.p4.5.m5.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p4.5.m5.1.1.2" xref="S4.SS3.SSS0.Px2.p4.5.m5.1.1.2.cmml">G</mi><mi id="S4.SS3.SSS0.Px2.p4.5.m5.1.1.3" xref="S4.SS3.SSS0.Px2.p4.5.m5.1.1.3.cmml">LP</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p4.5.m5.1b"><apply id="S4.SS3.SSS0.Px2.p4.5.m5.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.5.m5.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.5.m5.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.5.m5.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.5.m5.1.1.2">𝐺</ci><ci id="S4.SS3.SSS0.Px2.p4.5.m5.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.5.m5.1.1.3">LP</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.5.m5.1c">G_{\mathrm{LP}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.5.m5.1d">italic_G start_POSTSUBSCRIPT roman_LP end_POSTSUBSCRIPT</annotation></semantics></math> have biases <math alttext="\{b_{1},\ldots,b_{20}\}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.6.m6.3"><semantics id="S4.SS3.SSS0.Px2.p4.6.m6.3a"><mrow id="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2" xref="S4.SS3.SSS0.Px2.p4.6.m6.3.3.3.cmml"><mo id="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.3" stretchy="false" xref="S4.SS3.SSS0.Px2.p4.6.m6.3.3.3.cmml">{</mo><msub id="S4.SS3.SSS0.Px2.p4.6.m6.2.2.1.1" xref="S4.SS3.SSS0.Px2.p4.6.m6.2.2.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p4.6.m6.2.2.1.1.2" xref="S4.SS3.SSS0.Px2.p4.6.m6.2.2.1.1.2.cmml">b</mi><mn id="S4.SS3.SSS0.Px2.p4.6.m6.2.2.1.1.3" xref="S4.SS3.SSS0.Px2.p4.6.m6.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.4" xref="S4.SS3.SSS0.Px2.p4.6.m6.3.3.3.cmml">,</mo><mi id="S4.SS3.SSS0.Px2.p4.6.m6.1.1" mathvariant="normal" xref="S4.SS3.SSS0.Px2.p4.6.m6.1.1.cmml">…</mi><mo id="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.5" xref="S4.SS3.SSS0.Px2.p4.6.m6.3.3.3.cmml">,</mo><msub id="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.2" xref="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.2.cmml"><mi id="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.2.2" xref="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.2.2.cmml">b</mi><mn id="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.2.3" xref="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.2.3.cmml">20</mn></msub><mo id="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.6" stretchy="false" xref="S4.SS3.SSS0.Px2.p4.6.m6.3.3.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p4.6.m6.3b"><set id="S4.SS3.SSS0.Px2.p4.6.m6.3.3.3.cmml" xref="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2"><apply id="S4.SS3.SSS0.Px2.p4.6.m6.2.2.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.6.m6.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.6.m6.2.2.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.6.m6.2.2.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.6.m6.2.2.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.6.m6.2.2.1.1.2">𝑏</ci><cn id="S4.SS3.SSS0.Px2.p4.6.m6.2.2.1.1.3.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p4.6.m6.2.2.1.1.3">1</cn></apply><ci id="S4.SS3.SSS0.Px2.p4.6.m6.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.6.m6.1.1">…</ci><apply id="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.2.cmml" xref="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.2.1.cmml" xref="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.2">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.2.2.cmml" xref="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.2.2">𝑏</ci><cn id="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p4.6.m6.3.3.2.2.3">20</cn></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.6.m6.3c">\{b_{1},\ldots,b_{20}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.6.m6.3d">{ italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_b start_POSTSUBSCRIPT 20 end_POSTSUBSCRIPT }</annotation></semantics></math>, so we are interested in the selection probabilities <math alttext="\{\mathsf{PLSigmoid}_{b}(b_{i})\}_{i\in[20]}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.7.m7.2"><semantics id="S4.SS3.SSS0.Px2.p4.7.m7.2a"><msub id="S4.SS3.SSS0.Px2.p4.7.m7.2.2" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.cmml"><mrow id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.2.cmml"><mo id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.2" stretchy="false" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.2.cmml">{</mo><mrow id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.cmml"><msub id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.3" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.3.cmml"><mi id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.3.2" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.3.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.3.3" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.3.3.cmml">b</mi></msub><mo id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.2" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.1.cmml"><mo id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.2" stretchy="false" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.1" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.1.2" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.1.2.cmml">b</mi><mi id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.1.3" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.3" stretchy="false" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.3" stretchy="false" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.2.cmml">}</mo></mrow><mrow id="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1" xref="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.3" xref="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.3.cmml">i</mi><mo id="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.2" xref="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.2.cmml">∈</mo><mrow id="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.4.2" xref="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.4.1.cmml"><mo id="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.4.2.1" stretchy="false" xref="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.4.1.1.cmml">[</mo><mn id="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.1" xref="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.1.cmml">20</mn><mo id="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.4.2.2" stretchy="false" xref="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p4.7.m7.2b"><apply id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.2.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2">subscript</csymbol><set id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1"><apply id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1"><times id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.2"></times><apply id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.3">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.3.2.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.3.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.3.3.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.3.3">𝑏</ci></apply><apply id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.1.2">𝑏</ci><ci id="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.2.2.1.1.1.1.1.1.3">𝑖</ci></apply></apply></set><apply id="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1"><in id="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.2"></in><ci id="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.3">𝑖</ci><apply id="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.4.1.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.4.2"><csymbol cd="latexml" id="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.4.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.4.2.1">delimited-[]</csymbol><cn id="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.1.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p4.7.m7.1.1.1.1">20</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.7.m7.2c">\{\mathsf{PLSigmoid}_{b}(b_{i})\}_{i\in[20]}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.7.m7.2d">{ sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) } start_POSTSUBSCRIPT italic_i ∈ [ 20 ] end_POSTSUBSCRIPT</annotation></semantics></math>. For each bias <math alttext="b_{i}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.8.m8.1"><semantics id="S4.SS3.SSS0.Px2.p4.8.m8.1a"><msub id="S4.SS3.SSS0.Px2.p4.8.m8.1.1" xref="S4.SS3.SSS0.Px2.p4.8.m8.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p4.8.m8.1.1.2" xref="S4.SS3.SSS0.Px2.p4.8.m8.1.1.2.cmml">b</mi><mi id="S4.SS3.SSS0.Px2.p4.8.m8.1.1.3" xref="S4.SS3.SSS0.Px2.p4.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p4.8.m8.1b"><apply id="S4.SS3.SSS0.Px2.p4.8.m8.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.8.m8.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.8.m8.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.8.m8.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.8.m8.1.1.2">𝑏</ci><ci id="S4.SS3.SSS0.Px2.p4.8.m8.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.8.m8.1c">b_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.8.m8.1d">italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, if <math alttext="b_{i}\leq-b," class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.9.m9.1"><semantics id="S4.SS3.SSS0.Px2.p4.9.m9.1a"><mrow id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.cmml"><mrow id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.cmml"><msub id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.2" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.2.cmml"><mi id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.2.2" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.2.2.cmml">b</mi><mi id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.2.3" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.1" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.1.cmml">≤</mo><mrow id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.3" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.3.cmml"><mo id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.3a" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.3.cmml">−</mo><mi id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.3.2" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.3.2.cmml">b</mi></mrow></mrow><mo id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.2" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p4.9.m9.1b"><apply id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1"><leq id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.1"></leq><apply id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.2">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.2.2.cmml" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.2.2">𝑏</ci><ci id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.2.3.cmml" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.2.3">𝑖</ci></apply><apply id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.3"><minus id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.3"></minus><ci id="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.3.2.cmml" xref="S4.SS3.SSS0.Px2.p4.9.m9.1.1.1.1.3.2">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.9.m9.1c">b_{i}\leq-b,</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.9.m9.1d">italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ - italic_b ,</annotation></semantics></math> then <math alttext="\mathsf{PLSigmoid}_{b}(b_{i})=0" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.10.m10.1"><semantics id="S4.SS3.SSS0.Px2.p4.10.m10.1a"><mrow id="S4.SS3.SSS0.Px2.p4.10.m10.1.1" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.cmml"><mrow id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.cmml"><msub id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.3" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.3.cmml"><mi id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.3.2" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.3.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.3.3" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.3.3.cmml">b</mi></msub><mo id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.2" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.1.cmml"><mo id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.2" stretchy="false" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.1" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.1.2" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.1.2.cmml">b</mi><mi id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.1.3" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.3" stretchy="false" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.2" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.2.cmml">=</mo><mn id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.3" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p4.10.m10.1b"><apply id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1"><eq id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.2"></eq><apply id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1"><times id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.2"></times><apply id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.3">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.3.2.cmml" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.3.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.3.3.cmml" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.3.3">𝑏</ci></apply><apply id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.1.2">𝑏</ci><ci id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.1.1.1.1.3">𝑖</ci></apply></apply><cn id="S4.SS3.SSS0.Px2.p4.10.m10.1.1.3.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p4.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.10.m10.1c">\mathsf{PLSigmoid}_{b}(b_{i})=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.10.m10.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = 0</annotation></semantics></math> and if <math alttext="b_{i}\geq b," class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.11.m11.1"><semantics id="S4.SS3.SSS0.Px2.p4.11.m11.1a"><mrow id="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1" xref="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.cmml"><mrow id="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1" xref="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.cmml"><msub id="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.2" xref="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.2.cmml"><mi id="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.2.2" xref="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.2.2.cmml">b</mi><mi id="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.2.3" xref="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.1" xref="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.1.cmml">≥</mo><mi id="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.3" xref="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.3.cmml">b</mi></mrow><mo id="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.2" xref="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p4.11.m11.1b"><apply id="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1"><geq id="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.1"></geq><apply id="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.2">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.2.2.cmml" xref="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.2.2">𝑏</ci><ci id="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.2.3.cmml" xref="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.2.3">𝑖</ci></apply><ci id="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.11.m11.1.1.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.11.m11.1c">b_{i}\geq b,</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.11.m11.1d">italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ italic_b ,</annotation></semantics></math> then <math alttext="\mathsf{PLSigmoid}_{b}(b_{i})=1." class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.12.m12.1"><semantics id="S4.SS3.SSS0.Px2.p4.12.m12.1a"><mrow id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.cmml"><mrow id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.cmml"><mrow id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.cmml"><msub id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.3" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.3.cmml"><mi id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.3.2" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.3.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.3.3" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.3.3.cmml">b</mi></msub><mo id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.2" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.1" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.1.2" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.1.2.cmml">b</mi><mi id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.1.3" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.2" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.2.cmml">=</mo><mn id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.3" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.2" lspace="0em" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p4.12.m12.1b"><apply id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1"><eq id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.2"></eq><apply id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1"><times id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.2"></times><apply id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.3">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.3.2.cmml" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.3.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.3.3.cmml" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.3.3">𝑏</ci></apply><apply id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.1.2">𝑏</ci><ci id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply><cn id="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.3.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p4.12.m12.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.12.m12.1c">\mathsf{PLSigmoid}_{b}(b_{i})=1.</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.12.m12.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = 1 .</annotation></semantics></math> Otherwise, we can write</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex9"> <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="\mathsf{PLSigmoid}_{b}(b_{i})=\frac{b_{i}+b}{2b}=\frac{1}{2}+\frac{b_{i}}{2b}," class="ltx_Math" display="block" id="S4.Ex9.m1.1"><semantics id="S4.Ex9.m1.1a"><mrow id="S4.Ex9.m1.1.1.1" xref="S4.Ex9.m1.1.1.1.1.cmml"><mrow id="S4.Ex9.m1.1.1.1.1" xref="S4.Ex9.m1.1.1.1.1.cmml"><mrow id="S4.Ex9.m1.1.1.1.1.1" xref="S4.Ex9.m1.1.1.1.1.1.cmml"><msub id="S4.Ex9.m1.1.1.1.1.1.3" xref="S4.Ex9.m1.1.1.1.1.1.3.cmml"><mi id="S4.Ex9.m1.1.1.1.1.1.3.2" xref="S4.Ex9.m1.1.1.1.1.1.3.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S4.Ex9.m1.1.1.1.1.1.3.3" xref="S4.Ex9.m1.1.1.1.1.1.3.3.cmml">b</mi></msub><mo id="S4.Ex9.m1.1.1.1.1.1.2" xref="S4.Ex9.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex9.m1.1.1.1.1.1.1.1" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex9.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex9.m1.1.1.1.1.1.1.1.1" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex9.m1.1.1.1.1.1.1.1.1.2" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.2.cmml">b</mi><mi id="S4.Ex9.m1.1.1.1.1.1.1.1.1.3" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.Ex9.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex9.m1.1.1.1.1.3" xref="S4.Ex9.m1.1.1.1.1.3.cmml">=</mo><mfrac id="S4.Ex9.m1.1.1.1.1.4" xref="S4.Ex9.m1.1.1.1.1.4.cmml"><mrow id="S4.Ex9.m1.1.1.1.1.4.2" xref="S4.Ex9.m1.1.1.1.1.4.2.cmml"><msub id="S4.Ex9.m1.1.1.1.1.4.2.2" xref="S4.Ex9.m1.1.1.1.1.4.2.2.cmml"><mi id="S4.Ex9.m1.1.1.1.1.4.2.2.2" xref="S4.Ex9.m1.1.1.1.1.4.2.2.2.cmml">b</mi><mi id="S4.Ex9.m1.1.1.1.1.4.2.2.3" xref="S4.Ex9.m1.1.1.1.1.4.2.2.3.cmml">i</mi></msub><mo id="S4.Ex9.m1.1.1.1.1.4.2.1" xref="S4.Ex9.m1.1.1.1.1.4.2.1.cmml">+</mo><mi id="S4.Ex9.m1.1.1.1.1.4.2.3" xref="S4.Ex9.m1.1.1.1.1.4.2.3.cmml">b</mi></mrow><mrow id="S4.Ex9.m1.1.1.1.1.4.3" xref="S4.Ex9.m1.1.1.1.1.4.3.cmml"><mn id="S4.Ex9.m1.1.1.1.1.4.3.2" xref="S4.Ex9.m1.1.1.1.1.4.3.2.cmml">2</mn><mo id="S4.Ex9.m1.1.1.1.1.4.3.1" xref="S4.Ex9.m1.1.1.1.1.4.3.1.cmml">⁢</mo><mi id="S4.Ex9.m1.1.1.1.1.4.3.3" xref="S4.Ex9.m1.1.1.1.1.4.3.3.cmml">b</mi></mrow></mfrac><mo id="S4.Ex9.m1.1.1.1.1.5" xref="S4.Ex9.m1.1.1.1.1.5.cmml">=</mo><mrow id="S4.Ex9.m1.1.1.1.1.6" xref="S4.Ex9.m1.1.1.1.1.6.cmml"><mfrac id="S4.Ex9.m1.1.1.1.1.6.2" xref="S4.Ex9.m1.1.1.1.1.6.2.cmml"><mn id="S4.Ex9.m1.1.1.1.1.6.2.2" xref="S4.Ex9.m1.1.1.1.1.6.2.2.cmml">1</mn><mn id="S4.Ex9.m1.1.1.1.1.6.2.3" xref="S4.Ex9.m1.1.1.1.1.6.2.3.cmml">2</mn></mfrac><mo id="S4.Ex9.m1.1.1.1.1.6.1" xref="S4.Ex9.m1.1.1.1.1.6.1.cmml">+</mo><mfrac id="S4.Ex9.m1.1.1.1.1.6.3" xref="S4.Ex9.m1.1.1.1.1.6.3.cmml"><msub id="S4.Ex9.m1.1.1.1.1.6.3.2" xref="S4.Ex9.m1.1.1.1.1.6.3.2.cmml"><mi id="S4.Ex9.m1.1.1.1.1.6.3.2.2" xref="S4.Ex9.m1.1.1.1.1.6.3.2.2.cmml">b</mi><mi id="S4.Ex9.m1.1.1.1.1.6.3.2.3" xref="S4.Ex9.m1.1.1.1.1.6.3.2.3.cmml">i</mi></msub><mrow id="S4.Ex9.m1.1.1.1.1.6.3.3" xref="S4.Ex9.m1.1.1.1.1.6.3.3.cmml"><mn id="S4.Ex9.m1.1.1.1.1.6.3.3.2" xref="S4.Ex9.m1.1.1.1.1.6.3.3.2.cmml">2</mn><mo id="S4.Ex9.m1.1.1.1.1.6.3.3.1" xref="S4.Ex9.m1.1.1.1.1.6.3.3.1.cmml">⁢</mo><mi id="S4.Ex9.m1.1.1.1.1.6.3.3.3" xref="S4.Ex9.m1.1.1.1.1.6.3.3.3.cmml">b</mi></mrow></mfrac></mrow></mrow><mo id="S4.Ex9.m1.1.1.1.2" xref="S4.Ex9.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex9.m1.1b"><apply id="S4.Ex9.m1.1.1.1.1.cmml" xref="S4.Ex9.m1.1.1.1"><and id="S4.Ex9.m1.1.1.1.1a.cmml" xref="S4.Ex9.m1.1.1.1"></and><apply id="S4.Ex9.m1.1.1.1.1b.cmml" xref="S4.Ex9.m1.1.1.1"><eq id="S4.Ex9.m1.1.1.1.1.3.cmml" xref="S4.Ex9.m1.1.1.1.1.3"></eq><apply 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id="S4.Ex9.m1.1.1.1.1.4.2.cmml" xref="S4.Ex9.m1.1.1.1.1.4.2"><plus id="S4.Ex9.m1.1.1.1.1.4.2.1.cmml" xref="S4.Ex9.m1.1.1.1.1.4.2.1"></plus><apply id="S4.Ex9.m1.1.1.1.1.4.2.2.cmml" xref="S4.Ex9.m1.1.1.1.1.4.2.2"><csymbol cd="ambiguous" id="S4.Ex9.m1.1.1.1.1.4.2.2.1.cmml" xref="S4.Ex9.m1.1.1.1.1.4.2.2">subscript</csymbol><ci id="S4.Ex9.m1.1.1.1.1.4.2.2.2.cmml" xref="S4.Ex9.m1.1.1.1.1.4.2.2.2">𝑏</ci><ci id="S4.Ex9.m1.1.1.1.1.4.2.2.3.cmml" xref="S4.Ex9.m1.1.1.1.1.4.2.2.3">𝑖</ci></apply><ci id="S4.Ex9.m1.1.1.1.1.4.2.3.cmml" xref="S4.Ex9.m1.1.1.1.1.4.2.3">𝑏</ci></apply><apply id="S4.Ex9.m1.1.1.1.1.4.3.cmml" xref="S4.Ex9.m1.1.1.1.1.4.3"><times id="S4.Ex9.m1.1.1.1.1.4.3.1.cmml" xref="S4.Ex9.m1.1.1.1.1.4.3.1"></times><cn id="S4.Ex9.m1.1.1.1.1.4.3.2.cmml" type="integer" xref="S4.Ex9.m1.1.1.1.1.4.3.2">2</cn><ci id="S4.Ex9.m1.1.1.1.1.4.3.3.cmml" xref="S4.Ex9.m1.1.1.1.1.4.3.3">𝑏</ci></apply></apply></apply><apply id="S4.Ex9.m1.1.1.1.1c.cmml" xref="S4.Ex9.m1.1.1.1"><eq id="S4.Ex9.m1.1.1.1.1.5.cmml" xref="S4.Ex9.m1.1.1.1.1.5"></eq><share href="https://arxiv.org/html/2411.12976v1#S4.Ex9.m1.1.1.1.1.4.cmml" id="S4.Ex9.m1.1.1.1.1d.cmml" xref="S4.Ex9.m1.1.1.1"></share><apply id="S4.Ex9.m1.1.1.1.1.6.cmml" xref="S4.Ex9.m1.1.1.1.1.6"><plus id="S4.Ex9.m1.1.1.1.1.6.1.cmml" xref="S4.Ex9.m1.1.1.1.1.6.1"></plus><apply id="S4.Ex9.m1.1.1.1.1.6.2.cmml" xref="S4.Ex9.m1.1.1.1.1.6.2"><divide id="S4.Ex9.m1.1.1.1.1.6.2.1.cmml" xref="S4.Ex9.m1.1.1.1.1.6.2"></divide><cn id="S4.Ex9.m1.1.1.1.1.6.2.2.cmml" type="integer" xref="S4.Ex9.m1.1.1.1.1.6.2.2">1</cn><cn id="S4.Ex9.m1.1.1.1.1.6.2.3.cmml" type="integer" xref="S4.Ex9.m1.1.1.1.1.6.2.3">2</cn></apply><apply id="S4.Ex9.m1.1.1.1.1.6.3.cmml" xref="S4.Ex9.m1.1.1.1.1.6.3"><divide id="S4.Ex9.m1.1.1.1.1.6.3.1.cmml" xref="S4.Ex9.m1.1.1.1.1.6.3"></divide><apply id="S4.Ex9.m1.1.1.1.1.6.3.2.cmml" xref="S4.Ex9.m1.1.1.1.1.6.3.2"><csymbol cd="ambiguous" id="S4.Ex9.m1.1.1.1.1.6.3.2.1.cmml" xref="S4.Ex9.m1.1.1.1.1.6.3.2">subscript</csymbol><ci id="S4.Ex9.m1.1.1.1.1.6.3.2.2.cmml" xref="S4.Ex9.m1.1.1.1.1.6.3.2.2">𝑏</ci><ci id="S4.Ex9.m1.1.1.1.1.6.3.2.3.cmml" xref="S4.Ex9.m1.1.1.1.1.6.3.2.3">𝑖</ci></apply><apply id="S4.Ex9.m1.1.1.1.1.6.3.3.cmml" xref="S4.Ex9.m1.1.1.1.1.6.3.3"><times id="S4.Ex9.m1.1.1.1.1.6.3.3.1.cmml" xref="S4.Ex9.m1.1.1.1.1.6.3.3.1"></times><cn id="S4.Ex9.m1.1.1.1.1.6.3.3.2.cmml" type="integer" xref="S4.Ex9.m1.1.1.1.1.6.3.3.2">2</cn><ci id="S4.Ex9.m1.1.1.1.1.6.3.3.3.cmml" xref="S4.Ex9.m1.1.1.1.1.6.3.3.3">𝑏</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex9.m1.1c">\mathsf{PLSigmoid}_{b}(b_{i})=\frac{b_{i}+b}{2b}=\frac{1}{2}+\frac{b_{i}}{2b},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex9.m1.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = divide start_ARG italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_b end_ARG start_ARG 2 italic_b end_ARG = divide start_ARG 1 end_ARG start_ARG 2 end_ARG + divide start_ARG italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG 2 italic_b end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS3.SSS0.Px2.p4.19">a linear function in <math alttext="b^{-1}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.13.m1.1"><semantics id="S4.SS3.SSS0.Px2.p4.13.m1.1a"><msup id="S4.SS3.SSS0.Px2.p4.13.m1.1.1" xref="S4.SS3.SSS0.Px2.p4.13.m1.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p4.13.m1.1.1.2" xref="S4.SS3.SSS0.Px2.p4.13.m1.1.1.2.cmml">b</mi><mrow id="S4.SS3.SSS0.Px2.p4.13.m1.1.1.3" xref="S4.SS3.SSS0.Px2.p4.13.m1.1.1.3.cmml"><mo id="S4.SS3.SSS0.Px2.p4.13.m1.1.1.3a" xref="S4.SS3.SSS0.Px2.p4.13.m1.1.1.3.cmml">−</mo><mn id="S4.SS3.SSS0.Px2.p4.13.m1.1.1.3.2" xref="S4.SS3.SSS0.Px2.p4.13.m1.1.1.3.2.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p4.13.m1.1b"><apply id="S4.SS3.SSS0.Px2.p4.13.m1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.13.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.13.m1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.13.m1.1.1">superscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.13.m1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.13.m1.1.1.2">𝑏</ci><apply id="S4.SS3.SSS0.Px2.p4.13.m1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.13.m1.1.1.3"><minus id="S4.SS3.SSS0.Px2.p4.13.m1.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px2.p4.13.m1.1.1.3"></minus><cn id="S4.SS3.SSS0.Px2.p4.13.m1.1.1.3.2.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p4.13.m1.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.13.m1.1c">b^{-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.13.m1.1d">italic_b start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. Hence over any interval <math alttext="b\in[b_{i},b_{i+1}]" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.14.m2.2"><semantics id="S4.SS3.SSS0.Px2.p4.14.m2.2a"><mrow id="S4.SS3.SSS0.Px2.p4.14.m2.2.2" xref="S4.SS3.SSS0.Px2.p4.14.m2.2.2.cmml"><mi id="S4.SS3.SSS0.Px2.p4.14.m2.2.2.4" xref="S4.SS3.SSS0.Px2.p4.14.m2.2.2.4.cmml">b</mi><mo id="S4.SS3.SSS0.Px2.p4.14.m2.2.2.3" xref="S4.SS3.SSS0.Px2.p4.14.m2.2.2.3.cmml">∈</mo><mrow id="S4.SS3.SSS0.Px2.p4.14.m2.2.2.2.2" xref="S4.SS3.SSS0.Px2.p4.14.m2.2.2.2.3.cmml"><mo id="S4.SS3.SSS0.Px2.p4.14.m2.2.2.2.2.3" stretchy="false" xref="S4.SS3.SSS0.Px2.p4.14.m2.2.2.2.3.cmml">[</mo><msub id="S4.SS3.SSS0.Px2.p4.14.m2.1.1.1.1.1" xref="S4.SS3.SSS0.Px2.p4.14.m2.1.1.1.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p4.14.m2.1.1.1.1.1.2" xref="S4.SS3.SSS0.Px2.p4.14.m2.1.1.1.1.1.2.cmml">b</mi><mi id="S4.SS3.SSS0.Px2.p4.14.m2.1.1.1.1.1.3" xref="S4.SS3.SSS0.Px2.p4.14.m2.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.SS3.SSS0.Px2.p4.14.m2.2.2.2.2.4" 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xref="S4.SS3.SSS0.Px2.p4.14.m2.2.2.2.2.2.2">𝑏</ci><apply id="S4.SS3.SSS0.Px2.p4.14.m2.2.2.2.2.2.3.cmml" xref="S4.SS3.SSS0.Px2.p4.14.m2.2.2.2.2.2.3"><plus id="S4.SS3.SSS0.Px2.p4.14.m2.2.2.2.2.2.3.1.cmml" xref="S4.SS3.SSS0.Px2.p4.14.m2.2.2.2.2.2.3.1"></plus><ci id="S4.SS3.SSS0.Px2.p4.14.m2.2.2.2.2.2.3.2.cmml" xref="S4.SS3.SSS0.Px2.p4.14.m2.2.2.2.2.2.3.2">𝑖</ci><cn id="S4.SS3.SSS0.Px2.p4.14.m2.2.2.2.2.2.3.3.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p4.14.m2.2.2.2.2.2.3.3">1</cn></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.14.m2.2c">b\in[b_{i},b_{i+1}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.14.m2.2d">italic_b ∈ [ italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ]</annotation></semantics></math>, <math alttext="\mathsf{val}_{G_{\mathrm{LP}}}(\mathsf{PLSigmoid}_{b})" class="ltx_Math" display="inline" 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xref="S4.SS3.SSS0.Px2.p4.15.m3.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.15.m3.1.1.3.3.1.cmml" xref="S4.SS3.SSS0.Px2.p4.15.m3.1.1.3.3">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.15.m3.1.1.3.3.2.cmml" xref="S4.SS3.SSS0.Px2.p4.15.m3.1.1.3.3.2">𝐺</ci><ci id="S4.SS3.SSS0.Px2.p4.15.m3.1.1.3.3.3.cmml" xref="S4.SS3.SSS0.Px2.p4.15.m3.1.1.3.3.3">LP</ci></apply></apply><apply id="S4.SS3.SSS0.Px2.p4.15.m3.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.15.m3.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.15.m3.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.15.m3.1.1.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.15.m3.1.1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.15.m3.1.1.1.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S4.SS3.SSS0.Px2.p4.15.m3.1.1.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.15.m3.1.1.1.1.1.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.15.m3.1c">\mathsf{val}_{G_{\mathrm{LP}}}(\mathsf{PLSigmoid}_{b})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.15.m3.1d">sansserif_val start_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT roman_LP end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT )</annotation></semantics></math> is a quadratic function of <math alttext="b^{-1}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.16.m4.1"><semantics id="S4.SS3.SSS0.Px2.p4.16.m4.1a"><msup id="S4.SS3.SSS0.Px2.p4.16.m4.1.1" xref="S4.SS3.SSS0.Px2.p4.16.m4.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p4.16.m4.1.1.2" xref="S4.SS3.SSS0.Px2.p4.16.m4.1.1.2.cmml">b</mi><mrow id="S4.SS3.SSS0.Px2.p4.16.m4.1.1.3" xref="S4.SS3.SSS0.Px2.p4.16.m4.1.1.3.cmml"><mo id="S4.SS3.SSS0.Px2.p4.16.m4.1.1.3a" xref="S4.SS3.SSS0.Px2.p4.16.m4.1.1.3.cmml">−</mo><mn id="S4.SS3.SSS0.Px2.p4.16.m4.1.1.3.2" xref="S4.SS3.SSS0.Px2.p4.16.m4.1.1.3.2.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p4.16.m4.1b"><apply id="S4.SS3.SSS0.Px2.p4.16.m4.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.16.m4.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.16.m4.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.16.m4.1.1">superscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.16.m4.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.16.m4.1.1.2">𝑏</ci><apply id="S4.SS3.SSS0.Px2.p4.16.m4.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.16.m4.1.1.3"><minus id="S4.SS3.SSS0.Px2.p4.16.m4.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px2.p4.16.m4.1.1.3"></minus><cn id="S4.SS3.SSS0.Px2.p4.16.m4.1.1.3.2.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p4.16.m4.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.16.m4.1c">b^{-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.16.m4.1d">italic_b start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. We can therefore find the maximum <math alttext="b^{*}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.17.m5.1"><semantics id="S4.SS3.SSS0.Px2.p4.17.m5.1a"><msup id="S4.SS3.SSS0.Px2.p4.17.m5.1.1" xref="S4.SS3.SSS0.Px2.p4.17.m5.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p4.17.m5.1.1.2" xref="S4.SS3.SSS0.Px2.p4.17.m5.1.1.2.cmml">b</mi><mo id="S4.SS3.SSS0.Px2.p4.17.m5.1.1.3" xref="S4.SS3.SSS0.Px2.p4.17.m5.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p4.17.m5.1b"><apply id="S4.SS3.SSS0.Px2.p4.17.m5.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.17.m5.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.17.m5.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.17.m5.1.1">superscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.17.m5.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.17.m5.1.1.2">𝑏</ci><times id="S4.SS3.SSS0.Px2.p4.17.m5.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.17.m5.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.17.m5.1c">b^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.17.m5.1d">italic_b start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> of this quadratic function; if <math alttext="b^{*}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.18.m6.1"><semantics id="S4.SS3.SSS0.Px2.p4.18.m6.1a"><msup id="S4.SS3.SSS0.Px2.p4.18.m6.1.1" xref="S4.SS3.SSS0.Px2.p4.18.m6.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p4.18.m6.1.1.2" xref="S4.SS3.SSS0.Px2.p4.18.m6.1.1.2.cmml">b</mi><mo id="S4.SS3.SSS0.Px2.p4.18.m6.1.1.3" xref="S4.SS3.SSS0.Px2.p4.18.m6.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p4.18.m6.1b"><apply id="S4.SS3.SSS0.Px2.p4.18.m6.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.18.m6.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.18.m6.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.18.m6.1.1">superscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.18.m6.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p4.18.m6.1.1.2">𝑏</ci><times id="S4.SS3.SSS0.Px2.p4.18.m6.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p4.18.m6.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.18.m6.1c">b^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.18.m6.1d">italic_b start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is in the interval, we are done, otherwise we take the maximum value over <math alttext="b\in\{b_{i},b_{i+1}\}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p4.19.m7.2"><semantics id="S4.SS3.SSS0.Px2.p4.19.m7.2a"><mrow id="S4.SS3.SSS0.Px2.p4.19.m7.2.2" xref="S4.SS3.SSS0.Px2.p4.19.m7.2.2.cmml"><mi id="S4.SS3.SSS0.Px2.p4.19.m7.2.2.4" xref="S4.SS3.SSS0.Px2.p4.19.m7.2.2.4.cmml">b</mi><mo id="S4.SS3.SSS0.Px2.p4.19.m7.2.2.3" xref="S4.SS3.SSS0.Px2.p4.19.m7.2.2.3.cmml">∈</mo><mrow id="S4.SS3.SSS0.Px2.p4.19.m7.2.2.2.2" xref="S4.SS3.SSS0.Px2.p4.19.m7.2.2.2.3.cmml"><mo 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xref="S4.SS3.SSS0.Px2.p4.19.m7.2.2.2.2.2.3.1.cmml">+</mo><mn id="S4.SS3.SSS0.Px2.p4.19.m7.2.2.2.2.2.3.3" xref="S4.SS3.SSS0.Px2.p4.19.m7.2.2.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS3.SSS0.Px2.p4.19.m7.2.2.2.2.5" stretchy="false" xref="S4.SS3.SSS0.Px2.p4.19.m7.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p4.19.m7.2b"><apply id="S4.SS3.SSS0.Px2.p4.19.m7.2.2.cmml" xref="S4.SS3.SSS0.Px2.p4.19.m7.2.2"><in id="S4.SS3.SSS0.Px2.p4.19.m7.2.2.3.cmml" xref="S4.SS3.SSS0.Px2.p4.19.m7.2.2.3"></in><ci id="S4.SS3.SSS0.Px2.p4.19.m7.2.2.4.cmml" xref="S4.SS3.SSS0.Px2.p4.19.m7.2.2.4">𝑏</ci><set id="S4.SS3.SSS0.Px2.p4.19.m7.2.2.2.3.cmml" xref="S4.SS3.SSS0.Px2.p4.19.m7.2.2.2.2"><apply id="S4.SS3.SSS0.Px2.p4.19.m7.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.19.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p4.19.m7.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p4.19.m7.1.1.1.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p4.19.m7.1.1.1.1.1.2.cmml" 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encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p4.19.m7.2c">b\in\{b_{i},b_{i+1}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p4.19.m7.2d">italic_b ∈ { italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT }</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS3.SSS0.Px2.p5"> <p class="ltx_p" id="S4.SS3.SSS0.Px2.p5.13">We perform the explicit calculations for the edge-weights and the oblivious ratio in Mathematica (<a class="ltx_ref ltx_href ltx_font_typewriter" href="https://github.com/singerng/oblivious-csps/blob/main/plsigmoid_lb.nb" title="">plsigmoid_lb.nb</a> in the source repository) since the rational numbers have many digits. 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id="S4.SS3.SSS0.Px2.p5.4.m4.2c">[b_{17},b_{18}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p5.4.m4.2d">[ italic_b start_POSTSUBSCRIPT 17 end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT 18 end_POSTSUBSCRIPT ]</annotation></semantics></math>, <math alttext="[b_{16},b_{17}]" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p5.5.m5.2"><semantics id="S4.SS3.SSS0.Px2.p5.5.m5.2a"><mrow id="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2" xref="S4.SS3.SSS0.Px2.p5.5.m5.2.2.3.cmml"><mo id="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.3" stretchy="false" xref="S4.SS3.SSS0.Px2.p5.5.m5.2.2.3.cmml">[</mo><msub id="S4.SS3.SSS0.Px2.p5.5.m5.1.1.1.1" xref="S4.SS3.SSS0.Px2.p5.5.m5.1.1.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p5.5.m5.1.1.1.1.2" xref="S4.SS3.SSS0.Px2.p5.5.m5.1.1.1.1.2.cmml">b</mi><mn id="S4.SS3.SSS0.Px2.p5.5.m5.1.1.1.1.3" xref="S4.SS3.SSS0.Px2.p5.5.m5.1.1.1.1.3.cmml">16</mn></msub><mo id="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.4" xref="S4.SS3.SSS0.Px2.p5.5.m5.2.2.3.cmml">,</mo><msub id="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.2" xref="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.2.cmml"><mi id="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.2.2" xref="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.2.2.cmml">b</mi><mn id="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.2.3" xref="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.2.3.cmml">17</mn></msub><mo id="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.5" stretchy="false" xref="S4.SS3.SSS0.Px2.p5.5.m5.2.2.3.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p5.5.m5.2b"><interval closure="closed" id="S4.SS3.SSS0.Px2.p5.5.m5.2.2.3.cmml" xref="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2"><apply id="S4.SS3.SSS0.Px2.p5.5.m5.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p5.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p5.5.m5.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p5.5.m5.1.1.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p5.5.m5.1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p5.5.m5.1.1.1.1.2">𝑏</ci><cn id="S4.SS3.SSS0.Px2.p5.5.m5.1.1.1.1.3.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p5.5.m5.1.1.1.1.3">16</cn></apply><apply id="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.2.cmml" xref="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.2.1.cmml" xref="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.2">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.2.2.cmml" xref="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.2.2">𝑏</ci><cn id="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p5.5.m5.2.2.2.2.3">17</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p5.5.m5.2c">[b_{16},b_{17}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p5.5.m5.2d">[ italic_b start_POSTSUBSCRIPT 16 end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT 17 end_POSTSUBSCRIPT ]</annotation></semantics></math>, and <math alttext="[b_{15},b_{16}]" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p5.6.m6.2"><semantics id="S4.SS3.SSS0.Px2.p5.6.m6.2a"><mrow id="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2" xref="S4.SS3.SSS0.Px2.p5.6.m6.2.2.3.cmml"><mo id="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.3" stretchy="false" xref="S4.SS3.SSS0.Px2.p5.6.m6.2.2.3.cmml">[</mo><msub id="S4.SS3.SSS0.Px2.p5.6.m6.1.1.1.1" xref="S4.SS3.SSS0.Px2.p5.6.m6.1.1.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p5.6.m6.1.1.1.1.2" xref="S4.SS3.SSS0.Px2.p5.6.m6.1.1.1.1.2.cmml">b</mi><mn id="S4.SS3.SSS0.Px2.p5.6.m6.1.1.1.1.3" xref="S4.SS3.SSS0.Px2.p5.6.m6.1.1.1.1.3.cmml">15</mn></msub><mo id="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.4" xref="S4.SS3.SSS0.Px2.p5.6.m6.2.2.3.cmml">,</mo><msub id="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.2" xref="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.2.cmml"><mi id="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.2.2" xref="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.2.2.cmml">b</mi><mn id="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.2.3" xref="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.2.3.cmml">16</mn></msub><mo id="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.5" stretchy="false" xref="S4.SS3.SSS0.Px2.p5.6.m6.2.2.3.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p5.6.m6.2b"><interval closure="closed" id="S4.SS3.SSS0.Px2.p5.6.m6.2.2.3.cmml" xref="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2"><apply id="S4.SS3.SSS0.Px2.p5.6.m6.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p5.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p5.6.m6.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p5.6.m6.1.1.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p5.6.m6.1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p5.6.m6.1.1.1.1.2">𝑏</ci><cn id="S4.SS3.SSS0.Px2.p5.6.m6.1.1.1.1.3.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p5.6.m6.1.1.1.1.3">15</cn></apply><apply id="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.2.cmml" xref="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.2.1.cmml" xref="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.2">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.2.2.cmml" xref="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.2.2">𝑏</ci><cn id="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px2.p5.6.m6.2.2.2.2.3">16</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p5.6.m6.2c">[b_{15},b_{16}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p5.6.m6.2d">[ italic_b start_POSTSUBSCRIPT 15 end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT 16 end_POSTSUBSCRIPT ]</annotation></semantics></math> rounded up in the sixth decimal place are, respectively, <math alttext="0.485895" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p5.7.m7.1"><semantics id="S4.SS3.SSS0.Px2.p5.7.m7.1a"><mn id="S4.SS3.SSS0.Px2.p5.7.m7.1.1" xref="S4.SS3.SSS0.Px2.p5.7.m7.1.1.cmml">0.485895</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p5.7.m7.1b"><cn id="S4.SS3.SSS0.Px2.p5.7.m7.1.1.cmml" type="float" xref="S4.SS3.SSS0.Px2.p5.7.m7.1.1">0.485895</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p5.7.m7.1c">0.485895</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p5.7.m7.1d">0.485895</annotation></semantics></math>, <math alttext="0.485870" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p5.8.m8.1"><semantics id="S4.SS3.SSS0.Px2.p5.8.m8.1a"><mn id="S4.SS3.SSS0.Px2.p5.8.m8.1.1" xref="S4.SS3.SSS0.Px2.p5.8.m8.1.1.cmml">0.485870</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p5.8.m8.1b"><cn id="S4.SS3.SSS0.Px2.p5.8.m8.1.1.cmml" type="float" xref="S4.SS3.SSS0.Px2.p5.8.m8.1.1">0.485870</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p5.8.m8.1c">0.485870</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p5.8.m8.1d">0.485870</annotation></semantics></math>, <math alttext="0.485488" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p5.9.m9.1"><semantics id="S4.SS3.SSS0.Px2.p5.9.m9.1a"><mn id="S4.SS3.SSS0.Px2.p5.9.m9.1.1" xref="S4.SS3.SSS0.Px2.p5.9.m9.1.1.cmml">0.485488</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p5.9.m9.1b"><cn id="S4.SS3.SSS0.Px2.p5.9.m9.1.1.cmml" type="float" xref="S4.SS3.SSS0.Px2.p5.9.m9.1.1">0.485488</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p5.9.m9.1c">0.485488</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p5.9.m9.1d">0.485488</annotation></semantics></math>, <math alttext="0.484375" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p5.10.m10.1"><semantics id="S4.SS3.SSS0.Px2.p5.10.m10.1a"><mn id="S4.SS3.SSS0.Px2.p5.10.m10.1.1" xref="S4.SS3.SSS0.Px2.p5.10.m10.1.1.cmml">0.484375</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p5.10.m10.1b"><cn id="S4.SS3.SSS0.Px2.p5.10.m10.1.1.cmml" type="float" xref="S4.SS3.SSS0.Px2.p5.10.m10.1.1">0.484375</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p5.10.m10.1c">0.484375</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p5.10.m10.1d">0.484375</annotation></semantics></math>, <math alttext="0.482019" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p5.11.m11.1"><semantics id="S4.SS3.SSS0.Px2.p5.11.m11.1a"><mn id="S4.SS3.SSS0.Px2.p5.11.m11.1.1" xref="S4.SS3.SSS0.Px2.p5.11.m11.1.1.cmml">0.482019</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p5.11.m11.1b"><cn id="S4.SS3.SSS0.Px2.p5.11.m11.1.1.cmml" type="float" xref="S4.SS3.SSS0.Px2.p5.11.m11.1.1">0.482019</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p5.11.m11.1c">0.482019</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p5.11.m11.1d">0.482019</annotation></semantics></math>, and <math alttext="0.477739" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p5.12.m12.1"><semantics id="S4.SS3.SSS0.Px2.p5.12.m12.1a"><mn id="S4.SS3.SSS0.Px2.p5.12.m12.1.1" xref="S4.SS3.SSS0.Px2.p5.12.m12.1.1.cmml">0.477739</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p5.12.m12.1b"><cn id="S4.SS3.SSS0.Px2.p5.12.m12.1.1.cmml" type="float" xref="S4.SS3.SSS0.Px2.p5.12.m12.1.1">0.477739</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p5.12.m12.1c">0.477739</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p5.12.m12.1d">0.477739</annotation></semantics></math>; all are less than <math alttext="0.486" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p5.13.m13.1"><semantics id="S4.SS3.SSS0.Px2.p5.13.m13.1a"><mn id="S4.SS3.SSS0.Px2.p5.13.m13.1.1" xref="S4.SS3.SSS0.Px2.p5.13.m13.1.1.cmml">0.486</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p5.13.m13.1b"><cn id="S4.SS3.SSS0.Px2.p5.13.m13.1.1.cmml" type="float" xref="S4.SS3.SSS0.Px2.p5.13.m13.1.1">0.486</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p5.13.m13.1c">0.486</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p5.13.m13.1d">0.486</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS3.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Case 3: <math alttext="0\leq b\leq 0.225" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.1.m1.1"><semantics id="S4.SS3.SSS0.Px3.1.m1.1b"><mrow id="S4.SS3.SSS0.Px3.1.m1.1.1" xref="S4.SS3.SSS0.Px3.1.m1.1.1.cmml"><mn id="S4.SS3.SSS0.Px3.1.m1.1.1.2" xref="S4.SS3.SSS0.Px3.1.m1.1.1.2.cmml">0</mn><mo id="S4.SS3.SSS0.Px3.1.m1.1.1.3" xref="S4.SS3.SSS0.Px3.1.m1.1.1.3.cmml">≤</mo><mi id="S4.SS3.SSS0.Px3.1.m1.1.1.4" xref="S4.SS3.SSS0.Px3.1.m1.1.1.4.cmml">b</mi><mo id="S4.SS3.SSS0.Px3.1.m1.1.1.5" xref="S4.SS3.SSS0.Px3.1.m1.1.1.5.cmml">≤</mo><mn id="S4.SS3.SSS0.Px3.1.m1.1.1.6" xref="S4.SS3.SSS0.Px3.1.m1.1.1.6.cmml">0.225</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.1.m1.1c"><apply id="S4.SS3.SSS0.Px3.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px3.1.m1.1.1"><and id="S4.SS3.SSS0.Px3.1.m1.1.1a.cmml" xref="S4.SS3.SSS0.Px3.1.m1.1.1"></and><apply id="S4.SS3.SSS0.Px3.1.m1.1.1b.cmml" xref="S4.SS3.SSS0.Px3.1.m1.1.1"><leq id="S4.SS3.SSS0.Px3.1.m1.1.1.3.cmml" xref="S4.SS3.SSS0.Px3.1.m1.1.1.3"></leq><cn id="S4.SS3.SSS0.Px3.1.m1.1.1.2.cmml" type="integer" xref="S4.SS3.SSS0.Px3.1.m1.1.1.2">0</cn><ci id="S4.SS3.SSS0.Px3.1.m1.1.1.4.cmml" xref="S4.SS3.SSS0.Px3.1.m1.1.1.4">𝑏</ci></apply><apply id="S4.SS3.SSS0.Px3.1.m1.1.1c.cmml" xref="S4.SS3.SSS0.Px3.1.m1.1.1"><leq id="S4.SS3.SSS0.Px3.1.m1.1.1.5.cmml" xref="S4.SS3.SSS0.Px3.1.m1.1.1.5"></leq><share href="https://arxiv.org/html/2411.12976v1#S4.SS3.SSS0.Px3.1.m1.1.1.4.cmml" id="S4.SS3.SSS0.Px3.1.m1.1.1d.cmml" xref="S4.SS3.SSS0.Px3.1.m1.1.1"></share><cn id="S4.SS3.SSS0.Px3.1.m1.1.1.6.cmml" type="float" xref="S4.SS3.SSS0.Px3.1.m1.1.1.6">0.225</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.1.m1.1d">0\leq b\leq 0.225</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.1.m1.1e">0 ≤ italic_b ≤ 0.225</annotation></semantics></math>.</h4> <div class="ltx_para" id="S4.SS3.SSS0.Px3.p1"> <p class="ltx_p" id="S4.SS3.SSS0.Px3.p1.15">Now, consider the graph shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S5.F5" title="Figure 5 ‣ 5 Lower bound for arbitrary selection functions (Theorem 1.9) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">5</span></a> with <math alttext="c=\frac{1+b}{1-b}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.1.m1.1"><semantics id="S4.SS3.SSS0.Px3.p1.1.m1.1a"><mrow id="S4.SS3.SSS0.Px3.p1.1.m1.1.1" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.cmml"><mi id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.2" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.2.cmml">c</mi><mo id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.1" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.1.cmml">=</mo><mfrac id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.cmml"><mrow id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.cmml"><mn id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.2" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.1" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.1.cmml">+</mo><mi id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.3" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.3.cmml">b</mi></mrow><mrow id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.cmml"><mn id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.2" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.2.cmml">1</mn><mo id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.1" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.1.cmml">−</mo><mi id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.3" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.3.cmml">b</mi></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.1.m1.1b"><apply id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1"><eq id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.1"></eq><ci id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.2.cmml" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.2">𝑐</ci><apply id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.cmml" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3"><divide id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3"></divide><apply id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.cmml" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2"><plus id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.1.cmml" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.1"></plus><cn id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.2.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.2">1</cn><ci id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.3.cmml" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.2.3">𝑏</ci></apply><apply id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.cmml" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3"><minus id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.1.cmml" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.1"></minus><cn id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.2.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.2">1</cn><ci id="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.3.cmml" xref="S4.SS3.SSS0.Px3.p1.1.m1.1.1.3.3.3">𝑏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.1.m1.1c">c=\frac{1+b}{1-b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.1.m1.1d">italic_c = divide start_ARG 1 + italic_b end_ARG start_ARG 1 - italic_b end_ARG</annotation></semantics></math>. Vertices <math alttext="1" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.2.m2.1"><semantics id="S4.SS3.SSS0.Px3.p1.2.m2.1a"><mn id="S4.SS3.SSS0.Px3.p1.2.m2.1.1" xref="S4.SS3.SSS0.Px3.p1.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.2.m2.1b"><cn id="S4.SS3.SSS0.Px3.p1.2.m2.1.1.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.2.m2.1d">1</annotation></semantics></math> and <math alttext="2" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.3.m3.1"><semantics id="S4.SS3.SSS0.Px3.p1.3.m3.1a"><mn id="S4.SS3.SSS0.Px3.p1.3.m3.1.1" xref="S4.SS3.SSS0.Px3.p1.3.m3.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.3.m3.1b"><cn id="S4.SS3.SSS0.Px3.p1.3.m3.1.1.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.3.m3.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.3.m3.1c">2</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.3.m3.1d">2</annotation></semantics></math> have bias <math alttext="b" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.4.m4.1"><semantics id="S4.SS3.SSS0.Px3.p1.4.m4.1a"><mi id="S4.SS3.SSS0.Px3.p1.4.m4.1.1" xref="S4.SS3.SSS0.Px3.p1.4.m4.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.4.m4.1b"><ci id="S4.SS3.SSS0.Px3.p1.4.m4.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.4.m4.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.4.m4.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.4.m4.1d">italic_b</annotation></semantics></math> and vertices <math alttext="3" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.5.m5.1"><semantics id="S4.SS3.SSS0.Px3.p1.5.m5.1a"><mn id="S4.SS3.SSS0.Px3.p1.5.m5.1.1" xref="S4.SS3.SSS0.Px3.p1.5.m5.1.1.cmml">3</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.5.m5.1b"><cn id="S4.SS3.SSS0.Px3.p1.5.m5.1.1.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.5.m5.1.1">3</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.5.m5.1c">3</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.5.m5.1d">3</annotation></semantics></math> and <math alttext="4" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.6.m6.1"><semantics id="S4.SS3.SSS0.Px3.p1.6.m6.1a"><mn id="S4.SS3.SSS0.Px3.p1.6.m6.1.1" xref="S4.SS3.SSS0.Px3.p1.6.m6.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.6.m6.1b"><cn id="S4.SS3.SSS0.Px3.p1.6.m6.1.1.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.6.m6.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.6.m6.1c">4</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.6.m6.1d">4</annotation></semantics></math> have bias <math alttext="-b" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.7.m7.1"><semantics id="S4.SS3.SSS0.Px3.p1.7.m7.1a"><mrow id="S4.SS3.SSS0.Px3.p1.7.m7.1.1" xref="S4.SS3.SSS0.Px3.p1.7.m7.1.1.cmml"><mo id="S4.SS3.SSS0.Px3.p1.7.m7.1.1a" xref="S4.SS3.SSS0.Px3.p1.7.m7.1.1.cmml">−</mo><mi id="S4.SS3.SSS0.Px3.p1.7.m7.1.1.2" xref="S4.SS3.SSS0.Px3.p1.7.m7.1.1.2.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.7.m7.1b"><apply id="S4.SS3.SSS0.Px3.p1.7.m7.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.7.m7.1.1"><minus id="S4.SS3.SSS0.Px3.p1.7.m7.1.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.7.m7.1.1"></minus><ci id="S4.SS3.SSS0.Px3.p1.7.m7.1.1.2.cmml" xref="S4.SS3.SSS0.Px3.p1.7.m7.1.1.2">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.7.m7.1c">-b</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.7.m7.1d">- italic_b</annotation></semantics></math>. By the definition of <math alttext="\mathsf{PLSigmoid}_{b}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.8.m8.1"><semantics id="S4.SS3.SSS0.Px3.p1.8.m8.1a"><msub id="S4.SS3.SSS0.Px3.p1.8.m8.1.1" xref="S4.SS3.SSS0.Px3.p1.8.m8.1.1.cmml"><mi id="S4.SS3.SSS0.Px3.p1.8.m8.1.1.2" xref="S4.SS3.SSS0.Px3.p1.8.m8.1.1.2.cmml">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</mi><mi id="S4.SS3.SSS0.Px3.p1.8.m8.1.1.3" xref="S4.SS3.SSS0.Px3.p1.8.m8.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.8.m8.1b"><apply id="S4.SS3.SSS0.Px3.p1.8.m8.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px3.p1.8.m8.1.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.8.m8.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px3.p1.8.m8.1.1.2.cmml" xref="S4.SS3.SSS0.Px3.p1.8.m8.1.1.2">𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽</ci><ci id="S4.SS3.SSS0.Px3.p1.8.m8.1.1.3.cmml" xref="S4.SS3.SSS0.Px3.p1.8.m8.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.8.m8.1c">\mathsf{PLSigmoid}_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.8.m8.1d">sansserif_PLSigmoid start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>, it assigns <math alttext="\{1,2\}\to 1,\{3,4\}\to 0" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.9.m9.6"><semantics id="S4.SS3.SSS0.Px3.p1.9.m9.6a"><mrow id="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2" xref="S4.SS3.SSS0.Px3.p1.9.m9.6.6.3.cmml"><mrow id="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1" xref="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.cmml"><mrow id="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.2.2" xref="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.2.1.cmml"><mo id="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.2.2.1" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.2.1.cmml">{</mo><mn id="S4.SS3.SSS0.Px3.p1.9.m9.1.1" xref="S4.SS3.SSS0.Px3.p1.9.m9.1.1.cmml">1</mn><mo id="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.2.2.2" xref="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.2.1.cmml">,</mo><mn id="S4.SS3.SSS0.Px3.p1.9.m9.2.2" xref="S4.SS3.SSS0.Px3.p1.9.m9.2.2.cmml">2</mn><mo id="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.2.2.3" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.2.1.cmml">}</mo></mrow><mo id="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.1" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.1.cmml">→</mo><mn id="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.3" xref="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.3.cmml">1</mn></mrow><mo id="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.3" xref="S4.SS3.SSS0.Px3.p1.9.m9.6.6.3a.cmml">,</mo><mrow id="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2" xref="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.cmml"><mrow id="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.2.2" xref="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.2.1.cmml"><mo id="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.2.2.1" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.2.1.cmml">{</mo><mn id="S4.SS3.SSS0.Px3.p1.9.m9.3.3" xref="S4.SS3.SSS0.Px3.p1.9.m9.3.3.cmml">3</mn><mo id="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.2.2.2" xref="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.2.1.cmml">,</mo><mn id="S4.SS3.SSS0.Px3.p1.9.m9.4.4" xref="S4.SS3.SSS0.Px3.p1.9.m9.4.4.cmml">4</mn><mo id="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.2.2.3" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.2.1.cmml">}</mo></mrow><mo id="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.1" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.1.cmml">→</mo><mn id="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.3" xref="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.3.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.9.m9.6b"><apply id="S4.SS3.SSS0.Px3.p1.9.m9.6.6.3.cmml" xref="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px3.p1.9.m9.6.6.3a.cmml" xref="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.3">formulae-sequence</csymbol><apply id="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1"><ci id="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.1">→</ci><set id="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.2.2"><cn id="S4.SS3.SSS0.Px3.p1.9.m9.1.1.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.9.m9.1.1">1</cn><cn id="S4.SS3.SSS0.Px3.p1.9.m9.2.2.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.9.m9.2.2">2</cn></set><cn id="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.9.m9.5.5.1.1.3">1</cn></apply><apply id="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.cmml" xref="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2"><ci id="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.1.cmml" xref="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.1">→</ci><set id="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.2.1.cmml" xref="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.2.2"><cn id="S4.SS3.SSS0.Px3.p1.9.m9.3.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.9.m9.3.3">3</cn><cn id="S4.SS3.SSS0.Px3.p1.9.m9.4.4.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.9.m9.4.4">4</cn></set><cn id="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.9.m9.6.6.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.9.m9.6c">\{1,2\}\to 1,\{3,4\}\to 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.9.m9.6d">{ 1 , 2 } → 1 , { 3 , 4 } → 0</annotation></semantics></math> and the corresponding cut has value <math alttext="c^{2}-1" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.10.m10.1"><semantics id="S4.SS3.SSS0.Px3.p1.10.m10.1a"><mrow id="S4.SS3.SSS0.Px3.p1.10.m10.1.1" xref="S4.SS3.SSS0.Px3.p1.10.m10.1.1.cmml"><msup id="S4.SS3.SSS0.Px3.p1.10.m10.1.1.2" xref="S4.SS3.SSS0.Px3.p1.10.m10.1.1.2.cmml"><mi id="S4.SS3.SSS0.Px3.p1.10.m10.1.1.2.2" xref="S4.SS3.SSS0.Px3.p1.10.m10.1.1.2.2.cmml">c</mi><mn id="S4.SS3.SSS0.Px3.p1.10.m10.1.1.2.3" xref="S4.SS3.SSS0.Px3.p1.10.m10.1.1.2.3.cmml">2</mn></msup><mo id="S4.SS3.SSS0.Px3.p1.10.m10.1.1.1" xref="S4.SS3.SSS0.Px3.p1.10.m10.1.1.1.cmml">−</mo><mn id="S4.SS3.SSS0.Px3.p1.10.m10.1.1.3" xref="S4.SS3.SSS0.Px3.p1.10.m10.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.10.m10.1b"><apply id="S4.SS3.SSS0.Px3.p1.10.m10.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.10.m10.1.1"><minus id="S4.SS3.SSS0.Px3.p1.10.m10.1.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.10.m10.1.1.1"></minus><apply id="S4.SS3.SSS0.Px3.p1.10.m10.1.1.2.cmml" xref="S4.SS3.SSS0.Px3.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px3.p1.10.m10.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px3.p1.10.m10.1.1.2">superscript</csymbol><ci id="S4.SS3.SSS0.Px3.p1.10.m10.1.1.2.2.cmml" xref="S4.SS3.SSS0.Px3.p1.10.m10.1.1.2.2">𝑐</ci><cn id="S4.SS3.SSS0.Px3.p1.10.m10.1.1.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.10.m10.1.1.2.3">2</cn></apply><cn id="S4.SS3.SSS0.Px3.p1.10.m10.1.1.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.10.m10.1c">c^{2}-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.10.m10.1d">italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1</annotation></semantics></math>. On the other hand, the assignment <math alttext="\{1,3\}\to 1,\{2,4\}\to 0" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.11.m11.6"><semantics id="S4.SS3.SSS0.Px3.p1.11.m11.6a"><mrow id="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2" xref="S4.SS3.SSS0.Px3.p1.11.m11.6.6.3.cmml"><mrow id="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1" xref="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.cmml"><mrow id="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.2.2" xref="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.2.1.cmml"><mo id="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.2.2.1" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.2.1.cmml">{</mo><mn id="S4.SS3.SSS0.Px3.p1.11.m11.1.1" xref="S4.SS3.SSS0.Px3.p1.11.m11.1.1.cmml">1</mn><mo id="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.2.2.2" xref="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.2.1.cmml">,</mo><mn id="S4.SS3.SSS0.Px3.p1.11.m11.2.2" xref="S4.SS3.SSS0.Px3.p1.11.m11.2.2.cmml">3</mn><mo id="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.2.2.3" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.2.1.cmml">}</mo></mrow><mo id="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.1" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.1.cmml">→</mo><mn id="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.3" xref="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.3.cmml">1</mn></mrow><mo id="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.3" xref="S4.SS3.SSS0.Px3.p1.11.m11.6.6.3a.cmml">,</mo><mrow id="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2" xref="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.cmml"><mrow id="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.2.2" xref="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.2.1.cmml"><mo id="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.2.2.1" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.2.1.cmml">{</mo><mn id="S4.SS3.SSS0.Px3.p1.11.m11.3.3" xref="S4.SS3.SSS0.Px3.p1.11.m11.3.3.cmml">2</mn><mo id="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.2.2.2" xref="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.2.1.cmml">,</mo><mn id="S4.SS3.SSS0.Px3.p1.11.m11.4.4" xref="S4.SS3.SSS0.Px3.p1.11.m11.4.4.cmml">4</mn><mo id="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.2.2.3" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.2.1.cmml">}</mo></mrow><mo id="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.1" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.1.cmml">→</mo><mn id="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.3" xref="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.3.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.11.m11.6b"><apply id="S4.SS3.SSS0.Px3.p1.11.m11.6.6.3.cmml" xref="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px3.p1.11.m11.6.6.3a.cmml" xref="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.3">formulae-sequence</csymbol><apply id="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1"><ci id="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.1">→</ci><set id="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.2.2"><cn id="S4.SS3.SSS0.Px3.p1.11.m11.1.1.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.11.m11.1.1">1</cn><cn id="S4.SS3.SSS0.Px3.p1.11.m11.2.2.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.11.m11.2.2">3</cn></set><cn id="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.11.m11.5.5.1.1.3">1</cn></apply><apply id="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.cmml" xref="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2"><ci id="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.1.cmml" xref="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.1">→</ci><set id="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.2.1.cmml" xref="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.2.2"><cn id="S4.SS3.SSS0.Px3.p1.11.m11.3.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.11.m11.3.3">2</cn><cn id="S4.SS3.SSS0.Px3.p1.11.m11.4.4.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.11.m11.4.4">4</cn></set><cn id="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.11.m11.6.6.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.11.m11.6c">\{1,3\}\to 1,\{2,4\}\to 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.11.m11.6d">{ 1 , 3 } → 1 , { 2 , 4 } → 0</annotation></semantics></math> satisfies weight <math alttext="c^{2}-1+2\cdot 1=c^{2}+1" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.12.m12.1"><semantics id="S4.SS3.SSS0.Px3.p1.12.m12.1a"><mrow id="S4.SS3.SSS0.Px3.p1.12.m12.1.1" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.cmml"><mrow id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.cmml"><mrow id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.cmml"><msup id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.2" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.2.cmml"><mi id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.2.2" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.2.2.cmml">c</mi><mn id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.2.3" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.2.3.cmml">2</mn></msup><mo id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.1" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.1.cmml">−</mo><mn id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.3" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.3.cmml">1</mn></mrow><mo id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.1" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.1.cmml">+</mo><mrow id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3.cmml"><mn id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3.2" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3.2.cmml">2</mn><mo id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3.1.cmml">⋅</mo><mn id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3.3" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3.3.cmml">1</mn></mrow></mrow><mo id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.1" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.1.cmml">=</mo><mrow id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.cmml"><msup id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.2" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.2.cmml"><mi id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.2.2" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.2.2.cmml">c</mi><mn id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.2.3" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.2.3.cmml">2</mn></msup><mo id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.1" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.1.cmml">+</mo><mn id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.3" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.12.m12.1b"><apply id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1"><eq id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.1"></eq><apply id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2"><plus id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.1"></plus><apply id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2"><minus id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.1.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.1"></minus><apply id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.2.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.2.1.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.2">superscript</csymbol><ci id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.2.2.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.2.2">𝑐</ci><cn id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.2.3">2</cn></apply><cn id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.2.3">1</cn></apply><apply id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3"><ci id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3.1.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3.1">⋅</ci><cn id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3.2.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3.2">2</cn><cn id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.2.3.3">1</cn></apply></apply><apply id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3"><plus id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.1"></plus><apply id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.2.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.2.1.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.2">superscript</csymbol><ci id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.2.2.cmml" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.2.2">𝑐</ci><cn id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.2.3">2</cn></apply><cn id="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.12.m12.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.12.m12.1c">c^{2}-1+2\cdot 1=c^{2}+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.12.m12.1d">italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 + 2 ⋅ 1 = italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1</annotation></semantics></math>. Hence, the approximation ratio is at most <math alttext="(c^{2}-1)/(c^{2}+1)." class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.13.m13.1"><semantics id="S4.SS3.SSS0.Px3.p1.13.m13.1a"><mrow id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.cmml"><mrow id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.cmml"><mrow id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.cmml"><mo id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.cmml"><msup id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.2" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.2.cmml"><mi id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.2.2" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.2.2.cmml">c</mi><mn id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.2.3" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.2.3.cmml">2</mn></msup><mo id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.1" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.3" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.3" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.3.cmml">/</mo><mrow id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.cmml"><mo id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.2" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.cmml">(</mo><mrow id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.cmml"><msup id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.2" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.2.cmml"><mi id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.2.2" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.2.2.cmml">c</mi><mn id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.2.3" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.2.3.cmml">2</mn></msup><mo id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.1" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.1.cmml">+</mo><mn id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.3" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.3.cmml">1</mn></mrow><mo id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.3" stretchy="false" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.2" lspace="0em" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.13.m13.1b"><apply id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1"><divide id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.3.cmml" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.3"></divide><apply id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1"><minus id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.1"></minus><apply id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.2.cmml" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.2.2.cmml" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.2.2">𝑐</ci><cn id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.2.3">2</cn></apply><cn id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.1.1.1.3">1</cn></apply><apply id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1"><plus id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.1"></plus><apply id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.2.cmml" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.2">superscript</csymbol><ci id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.2.2.cmml" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.2.2">𝑐</ci><cn id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.2.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.2.3">2</cn></apply><cn id="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.3.cmml" type="integer" xref="S4.SS3.SSS0.Px3.p1.13.m13.1.1.1.1.2.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.13.m13.1c">(c^{2}-1)/(c^{2}+1).</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.13.m13.1d">( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) / ( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1 ) .</annotation></semantics></math> For <math alttext="b\leq 0.225" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.14.m14.1"><semantics id="S4.SS3.SSS0.Px3.p1.14.m14.1a"><mrow id="S4.SS3.SSS0.Px3.p1.14.m14.1.1" xref="S4.SS3.SSS0.Px3.p1.14.m14.1.1.cmml"><mi id="S4.SS3.SSS0.Px3.p1.14.m14.1.1.2" xref="S4.SS3.SSS0.Px3.p1.14.m14.1.1.2.cmml">b</mi><mo id="S4.SS3.SSS0.Px3.p1.14.m14.1.1.1" xref="S4.SS3.SSS0.Px3.p1.14.m14.1.1.1.cmml">≤</mo><mn id="S4.SS3.SSS0.Px3.p1.14.m14.1.1.3" xref="S4.SS3.SSS0.Px3.p1.14.m14.1.1.3.cmml">0.225</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.14.m14.1b"><apply id="S4.SS3.SSS0.Px3.p1.14.m14.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.14.m14.1.1"><leq id="S4.SS3.SSS0.Px3.p1.14.m14.1.1.1.cmml" xref="S4.SS3.SSS0.Px3.p1.14.m14.1.1.1"></leq><ci id="S4.SS3.SSS0.Px3.p1.14.m14.1.1.2.cmml" xref="S4.SS3.SSS0.Px3.p1.14.m14.1.1.2">𝑏</ci><cn id="S4.SS3.SSS0.Px3.p1.14.m14.1.1.3.cmml" type="float" xref="S4.SS3.SSS0.Px3.p1.14.m14.1.1.3">0.225</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.14.m14.1c">b\leq 0.225</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.14.m14.1d">italic_b ≤ 0.225</annotation></semantics></math>, this value is less than <math alttext="0.486" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p1.15.m15.1"><semantics id="S4.SS3.SSS0.Px3.p1.15.m15.1a"><mn id="S4.SS3.SSS0.Px3.p1.15.m15.1.1" xref="S4.SS3.SSS0.Px3.p1.15.m15.1.1.cmml">0.486</mn><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p1.15.m15.1b"><cn id="S4.SS3.SSS0.Px3.p1.15.m15.1.1.cmml" type="float" xref="S4.SS3.SSS0.Px3.p1.15.m15.1.1">0.486</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p1.15.m15.1c">0.486</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p1.15.m15.1d">0.486</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS3.SSS0.Px3.p2"> <p class="ltx_p" id="S4.SS3.SSS0.Px3.p2.1">We conclude that no PL sigmoid function can achieve an approximation ratio of <math alttext="0.486," class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px3.p2.1.m1.1"><semantics id="S4.SS3.SSS0.Px3.p2.1.m1.1a"><mrow id="S4.SS3.SSS0.Px3.p2.1.m1.1.2.2"><mn id="S4.SS3.SSS0.Px3.p2.1.m1.1.1" xref="S4.SS3.SSS0.Px3.p2.1.m1.1.1.cmml">0.486</mn><mo id="S4.SS3.SSS0.Px3.p2.1.m1.1.2.2.1">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px3.p2.1.m1.1b"><cn id="S4.SS3.SSS0.Px3.p2.1.m1.1.1.cmml" type="float" xref="S4.SS3.SSS0.Px3.p2.1.m1.1.1">0.486</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px3.p2.1.m1.1c">0.486,</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px3.p2.1.m1.1d">0.486 ,</annotation></semantics></math> as desired. ∎</p> </div> </section> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Lower bound for arbitrary selection functions (<a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem9" title="Theorem 1.9 (Lower bound for general selection). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.9</span></a>)</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">In this section, we prove <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem9" title="Theorem 1.9 (Lower bound for general selection). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.9</span></a>:</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1">See <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem9" title="Theorem 1.9 (Lower bound for general selection). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">1.9</span></a></p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.1">This gives a lower bound on the ratio achievable by <em class="ltx_emph ltx_font_italic" id="S5.p3.1.1">any</em> (not necessarily antisymmetric) selection function. We use the graph in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S4.F3" title="In 4.2 Bounds against 𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽_{1/2} (Theorem 1.6) ‣ 4 Lower bounds ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Fig.</span> <span class="ltx_text ltx_ref_tag">3</span></a> above as well as <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S5.F5" title="In 5 Lower bound for arbitrary selection functions (Theorem 1.9) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Fig.</span> <span class="ltx_text ltx_ref_tag">5</span></a> below:</p> </div> <figure class="ltx_figure" id="S5.F5"><svg class="ltx_picture ltx_centering" height="142.98" id="S5.F5.pic1" overflow="visible" version="1.1" width="168.46"><g fill="#000000" stroke="#000000" transform="translate(0,142.98) matrix(1 0 0 -1 0 0) translate(-92.73,0) translate(0,12.43)"><g stroke-width="0.4pt"><g 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class="ltx_text" id="S5.F5.25.11.1" style="font-size:90%;">Figure 5</span>: </span><em class="ltx_emph ltx_font_italic" id="S5.F5.26.12" style="font-size:90%;">Parameter:</em><span class="ltx_text" id="S5.F5.20.10" style="font-size:90%;"> <math alttext="c>1" class="ltx_Math" display="inline" id="S5.F5.11.1.m1.1"><semantics id="S5.F5.11.1.m1.1b"><mrow id="S5.F5.11.1.m1.1.1" xref="S5.F5.11.1.m1.1.1.cmml"><mi id="S5.F5.11.1.m1.1.1.2" xref="S5.F5.11.1.m1.1.1.2.cmml">c</mi><mo id="S5.F5.11.1.m1.1.1.1" xref="S5.F5.11.1.m1.1.1.1.cmml">></mo><mn id="S5.F5.11.1.m1.1.1.3" xref="S5.F5.11.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.F5.11.1.m1.1c"><apply id="S5.F5.11.1.m1.1.1.cmml" xref="S5.F5.11.1.m1.1.1"><gt id="S5.F5.11.1.m1.1.1.1.cmml" xref="S5.F5.11.1.m1.1.1.1"></gt><ci id="S5.F5.11.1.m1.1.1.2.cmml" xref="S5.F5.11.1.m1.1.1.2">𝑐</ci><cn id="S5.F5.11.1.m1.1.1.3.cmml" type="integer" xref="S5.F5.11.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F5.11.1.m1.1d">c>1</annotation><annotation encoding="application/x-llamapun" id="S5.F5.11.1.m1.1e">italic_c > 1</annotation></semantics></math>. The <span class="ltx_text" id="S5.F5.20.10.1" style="background-color:#ABDEE6;">LIGHT BLUE</span> vertices (<math alttext="\{1,2\}" class="ltx_Math" display="inline" id="S5.F5.12.2.m2.2"><semantics id="S5.F5.12.2.m2.2b"><mrow id="S5.F5.12.2.m2.2.3.2" xref="S5.F5.12.2.m2.2.3.1.cmml"><mo id="S5.F5.12.2.m2.2.3.2.1" stretchy="false" xref="S5.F5.12.2.m2.2.3.1.cmml">{</mo><mn id="S5.F5.12.2.m2.1.1" xref="S5.F5.12.2.m2.1.1.cmml">1</mn><mo id="S5.F5.12.2.m2.2.3.2.2" xref="S5.F5.12.2.m2.2.3.1.cmml">,</mo><mn id="S5.F5.12.2.m2.2.2" xref="S5.F5.12.2.m2.2.2.cmml">2</mn><mo id="S5.F5.12.2.m2.2.3.2.3" stretchy="false" xref="S5.F5.12.2.m2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.F5.12.2.m2.2c"><set id="S5.F5.12.2.m2.2.3.1.cmml" xref="S5.F5.12.2.m2.2.3.2"><cn id="S5.F5.12.2.m2.1.1.cmml" type="integer" xref="S5.F5.12.2.m2.1.1">1</cn><cn id="S5.F5.12.2.m2.2.2.cmml" type="integer" xref="S5.F5.12.2.m2.2.2">2</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S5.F5.12.2.m2.2d">\{1,2\}</annotation><annotation encoding="application/x-llamapun" id="S5.F5.12.2.m2.2e">{ 1 , 2 }</annotation></semantics></math>) have bias <math alttext="+\frac{c-1}{c+1}" class="ltx_Math" display="inline" id="S5.F5.13.3.m3.1"><semantics id="S5.F5.13.3.m3.1b"><mrow id="S5.F5.13.3.m3.1.1" xref="S5.F5.13.3.m3.1.1.cmml"><mo id="S5.F5.13.3.m3.1.1b" xref="S5.F5.13.3.m3.1.1.cmml">+</mo><mfrac id="S5.F5.13.3.m3.1.1.2" xref="S5.F5.13.3.m3.1.1.2.cmml"><mrow id="S5.F5.13.3.m3.1.1.2.2" xref="S5.F5.13.3.m3.1.1.2.2.cmml"><mi id="S5.F5.13.3.m3.1.1.2.2.2" xref="S5.F5.13.3.m3.1.1.2.2.2.cmml">c</mi><mo id="S5.F5.13.3.m3.1.1.2.2.1" xref="S5.F5.13.3.m3.1.1.2.2.1.cmml">−</mo><mn id="S5.F5.13.3.m3.1.1.2.2.3" xref="S5.F5.13.3.m3.1.1.2.2.3.cmml">1</mn></mrow><mrow id="S5.F5.13.3.m3.1.1.2.3" xref="S5.F5.13.3.m3.1.1.2.3.cmml"><mi id="S5.F5.13.3.m3.1.1.2.3.2" xref="S5.F5.13.3.m3.1.1.2.3.2.cmml">c</mi><mo id="S5.F5.13.3.m3.1.1.2.3.1" xref="S5.F5.13.3.m3.1.1.2.3.1.cmml">+</mo><mn id="S5.F5.13.3.m3.1.1.2.3.3" xref="S5.F5.13.3.m3.1.1.2.3.3.cmml">1</mn></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.F5.13.3.m3.1c"><apply id="S5.F5.13.3.m3.1.1.cmml" xref="S5.F5.13.3.m3.1.1"><plus id="S5.F5.13.3.m3.1.1.1.cmml" xref="S5.F5.13.3.m3.1.1"></plus><apply id="S5.F5.13.3.m3.1.1.2.cmml" xref="S5.F5.13.3.m3.1.1.2"><divide id="S5.F5.13.3.m3.1.1.2.1.cmml" xref="S5.F5.13.3.m3.1.1.2"></divide><apply id="S5.F5.13.3.m3.1.1.2.2.cmml" xref="S5.F5.13.3.m3.1.1.2.2"><minus id="S5.F5.13.3.m3.1.1.2.2.1.cmml" xref="S5.F5.13.3.m3.1.1.2.2.1"></minus><ci id="S5.F5.13.3.m3.1.1.2.2.2.cmml" xref="S5.F5.13.3.m3.1.1.2.2.2">𝑐</ci><cn id="S5.F5.13.3.m3.1.1.2.2.3.cmml" type="integer" xref="S5.F5.13.3.m3.1.1.2.2.3">1</cn></apply><apply id="S5.F5.13.3.m3.1.1.2.3.cmml" xref="S5.F5.13.3.m3.1.1.2.3"><plus id="S5.F5.13.3.m3.1.1.2.3.1.cmml" xref="S5.F5.13.3.m3.1.1.2.3.1"></plus><ci id="S5.F5.13.3.m3.1.1.2.3.2.cmml" xref="S5.F5.13.3.m3.1.1.2.3.2">𝑐</ci><cn id="S5.F5.13.3.m3.1.1.2.3.3.cmml" type="integer" xref="S5.F5.13.3.m3.1.1.2.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F5.13.3.m3.1d">+\frac{c-1}{c+1}</annotation><annotation encoding="application/x-llamapun" id="S5.F5.13.3.m3.1e">+ divide start_ARG italic_c - 1 end_ARG start_ARG italic_c + 1 end_ARG</annotation></semantics></math> (note that <math alttext="\frac{(c^{2}-1)+1-c}{(c^{2}-1)+1+c}=\frac{c(c-1)}{c(c+1)}=\frac{c-1}{c+1}" class="ltx_Math" display="inline" id="S5.F5.14.4.m4.4"><semantics id="S5.F5.14.4.m4.4b"><mrow id="S5.F5.14.4.m4.4.5" xref="S5.F5.14.4.m4.4.5.cmml"><mfrac id="S5.F5.14.4.m4.2.2" xref="S5.F5.14.4.m4.2.2.cmml"><mrow id="S5.F5.14.4.m4.1.1.1" xref="S5.F5.14.4.m4.1.1.1.cmml"><mrow id="S5.F5.14.4.m4.1.1.1.1" xref="S5.F5.14.4.m4.1.1.1.1.cmml"><mrow id="S5.F5.14.4.m4.1.1.1.1.1.1" xref="S5.F5.14.4.m4.1.1.1.1.1.1.1.cmml"><mo id="S5.F5.14.4.m4.1.1.1.1.1.1.2" stretchy="false" xref="S5.F5.14.4.m4.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.F5.14.4.m4.1.1.1.1.1.1.1" xref="S5.F5.14.4.m4.1.1.1.1.1.1.1.cmml"><msup id="S5.F5.14.4.m4.1.1.1.1.1.1.1.2" xref="S5.F5.14.4.m4.1.1.1.1.1.1.1.2.cmml"><mi id="S5.F5.14.4.m4.1.1.1.1.1.1.1.2.2" xref="S5.F5.14.4.m4.1.1.1.1.1.1.1.2.2.cmml">c</mi><mn id="S5.F5.14.4.m4.1.1.1.1.1.1.1.2.3" xref="S5.F5.14.4.m4.1.1.1.1.1.1.1.2.3.cmml">2</mn></msup><mo id="S5.F5.14.4.m4.1.1.1.1.1.1.1.1" xref="S5.F5.14.4.m4.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S5.F5.14.4.m4.1.1.1.1.1.1.1.3" xref="S5.F5.14.4.m4.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S5.F5.14.4.m4.1.1.1.1.1.1.3" stretchy="false" xref="S5.F5.14.4.m4.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.F5.14.4.m4.1.1.1.1.2" xref="S5.F5.14.4.m4.1.1.1.1.2.cmml">+</mo><mn id="S5.F5.14.4.m4.1.1.1.1.3" xref="S5.F5.14.4.m4.1.1.1.1.3.cmml">1</mn></mrow><mo id="S5.F5.14.4.m4.1.1.1.2" xref="S5.F5.14.4.m4.1.1.1.2.cmml">−</mo><mi id="S5.F5.14.4.m4.1.1.1.3" xref="S5.F5.14.4.m4.1.1.1.3.cmml">c</mi></mrow><mrow id="S5.F5.14.4.m4.2.2.2" xref="S5.F5.14.4.m4.2.2.2.cmml"><mrow id="S5.F5.14.4.m4.2.2.2.1.1" xref="S5.F5.14.4.m4.2.2.2.1.1.1.cmml"><mo id="S5.F5.14.4.m4.2.2.2.1.1.2" stretchy="false" xref="S5.F5.14.4.m4.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.F5.14.4.m4.2.2.2.1.1.1" xref="S5.F5.14.4.m4.2.2.2.1.1.1.cmml"><msup id="S5.F5.14.4.m4.2.2.2.1.1.1.2" xref="S5.F5.14.4.m4.2.2.2.1.1.1.2.cmml"><mi id="S5.F5.14.4.m4.2.2.2.1.1.1.2.2" xref="S5.F5.14.4.m4.2.2.2.1.1.1.2.2.cmml">c</mi><mn id="S5.F5.14.4.m4.2.2.2.1.1.1.2.3" xref="S5.F5.14.4.m4.2.2.2.1.1.1.2.3.cmml">2</mn></msup><mo id="S5.F5.14.4.m4.2.2.2.1.1.1.1" xref="S5.F5.14.4.m4.2.2.2.1.1.1.1.cmml">−</mo><mn id="S5.F5.14.4.m4.2.2.2.1.1.1.3" xref="S5.F5.14.4.m4.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S5.F5.14.4.m4.2.2.2.1.1.3" stretchy="false" xref="S5.F5.14.4.m4.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="S5.F5.14.4.m4.2.2.2.2" xref="S5.F5.14.4.m4.2.2.2.2.cmml">+</mo><mn id="S5.F5.14.4.m4.2.2.2.3" xref="S5.F5.14.4.m4.2.2.2.3.cmml">1</mn><mo id="S5.F5.14.4.m4.2.2.2.2b" xref="S5.F5.14.4.m4.2.2.2.2.cmml">+</mo><mi id="S5.F5.14.4.m4.2.2.2.4" xref="S5.F5.14.4.m4.2.2.2.4.cmml">c</mi></mrow></mfrac><mo id="S5.F5.14.4.m4.4.5.2" xref="S5.F5.14.4.m4.4.5.2.cmml">=</mo><mfrac id="S5.F5.14.4.m4.4.4" xref="S5.F5.14.4.m4.4.4.cmml"><mrow id="S5.F5.14.4.m4.3.3.1" xref="S5.F5.14.4.m4.3.3.1.cmml"><mi id="S5.F5.14.4.m4.3.3.1.3" xref="S5.F5.14.4.m4.3.3.1.3.cmml">c</mi><mo id="S5.F5.14.4.m4.3.3.1.2" xref="S5.F5.14.4.m4.3.3.1.2.cmml">⁢</mo><mrow id="S5.F5.14.4.m4.3.3.1.1.1" xref="S5.F5.14.4.m4.3.3.1.1.1.1.cmml"><mo id="S5.F5.14.4.m4.3.3.1.1.1.2" stretchy="false" xref="S5.F5.14.4.m4.3.3.1.1.1.1.cmml">(</mo><mrow id="S5.F5.14.4.m4.3.3.1.1.1.1" xref="S5.F5.14.4.m4.3.3.1.1.1.1.cmml"><mi id="S5.F5.14.4.m4.3.3.1.1.1.1.2" xref="S5.F5.14.4.m4.3.3.1.1.1.1.2.cmml">c</mi><mo id="S5.F5.14.4.m4.3.3.1.1.1.1.1" xref="S5.F5.14.4.m4.3.3.1.1.1.1.1.cmml">−</mo><mn id="S5.F5.14.4.m4.3.3.1.1.1.1.3" xref="S5.F5.14.4.m4.3.3.1.1.1.1.3.cmml">1</mn></mrow><mo id="S5.F5.14.4.m4.3.3.1.1.1.3" stretchy="false" xref="S5.F5.14.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S5.F5.14.4.m4.4.4.2" xref="S5.F5.14.4.m4.4.4.2.cmml"><mi id="S5.F5.14.4.m4.4.4.2.3" xref="S5.F5.14.4.m4.4.4.2.3.cmml">c</mi><mo id="S5.F5.14.4.m4.4.4.2.2" xref="S5.F5.14.4.m4.4.4.2.2.cmml">⁢</mo><mrow id="S5.F5.14.4.m4.4.4.2.1.1" xref="S5.F5.14.4.m4.4.4.2.1.1.1.cmml"><mo id="S5.F5.14.4.m4.4.4.2.1.1.2" stretchy="false" xref="S5.F5.14.4.m4.4.4.2.1.1.1.cmml">(</mo><mrow id="S5.F5.14.4.m4.4.4.2.1.1.1" xref="S5.F5.14.4.m4.4.4.2.1.1.1.cmml"><mi id="S5.F5.14.4.m4.4.4.2.1.1.1.2" xref="S5.F5.14.4.m4.4.4.2.1.1.1.2.cmml">c</mi><mo id="S5.F5.14.4.m4.4.4.2.1.1.1.1" xref="S5.F5.14.4.m4.4.4.2.1.1.1.1.cmml">+</mo><mn id="S5.F5.14.4.m4.4.4.2.1.1.1.3" xref="S5.F5.14.4.m4.4.4.2.1.1.1.3.cmml">1</mn></mrow><mo id="S5.F5.14.4.m4.4.4.2.1.1.3" stretchy="false" xref="S5.F5.14.4.m4.4.4.2.1.1.1.cmml">)</mo></mrow></mrow></mfrac><mo id="S5.F5.14.4.m4.4.5.3" 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xref="S5.F5.14.4.m4.3.3.1.1.1.1.3">1</cn></apply></apply><apply id="S5.F5.14.4.m4.4.4.2.cmml" xref="S5.F5.14.4.m4.4.4.2"><times id="S5.F5.14.4.m4.4.4.2.2.cmml" xref="S5.F5.14.4.m4.4.4.2.2"></times><ci id="S5.F5.14.4.m4.4.4.2.3.cmml" xref="S5.F5.14.4.m4.4.4.2.3">𝑐</ci><apply id="S5.F5.14.4.m4.4.4.2.1.1.1.cmml" xref="S5.F5.14.4.m4.4.4.2.1.1"><plus id="S5.F5.14.4.m4.4.4.2.1.1.1.1.cmml" xref="S5.F5.14.4.m4.4.4.2.1.1.1.1"></plus><ci id="S5.F5.14.4.m4.4.4.2.1.1.1.2.cmml" xref="S5.F5.14.4.m4.4.4.2.1.1.1.2">𝑐</ci><cn id="S5.F5.14.4.m4.4.4.2.1.1.1.3.cmml" type="integer" xref="S5.F5.14.4.m4.4.4.2.1.1.1.3">1</cn></apply></apply></apply></apply><apply id="S5.F5.14.4.m4.4.5c.cmml" xref="S5.F5.14.4.m4.4.5"><eq id="S5.F5.14.4.m4.4.5.3.cmml" xref="S5.F5.14.4.m4.4.5.3"></eq><share href="https://arxiv.org/html/2411.12976v1#S5.F5.14.4.m4.4.4.cmml" id="S5.F5.14.4.m4.4.5d.cmml" xref="S5.F5.14.4.m4.4.5"></share><apply id="S5.F5.14.4.m4.4.5.4.cmml" xref="S5.F5.14.4.m4.4.5.4"><divide id="S5.F5.14.4.m4.4.5.4.1.cmml" xref="S5.F5.14.4.m4.4.5.4"></divide><apply id="S5.F5.14.4.m4.4.5.4.2.cmml" xref="S5.F5.14.4.m4.4.5.4.2"><minus id="S5.F5.14.4.m4.4.5.4.2.1.cmml" xref="S5.F5.14.4.m4.4.5.4.2.1"></minus><ci id="S5.F5.14.4.m4.4.5.4.2.2.cmml" xref="S5.F5.14.4.m4.4.5.4.2.2">𝑐</ci><cn id="S5.F5.14.4.m4.4.5.4.2.3.cmml" type="integer" xref="S5.F5.14.4.m4.4.5.4.2.3">1</cn></apply><apply id="S5.F5.14.4.m4.4.5.4.3.cmml" xref="S5.F5.14.4.m4.4.5.4.3"><plus id="S5.F5.14.4.m4.4.5.4.3.1.cmml" xref="S5.F5.14.4.m4.4.5.4.3.1"></plus><ci id="S5.F5.14.4.m4.4.5.4.3.2.cmml" xref="S5.F5.14.4.m4.4.5.4.3.2">𝑐</ci><cn id="S5.F5.14.4.m4.4.5.4.3.3.cmml" type="integer" xref="S5.F5.14.4.m4.4.5.4.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F5.14.4.m4.4d">\frac{(c^{2}-1)+1-c}{(c^{2}-1)+1+c}=\frac{c(c-1)}{c(c+1)}=\frac{c-1}{c+1}</annotation><annotation encoding="application/x-llamapun" id="S5.F5.14.4.m4.4e">divide start_ARG ( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) + 1 - italic_c end_ARG start_ARG ( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) + 1 + italic_c end_ARG = divide start_ARG italic_c ( italic_c - 1 ) end_ARG start_ARG italic_c ( italic_c + 1 ) end_ARG = divide start_ARG italic_c - 1 end_ARG start_ARG italic_c + 1 end_ARG</annotation></semantics></math>). The <span class="ltx_text" id="S5.F5.20.10.2" style="background-color:#FEE1E8;">PINK</span> vertices (<math alttext="\{3,4\}" class="ltx_Math" display="inline" id="S5.F5.15.5.m5.2"><semantics id="S5.F5.15.5.m5.2b"><mrow id="S5.F5.15.5.m5.2.3.2" xref="S5.F5.15.5.m5.2.3.1.cmml"><mo id="S5.F5.15.5.m5.2.3.2.1" stretchy="false" xref="S5.F5.15.5.m5.2.3.1.cmml">{</mo><mn id="S5.F5.15.5.m5.1.1" xref="S5.F5.15.5.m5.1.1.cmml">3</mn><mo id="S5.F5.15.5.m5.2.3.2.2" xref="S5.F5.15.5.m5.2.3.1.cmml">,</mo><mn id="S5.F5.15.5.m5.2.2" xref="S5.F5.15.5.m5.2.2.cmml">4</mn><mo id="S5.F5.15.5.m5.2.3.2.3" stretchy="false" xref="S5.F5.15.5.m5.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.F5.15.5.m5.2c"><set id="S5.F5.15.5.m5.2.3.1.cmml" xref="S5.F5.15.5.m5.2.3.2"><cn id="S5.F5.15.5.m5.1.1.cmml" type="integer" xref="S5.F5.15.5.m5.1.1">3</cn><cn id="S5.F5.15.5.m5.2.2.cmml" type="integer" xref="S5.F5.15.5.m5.2.2">4</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S5.F5.15.5.m5.2d">\{3,4\}</annotation><annotation encoding="application/x-llamapun" id="S5.F5.15.5.m5.2e">{ 3 , 4 }</annotation></semantics></math>) have bias <math alttext="-\frac{c-1}{c+1}" class="ltx_Math" display="inline" id="S5.F5.16.6.m6.1"><semantics id="S5.F5.16.6.m6.1b"><mrow id="S5.F5.16.6.m6.1.1" xref="S5.F5.16.6.m6.1.1.cmml"><mo id="S5.F5.16.6.m6.1.1b" xref="S5.F5.16.6.m6.1.1.cmml">−</mo><mfrac id="S5.F5.16.6.m6.1.1.2" xref="S5.F5.16.6.m6.1.1.2.cmml"><mrow id="S5.F5.16.6.m6.1.1.2.2" xref="S5.F5.16.6.m6.1.1.2.2.cmml"><mi id="S5.F5.16.6.m6.1.1.2.2.2" xref="S5.F5.16.6.m6.1.1.2.2.2.cmml">c</mi><mo id="S5.F5.16.6.m6.1.1.2.2.1" xref="S5.F5.16.6.m6.1.1.2.2.1.cmml">−</mo><mn id="S5.F5.16.6.m6.1.1.2.2.3" xref="S5.F5.16.6.m6.1.1.2.2.3.cmml">1</mn></mrow><mrow id="S5.F5.16.6.m6.1.1.2.3" xref="S5.F5.16.6.m6.1.1.2.3.cmml"><mi id="S5.F5.16.6.m6.1.1.2.3.2" xref="S5.F5.16.6.m6.1.1.2.3.2.cmml">c</mi><mo id="S5.F5.16.6.m6.1.1.2.3.1" xref="S5.F5.16.6.m6.1.1.2.3.1.cmml">+</mo><mn id="S5.F5.16.6.m6.1.1.2.3.3" xref="S5.F5.16.6.m6.1.1.2.3.3.cmml">1</mn></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.F5.16.6.m6.1c"><apply id="S5.F5.16.6.m6.1.1.cmml" xref="S5.F5.16.6.m6.1.1"><minus id="S5.F5.16.6.m6.1.1.1.cmml" xref="S5.F5.16.6.m6.1.1"></minus><apply id="S5.F5.16.6.m6.1.1.2.cmml" xref="S5.F5.16.6.m6.1.1.2"><divide id="S5.F5.16.6.m6.1.1.2.1.cmml" xref="S5.F5.16.6.m6.1.1.2"></divide><apply id="S5.F5.16.6.m6.1.1.2.2.cmml" xref="S5.F5.16.6.m6.1.1.2.2"><minus id="S5.F5.16.6.m6.1.1.2.2.1.cmml" xref="S5.F5.16.6.m6.1.1.2.2.1"></minus><ci id="S5.F5.16.6.m6.1.1.2.2.2.cmml" xref="S5.F5.16.6.m6.1.1.2.2.2">𝑐</ci><cn id="S5.F5.16.6.m6.1.1.2.2.3.cmml" type="integer" xref="S5.F5.16.6.m6.1.1.2.2.3">1</cn></apply><apply id="S5.F5.16.6.m6.1.1.2.3.cmml" xref="S5.F5.16.6.m6.1.1.2.3"><plus id="S5.F5.16.6.m6.1.1.2.3.1.cmml" xref="S5.F5.16.6.m6.1.1.2.3.1"></plus><ci id="S5.F5.16.6.m6.1.1.2.3.2.cmml" xref="S5.F5.16.6.m6.1.1.2.3.2">𝑐</ci><cn id="S5.F5.16.6.m6.1.1.2.3.3.cmml" type="integer" xref="S5.F5.16.6.m6.1.1.2.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F5.16.6.m6.1d">-\frac{c-1}{c+1}</annotation><annotation encoding="application/x-llamapun" id="S5.F5.16.6.m6.1e">- divide start_ARG italic_c - 1 end_ARG start_ARG italic_c + 1 end_ARG</annotation></semantics></math>. The assignment <math alttext="\{1,3\}\to 1,\{2,4\}\to 0" class="ltx_Math" display="inline" id="S5.F5.17.7.m7.6"><semantics id="S5.F5.17.7.m7.6b"><mrow id="S5.F5.17.7.m7.6.6.2" xref="S5.F5.17.7.m7.6.6.3.cmml"><mrow id="S5.F5.17.7.m7.5.5.1.1" xref="S5.F5.17.7.m7.5.5.1.1.cmml"><mrow id="S5.F5.17.7.m7.5.5.1.1.2.2" xref="S5.F5.17.7.m7.5.5.1.1.2.1.cmml"><mo id="S5.F5.17.7.m7.5.5.1.1.2.2.1" stretchy="false" xref="S5.F5.17.7.m7.5.5.1.1.2.1.cmml">{</mo><mn id="S5.F5.17.7.m7.1.1" xref="S5.F5.17.7.m7.1.1.cmml">1</mn><mo id="S5.F5.17.7.m7.5.5.1.1.2.2.2" xref="S5.F5.17.7.m7.5.5.1.1.2.1.cmml">,</mo><mn id="S5.F5.17.7.m7.2.2" xref="S5.F5.17.7.m7.2.2.cmml">3</mn><mo id="S5.F5.17.7.m7.5.5.1.1.2.2.3" stretchy="false" xref="S5.F5.17.7.m7.5.5.1.1.2.1.cmml">}</mo></mrow><mo id="S5.F5.17.7.m7.5.5.1.1.1" stretchy="false" xref="S5.F5.17.7.m7.5.5.1.1.1.cmml">→</mo><mn id="S5.F5.17.7.m7.5.5.1.1.3" xref="S5.F5.17.7.m7.5.5.1.1.3.cmml">1</mn></mrow><mo id="S5.F5.17.7.m7.6.6.2.3" xref="S5.F5.17.7.m7.6.6.3a.cmml">,</mo><mrow id="S5.F5.17.7.m7.6.6.2.2" xref="S5.F5.17.7.m7.6.6.2.2.cmml"><mrow id="S5.F5.17.7.m7.6.6.2.2.2.2" xref="S5.F5.17.7.m7.6.6.2.2.2.1.cmml"><mo id="S5.F5.17.7.m7.6.6.2.2.2.2.1" stretchy="false" xref="S5.F5.17.7.m7.6.6.2.2.2.1.cmml">{</mo><mn id="S5.F5.17.7.m7.3.3" xref="S5.F5.17.7.m7.3.3.cmml">2</mn><mo id="S5.F5.17.7.m7.6.6.2.2.2.2.2" xref="S5.F5.17.7.m7.6.6.2.2.2.1.cmml">,</mo><mn id="S5.F5.17.7.m7.4.4" xref="S5.F5.17.7.m7.4.4.cmml">4</mn><mo id="S5.F5.17.7.m7.6.6.2.2.2.2.3" stretchy="false" xref="S5.F5.17.7.m7.6.6.2.2.2.1.cmml">}</mo></mrow><mo id="S5.F5.17.7.m7.6.6.2.2.1" stretchy="false" xref="S5.F5.17.7.m7.6.6.2.2.1.cmml">→</mo><mn id="S5.F5.17.7.m7.6.6.2.2.3" xref="S5.F5.17.7.m7.6.6.2.2.3.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.F5.17.7.m7.6c"><apply id="S5.F5.17.7.m7.6.6.3.cmml" xref="S5.F5.17.7.m7.6.6.2"><csymbol cd="ambiguous" id="S5.F5.17.7.m7.6.6.3a.cmml" xref="S5.F5.17.7.m7.6.6.2.3">formulae-sequence</csymbol><apply id="S5.F5.17.7.m7.5.5.1.1.cmml" xref="S5.F5.17.7.m7.5.5.1.1"><ci id="S5.F5.17.7.m7.5.5.1.1.1.cmml" xref="S5.F5.17.7.m7.5.5.1.1.1">→</ci><set id="S5.F5.17.7.m7.5.5.1.1.2.1.cmml" xref="S5.F5.17.7.m7.5.5.1.1.2.2"><cn id="S5.F5.17.7.m7.1.1.cmml" type="integer" xref="S5.F5.17.7.m7.1.1">1</cn><cn id="S5.F5.17.7.m7.2.2.cmml" type="integer" xref="S5.F5.17.7.m7.2.2">3</cn></set><cn id="S5.F5.17.7.m7.5.5.1.1.3.cmml" type="integer" xref="S5.F5.17.7.m7.5.5.1.1.3">1</cn></apply><apply id="S5.F5.17.7.m7.6.6.2.2.cmml" xref="S5.F5.17.7.m7.6.6.2.2"><ci id="S5.F5.17.7.m7.6.6.2.2.1.cmml" xref="S5.F5.17.7.m7.6.6.2.2.1">→</ci><set id="S5.F5.17.7.m7.6.6.2.2.2.1.cmml" xref="S5.F5.17.7.m7.6.6.2.2.2.2"><cn id="S5.F5.17.7.m7.3.3.cmml" type="integer" xref="S5.F5.17.7.m7.3.3">2</cn><cn id="S5.F5.17.7.m7.4.4.cmml" type="integer" xref="S5.F5.17.7.m7.4.4">4</cn></set><cn id="S5.F5.17.7.m7.6.6.2.2.3.cmml" type="integer" xref="S5.F5.17.7.m7.6.6.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F5.17.7.m7.6d">\{1,3\}\to 1,\{2,4\}\to 0</annotation><annotation encoding="application/x-llamapun" id="S5.F5.17.7.m7.6e">{ 1 , 3 } → 1 , { 2 , 4 } → 0</annotation></semantics></math> satisfies weight <math alttext="c^{2}-1+2\cdot 1=c^{2}+1" class="ltx_Math" display="inline" id="S5.F5.18.8.m8.1"><semantics id="S5.F5.18.8.m8.1b"><mrow id="S5.F5.18.8.m8.1.1" xref="S5.F5.18.8.m8.1.1.cmml"><mrow id="S5.F5.18.8.m8.1.1.2" xref="S5.F5.18.8.m8.1.1.2.cmml"><mrow id="S5.F5.18.8.m8.1.1.2.2" xref="S5.F5.18.8.m8.1.1.2.2.cmml"><msup id="S5.F5.18.8.m8.1.1.2.2.2" xref="S5.F5.18.8.m8.1.1.2.2.2.cmml"><mi id="S5.F5.18.8.m8.1.1.2.2.2.2" xref="S5.F5.18.8.m8.1.1.2.2.2.2.cmml">c</mi><mn id="S5.F5.18.8.m8.1.1.2.2.2.3" xref="S5.F5.18.8.m8.1.1.2.2.2.3.cmml">2</mn></msup><mo id="S5.F5.18.8.m8.1.1.2.2.1" xref="S5.F5.18.8.m8.1.1.2.2.1.cmml">−</mo><mn id="S5.F5.18.8.m8.1.1.2.2.3" xref="S5.F5.18.8.m8.1.1.2.2.3.cmml">1</mn></mrow><mo id="S5.F5.18.8.m8.1.1.2.1" xref="S5.F5.18.8.m8.1.1.2.1.cmml">+</mo><mrow id="S5.F5.18.8.m8.1.1.2.3" xref="S5.F5.18.8.m8.1.1.2.3.cmml"><mn id="S5.F5.18.8.m8.1.1.2.3.2" xref="S5.F5.18.8.m8.1.1.2.3.2.cmml">2</mn><mo id="S5.F5.18.8.m8.1.1.2.3.1" lspace="0.222em" rspace="0.222em" xref="S5.F5.18.8.m8.1.1.2.3.1.cmml">⋅</mo><mn id="S5.F5.18.8.m8.1.1.2.3.3" xref="S5.F5.18.8.m8.1.1.2.3.3.cmml">1</mn></mrow></mrow><mo id="S5.F5.18.8.m8.1.1.1" xref="S5.F5.18.8.m8.1.1.1.cmml">=</mo><mrow id="S5.F5.18.8.m8.1.1.3" xref="S5.F5.18.8.m8.1.1.3.cmml"><msup id="S5.F5.18.8.m8.1.1.3.2" xref="S5.F5.18.8.m8.1.1.3.2.cmml"><mi id="S5.F5.18.8.m8.1.1.3.2.2" xref="S5.F5.18.8.m8.1.1.3.2.2.cmml">c</mi><mn id="S5.F5.18.8.m8.1.1.3.2.3" xref="S5.F5.18.8.m8.1.1.3.2.3.cmml">2</mn></msup><mo id="S5.F5.18.8.m8.1.1.3.1" xref="S5.F5.18.8.m8.1.1.3.1.cmml">+</mo><mn id="S5.F5.18.8.m8.1.1.3.3" xref="S5.F5.18.8.m8.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.F5.18.8.m8.1c"><apply id="S5.F5.18.8.m8.1.1.cmml" xref="S5.F5.18.8.m8.1.1"><eq id="S5.F5.18.8.m8.1.1.1.cmml" xref="S5.F5.18.8.m8.1.1.1"></eq><apply id="S5.F5.18.8.m8.1.1.2.cmml" xref="S5.F5.18.8.m8.1.1.2"><plus id="S5.F5.18.8.m8.1.1.2.1.cmml" xref="S5.F5.18.8.m8.1.1.2.1"></plus><apply id="S5.F5.18.8.m8.1.1.2.2.cmml" xref="S5.F5.18.8.m8.1.1.2.2"><minus id="S5.F5.18.8.m8.1.1.2.2.1.cmml" xref="S5.F5.18.8.m8.1.1.2.2.1"></minus><apply id="S5.F5.18.8.m8.1.1.2.2.2.cmml" xref="S5.F5.18.8.m8.1.1.2.2.2"><csymbol cd="ambiguous" id="S5.F5.18.8.m8.1.1.2.2.2.1.cmml" xref="S5.F5.18.8.m8.1.1.2.2.2">superscript</csymbol><ci id="S5.F5.18.8.m8.1.1.2.2.2.2.cmml" xref="S5.F5.18.8.m8.1.1.2.2.2.2">𝑐</ci><cn id="S5.F5.18.8.m8.1.1.2.2.2.3.cmml" type="integer" xref="S5.F5.18.8.m8.1.1.2.2.2.3">2</cn></apply><cn id="S5.F5.18.8.m8.1.1.2.2.3.cmml" type="integer" xref="S5.F5.18.8.m8.1.1.2.2.3">1</cn></apply><apply id="S5.F5.18.8.m8.1.1.2.3.cmml" xref="S5.F5.18.8.m8.1.1.2.3"><ci id="S5.F5.18.8.m8.1.1.2.3.1.cmml" xref="S5.F5.18.8.m8.1.1.2.3.1">⋅</ci><cn id="S5.F5.18.8.m8.1.1.2.3.2.cmml" type="integer" xref="S5.F5.18.8.m8.1.1.2.3.2">2</cn><cn id="S5.F5.18.8.m8.1.1.2.3.3.cmml" type="integer" xref="S5.F5.18.8.m8.1.1.2.3.3">1</cn></apply></apply><apply id="S5.F5.18.8.m8.1.1.3.cmml" xref="S5.F5.18.8.m8.1.1.3"><plus id="S5.F5.18.8.m8.1.1.3.1.cmml" xref="S5.F5.18.8.m8.1.1.3.1"></plus><apply id="S5.F5.18.8.m8.1.1.3.2.cmml" xref="S5.F5.18.8.m8.1.1.3.2"><csymbol cd="ambiguous" id="S5.F5.18.8.m8.1.1.3.2.1.cmml" xref="S5.F5.18.8.m8.1.1.3.2">superscript</csymbol><ci id="S5.F5.18.8.m8.1.1.3.2.2.cmml" xref="S5.F5.18.8.m8.1.1.3.2.2">𝑐</ci><cn id="S5.F5.18.8.m8.1.1.3.2.3.cmml" type="integer" xref="S5.F5.18.8.m8.1.1.3.2.3">2</cn></apply><cn id="S5.F5.18.8.m8.1.1.3.3.cmml" type="integer" xref="S5.F5.18.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F5.18.8.m8.1d">c^{2}-1+2\cdot 1=c^{2}+1</annotation><annotation encoding="application/x-llamapun" id="S5.F5.18.8.m8.1e">italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 + 2 ⋅ 1 = italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1</annotation></semantics></math>. An oblivious assignment <math alttext="\{1,2\}\to p,\{3,4\}\to q" class="ltx_Math" display="inline" id="S5.F5.19.9.m9.6"><semantics id="S5.F5.19.9.m9.6b"><mrow id="S5.F5.19.9.m9.6.6.2" xref="S5.F5.19.9.m9.6.6.3.cmml"><mrow id="S5.F5.19.9.m9.5.5.1.1" xref="S5.F5.19.9.m9.5.5.1.1.cmml"><mrow id="S5.F5.19.9.m9.5.5.1.1.2.2" xref="S5.F5.19.9.m9.5.5.1.1.2.1.cmml"><mo id="S5.F5.19.9.m9.5.5.1.1.2.2.1" stretchy="false" xref="S5.F5.19.9.m9.5.5.1.1.2.1.cmml">{</mo><mn id="S5.F5.19.9.m9.1.1" xref="S5.F5.19.9.m9.1.1.cmml">1</mn><mo id="S5.F5.19.9.m9.5.5.1.1.2.2.2" xref="S5.F5.19.9.m9.5.5.1.1.2.1.cmml">,</mo><mn id="S5.F5.19.9.m9.2.2" xref="S5.F5.19.9.m9.2.2.cmml">2</mn><mo id="S5.F5.19.9.m9.5.5.1.1.2.2.3" stretchy="false" xref="S5.F5.19.9.m9.5.5.1.1.2.1.cmml">}</mo></mrow><mo id="S5.F5.19.9.m9.5.5.1.1.1" stretchy="false" xref="S5.F5.19.9.m9.5.5.1.1.1.cmml">→</mo><mi id="S5.F5.19.9.m9.5.5.1.1.3" xref="S5.F5.19.9.m9.5.5.1.1.3.cmml">p</mi></mrow><mo id="S5.F5.19.9.m9.6.6.2.3" xref="S5.F5.19.9.m9.6.6.3a.cmml">,</mo><mrow id="S5.F5.19.9.m9.6.6.2.2" xref="S5.F5.19.9.m9.6.6.2.2.cmml"><mrow id="S5.F5.19.9.m9.6.6.2.2.2.2" xref="S5.F5.19.9.m9.6.6.2.2.2.1.cmml"><mo id="S5.F5.19.9.m9.6.6.2.2.2.2.1" stretchy="false" xref="S5.F5.19.9.m9.6.6.2.2.2.1.cmml">{</mo><mn id="S5.F5.19.9.m9.3.3" xref="S5.F5.19.9.m9.3.3.cmml">3</mn><mo id="S5.F5.19.9.m9.6.6.2.2.2.2.2" xref="S5.F5.19.9.m9.6.6.2.2.2.1.cmml">,</mo><mn id="S5.F5.19.9.m9.4.4" xref="S5.F5.19.9.m9.4.4.cmml">4</mn><mo id="S5.F5.19.9.m9.6.6.2.2.2.2.3" stretchy="false" xref="S5.F5.19.9.m9.6.6.2.2.2.1.cmml">}</mo></mrow><mo id="S5.F5.19.9.m9.6.6.2.2.1" stretchy="false" xref="S5.F5.19.9.m9.6.6.2.2.1.cmml">→</mo><mi id="S5.F5.19.9.m9.6.6.2.2.3" xref="S5.F5.19.9.m9.6.6.2.2.3.cmml">q</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.F5.19.9.m9.6c"><apply id="S5.F5.19.9.m9.6.6.3.cmml" xref="S5.F5.19.9.m9.6.6.2"><csymbol cd="ambiguous" id="S5.F5.19.9.m9.6.6.3a.cmml" xref="S5.F5.19.9.m9.6.6.2.3">formulae-sequence</csymbol><apply id="S5.F5.19.9.m9.5.5.1.1.cmml" xref="S5.F5.19.9.m9.5.5.1.1"><ci id="S5.F5.19.9.m9.5.5.1.1.1.cmml" xref="S5.F5.19.9.m9.5.5.1.1.1">→</ci><set id="S5.F5.19.9.m9.5.5.1.1.2.1.cmml" xref="S5.F5.19.9.m9.5.5.1.1.2.2"><cn id="S5.F5.19.9.m9.1.1.cmml" type="integer" xref="S5.F5.19.9.m9.1.1">1</cn><cn id="S5.F5.19.9.m9.2.2.cmml" type="integer" xref="S5.F5.19.9.m9.2.2">2</cn></set><ci id="S5.F5.19.9.m9.5.5.1.1.3.cmml" xref="S5.F5.19.9.m9.5.5.1.1.3">𝑝</ci></apply><apply id="S5.F5.19.9.m9.6.6.2.2.cmml" xref="S5.F5.19.9.m9.6.6.2.2"><ci id="S5.F5.19.9.m9.6.6.2.2.1.cmml" xref="S5.F5.19.9.m9.6.6.2.2.1">→</ci><set id="S5.F5.19.9.m9.6.6.2.2.2.1.cmml" xref="S5.F5.19.9.m9.6.6.2.2.2.2"><cn id="S5.F5.19.9.m9.3.3.cmml" type="integer" xref="S5.F5.19.9.m9.3.3">3</cn><cn id="S5.F5.19.9.m9.4.4.cmml" type="integer" xref="S5.F5.19.9.m9.4.4">4</cn></set><ci id="S5.F5.19.9.m9.6.6.2.2.3.cmml" xref="S5.F5.19.9.m9.6.6.2.2.3">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F5.19.9.m9.6d">\{1,2\}\to p,\{3,4\}\to q</annotation><annotation encoding="application/x-llamapun" id="S5.F5.19.9.m9.6e">{ 1 , 2 } → italic_p , { 3 , 4 } → italic_q</annotation></semantics></math> satisfies weight <math alttext="p(1-p)(c+1)+q(1-q)(c+1)+p(1-q)(c^{2}-1)" class="ltx_Math" display="inline" id="S5.F5.20.10.m10.6"><semantics id="S5.F5.20.10.m10.6b"><mrow id="S5.F5.20.10.m10.6.6" xref="S5.F5.20.10.m10.6.6.cmml"><mrow id="S5.F5.20.10.m10.2.2.2" xref="S5.F5.20.10.m10.2.2.2.cmml"><mi id="S5.F5.20.10.m10.2.2.2.4" xref="S5.F5.20.10.m10.2.2.2.4.cmml">p</mi><mo id="S5.F5.20.10.m10.2.2.2.3" xref="S5.F5.20.10.m10.2.2.2.3.cmml">⁢</mo><mrow id="S5.F5.20.10.m10.1.1.1.1.1" xref="S5.F5.20.10.m10.1.1.1.1.1.1.cmml"><mo id="S5.F5.20.10.m10.1.1.1.1.1.2" stretchy="false" xref="S5.F5.20.10.m10.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.F5.20.10.m10.1.1.1.1.1.1" xref="S5.F5.20.10.m10.1.1.1.1.1.1.cmml"><mn id="S5.F5.20.10.m10.1.1.1.1.1.1.2" xref="S5.F5.20.10.m10.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.F5.20.10.m10.1.1.1.1.1.1.1" xref="S5.F5.20.10.m10.1.1.1.1.1.1.1.cmml">−</mo><mi id="S5.F5.20.10.m10.1.1.1.1.1.1.3" xref="S5.F5.20.10.m10.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="S5.F5.20.10.m10.1.1.1.1.1.3" stretchy="false" xref="S5.F5.20.10.m10.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.F5.20.10.m10.2.2.2.3b" xref="S5.F5.20.10.m10.2.2.2.3.cmml">⁢</mo><mrow id="S5.F5.20.10.m10.2.2.2.2.1" xref="S5.F5.20.10.m10.2.2.2.2.1.1.cmml"><mo id="S5.F5.20.10.m10.2.2.2.2.1.2" stretchy="false" xref="S5.F5.20.10.m10.2.2.2.2.1.1.cmml">(</mo><mrow id="S5.F5.20.10.m10.2.2.2.2.1.1" xref="S5.F5.20.10.m10.2.2.2.2.1.1.cmml"><mi id="S5.F5.20.10.m10.2.2.2.2.1.1.2" xref="S5.F5.20.10.m10.2.2.2.2.1.1.2.cmml">c</mi><mo id="S5.F5.20.10.m10.2.2.2.2.1.1.1" xref="S5.F5.20.10.m10.2.2.2.2.1.1.1.cmml">+</mo><mn id="S5.F5.20.10.m10.2.2.2.2.1.1.3" xref="S5.F5.20.10.m10.2.2.2.2.1.1.3.cmml">1</mn></mrow><mo id="S5.F5.20.10.m10.2.2.2.2.1.3" stretchy="false" xref="S5.F5.20.10.m10.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S5.F5.20.10.m10.6.6.7" xref="S5.F5.20.10.m10.6.6.7.cmml">+</mo><mrow id="S5.F5.20.10.m10.4.4.4" xref="S5.F5.20.10.m10.4.4.4.cmml"><mi id="S5.F5.20.10.m10.4.4.4.4" xref="S5.F5.20.10.m10.4.4.4.4.cmml">q</mi><mo id="S5.F5.20.10.m10.4.4.4.3" xref="S5.F5.20.10.m10.4.4.4.3.cmml">⁢</mo><mrow id="S5.F5.20.10.m10.3.3.3.1.1" xref="S5.F5.20.10.m10.3.3.3.1.1.1.cmml"><mo id="S5.F5.20.10.m10.3.3.3.1.1.2" stretchy="false" xref="S5.F5.20.10.m10.3.3.3.1.1.1.cmml">(</mo><mrow id="S5.F5.20.10.m10.3.3.3.1.1.1" xref="S5.F5.20.10.m10.3.3.3.1.1.1.cmml"><mn id="S5.F5.20.10.m10.3.3.3.1.1.1.2" xref="S5.F5.20.10.m10.3.3.3.1.1.1.2.cmml">1</mn><mo id="S5.F5.20.10.m10.3.3.3.1.1.1.1" xref="S5.F5.20.10.m10.3.3.3.1.1.1.1.cmml">−</mo><mi id="S5.F5.20.10.m10.3.3.3.1.1.1.3" xref="S5.F5.20.10.m10.3.3.3.1.1.1.3.cmml">q</mi></mrow><mo id="S5.F5.20.10.m10.3.3.3.1.1.3" stretchy="false" xref="S5.F5.20.10.m10.3.3.3.1.1.1.cmml">)</mo></mrow><mo id="S5.F5.20.10.m10.4.4.4.3b" xref="S5.F5.20.10.m10.4.4.4.3.cmml">⁢</mo><mrow id="S5.F5.20.10.m10.4.4.4.2.1" xref="S5.F5.20.10.m10.4.4.4.2.1.1.cmml"><mo id="S5.F5.20.10.m10.4.4.4.2.1.2" stretchy="false" xref="S5.F5.20.10.m10.4.4.4.2.1.1.cmml">(</mo><mrow id="S5.F5.20.10.m10.4.4.4.2.1.1" xref="S5.F5.20.10.m10.4.4.4.2.1.1.cmml"><mi id="S5.F5.20.10.m10.4.4.4.2.1.1.2" xref="S5.F5.20.10.m10.4.4.4.2.1.1.2.cmml">c</mi><mo id="S5.F5.20.10.m10.4.4.4.2.1.1.1" xref="S5.F5.20.10.m10.4.4.4.2.1.1.1.cmml">+</mo><mn id="S5.F5.20.10.m10.4.4.4.2.1.1.3" xref="S5.F5.20.10.m10.4.4.4.2.1.1.3.cmml">1</mn></mrow><mo id="S5.F5.20.10.m10.4.4.4.2.1.3" stretchy="false" xref="S5.F5.20.10.m10.4.4.4.2.1.1.cmml">)</mo></mrow></mrow><mo id="S5.F5.20.10.m10.6.6.7b" xref="S5.F5.20.10.m10.6.6.7.cmml">+</mo><mrow id="S5.F5.20.10.m10.6.6.6" xref="S5.F5.20.10.m10.6.6.6.cmml"><mi id="S5.F5.20.10.m10.6.6.6.4" xref="S5.F5.20.10.m10.6.6.6.4.cmml">p</mi><mo id="S5.F5.20.10.m10.6.6.6.3" xref="S5.F5.20.10.m10.6.6.6.3.cmml">⁢</mo><mrow id="S5.F5.20.10.m10.5.5.5.1.1" xref="S5.F5.20.10.m10.5.5.5.1.1.1.cmml"><mo id="S5.F5.20.10.m10.5.5.5.1.1.2" stretchy="false" xref="S5.F5.20.10.m10.5.5.5.1.1.1.cmml">(</mo><mrow id="S5.F5.20.10.m10.5.5.5.1.1.1" xref="S5.F5.20.10.m10.5.5.5.1.1.1.cmml"><mn id="S5.F5.20.10.m10.5.5.5.1.1.1.2" xref="S5.F5.20.10.m10.5.5.5.1.1.1.2.cmml">1</mn><mo id="S5.F5.20.10.m10.5.5.5.1.1.1.1" xref="S5.F5.20.10.m10.5.5.5.1.1.1.1.cmml">−</mo><mi id="S5.F5.20.10.m10.5.5.5.1.1.1.3" xref="S5.F5.20.10.m10.5.5.5.1.1.1.3.cmml">q</mi></mrow><mo id="S5.F5.20.10.m10.5.5.5.1.1.3" stretchy="false" xref="S5.F5.20.10.m10.5.5.5.1.1.1.cmml">)</mo></mrow><mo id="S5.F5.20.10.m10.6.6.6.3b" xref="S5.F5.20.10.m10.6.6.6.3.cmml">⁢</mo><mrow id="S5.F5.20.10.m10.6.6.6.2.1" xref="S5.F5.20.10.m10.6.6.6.2.1.1.cmml"><mo id="S5.F5.20.10.m10.6.6.6.2.1.2" stretchy="false" xref="S5.F5.20.10.m10.6.6.6.2.1.1.cmml">(</mo><mrow id="S5.F5.20.10.m10.6.6.6.2.1.1" xref="S5.F5.20.10.m10.6.6.6.2.1.1.cmml"><msup id="S5.F5.20.10.m10.6.6.6.2.1.1.2" xref="S5.F5.20.10.m10.6.6.6.2.1.1.2.cmml"><mi id="S5.F5.20.10.m10.6.6.6.2.1.1.2.2" xref="S5.F5.20.10.m10.6.6.6.2.1.1.2.2.cmml">c</mi><mn id="S5.F5.20.10.m10.6.6.6.2.1.1.2.3" xref="S5.F5.20.10.m10.6.6.6.2.1.1.2.3.cmml">2</mn></msup><mo id="S5.F5.20.10.m10.6.6.6.2.1.1.1" xref="S5.F5.20.10.m10.6.6.6.2.1.1.1.cmml">−</mo><mn id="S5.F5.20.10.m10.6.6.6.2.1.1.3" xref="S5.F5.20.10.m10.6.6.6.2.1.1.3.cmml">1</mn></mrow><mo id="S5.F5.20.10.m10.6.6.6.2.1.3" stretchy="false" xref="S5.F5.20.10.m10.6.6.6.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.F5.20.10.m10.6c"><apply id="S5.F5.20.10.m10.6.6.cmml" xref="S5.F5.20.10.m10.6.6"><plus id="S5.F5.20.10.m10.6.6.7.cmml" xref="S5.F5.20.10.m10.6.6.7"></plus><apply id="S5.F5.20.10.m10.2.2.2.cmml" xref="S5.F5.20.10.m10.2.2.2"><times id="S5.F5.20.10.m10.2.2.2.3.cmml" xref="S5.F5.20.10.m10.2.2.2.3"></times><ci id="S5.F5.20.10.m10.2.2.2.4.cmml" xref="S5.F5.20.10.m10.2.2.2.4">𝑝</ci><apply id="S5.F5.20.10.m10.1.1.1.1.1.1.cmml" xref="S5.F5.20.10.m10.1.1.1.1.1"><minus id="S5.F5.20.10.m10.1.1.1.1.1.1.1.cmml" xref="S5.F5.20.10.m10.1.1.1.1.1.1.1"></minus><cn id="S5.F5.20.10.m10.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.F5.20.10.m10.1.1.1.1.1.1.2">1</cn><ci id="S5.F5.20.10.m10.1.1.1.1.1.1.3.cmml" xref="S5.F5.20.10.m10.1.1.1.1.1.1.3">𝑝</ci></apply><apply id="S5.F5.20.10.m10.2.2.2.2.1.1.cmml" xref="S5.F5.20.10.m10.2.2.2.2.1"><plus id="S5.F5.20.10.m10.2.2.2.2.1.1.1.cmml" xref="S5.F5.20.10.m10.2.2.2.2.1.1.1"></plus><ci id="S5.F5.20.10.m10.2.2.2.2.1.1.2.cmml" xref="S5.F5.20.10.m10.2.2.2.2.1.1.2">𝑐</ci><cn id="S5.F5.20.10.m10.2.2.2.2.1.1.3.cmml" type="integer" xref="S5.F5.20.10.m10.2.2.2.2.1.1.3">1</cn></apply></apply><apply id="S5.F5.20.10.m10.4.4.4.cmml" xref="S5.F5.20.10.m10.4.4.4"><times id="S5.F5.20.10.m10.4.4.4.3.cmml" xref="S5.F5.20.10.m10.4.4.4.3"></times><ci id="S5.F5.20.10.m10.4.4.4.4.cmml" xref="S5.F5.20.10.m10.4.4.4.4">𝑞</ci><apply id="S5.F5.20.10.m10.3.3.3.1.1.1.cmml" xref="S5.F5.20.10.m10.3.3.3.1.1"><minus id="S5.F5.20.10.m10.3.3.3.1.1.1.1.cmml" xref="S5.F5.20.10.m10.3.3.3.1.1.1.1"></minus><cn id="S5.F5.20.10.m10.3.3.3.1.1.1.2.cmml" type="integer" xref="S5.F5.20.10.m10.3.3.3.1.1.1.2">1</cn><ci id="S5.F5.20.10.m10.3.3.3.1.1.1.3.cmml" xref="S5.F5.20.10.m10.3.3.3.1.1.1.3">𝑞</ci></apply><apply id="S5.F5.20.10.m10.4.4.4.2.1.1.cmml" xref="S5.F5.20.10.m10.4.4.4.2.1"><plus id="S5.F5.20.10.m10.4.4.4.2.1.1.1.cmml" xref="S5.F5.20.10.m10.4.4.4.2.1.1.1"></plus><ci id="S5.F5.20.10.m10.4.4.4.2.1.1.2.cmml" xref="S5.F5.20.10.m10.4.4.4.2.1.1.2">𝑐</ci><cn id="S5.F5.20.10.m10.4.4.4.2.1.1.3.cmml" type="integer" xref="S5.F5.20.10.m10.4.4.4.2.1.1.3">1</cn></apply></apply><apply id="S5.F5.20.10.m10.6.6.6.cmml" xref="S5.F5.20.10.m10.6.6.6"><times id="S5.F5.20.10.m10.6.6.6.3.cmml" xref="S5.F5.20.10.m10.6.6.6.3"></times><ci id="S5.F5.20.10.m10.6.6.6.4.cmml" xref="S5.F5.20.10.m10.6.6.6.4">𝑝</ci><apply id="S5.F5.20.10.m10.5.5.5.1.1.1.cmml" xref="S5.F5.20.10.m10.5.5.5.1.1"><minus id="S5.F5.20.10.m10.5.5.5.1.1.1.1.cmml" xref="S5.F5.20.10.m10.5.5.5.1.1.1.1"></minus><cn id="S5.F5.20.10.m10.5.5.5.1.1.1.2.cmml" type="integer" xref="S5.F5.20.10.m10.5.5.5.1.1.1.2">1</cn><ci id="S5.F5.20.10.m10.5.5.5.1.1.1.3.cmml" xref="S5.F5.20.10.m10.5.5.5.1.1.1.3">𝑞</ci></apply><apply id="S5.F5.20.10.m10.6.6.6.2.1.1.cmml" xref="S5.F5.20.10.m10.6.6.6.2.1"><minus id="S5.F5.20.10.m10.6.6.6.2.1.1.1.cmml" xref="S5.F5.20.10.m10.6.6.6.2.1.1.1"></minus><apply id="S5.F5.20.10.m10.6.6.6.2.1.1.2.cmml" xref="S5.F5.20.10.m10.6.6.6.2.1.1.2"><csymbol cd="ambiguous" id="S5.F5.20.10.m10.6.6.6.2.1.1.2.1.cmml" xref="S5.F5.20.10.m10.6.6.6.2.1.1.2">superscript</csymbol><ci id="S5.F5.20.10.m10.6.6.6.2.1.1.2.2.cmml" xref="S5.F5.20.10.m10.6.6.6.2.1.1.2.2">𝑐</ci><cn id="S5.F5.20.10.m10.6.6.6.2.1.1.2.3.cmml" type="integer" xref="S5.F5.20.10.m10.6.6.6.2.1.1.2.3">2</cn></apply><cn id="S5.F5.20.10.m10.6.6.6.2.1.1.3.cmml" type="integer" xref="S5.F5.20.10.m10.6.6.6.2.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F5.20.10.m10.6d">p(1-p)(c+1)+q(1-q)(c+1)+p(1-q)(c^{2}-1)</annotation><annotation encoding="application/x-llamapun" id="S5.F5.20.10.m10.6e">italic_p ( 1 - italic_p ) ( italic_c + 1 ) + italic_q ( 1 - italic_q ) ( italic_c + 1 ) + italic_p ( 1 - italic_q ) ( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 )</annotation></semantics></math>.</span></figcaption> </figure> <div class="ltx_proof" id="S5.2"> <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.p1.7">Let <math alttext="0\leq\lambda<1" class="ltx_Math" display="inline" id="S5.1.p1.1.m1.1"><semantics id="S5.1.p1.1.m1.1a"><mrow id="S5.1.p1.1.m1.1.1" xref="S5.1.p1.1.m1.1.1.cmml"><mn id="S5.1.p1.1.m1.1.1.2" xref="S5.1.p1.1.m1.1.1.2.cmml">0</mn><mo id="S5.1.p1.1.m1.1.1.3" xref="S5.1.p1.1.m1.1.1.3.cmml">≤</mo><mi id="S5.1.p1.1.m1.1.1.4" xref="S5.1.p1.1.m1.1.1.4.cmml">λ</mi><mo id="S5.1.p1.1.m1.1.1.5" xref="S5.1.p1.1.m1.1.1.5.cmml"><</mo><mn id="S5.1.p1.1.m1.1.1.6" xref="S5.1.p1.1.m1.1.1.6.cmml">1</mn></mrow><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"><and id="S5.1.p1.1.m1.1.1a.cmml" xref="S5.1.p1.1.m1.1.1"></and><apply id="S5.1.p1.1.m1.1.1b.cmml" xref="S5.1.p1.1.m1.1.1"><leq id="S5.1.p1.1.m1.1.1.3.cmml" xref="S5.1.p1.1.m1.1.1.3"></leq><cn id="S5.1.p1.1.m1.1.1.2.cmml" type="integer" xref="S5.1.p1.1.m1.1.1.2">0</cn><ci id="S5.1.p1.1.m1.1.1.4.cmml" xref="S5.1.p1.1.m1.1.1.4">𝜆</ci></apply><apply id="S5.1.p1.1.m1.1.1c.cmml" xref="S5.1.p1.1.m1.1.1"><lt id="S5.1.p1.1.m1.1.1.5.cmml" xref="S5.1.p1.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2411.12976v1#S5.1.p1.1.m1.1.1.4.cmml" id="S5.1.p1.1.m1.1.1d.cmml" xref="S5.1.p1.1.m1.1.1"></share><cn id="S5.1.p1.1.m1.1.1.6.cmml" type="integer" xref="S5.1.p1.1.m1.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.1.m1.1c">0\leq\lambda<1</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.1.m1.1d">0 ≤ italic_λ < 1</annotation></semantics></math> and <math alttext="c>1" 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"><mi id="S5.1.p1.2.m2.1.1.2" xref="S5.1.p1.2.m2.1.1.2.cmml">c</mi><mo id="S5.1.p1.2.m2.1.1.1" xref="S5.1.p1.2.m2.1.1.1.cmml">></mo><mn id="S5.1.p1.2.m2.1.1.3" xref="S5.1.p1.2.m2.1.1.3.cmml">1</mn></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"><gt id="S5.1.p1.2.m2.1.1.1.cmml" xref="S5.1.p1.2.m2.1.1.1"></gt><ci id="S5.1.p1.2.m2.1.1.2.cmml" xref="S5.1.p1.2.m2.1.1.2">𝑐</ci><cn id="S5.1.p1.2.m2.1.1.3.cmml" type="integer" xref="S5.1.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.2.m2.1c">c>1</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.2.m2.1d">italic_c > 1</annotation></semantics></math> be parameters. Consider a graph <math alttext="G" class="ltx_Math" display="inline" id="S5.1.p1.3.m3.1"><semantics id="S5.1.p1.3.m3.1a"><mi id="S5.1.p1.3.m3.1.1" xref="S5.1.p1.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.3.m3.1b"><ci id="S5.1.p1.3.m3.1.1.cmml" xref="S5.1.p1.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.3.m3.1d">italic_G</annotation></semantics></math> consisting of disjoint copies of the graphs in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S4.F3" title="In 4.2 Bounds against 𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽_{1/2} (Theorem 1.6) ‣ 4 Lower bounds ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Figs.</span> <span class="ltx_text ltx_ref_tag">3</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S5.F5" title="Figure 5 ‣ 5 Lower bound for arbitrary selection functions (Theorem 1.9) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">5</span></a> weighted by <math alttext="\lambda" class="ltx_Math" display="inline" id="S5.1.p1.4.m4.1"><semantics id="S5.1.p1.4.m4.1a"><mi id="S5.1.p1.4.m4.1.1" xref="S5.1.p1.4.m4.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.4.m4.1b"><ci id="S5.1.p1.4.m4.1.1.cmml" xref="S5.1.p1.4.m4.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.4.m4.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.4.m4.1d">italic_λ</annotation></semantics></math> and <math alttext="1-\lambda" class="ltx_Math" display="inline" id="S5.1.p1.5.m5.1"><semantics id="S5.1.p1.5.m5.1a"><mrow id="S5.1.p1.5.m5.1.1" xref="S5.1.p1.5.m5.1.1.cmml"><mn id="S5.1.p1.5.m5.1.1.2" xref="S5.1.p1.5.m5.1.1.2.cmml">1</mn><mo id="S5.1.p1.5.m5.1.1.1" xref="S5.1.p1.5.m5.1.1.1.cmml">−</mo><mi id="S5.1.p1.5.m5.1.1.3" xref="S5.1.p1.5.m5.1.1.3.cmml">λ</mi></mrow><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"><minus id="S5.1.p1.5.m5.1.1.1.cmml" xref="S5.1.p1.5.m5.1.1.1"></minus><cn id="S5.1.p1.5.m5.1.1.2.cmml" type="integer" xref="S5.1.p1.5.m5.1.1.2">1</cn><ci id="S5.1.p1.5.m5.1.1.3.cmml" xref="S5.1.p1.5.m5.1.1.3">𝜆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.5.m5.1c">1-\lambda</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.5.m5.1d">1 - italic_λ</annotation></semantics></math>, respectively. Examining these graphs, we see that all vertices in <math alttext="G" 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">G</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">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.6.m6.1d">italic_G</annotation></semantics></math> have bias <math alttext="\pm\frac{c-1}{c+1}" class="ltx_Math" display="inline" id="S5.1.p1.7.m7.1"><semantics id="S5.1.p1.7.m7.1a"><mrow id="S5.1.p1.7.m7.1.1" xref="S5.1.p1.7.m7.1.1.cmml"><mo id="S5.1.p1.7.m7.1.1a" xref="S5.1.p1.7.m7.1.1.cmml">±</mo><mfrac id="S5.1.p1.7.m7.1.1.2" xref="S5.1.p1.7.m7.1.1.2.cmml"><mrow id="S5.1.p1.7.m7.1.1.2.2" xref="S5.1.p1.7.m7.1.1.2.2.cmml"><mi id="S5.1.p1.7.m7.1.1.2.2.2" xref="S5.1.p1.7.m7.1.1.2.2.2.cmml">c</mi><mo id="S5.1.p1.7.m7.1.1.2.2.1" xref="S5.1.p1.7.m7.1.1.2.2.1.cmml">−</mo><mn id="S5.1.p1.7.m7.1.1.2.2.3" xref="S5.1.p1.7.m7.1.1.2.2.3.cmml">1</mn></mrow><mrow id="S5.1.p1.7.m7.1.1.2.3" xref="S5.1.p1.7.m7.1.1.2.3.cmml"><mi id="S5.1.p1.7.m7.1.1.2.3.2" xref="S5.1.p1.7.m7.1.1.2.3.2.cmml">c</mi><mo id="S5.1.p1.7.m7.1.1.2.3.1" xref="S5.1.p1.7.m7.1.1.2.3.1.cmml">+</mo><mn id="S5.1.p1.7.m7.1.1.2.3.3" xref="S5.1.p1.7.m7.1.1.2.3.3.cmml">1</mn></mrow></mfrac></mrow><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="latexml" id="S5.1.p1.7.m7.1.1.1.cmml" xref="S5.1.p1.7.m7.1.1">plus-or-minus</csymbol><apply id="S5.1.p1.7.m7.1.1.2.cmml" xref="S5.1.p1.7.m7.1.1.2"><divide id="S5.1.p1.7.m7.1.1.2.1.cmml" xref="S5.1.p1.7.m7.1.1.2"></divide><apply id="S5.1.p1.7.m7.1.1.2.2.cmml" xref="S5.1.p1.7.m7.1.1.2.2"><minus id="S5.1.p1.7.m7.1.1.2.2.1.cmml" xref="S5.1.p1.7.m7.1.1.2.2.1"></minus><ci id="S5.1.p1.7.m7.1.1.2.2.2.cmml" xref="S5.1.p1.7.m7.1.1.2.2.2">𝑐</ci><cn id="S5.1.p1.7.m7.1.1.2.2.3.cmml" type="integer" xref="S5.1.p1.7.m7.1.1.2.2.3">1</cn></apply><apply id="S5.1.p1.7.m7.1.1.2.3.cmml" xref="S5.1.p1.7.m7.1.1.2.3"><plus id="S5.1.p1.7.m7.1.1.2.3.1.cmml" xref="S5.1.p1.7.m7.1.1.2.3.1"></plus><ci id="S5.1.p1.7.m7.1.1.2.3.2.cmml" xref="S5.1.p1.7.m7.1.1.2.3.2">𝑐</ci><cn id="S5.1.p1.7.m7.1.1.2.3.3.cmml" type="integer" xref="S5.1.p1.7.m7.1.1.2.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.7.m7.1c">\pm\frac{c-1}{c+1}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.7.m7.1d">± divide start_ARG italic_c - 1 end_ARG start_ARG italic_c + 1 end_ARG</annotation></semantics></math>. Further, there exists a cut cutting weight</p> <table class="ltx_equation ltx_eqn_table" id="S5.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="\lambda c+(1-\lambda)(c^{2}+1)" class="ltx_Math" display="block" id="S5.E1.m1.2"><semantics id="S5.E1.m1.2a"><mrow id="S5.E1.m1.2.2" xref="S5.E1.m1.2.2.cmml"><mrow id="S5.E1.m1.2.2.4" xref="S5.E1.m1.2.2.4.cmml"><mi id="S5.E1.m1.2.2.4.2" xref="S5.E1.m1.2.2.4.2.cmml">λ</mi><mo id="S5.E1.m1.2.2.4.1" xref="S5.E1.m1.2.2.4.1.cmml">⁢</mo><mi id="S5.E1.m1.2.2.4.3" xref="S5.E1.m1.2.2.4.3.cmml">c</mi></mrow><mo id="S5.E1.m1.2.2.3" xref="S5.E1.m1.2.2.3.cmml">+</mo><mrow id="S5.E1.m1.2.2.2" xref="S5.E1.m1.2.2.2.cmml"><mrow id="S5.E1.m1.1.1.1.1.1" xref="S5.E1.m1.1.1.1.1.1.1.cmml"><mo id="S5.E1.m1.1.1.1.1.1.2" stretchy="false" xref="S5.E1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.E1.m1.1.1.1.1.1.1" xref="S5.E1.m1.1.1.1.1.1.1.cmml"><mn id="S5.E1.m1.1.1.1.1.1.1.2" xref="S5.E1.m1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.E1.m1.1.1.1.1.1.1.1" xref="S5.E1.m1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S5.E1.m1.1.1.1.1.1.1.3" xref="S5.E1.m1.1.1.1.1.1.1.3.cmml">λ</mi></mrow><mo id="S5.E1.m1.1.1.1.1.1.3" stretchy="false" xref="S5.E1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.E1.m1.2.2.2.3" xref="S5.E1.m1.2.2.2.3.cmml">⁢</mo><mrow id="S5.E1.m1.2.2.2.2.1" xref="S5.E1.m1.2.2.2.2.1.1.cmml"><mo id="S5.E1.m1.2.2.2.2.1.2" stretchy="false" xref="S5.E1.m1.2.2.2.2.1.1.cmml">(</mo><mrow id="S5.E1.m1.2.2.2.2.1.1" xref="S5.E1.m1.2.2.2.2.1.1.cmml"><msup id="S5.E1.m1.2.2.2.2.1.1.2" xref="S5.E1.m1.2.2.2.2.1.1.2.cmml"><mi id="S5.E1.m1.2.2.2.2.1.1.2.2" xref="S5.E1.m1.2.2.2.2.1.1.2.2.cmml">c</mi><mn id="S5.E1.m1.2.2.2.2.1.1.2.3" xref="S5.E1.m1.2.2.2.2.1.1.2.3.cmml">2</mn></msup><mo id="S5.E1.m1.2.2.2.2.1.1.1" xref="S5.E1.m1.2.2.2.2.1.1.1.cmml">+</mo><mn id="S5.E1.m1.2.2.2.2.1.1.3" xref="S5.E1.m1.2.2.2.2.1.1.3.cmml">1</mn></mrow><mo id="S5.E1.m1.2.2.2.2.1.3" stretchy="false" xref="S5.E1.m1.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.E1.m1.2b"><apply id="S5.E1.m1.2.2.cmml" xref="S5.E1.m1.2.2"><plus id="S5.E1.m1.2.2.3.cmml" xref="S5.E1.m1.2.2.3"></plus><apply id="S5.E1.m1.2.2.4.cmml" xref="S5.E1.m1.2.2.4"><times id="S5.E1.m1.2.2.4.1.cmml" xref="S5.E1.m1.2.2.4.1"></times><ci id="S5.E1.m1.2.2.4.2.cmml" xref="S5.E1.m1.2.2.4.2">𝜆</ci><ci id="S5.E1.m1.2.2.4.3.cmml" xref="S5.E1.m1.2.2.4.3">𝑐</ci></apply><apply id="S5.E1.m1.2.2.2.cmml" xref="S5.E1.m1.2.2.2"><times id="S5.E1.m1.2.2.2.3.cmml" xref="S5.E1.m1.2.2.2.3"></times><apply id="S5.E1.m1.1.1.1.1.1.1.cmml" xref="S5.E1.m1.1.1.1.1.1"><minus id="S5.E1.m1.1.1.1.1.1.1.1.cmml" xref="S5.E1.m1.1.1.1.1.1.1.1"></minus><cn id="S5.E1.m1.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.E1.m1.1.1.1.1.1.1.2">1</cn><ci id="S5.E1.m1.1.1.1.1.1.1.3.cmml" xref="S5.E1.m1.1.1.1.1.1.1.3">𝜆</ci></apply><apply id="S5.E1.m1.2.2.2.2.1.1.cmml" xref="S5.E1.m1.2.2.2.2.1"><plus id="S5.E1.m1.2.2.2.2.1.1.1.cmml" xref="S5.E1.m1.2.2.2.2.1.1.1"></plus><apply id="S5.E1.m1.2.2.2.2.1.1.2.cmml" xref="S5.E1.m1.2.2.2.2.1.1.2"><csymbol cd="ambiguous" id="S5.E1.m1.2.2.2.2.1.1.2.1.cmml" xref="S5.E1.m1.2.2.2.2.1.1.2">superscript</csymbol><ci id="S5.E1.m1.2.2.2.2.1.1.2.2.cmml" xref="S5.E1.m1.2.2.2.2.1.1.2.2">𝑐</ci><cn id="S5.E1.m1.2.2.2.2.1.1.2.3.cmml" type="integer" xref="S5.E1.m1.2.2.2.2.1.1.2.3">2</cn></apply><cn id="S5.E1.m1.2.2.2.2.1.1.3.cmml" type="integer" xref="S5.E1.m1.2.2.2.2.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E1.m1.2c">\lambda c+(1-\lambda)(c^{2}+1)</annotation><annotation encoding="application/x-llamapun" id="S5.E1.m1.2d">italic_λ italic_c + ( 1 - italic_λ ) ( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1 )</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">(5.1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.1.p1.11">while an oblivious assignment assigning the positive-bias vertices to <math alttext="1" class="ltx_Math" display="inline" id="S5.1.p1.8.m1.1"><semantics id="S5.1.p1.8.m1.1a"><mn id="S5.1.p1.8.m1.1.1" xref="S5.1.p1.8.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.1.p1.8.m1.1b"><cn id="S5.1.p1.8.m1.1.1.cmml" type="integer" xref="S5.1.p1.8.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.8.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.8.m1.1d">1</annotation></semantics></math> w.p. <math alttext="p" class="ltx_Math" display="inline" id="S5.1.p1.9.m2.1"><semantics id="S5.1.p1.9.m2.1a"><mi id="S5.1.p1.9.m2.1.1" xref="S5.1.p1.9.m2.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.9.m2.1b"><ci id="S5.1.p1.9.m2.1.1.cmml" xref="S5.1.p1.9.m2.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.9.m2.1c">p</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.9.m2.1d">italic_p</annotation></semantics></math> and the negative-bias vertices to <math alttext="1" class="ltx_Math" display="inline" id="S5.1.p1.10.m3.1"><semantics id="S5.1.p1.10.m3.1a"><mn id="S5.1.p1.10.m3.1.1" xref="S5.1.p1.10.m3.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.1.p1.10.m3.1b"><cn id="S5.1.p1.10.m3.1.1.cmml" type="integer" xref="S5.1.p1.10.m3.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.10.m3.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.10.m3.1d">1</annotation></semantics></math> w.p. <math alttext="q" class="ltx_Math" display="inline" id="S5.1.p1.11.m4.1"><semantics id="S5.1.p1.11.m4.1a"><mi id="S5.1.p1.11.m4.1.1" xref="S5.1.p1.11.m4.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.11.m4.1b"><ci id="S5.1.p1.11.m4.1.1.cmml" xref="S5.1.p1.11.m4.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.11.m4.1c">q</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.11.m4.1d">italic_q</annotation></semantics></math> achieves value</p> <table class="ltx_equation ltx_eqn_table" id="S5.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="\lambda(p(1-q)c+q(1-p))+(1-\lambda)p(1-p)(c+1)+q(1-q)(c+1)+p(1-q)(c^{2}-1)." class="ltx_Math" display="block" id="S5.E2.m1.1"><semantics id="S5.E2.m1.1a"><mrow id="S5.E2.m1.1.1.1" xref="S5.E2.m1.1.1.1.1.cmml"><mrow id="S5.E2.m1.1.1.1.1" xref="S5.E2.m1.1.1.1.1.cmml"><mrow id="S5.E2.m1.1.1.1.1.1" xref="S5.E2.m1.1.1.1.1.1.cmml"><mi id="S5.E2.m1.1.1.1.1.1.3" xref="S5.E2.m1.1.1.1.1.1.3.cmml">λ</mi><mo id="S5.E2.m1.1.1.1.1.1.2" xref="S5.E2.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.E2.m1.1.1.1.1.1.1.1" xref="S5.E2.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.E2.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.E2.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.E2.m1.1.1.1.1.1.1.1.1" xref="S5.E2.m1.1.1.1.1.1.1.1.1.cmml"><mrow id="S5.E2.m1.1.1.1.1.1.1.1.1.1" xref="S5.E2.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S5.E2.m1.1.1.1.1.1.1.1.1.1.3" xref="S5.E2.m1.1.1.1.1.1.1.1.1.1.3.cmml">p</mi><mo id="S5.E2.m1.1.1.1.1.1.1.1.1.1.2" xref="S5.E2.m1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.E2.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S5.E2.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.E2.m1.1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.E2.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.E2.m1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S5.E2.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mn id="S5.E2.m1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S5.E2.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo 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italic_p ) ( italic_c + 1 ) + italic_q ( 1 - italic_q ) ( italic_c + 1 ) + italic_p ( 1 - italic_q ) ( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) .</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">(5.2)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S5.2.p2"> <p class="ltx_p" id="S5.2.p2.6">At <math alttext="\lambda=\frac{15}{32}" class="ltx_Math" display="inline" id="S5.2.p2.1.m1.1"><semantics id="S5.2.p2.1.m1.1a"><mrow id="S5.2.p2.1.m1.1.1" xref="S5.2.p2.1.m1.1.1.cmml"><mi id="S5.2.p2.1.m1.1.1.2" xref="S5.2.p2.1.m1.1.1.2.cmml">λ</mi><mo id="S5.2.p2.1.m1.1.1.1" xref="S5.2.p2.1.m1.1.1.1.cmml">=</mo><mfrac id="S5.2.p2.1.m1.1.1.3" xref="S5.2.p2.1.m1.1.1.3.cmml"><mn id="S5.2.p2.1.m1.1.1.3.2" xref="S5.2.p2.1.m1.1.1.3.2.cmml">15</mn><mn id="S5.2.p2.1.m1.1.1.3.3" xref="S5.2.p2.1.m1.1.1.3.3.cmml">32</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.1.m1.1b"><apply id="S5.2.p2.1.m1.1.1.cmml" xref="S5.2.p2.1.m1.1.1"><eq id="S5.2.p2.1.m1.1.1.1.cmml" xref="S5.2.p2.1.m1.1.1.1"></eq><ci id="S5.2.p2.1.m1.1.1.2.cmml" xref="S5.2.p2.1.m1.1.1.2">𝜆</ci><apply id="S5.2.p2.1.m1.1.1.3.cmml" xref="S5.2.p2.1.m1.1.1.3"><divide id="S5.2.p2.1.m1.1.1.3.1.cmml" xref="S5.2.p2.1.m1.1.1.3"></divide><cn id="S5.2.p2.1.m1.1.1.3.2.cmml" type="integer" xref="S5.2.p2.1.m1.1.1.3.2">15</cn><cn id="S5.2.p2.1.m1.1.1.3.3.cmml" type="integer" xref="S5.2.p2.1.m1.1.1.3.3">32</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.1.m1.1c">\lambda=\frac{15}{32}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.1.m1.1d">italic_λ = divide start_ARG 15 end_ARG start_ARG 32 end_ARG</annotation></semantics></math> and <math alttext="c=\frac{9}{8}" class="ltx_Math" display="inline" id="S5.2.p2.2.m2.1"><semantics id="S5.2.p2.2.m2.1a"><mrow id="S5.2.p2.2.m2.1.1" xref="S5.2.p2.2.m2.1.1.cmml"><mi id="S5.2.p2.2.m2.1.1.2" xref="S5.2.p2.2.m2.1.1.2.cmml">c</mi><mo id="S5.2.p2.2.m2.1.1.1" xref="S5.2.p2.2.m2.1.1.1.cmml">=</mo><mfrac id="S5.2.p2.2.m2.1.1.3" xref="S5.2.p2.2.m2.1.1.3.cmml"><mn id="S5.2.p2.2.m2.1.1.3.2" xref="S5.2.p2.2.m2.1.1.3.2.cmml">9</mn><mn id="S5.2.p2.2.m2.1.1.3.3" xref="S5.2.p2.2.m2.1.1.3.3.cmml">8</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.2.m2.1b"><apply id="S5.2.p2.2.m2.1.1.cmml" xref="S5.2.p2.2.m2.1.1"><eq id="S5.2.p2.2.m2.1.1.1.cmml" xref="S5.2.p2.2.m2.1.1.1"></eq><ci id="S5.2.p2.2.m2.1.1.2.cmml" xref="S5.2.p2.2.m2.1.1.2">𝑐</ci><apply id="S5.2.p2.2.m2.1.1.3.cmml" xref="S5.2.p2.2.m2.1.1.3"><divide id="S5.2.p2.2.m2.1.1.3.1.cmml" xref="S5.2.p2.2.m2.1.1.3"></divide><cn id="S5.2.p2.2.m2.1.1.3.2.cmml" type="integer" xref="S5.2.p2.2.m2.1.1.3.2">9</cn><cn id="S5.2.p2.2.m2.1.1.3.3.cmml" type="integer" xref="S5.2.p2.2.m2.1.1.3.3">8</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.2.m2.1c">c=\frac{9}{8}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.2.m2.1d">italic_c = divide start_ARG 9 end_ARG start_ARG 8 end_ARG</annotation></semantics></math>,<span class="ltx_note ltx_role_footnote" id="footnote7"><sup class="ltx_note_mark">7</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">7</sup><span class="ltx_tag ltx_tag_note">7</span>These fractions were suggested by numerical search on a computer; they are almost certainly non-optimal, but we include fractions so that the calculations in this proof can be checked exactly. For simplicity’s sake, we mention that a ratio strictly below <math alttext="\frac{1}{2}" class="ltx_Math" display="inline" id="footnote7.m1.1"><semantics id="footnote7.m1.1b"><mfrac id="footnote7.m1.1.1" xref="footnote7.m1.1.1.cmml"><mn id="footnote7.m1.1.1.2" xref="footnote7.m1.1.1.2.cmml">1</mn><mn id="footnote7.m1.1.1.3" xref="footnote7.m1.1.1.3.cmml">2</mn></mfrac><annotation-xml encoding="MathML-Content" id="footnote7.m1.1c"><apply id="footnote7.m1.1.1.cmml" xref="footnote7.m1.1.1"><divide id="footnote7.m1.1.1.1.cmml" xref="footnote7.m1.1.1"></divide><cn id="footnote7.m1.1.1.2.cmml" type="integer" xref="footnote7.m1.1.1.2">1</cn><cn id="footnote7.m1.1.1.3.cmml" type="integer" xref="footnote7.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote7.m1.1d">\frac{1}{2}</annotation><annotation encoding="application/x-llamapun" id="footnote7.m1.1e">divide start_ARG 1 end_ARG start_ARG 2 end_ARG</annotation></semantics></math> (and indeed, better than that of <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem3" title="Theorem 1.3 (Prior lower bound for general selection, [FJ15, Thm. 1.5]). ‣ 1.1 Background: Directed cuts, bias, and oblivious algorithms ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.3</span></a>) is achieved by redoing this calculation at <math alttext="\lambda=\frac{1}{3}" class="ltx_Math" display="inline" id="footnote7.m2.1"><semantics id="footnote7.m2.1b"><mrow id="footnote7.m2.1.1" xref="footnote7.m2.1.1.cmml"><mi id="footnote7.m2.1.1.2" xref="footnote7.m2.1.1.2.cmml">λ</mi><mo id="footnote7.m2.1.1.1" xref="footnote7.m2.1.1.1.cmml">=</mo><mfrac id="footnote7.m2.1.1.3" xref="footnote7.m2.1.1.3.cmml"><mn id="footnote7.m2.1.1.3.2" xref="footnote7.m2.1.1.3.2.cmml">1</mn><mn id="footnote7.m2.1.1.3.3" xref="footnote7.m2.1.1.3.3.cmml">3</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="footnote7.m2.1c"><apply id="footnote7.m2.1.1.cmml" xref="footnote7.m2.1.1"><eq id="footnote7.m2.1.1.1.cmml" xref="footnote7.m2.1.1.1"></eq><ci id="footnote7.m2.1.1.2.cmml" xref="footnote7.m2.1.1.2">𝜆</ci><apply id="footnote7.m2.1.1.3.cmml" xref="footnote7.m2.1.1.3"><divide id="footnote7.m2.1.1.3.1.cmml" xref="footnote7.m2.1.1.3"></divide><cn id="footnote7.m2.1.1.3.2.cmml" type="integer" xref="footnote7.m2.1.1.3.2">1</cn><cn id="footnote7.m2.1.1.3.3.cmml" type="integer" xref="footnote7.m2.1.1.3.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote7.m2.1d">\lambda=\frac{1}{3}</annotation><annotation encoding="application/x-llamapun" id="footnote7.m2.1e">italic_λ = divide start_ARG 1 end_ARG start_ARG 3 end_ARG</annotation></semantics></math> and <math alttext="c=\frac{5}{4}" class="ltx_Math" display="inline" id="footnote7.m3.1"><semantics id="footnote7.m3.1b"><mrow id="footnote7.m3.1.1" xref="footnote7.m3.1.1.cmml"><mi id="footnote7.m3.1.1.2" xref="footnote7.m3.1.1.2.cmml">c</mi><mo id="footnote7.m3.1.1.1" xref="footnote7.m3.1.1.1.cmml">=</mo><mfrac id="footnote7.m3.1.1.3" xref="footnote7.m3.1.1.3.cmml"><mn id="footnote7.m3.1.1.3.2" xref="footnote7.m3.1.1.3.2.cmml">5</mn><mn id="footnote7.m3.1.1.3.3" xref="footnote7.m3.1.1.3.3.cmml">4</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="footnote7.m3.1c"><apply id="footnote7.m3.1.1.cmml" xref="footnote7.m3.1.1"><eq id="footnote7.m3.1.1.1.cmml" xref="footnote7.m3.1.1.1"></eq><ci id="footnote7.m3.1.1.2.cmml" xref="footnote7.m3.1.1.2">𝑐</ci><apply id="footnote7.m3.1.1.3.cmml" xref="footnote7.m3.1.1.3"><divide id="footnote7.m3.1.1.3.1.cmml" xref="footnote7.m3.1.1.3"></divide><cn id="footnote7.m3.1.1.3.2.cmml" type="integer" xref="footnote7.m3.1.1.3.2">5</cn><cn id="footnote7.m3.1.1.3.3.cmml" type="integer" xref="footnote7.m3.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote7.m3.1d">c=\frac{5}{4}</annotation><annotation encoding="application/x-llamapun" id="footnote7.m3.1e">italic_c = divide start_ARG 5 end_ARG start_ARG 4 end_ARG</annotation></semantics></math>.</span></span></span> the quantity in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S5.E2" title="In Proof. ‣ 5 Lower bound for arbitrary selection functions (Theorem 1.9) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Eq.</span> <span class="ltx_text ltx_ref_tag">5.2</span></a> is optimized at <math alttext="(p,q)=(\frac{1352}{2295},\frac{943}{2295})" class="ltx_Math" display="inline" id="S5.2.p2.3.m3.4"><semantics id="S5.2.p2.3.m3.4a"><mrow id="S5.2.p2.3.m3.4.5" xref="S5.2.p2.3.m3.4.5.cmml"><mrow id="S5.2.p2.3.m3.4.5.2.2" xref="S5.2.p2.3.m3.4.5.2.1.cmml"><mo id="S5.2.p2.3.m3.4.5.2.2.1" stretchy="false" xref="S5.2.p2.3.m3.4.5.2.1.cmml">(</mo><mi id="S5.2.p2.3.m3.1.1" xref="S5.2.p2.3.m3.1.1.cmml">p</mi><mo id="S5.2.p2.3.m3.4.5.2.2.2" xref="S5.2.p2.3.m3.4.5.2.1.cmml">,</mo><mi id="S5.2.p2.3.m3.2.2" xref="S5.2.p2.3.m3.2.2.cmml">q</mi><mo id="S5.2.p2.3.m3.4.5.2.2.3" stretchy="false" xref="S5.2.p2.3.m3.4.5.2.1.cmml">)</mo></mrow><mo id="S5.2.p2.3.m3.4.5.1" xref="S5.2.p2.3.m3.4.5.1.cmml">=</mo><mrow id="S5.2.p2.3.m3.4.5.3.2" xref="S5.2.p2.3.m3.4.5.3.1.cmml"><mo id="S5.2.p2.3.m3.4.5.3.2.1" stretchy="false" xref="S5.2.p2.3.m3.4.5.3.1.cmml">(</mo><mfrac id="S5.2.p2.3.m3.3.3" xref="S5.2.p2.3.m3.3.3.cmml"><mn id="S5.2.p2.3.m3.3.3.2" xref="S5.2.p2.3.m3.3.3.2.cmml">1352</mn><mn id="S5.2.p2.3.m3.3.3.3" xref="S5.2.p2.3.m3.3.3.3.cmml">2295</mn></mfrac><mo id="S5.2.p2.3.m3.4.5.3.2.2" xref="S5.2.p2.3.m3.4.5.3.1.cmml">,</mo><mfrac id="S5.2.p2.3.m3.4.4" xref="S5.2.p2.3.m3.4.4.cmml"><mn id="S5.2.p2.3.m3.4.4.2" xref="S5.2.p2.3.m3.4.4.2.cmml">943</mn><mn id="S5.2.p2.3.m3.4.4.3" xref="S5.2.p2.3.m3.4.4.3.cmml">2295</mn></mfrac><mo id="S5.2.p2.3.m3.4.5.3.2.3" stretchy="false" xref="S5.2.p2.3.m3.4.5.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.3.m3.4b"><apply id="S5.2.p2.3.m3.4.5.cmml" xref="S5.2.p2.3.m3.4.5"><eq id="S5.2.p2.3.m3.4.5.1.cmml" xref="S5.2.p2.3.m3.4.5.1"></eq><interval closure="open" id="S5.2.p2.3.m3.4.5.2.1.cmml" xref="S5.2.p2.3.m3.4.5.2.2"><ci id="S5.2.p2.3.m3.1.1.cmml" xref="S5.2.p2.3.m3.1.1">𝑝</ci><ci id="S5.2.p2.3.m3.2.2.cmml" xref="S5.2.p2.3.m3.2.2">𝑞</ci></interval><interval closure="open" id="S5.2.p2.3.m3.4.5.3.1.cmml" xref="S5.2.p2.3.m3.4.5.3.2"><apply id="S5.2.p2.3.m3.3.3.cmml" xref="S5.2.p2.3.m3.3.3"><divide id="S5.2.p2.3.m3.3.3.1.cmml" xref="S5.2.p2.3.m3.3.3"></divide><cn id="S5.2.p2.3.m3.3.3.2.cmml" type="integer" xref="S5.2.p2.3.m3.3.3.2">1352</cn><cn id="S5.2.p2.3.m3.3.3.3.cmml" type="integer" xref="S5.2.p2.3.m3.3.3.3">2295</cn></apply><apply id="S5.2.p2.3.m3.4.4.cmml" xref="S5.2.p2.3.m3.4.4"><divide id="S5.2.p2.3.m3.4.4.1.cmml" xref="S5.2.p2.3.m3.4.4"></divide><cn id="S5.2.p2.3.m3.4.4.2.cmml" type="integer" xref="S5.2.p2.3.m3.4.4.2">943</cn><cn id="S5.2.p2.3.m3.4.4.3.cmml" type="integer" xref="S5.2.p2.3.m3.4.4.3">2295</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.3.m3.4c">(p,q)=(\frac{1352}{2295},\frac{943}{2295})</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.3.m3.4d">( italic_p , italic_q ) = ( divide start_ARG 1352 end_ARG start_ARG 2295 end_ARG , divide start_ARG 943 end_ARG start_ARG 2295 end_ARG )</annotation></semantics></math>. Dividing this value by the value of <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S5.E1" title="In Proof. ‣ 5 Lower bound for arbitrary selection functions (Theorem 1.9) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Eq.</span> <span class="ltx_text ltx_ref_tag">5.1</span></a> at the same values of <math alttext="\lambda" class="ltx_Math" display="inline" id="S5.2.p2.4.m4.1"><semantics id="S5.2.p2.4.m4.1a"><mi id="S5.2.p2.4.m4.1.1" xref="S5.2.p2.4.m4.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.4.m4.1b"><ci id="S5.2.p2.4.m4.1.1.cmml" xref="S5.2.p2.4.m4.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.4.m4.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.4.m4.1d">italic_λ</annotation></semantics></math> and <math alttext="c" class="ltx_Math" display="inline" id="S5.2.p2.5.m5.1"><semantics id="S5.2.p2.5.m5.1a"><mi id="S5.2.p2.5.m5.1.1" xref="S5.2.p2.5.m5.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.5.m5.1b"><ci id="S5.2.p2.5.m5.1.1.cmml" xref="S5.2.p2.5.m5.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.5.m5.1c">c</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.5.m5.1d">italic_c</annotation></semantics></math>, we arrive at a ratio of <math alttext="\frac{4031104}{8135775}\approx 0.4955" class="ltx_Math" display="inline" id="S5.2.p2.6.m6.1"><semantics id="S5.2.p2.6.m6.1a"><mrow id="S5.2.p2.6.m6.1.1" xref="S5.2.p2.6.m6.1.1.cmml"><mfrac id="S5.2.p2.6.m6.1.1.2" xref="S5.2.p2.6.m6.1.1.2.cmml"><mn id="S5.2.p2.6.m6.1.1.2.2" xref="S5.2.p2.6.m6.1.1.2.2.cmml">4031104</mn><mn id="S5.2.p2.6.m6.1.1.2.3" xref="S5.2.p2.6.m6.1.1.2.3.cmml">8135775</mn></mfrac><mo id="S5.2.p2.6.m6.1.1.1" xref="S5.2.p2.6.m6.1.1.1.cmml">≈</mo><mn id="S5.2.p2.6.m6.1.1.3" xref="S5.2.p2.6.m6.1.1.3.cmml">0.4955</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.6.m6.1b"><apply id="S5.2.p2.6.m6.1.1.cmml" xref="S5.2.p2.6.m6.1.1"><approx id="S5.2.p2.6.m6.1.1.1.cmml" xref="S5.2.p2.6.m6.1.1.1"></approx><apply id="S5.2.p2.6.m6.1.1.2.cmml" xref="S5.2.p2.6.m6.1.1.2"><divide id="S5.2.p2.6.m6.1.1.2.1.cmml" xref="S5.2.p2.6.m6.1.1.2"></divide><cn id="S5.2.p2.6.m6.1.1.2.2.cmml" type="integer" xref="S5.2.p2.6.m6.1.1.2.2">4031104</cn><cn id="S5.2.p2.6.m6.1.1.2.3.cmml" type="integer" xref="S5.2.p2.6.m6.1.1.2.3">8135775</cn></apply><cn id="S5.2.p2.6.m6.1.1.3.cmml" type="float" xref="S5.2.p2.6.m6.1.1.3">0.4955</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.6.m6.1c">\frac{4031104}{8135775}\approx 0.4955</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.6.m6.1d">divide start_ARG 4031104 end_ARG start_ARG 8135775 end_ARG ≈ 0.4955</annotation></semantics></math>. ∎</p> </div> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6 </span>Lower bounds for antisymmetric selection functions (<a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem8" title="Theorem 1.8 (Lower bound for symmetric selection). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.8</span></a>)</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">In this section, we prove <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem8" title="Theorem 1.8 (Lower bound for symmetric selection). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.8</span></a>:</p> </div> <div class="ltx_para" id="S6.p2"> <p class="ltx_p" id="S6.p2.1">See <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem8" title="Theorem 1.8 (Lower bound for symmetric selection). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">1.8</span></a></p> </div> <div class="ltx_para" id="S6.p3"> <p class="ltx_p" id="S6.p3.1">This theorem improves on the bound of <span class="ltx_ERROR undefined" id="S6.p3.1.1">\textcite</span>FJ15 (see <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem2" title="Theorem 1.2 (Prior lower bound for antisymmetric selection, [FJ15, Thm. 1.4]). ‣ 1.1 Background: Directed cuts, bias, and oblivious algorithms ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.2</span></a> above). The proof uses the following single graph:</p> </div> <figure class="ltx_figure" id="S6.F6"><svg class="ltx_picture ltx_centering" height="221.72" id="S6.F6.pic1" overflow="visible" version="1.1" width="659.33"><g fill="#000000" stroke="#000000" transform="translate(0,221.72) matrix(1 0 0 -1 0 0) translate(63.92,0) translate(0,12.43)"><g stroke-width="0.4pt"><g fill="#FEE1E8"><path d="M 12.16 196.85 C 12.16 203.56 6.71 209.01 0 209.01 C -6.71 209.01 -12.16 203.56 -12.16 196.85 C -12.16 190.14 -6.71 184.69 0 184.69 C 6.71 184.69 12.16 190.14 12.16 196.85 Z M 0 196.85"></path></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 -3.46 192.39)"><foreignobject height="8.92" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="6.92"><math alttext="1" class="ltx_Math" display="inline" id="S6.F6.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F6.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mn 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0 Z"></path></g><g fill="#E6E6E6" stroke="#000000"><path d="M 50.77 101.32 h 50.04 v 17.25 h -50.04 Z"></path></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 55.38 105.93)"><foreignobject height="8.03" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="40.82"><span class="ltx_text" id="S6.F6.pic1.16.16.16.1.1.1" style="font-size:90%;">118.215</span></foreignobject></g></g><g stroke-width="0.85358pt"><path d="M 525.28 10.77 C 478.4 92.02 440.27 134.39 370.81 184.88" style="fill:none"></path><g stroke-dasharray="none" stroke-dashoffset="0.0pt" stroke-linejoin="miter" transform="matrix(-0.80888 0.58797 -0.58797 -0.80888 370.81 184.88)"><path d="M 6.4 0 C 5.61 0.28 2.16 1.87 0 3.6 L 0 -3.6 C 2.16 -1.87 5.61 -0.28 6.4 0 Z"></path></g><g fill="#E6E6E6" stroke="#000000"><path d="M 433.8 101.32 h 43.82 v 17.25 h -43.82 Z"></path></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 438.41 105.93)"><foreignobject height="8.03" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="34.59"><span class="ltx_text" id="S6.F6.pic1.17.17.17.1.1.1" style="font-size:90%;">22.005</span></foreignobject></g></g><g stroke-width="0.85358pt"><path d="M 360.55 186.08 C 407.43 104.83 445.56 62.46 515.02 11.98" style="fill:none"></path><g stroke-dasharray="none" stroke-dashoffset="0.0pt" stroke-linejoin="miter" transform="matrix(0.80888 -0.58797 0.58797 0.80888 515.02 11.98)"><path d="M 6.4 0 C 5.61 0.28 2.16 1.87 0 3.6 L 0 -3.6 C 2.16 -1.87 5.61 -0.28 6.4 0 Z"></path></g><g fill="#E6E6E6" stroke="#000000"><path d="M 408.21 78.29 h 43.82 v 17.25 h -43.82 Z"></path></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 412.82 82.9)"><foreignobject height="8.03" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="34.59"><span class="ltx_text" id="S6.F6.pic1.18.18.18.1.1.1" style="font-size:90%;">13.995</span></foreignobject></g></g><g stroke-width="0.85358pt"><path d="M 187.22 7.31 C 263.1 62.46 301.23 104.83 344.15 179.21" style="fill:none"></path><g stroke-dasharray="none" stroke-dashoffset="0.0pt" stroke-linejoin="miter" transform="matrix(0.49982 0.86613 -0.86613 0.49982 344.15 179.21)"><path d="M 6.4 0 C 5.61 0.28 2.16 1.87 0 3.6 L 0 -3.6 C 2.16 -1.87 5.61 -0.28 6.4 0 Z"></path></g><g fill="#E6E6E6" stroke="#000000"><path d="M 259.75 78.29 h 37.59 v 17.25 h -37.59 Z"></path></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 264.36 82.9)"><foreignobject height="8.03" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="28.37"><span class="ltx_text" id="S6.F6.pic1.19.19.19.1.1.1" style="font-size:90%;">1.035</span></foreignobject></g></g><g stroke-width="0.85358pt"><path d="M 344.27 189.54 C 268.4 134.39 230.26 92.02 187.34 17.64" style="fill:none"></path><g stroke-dasharray="none" stroke-dashoffset="0.0pt" stroke-linejoin="miter" transform="matrix(-0.49982 -0.86613 0.86613 -0.49982 187.35 17.64)"><path d="M 6.4 0 C 5.61 0.28 2.16 1.87 0 3.6 L 0 -3.6 C 2.16 -1.87 5.61 -0.28 6.4 0 Z"></path></g><g fill="#E6E6E6" stroke="#000000"><path d="M 231.05 101.32 h 43.82 v 17.25 h -43.82 Z"></path></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 235.66 105.93)"><foreignobject height="8.03" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="34.59"><span class="ltx_text" id="S6.F6.pic1.20.20.20.1.1.1" style="font-size:90%;">25.065</span></foreignobject></g></g><g stroke-width="0.85358pt"><path d="M 177.17 12.43 L 177.17 176.48" style="fill:none"></path><g stroke-dasharray="none" stroke-dashoffset="0.0pt" stroke-linejoin="miter" transform="matrix(0.0 1.0 -1.0 0.0 177.17 176.48)"><path d="M 6.4 0 C 5.61 0.28 2.16 1.87 0 3.6 L 0 -3.6 C 2.16 -1.87 5.61 -0.28 6.4 0 Z"></path></g><g fill="#E6E6E6" stroke="#000000"><path d="M 158.37 89.8 h 37.59 v 17.25 h -37.59 Z"></path></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 162.98 94.41)"><foreignobject height="8.03" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="28.37"><span class="ltx_text" id="S6.F6.pic1.21.21.21.1.1.1" style="font-size:90%;">24.03</span></foreignobject></g></g></g></svg> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S6.F6.46.21.1" style="font-size:90%;">Figure 6</span>: </span><span class="ltx_text" id="S6.F6.40.20" style="font-size:90%;">Graph giving an improved lower bound against antisymmetric selection functions. The <span class="ltx_text" id="S6.F6.40.20.1" style="background-color:#FEE1E8;">PINK</span> vertices (<math alttext="1" class="ltx_Math" display="inline" id="S6.F6.21.1.m1.1"><semantics id="S6.F6.21.1.m1.1b"><mn id="S6.F6.21.1.m1.1.1" xref="S6.F6.21.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S6.F6.21.1.m1.1c"><cn id="S6.F6.21.1.m1.1.1.cmml" type="integer" xref="S6.F6.21.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.21.1.m1.1d">1</annotation><annotation encoding="application/x-llamapun" id="S6.F6.21.1.m1.1e">1</annotation></semantics></math> and <math alttext="2" class="ltx_Math" display="inline" id="S6.F6.22.2.m2.1"><semantics id="S6.F6.22.2.m2.1b"><mn id="S6.F6.22.2.m2.1.1" xref="S6.F6.22.2.m2.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S6.F6.22.2.m2.1c"><cn id="S6.F6.22.2.m2.1.1.cmml" type="integer" xref="S6.F6.22.2.m2.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.22.2.m2.1d">2</annotation><annotation encoding="application/x-llamapun" id="S6.F6.22.2.m2.1e">2</annotation></semantics></math>), the <span class="ltx_text" id="S6.F6.40.20.2" style="background-color:#CBAACB;">PURPLE</span> vertices (<math alttext="3" class="ltx_Math" display="inline" id="S6.F6.23.3.m3.1"><semantics id="S6.F6.23.3.m3.1b"><mn id="S6.F6.23.3.m3.1.1" xref="S6.F6.23.3.m3.1.1.cmml">3</mn><annotation-xml encoding="MathML-Content" id="S6.F6.23.3.m3.1c"><cn id="S6.F6.23.3.m3.1.1.cmml" type="integer" xref="S6.F6.23.3.m3.1.1">3</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.23.3.m3.1d">3</annotation><annotation encoding="application/x-llamapun" id="S6.F6.23.3.m3.1e">3</annotation></semantics></math> and <math alttext="4" class="ltx_Math" display="inline" id="S6.F6.24.4.m4.1"><semantics id="S6.F6.24.4.m4.1b"><mn id="S6.F6.24.4.m4.1.1" xref="S6.F6.24.4.m4.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="S6.F6.24.4.m4.1c"><cn id="S6.F6.24.4.m4.1.1.cmml" type="integer" xref="S6.F6.24.4.m4.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.24.4.m4.1d">4</annotation><annotation encoding="application/x-llamapun" id="S6.F6.24.4.m4.1e">4</annotation></semantics></math>), the <span class="ltx_text" id="S6.F6.40.20.3" style="background-color:#ABDEE6;">LIGHT BLUE</span> vertices (<math alttext="7" class="ltx_Math" display="inline" id="S6.F6.25.5.m5.1"><semantics id="S6.F6.25.5.m5.1b"><mn id="S6.F6.25.5.m5.1.1" xref="S6.F6.25.5.m5.1.1.cmml">7</mn><annotation-xml encoding="MathML-Content" id="S6.F6.25.5.m5.1c"><cn id="S6.F6.25.5.m5.1.1.cmml" type="integer" xref="S6.F6.25.5.m5.1.1">7</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.25.5.m5.1d">7</annotation><annotation encoding="application/x-llamapun" id="S6.F6.25.5.m5.1e">7</annotation></semantics></math> and <math alttext="8" class="ltx_Math" display="inline" id="S6.F6.26.6.m6.1"><semantics id="S6.F6.26.6.m6.1b"><mn id="S6.F6.26.6.m6.1.1" xref="S6.F6.26.6.m6.1.1.cmml">8</mn><annotation-xml encoding="MathML-Content" id="S6.F6.26.6.m6.1c"><cn id="S6.F6.26.6.m6.1.1.cmml" type="integer" xref="S6.F6.26.6.m6.1.1">8</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.26.6.m6.1d">8</annotation><annotation encoding="application/x-llamapun" id="S6.F6.26.6.m6.1e">8</annotation></semantics></math>), and the <span class="ltx_text" id="S6.F6.40.20.4" style="background-color:#8FAAE3;">DARK BLUE</span> vertices (<math alttext="5" class="ltx_Math" display="inline" id="S6.F6.27.7.m7.1"><semantics id="S6.F6.27.7.m7.1b"><mn id="S6.F6.27.7.m7.1.1" xref="S6.F6.27.7.m7.1.1.cmml">5</mn><annotation-xml encoding="MathML-Content" id="S6.F6.27.7.m7.1c"><cn id="S6.F6.27.7.m7.1.1.cmml" type="integer" xref="S6.F6.27.7.m7.1.1">5</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.27.7.m7.1d">5</annotation><annotation encoding="application/x-llamapun" id="S6.F6.27.7.m7.1e">5</annotation></semantics></math> and <math alttext="6" class="ltx_Math" display="inline" id="S6.F6.28.8.m8.1"><semantics id="S6.F6.28.8.m8.1b"><mn id="S6.F6.28.8.m8.1.1" xref="S6.F6.28.8.m8.1.1.cmml">6</mn><annotation-xml encoding="MathML-Content" id="S6.F6.28.8.m8.1c"><cn id="S6.F6.28.8.m8.1.1.cmml" type="integer" xref="S6.F6.28.8.m8.1.1">6</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.28.8.m8.1d">6</annotation><annotation encoding="application/x-llamapun" id="S6.F6.28.8.m8.1e">6</annotation></semantics></math>) have biases <math alttext="-0.1" class="ltx_Math" display="inline" id="S6.F6.29.9.m9.1"><semantics id="S6.F6.29.9.m9.1b"><mrow id="S6.F6.29.9.m9.1.1" xref="S6.F6.29.9.m9.1.1.cmml"><mo id="S6.F6.29.9.m9.1.1b" xref="S6.F6.29.9.m9.1.1.cmml">−</mo><mn id="S6.F6.29.9.m9.1.1.2" xref="S6.F6.29.9.m9.1.1.2.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.F6.29.9.m9.1c"><apply id="S6.F6.29.9.m9.1.1.cmml" xref="S6.F6.29.9.m9.1.1"><minus id="S6.F6.29.9.m9.1.1.1.cmml" xref="S6.F6.29.9.m9.1.1"></minus><cn id="S6.F6.29.9.m9.1.1.2.cmml" type="float" xref="S6.F6.29.9.m9.1.1.2">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.29.9.m9.1d">-0.1</annotation><annotation encoding="application/x-llamapun" id="S6.F6.29.9.m9.1e">- 0.1</annotation></semantics></math>, <math alttext="0" class="ltx_Math" display="inline" id="S6.F6.30.10.m10.1"><semantics id="S6.F6.30.10.m10.1b"><mn id="S6.F6.30.10.m10.1.1" xref="S6.F6.30.10.m10.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S6.F6.30.10.m10.1c"><cn id="S6.F6.30.10.m10.1.1.cmml" type="integer" xref="S6.F6.30.10.m10.1.1">0</cn></annotation-xml></semantics></math>, <math alttext="+0.1," class="ltx_Math" display="inline" id="S6.F6.31.11.m11.1"><semantics id="S6.F6.31.11.m11.1b"><mrow id="S6.F6.31.11.m11.1.1.1" xref="S6.F6.31.11.m11.1.1.1.1.cmml"><mrow id="S6.F6.31.11.m11.1.1.1.1" xref="S6.F6.31.11.m11.1.1.1.1.cmml"><mo id="S6.F6.31.11.m11.1.1.1.1b" xref="S6.F6.31.11.m11.1.1.1.1.cmml">+</mo><mn id="S6.F6.31.11.m11.1.1.1.1.2" xref="S6.F6.31.11.m11.1.1.1.1.2.cmml">0.1</mn></mrow><mo id="S6.F6.31.11.m11.1.1.1.2" xref="S6.F6.31.11.m11.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.F6.31.11.m11.1c"><apply id="S6.F6.31.11.m11.1.1.1.1.cmml" xref="S6.F6.31.11.m11.1.1.1"><plus id="S6.F6.31.11.m11.1.1.1.1.1.cmml" xref="S6.F6.31.11.m11.1.1.1"></plus><cn id="S6.F6.31.11.m11.1.1.1.1.2.cmml" type="float" xref="S6.F6.31.11.m11.1.1.1.1.2">0.1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.31.11.m11.1d">+0.1,</annotation><annotation encoding="application/x-llamapun" id="S6.F6.31.11.m11.1e">+ 0.1 ,</annotation></semantics></math> and <math alttext="+0.2" class="ltx_Math" display="inline" id="S6.F6.32.12.m12.1"><semantics id="S6.F6.32.12.m12.1b"><mrow id="S6.F6.32.12.m12.1.1" xref="S6.F6.32.12.m12.1.1.cmml"><mo id="S6.F6.32.12.m12.1.1b" xref="S6.F6.32.12.m12.1.1.cmml">+</mo><mn id="S6.F6.32.12.m12.1.1.2" xref="S6.F6.32.12.m12.1.1.2.cmml">0.2</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.F6.32.12.m12.1c"><apply id="S6.F6.32.12.m12.1.1.cmml" xref="S6.F6.32.12.m12.1.1"><plus id="S6.F6.32.12.m12.1.1.1.cmml" xref="S6.F6.32.12.m12.1.1"></plus><cn id="S6.F6.32.12.m12.1.1.2.cmml" type="float" xref="S6.F6.32.12.m12.1.1.2">0.2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.32.12.m12.1d">+0.2</annotation><annotation encoding="application/x-llamapun" id="S6.F6.32.12.m12.1e">+ 0.2</annotation></semantics></math>, respectively. The cut assigning <math alttext="\{1,3,5,7\}\to 1" class="ltx_Math" display="inline" id="S6.F6.33.13.m13.4"><semantics id="S6.F6.33.13.m13.4b"><mrow id="S6.F6.33.13.m13.4.5" xref="S6.F6.33.13.m13.4.5.cmml"><mrow id="S6.F6.33.13.m13.4.5.2.2" xref="S6.F6.33.13.m13.4.5.2.1.cmml"><mo id="S6.F6.33.13.m13.4.5.2.2.1" stretchy="false" xref="S6.F6.33.13.m13.4.5.2.1.cmml">{</mo><mn id="S6.F6.33.13.m13.1.1" xref="S6.F6.33.13.m13.1.1.cmml">1</mn><mo id="S6.F6.33.13.m13.4.5.2.2.2" xref="S6.F6.33.13.m13.4.5.2.1.cmml">,</mo><mn id="S6.F6.33.13.m13.2.2" xref="S6.F6.33.13.m13.2.2.cmml">3</mn><mo id="S6.F6.33.13.m13.4.5.2.2.3" xref="S6.F6.33.13.m13.4.5.2.1.cmml">,</mo><mn id="S6.F6.33.13.m13.3.3" xref="S6.F6.33.13.m13.3.3.cmml">5</mn><mo id="S6.F6.33.13.m13.4.5.2.2.4" xref="S6.F6.33.13.m13.4.5.2.1.cmml">,</mo><mn id="S6.F6.33.13.m13.4.4" xref="S6.F6.33.13.m13.4.4.cmml">7</mn><mo id="S6.F6.33.13.m13.4.5.2.2.5" stretchy="false" xref="S6.F6.33.13.m13.4.5.2.1.cmml">}</mo></mrow><mo id="S6.F6.33.13.m13.4.5.1" stretchy="false" xref="S6.F6.33.13.m13.4.5.1.cmml">→</mo><mn id="S6.F6.33.13.m13.4.5.3" xref="S6.F6.33.13.m13.4.5.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.F6.33.13.m13.4c"><apply id="S6.F6.33.13.m13.4.5.cmml" xref="S6.F6.33.13.m13.4.5"><ci id="S6.F6.33.13.m13.4.5.1.cmml" xref="S6.F6.33.13.m13.4.5.1">→</ci><set id="S6.F6.33.13.m13.4.5.2.1.cmml" xref="S6.F6.33.13.m13.4.5.2.2"><cn id="S6.F6.33.13.m13.1.1.cmml" type="integer" xref="S6.F6.33.13.m13.1.1">1</cn><cn id="S6.F6.33.13.m13.2.2.cmml" type="integer" xref="S6.F6.33.13.m13.2.2">3</cn><cn id="S6.F6.33.13.m13.3.3.cmml" type="integer" xref="S6.F6.33.13.m13.3.3">5</cn><cn id="S6.F6.33.13.m13.4.4.cmml" type="integer" xref="S6.F6.33.13.m13.4.4">7</cn></set><cn id="S6.F6.33.13.m13.4.5.3.cmml" type="integer" xref="S6.F6.33.13.m13.4.5.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.33.13.m13.4d">\{1,3,5,7\}\to 1</annotation><annotation encoding="application/x-llamapun" id="S6.F6.33.13.m13.4e">{ 1 , 3 , 5 , 7 } → 1</annotation></semantics></math> and <math alttext="\{2,4,6,8\}\to 0" class="ltx_Math" display="inline" id="S6.F6.34.14.m14.4"><semantics id="S6.F6.34.14.m14.4b"><mrow id="S6.F6.34.14.m14.4.5" xref="S6.F6.34.14.m14.4.5.cmml"><mrow id="S6.F6.34.14.m14.4.5.2.2" xref="S6.F6.34.14.m14.4.5.2.1.cmml"><mo id="S6.F6.34.14.m14.4.5.2.2.1" stretchy="false" xref="S6.F6.34.14.m14.4.5.2.1.cmml">{</mo><mn id="S6.F6.34.14.m14.1.1" xref="S6.F6.34.14.m14.1.1.cmml">2</mn><mo id="S6.F6.34.14.m14.4.5.2.2.2" xref="S6.F6.34.14.m14.4.5.2.1.cmml">,</mo><mn id="S6.F6.34.14.m14.2.2" xref="S6.F6.34.14.m14.2.2.cmml">4</mn><mo id="S6.F6.34.14.m14.4.5.2.2.3" xref="S6.F6.34.14.m14.4.5.2.1.cmml">,</mo><mn id="S6.F6.34.14.m14.3.3" xref="S6.F6.34.14.m14.3.3.cmml">6</mn><mo id="S6.F6.34.14.m14.4.5.2.2.4" xref="S6.F6.34.14.m14.4.5.2.1.cmml">,</mo><mn id="S6.F6.34.14.m14.4.4" xref="S6.F6.34.14.m14.4.4.cmml">8</mn><mo id="S6.F6.34.14.m14.4.5.2.2.5" stretchy="false" xref="S6.F6.34.14.m14.4.5.2.1.cmml">}</mo></mrow><mo id="S6.F6.34.14.m14.4.5.1" stretchy="false" xref="S6.F6.34.14.m14.4.5.1.cmml">→</mo><mn id="S6.F6.34.14.m14.4.5.3" xref="S6.F6.34.14.m14.4.5.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.F6.34.14.m14.4c"><apply id="S6.F6.34.14.m14.4.5.cmml" xref="S6.F6.34.14.m14.4.5"><ci id="S6.F6.34.14.m14.4.5.1.cmml" xref="S6.F6.34.14.m14.4.5.1">→</ci><set id="S6.F6.34.14.m14.4.5.2.1.cmml" xref="S6.F6.34.14.m14.4.5.2.2"><cn id="S6.F6.34.14.m14.1.1.cmml" type="integer" xref="S6.F6.34.14.m14.1.1">2</cn><cn id="S6.F6.34.14.m14.2.2.cmml" type="integer" xref="S6.F6.34.14.m14.2.2">4</cn><cn id="S6.F6.34.14.m14.3.3.cmml" type="integer" xref="S6.F6.34.14.m14.3.3">6</cn><cn id="S6.F6.34.14.m14.4.4.cmml" type="integer" xref="S6.F6.34.14.m14.4.4">8</cn></set><cn id="S6.F6.34.14.m14.4.5.3.cmml" type="integer" xref="S6.F6.34.14.m14.4.5.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.34.14.m14.4d">\{2,4,6,8\}\to 0</annotation><annotation encoding="application/x-llamapun" id="S6.F6.34.14.m14.4e">{ 2 , 4 , 6 , 8 } → 0</annotation></semantics></math> has weight <math alttext="6.2775+118.215+25.065+3.6+13.995+12.1275=179.28" class="ltx_Math" display="inline" id="S6.F6.35.15.m15.1"><semantics id="S6.F6.35.15.m15.1b"><mrow id="S6.F6.35.15.m15.1.1" xref="S6.F6.35.15.m15.1.1.cmml"><mrow id="S6.F6.35.15.m15.1.1.2" xref="S6.F6.35.15.m15.1.1.2.cmml"><mn id="S6.F6.35.15.m15.1.1.2.2" xref="S6.F6.35.15.m15.1.1.2.2.cmml">6.2775</mn><mo id="S6.F6.35.15.m15.1.1.2.1" xref="S6.F6.35.15.m15.1.1.2.1.cmml">+</mo><mn id="S6.F6.35.15.m15.1.1.2.3" xref="S6.F6.35.15.m15.1.1.2.3.cmml">118.215</mn><mo id="S6.F6.35.15.m15.1.1.2.1b" xref="S6.F6.35.15.m15.1.1.2.1.cmml">+</mo><mn id="S6.F6.35.15.m15.1.1.2.4" xref="S6.F6.35.15.m15.1.1.2.4.cmml">25.065</mn><mo id="S6.F6.35.15.m15.1.1.2.1c" xref="S6.F6.35.15.m15.1.1.2.1.cmml">+</mo><mn id="S6.F6.35.15.m15.1.1.2.5" xref="S6.F6.35.15.m15.1.1.2.5.cmml">3.6</mn><mo id="S6.F6.35.15.m15.1.1.2.1d" xref="S6.F6.35.15.m15.1.1.2.1.cmml">+</mo><mn id="S6.F6.35.15.m15.1.1.2.6" xref="S6.F6.35.15.m15.1.1.2.6.cmml">13.995</mn><mo id="S6.F6.35.15.m15.1.1.2.1e" xref="S6.F6.35.15.m15.1.1.2.1.cmml">+</mo><mn id="S6.F6.35.15.m15.1.1.2.7" xref="S6.F6.35.15.m15.1.1.2.7.cmml">12.1275</mn></mrow><mo id="S6.F6.35.15.m15.1.1.1" xref="S6.F6.35.15.m15.1.1.1.cmml">=</mo><mn id="S6.F6.35.15.m15.1.1.3" xref="S6.F6.35.15.m15.1.1.3.cmml">179.28</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.F6.35.15.m15.1c"><apply id="S6.F6.35.15.m15.1.1.cmml" xref="S6.F6.35.15.m15.1.1"><eq id="S6.F6.35.15.m15.1.1.1.cmml" xref="S6.F6.35.15.m15.1.1.1"></eq><apply id="S6.F6.35.15.m15.1.1.2.cmml" xref="S6.F6.35.15.m15.1.1.2"><plus id="S6.F6.35.15.m15.1.1.2.1.cmml" xref="S6.F6.35.15.m15.1.1.2.1"></plus><cn id="S6.F6.35.15.m15.1.1.2.2.cmml" type="float" xref="S6.F6.35.15.m15.1.1.2.2">6.2775</cn><cn id="S6.F6.35.15.m15.1.1.2.3.cmml" type="float" xref="S6.F6.35.15.m15.1.1.2.3">118.215</cn><cn id="S6.F6.35.15.m15.1.1.2.4.cmml" type="float" xref="S6.F6.35.15.m15.1.1.2.4">25.065</cn><cn id="S6.F6.35.15.m15.1.1.2.5.cmml" type="float" xref="S6.F6.35.15.m15.1.1.2.5">3.6</cn><cn id="S6.F6.35.15.m15.1.1.2.6.cmml" type="float" xref="S6.F6.35.15.m15.1.1.2.6">13.995</cn><cn id="S6.F6.35.15.m15.1.1.2.7.cmml" type="float" xref="S6.F6.35.15.m15.1.1.2.7">12.1275</cn></apply><cn id="S6.F6.35.15.m15.1.1.3.cmml" type="float" xref="S6.F6.35.15.m15.1.1.3">179.28</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.35.15.m15.1d">6.2775+118.215+25.065+3.6+13.995+12.1275=179.28</annotation><annotation encoding="application/x-llamapun" id="S6.F6.35.15.m15.1e">6.2775 + 118.215 + 25.065 + 3.6 + 13.995 + 12.1275 = 179.28</annotation></semantics></math>. An antisymmetric oblivious cut assigning <math alttext="\{1,2\}\to 1-p" class="ltx_Math" display="inline" id="S6.F6.36.16.m16.2"><semantics id="S6.F6.36.16.m16.2b"><mrow id="S6.F6.36.16.m16.2.3" xref="S6.F6.36.16.m16.2.3.cmml"><mrow id="S6.F6.36.16.m16.2.3.2.2" xref="S6.F6.36.16.m16.2.3.2.1.cmml"><mo id="S6.F6.36.16.m16.2.3.2.2.1" stretchy="false" xref="S6.F6.36.16.m16.2.3.2.1.cmml">{</mo><mn id="S6.F6.36.16.m16.1.1" xref="S6.F6.36.16.m16.1.1.cmml">1</mn><mo id="S6.F6.36.16.m16.2.3.2.2.2" xref="S6.F6.36.16.m16.2.3.2.1.cmml">,</mo><mn id="S6.F6.36.16.m16.2.2" xref="S6.F6.36.16.m16.2.2.cmml">2</mn><mo id="S6.F6.36.16.m16.2.3.2.2.3" stretchy="false" xref="S6.F6.36.16.m16.2.3.2.1.cmml">}</mo></mrow><mo id="S6.F6.36.16.m16.2.3.1" stretchy="false" xref="S6.F6.36.16.m16.2.3.1.cmml">→</mo><mrow id="S6.F6.36.16.m16.2.3.3" xref="S6.F6.36.16.m16.2.3.3.cmml"><mn id="S6.F6.36.16.m16.2.3.3.2" xref="S6.F6.36.16.m16.2.3.3.2.cmml">1</mn><mo id="S6.F6.36.16.m16.2.3.3.1" xref="S6.F6.36.16.m16.2.3.3.1.cmml">−</mo><mi id="S6.F6.36.16.m16.2.3.3.3" xref="S6.F6.36.16.m16.2.3.3.3.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.F6.36.16.m16.2c"><apply id="S6.F6.36.16.m16.2.3.cmml" xref="S6.F6.36.16.m16.2.3"><ci id="S6.F6.36.16.m16.2.3.1.cmml" xref="S6.F6.36.16.m16.2.3.1">→</ci><set id="S6.F6.36.16.m16.2.3.2.1.cmml" xref="S6.F6.36.16.m16.2.3.2.2"><cn id="S6.F6.36.16.m16.1.1.cmml" type="integer" xref="S6.F6.36.16.m16.1.1">1</cn><cn id="S6.F6.36.16.m16.2.2.cmml" type="integer" xref="S6.F6.36.16.m16.2.2">2</cn></set><apply id="S6.F6.36.16.m16.2.3.3.cmml" xref="S6.F6.36.16.m16.2.3.3"><minus id="S6.F6.36.16.m16.2.3.3.1.cmml" xref="S6.F6.36.16.m16.2.3.3.1"></minus><cn id="S6.F6.36.16.m16.2.3.3.2.cmml" type="integer" xref="S6.F6.36.16.m16.2.3.3.2">1</cn><ci id="S6.F6.36.16.m16.2.3.3.3.cmml" xref="S6.F6.36.16.m16.2.3.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.36.16.m16.2d">\{1,2\}\to 1-p</annotation><annotation encoding="application/x-llamapun" id="S6.F6.36.16.m16.2e">{ 1 , 2 } → 1 - italic_p</annotation></semantics></math>, <math alttext="\{3,4\}\to\frac{1}{2}" class="ltx_Math" display="inline" id="S6.F6.37.17.m17.2"><semantics id="S6.F6.37.17.m17.2b"><mrow id="S6.F6.37.17.m17.2.3" xref="S6.F6.37.17.m17.2.3.cmml"><mrow id="S6.F6.37.17.m17.2.3.2.2" xref="S6.F6.37.17.m17.2.3.2.1.cmml"><mo id="S6.F6.37.17.m17.2.3.2.2.1" stretchy="false" xref="S6.F6.37.17.m17.2.3.2.1.cmml">{</mo><mn id="S6.F6.37.17.m17.1.1" xref="S6.F6.37.17.m17.1.1.cmml">3</mn><mo id="S6.F6.37.17.m17.2.3.2.2.2" xref="S6.F6.37.17.m17.2.3.2.1.cmml">,</mo><mn id="S6.F6.37.17.m17.2.2" xref="S6.F6.37.17.m17.2.2.cmml">4</mn><mo id="S6.F6.37.17.m17.2.3.2.2.3" stretchy="false" xref="S6.F6.37.17.m17.2.3.2.1.cmml">}</mo></mrow><mo id="S6.F6.37.17.m17.2.3.1" stretchy="false" xref="S6.F6.37.17.m17.2.3.1.cmml">→</mo><mfrac id="S6.F6.37.17.m17.2.3.3" xref="S6.F6.37.17.m17.2.3.3.cmml"><mn id="S6.F6.37.17.m17.2.3.3.2" xref="S6.F6.37.17.m17.2.3.3.2.cmml">1</mn><mn id="S6.F6.37.17.m17.2.3.3.3" xref="S6.F6.37.17.m17.2.3.3.3.cmml">2</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S6.F6.37.17.m17.2c"><apply id="S6.F6.37.17.m17.2.3.cmml" xref="S6.F6.37.17.m17.2.3"><ci id="S6.F6.37.17.m17.2.3.1.cmml" xref="S6.F6.37.17.m17.2.3.1">→</ci><set id="S6.F6.37.17.m17.2.3.2.1.cmml" xref="S6.F6.37.17.m17.2.3.2.2"><cn id="S6.F6.37.17.m17.1.1.cmml" type="integer" xref="S6.F6.37.17.m17.1.1">3</cn><cn id="S6.F6.37.17.m17.2.2.cmml" type="integer" xref="S6.F6.37.17.m17.2.2">4</cn></set><apply id="S6.F6.37.17.m17.2.3.3.cmml" xref="S6.F6.37.17.m17.2.3.3"><divide id="S6.F6.37.17.m17.2.3.3.1.cmml" xref="S6.F6.37.17.m17.2.3.3"></divide><cn id="S6.F6.37.17.m17.2.3.3.2.cmml" type="integer" xref="S6.F6.37.17.m17.2.3.3.2">1</cn><cn id="S6.F6.37.17.m17.2.3.3.3.cmml" type="integer" xref="S6.F6.37.17.m17.2.3.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.37.17.m17.2d">\{3,4\}\to\frac{1}{2}</annotation><annotation encoding="application/x-llamapun" id="S6.F6.37.17.m17.2e">{ 3 , 4 } → divide start_ARG 1 end_ARG start_ARG 2 end_ARG</annotation></semantics></math>, <math alttext="\{5,6\}\to q" class="ltx_Math" display="inline" id="S6.F6.38.18.m18.2"><semantics id="S6.F6.38.18.m18.2b"><mrow id="S6.F6.38.18.m18.2.3" xref="S6.F6.38.18.m18.2.3.cmml"><mrow id="S6.F6.38.18.m18.2.3.2.2" xref="S6.F6.38.18.m18.2.3.2.1.cmml"><mo id="S6.F6.38.18.m18.2.3.2.2.1" stretchy="false" xref="S6.F6.38.18.m18.2.3.2.1.cmml">{</mo><mn id="S6.F6.38.18.m18.1.1" xref="S6.F6.38.18.m18.1.1.cmml">5</mn><mo id="S6.F6.38.18.m18.2.3.2.2.2" xref="S6.F6.38.18.m18.2.3.2.1.cmml">,</mo><mn id="S6.F6.38.18.m18.2.2" xref="S6.F6.38.18.m18.2.2.cmml">6</mn><mo id="S6.F6.38.18.m18.2.3.2.2.3" stretchy="false" xref="S6.F6.38.18.m18.2.3.2.1.cmml">}</mo></mrow><mo id="S6.F6.38.18.m18.2.3.1" stretchy="false" xref="S6.F6.38.18.m18.2.3.1.cmml">→</mo><mi id="S6.F6.38.18.m18.2.3.3" xref="S6.F6.38.18.m18.2.3.3.cmml">q</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.F6.38.18.m18.2c"><apply id="S6.F6.38.18.m18.2.3.cmml" xref="S6.F6.38.18.m18.2.3"><ci id="S6.F6.38.18.m18.2.3.1.cmml" xref="S6.F6.38.18.m18.2.3.1">→</ci><set id="S6.F6.38.18.m18.2.3.2.1.cmml" xref="S6.F6.38.18.m18.2.3.2.2"><cn id="S6.F6.38.18.m18.1.1.cmml" type="integer" xref="S6.F6.38.18.m18.1.1">5</cn><cn id="S6.F6.38.18.m18.2.2.cmml" type="integer" xref="S6.F6.38.18.m18.2.2">6</cn></set><ci id="S6.F6.38.18.m18.2.3.3.cmml" xref="S6.F6.38.18.m18.2.3.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.38.18.m18.2d">\{5,6\}\to q</annotation><annotation encoding="application/x-llamapun" id="S6.F6.38.18.m18.2e">{ 5 , 6 } → italic_q</annotation></semantics></math>, and <math alttext="\{7,8\}\to p" class="ltx_Math" display="inline" id="S6.F6.39.19.m19.2"><semantics id="S6.F6.39.19.m19.2b"><mrow id="S6.F6.39.19.m19.2.3" xref="S6.F6.39.19.m19.2.3.cmml"><mrow id="S6.F6.39.19.m19.2.3.2.2" xref="S6.F6.39.19.m19.2.3.2.1.cmml"><mo id="S6.F6.39.19.m19.2.3.2.2.1" stretchy="false" xref="S6.F6.39.19.m19.2.3.2.1.cmml">{</mo><mn id="S6.F6.39.19.m19.1.1" xref="S6.F6.39.19.m19.1.1.cmml">7</mn><mo id="S6.F6.39.19.m19.2.3.2.2.2" xref="S6.F6.39.19.m19.2.3.2.1.cmml">,</mo><mn id="S6.F6.39.19.m19.2.2" xref="S6.F6.39.19.m19.2.2.cmml">8</mn><mo id="S6.F6.39.19.m19.2.3.2.2.3" stretchy="false" xref="S6.F6.39.19.m19.2.3.2.1.cmml">}</mo></mrow><mo id="S6.F6.39.19.m19.2.3.1" stretchy="false" xref="S6.F6.39.19.m19.2.3.1.cmml">→</mo><mi id="S6.F6.39.19.m19.2.3.3" xref="S6.F6.39.19.m19.2.3.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.F6.39.19.m19.2c"><apply id="S6.F6.39.19.m19.2.3.cmml" xref="S6.F6.39.19.m19.2.3"><ci id="S6.F6.39.19.m19.2.3.1.cmml" xref="S6.F6.39.19.m19.2.3.1">→</ci><set id="S6.F6.39.19.m19.2.3.2.1.cmml" xref="S6.F6.39.19.m19.2.3.2.2"><cn id="S6.F6.39.19.m19.1.1.cmml" type="integer" xref="S6.F6.39.19.m19.1.1">7</cn><cn id="S6.F6.39.19.m19.2.2.cmml" type="integer" xref="S6.F6.39.19.m19.2.2">8</cn></set><ci id="S6.F6.39.19.m19.2.3.3.cmml" xref="S6.F6.39.19.m19.2.3.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F6.39.19.m19.2d">\{7,8\}\to p</annotation><annotation encoding="application/x-llamapun" id="S6.F6.39.19.m19.2e">{ 7 , 8 } → italic_p</annotation></semantics></math> has value <math alttext="(6.2775+7.6725)p(1-p)+94.185\cdot\frac{1}{2}(1-p)+118.215\cdot\frac{1}{2}p+24.% 03\cdot\frac{1}{4}+1.035\cdot\frac{1}{2}(1-q)+25.065\cdot\frac{1}{2}q+22.005p(% 1-q)+13.995q(1-p)+(3.6+5.4)q(1-q)+(12.1275+9.9225)\cdot p(1-p)=53.6175-36p^{2}% +p(70.02-36q)+35.01q-9q^{2}" class="ltx_Math" display="inline" id="S6.F6.40.20.m20.11"><semantics id="S6.F6.40.20.m20.11b"><mrow id="S6.F6.40.20.m20.11.11" xref="S6.F6.40.20.m20.11.11.cmml"><mrow id="S6.F6.40.20.m20.10.10.10" xref="S6.F6.40.20.m20.10.10.10.cmml"><mrow id="S6.F6.40.20.m20.2.2.2.2" xref="S6.F6.40.20.m20.2.2.2.2.cmml"><mrow id="S6.F6.40.20.m20.1.1.1.1.1.1" xref="S6.F6.40.20.m20.1.1.1.1.1.1.1.cmml"><mo id="S6.F6.40.20.m20.1.1.1.1.1.1.2" stretchy="false" xref="S6.F6.40.20.m20.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.F6.40.20.m20.1.1.1.1.1.1.1" xref="S6.F6.40.20.m20.1.1.1.1.1.1.1.cmml"><mn id="S6.F6.40.20.m20.1.1.1.1.1.1.1.2" xref="S6.F6.40.20.m20.1.1.1.1.1.1.1.2.cmml">6.2775</mn><mo id="S6.F6.40.20.m20.1.1.1.1.1.1.1.1" xref="S6.F6.40.20.m20.1.1.1.1.1.1.1.1.cmml">+</mo><mn id="S6.F6.40.20.m20.1.1.1.1.1.1.1.3" xref="S6.F6.40.20.m20.1.1.1.1.1.1.1.3.cmml">7.6725</mn></mrow><mo id="S6.F6.40.20.m20.1.1.1.1.1.1.3" stretchy="false" xref="S6.F6.40.20.m20.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.F6.40.20.m20.2.2.2.2.3" xref="S6.F6.40.20.m20.2.2.2.2.3.cmml">⁢</mo><mi id="S6.F6.40.20.m20.2.2.2.2.4" xref="S6.F6.40.20.m20.2.2.2.2.4.cmml">p</mi><mo id="S6.F6.40.20.m20.2.2.2.2.3b" xref="S6.F6.40.20.m20.2.2.2.2.3.cmml">⁢</mo><mrow id="S6.F6.40.20.m20.2.2.2.2.2.1" xref="S6.F6.40.20.m20.2.2.2.2.2.1.1.cmml"><mo id="S6.F6.40.20.m20.2.2.2.2.2.1.2" stretchy="false" xref="S6.F6.40.20.m20.2.2.2.2.2.1.1.cmml">(</mo><mrow id="S6.F6.40.20.m20.2.2.2.2.2.1.1" xref="S6.F6.40.20.m20.2.2.2.2.2.1.1.cmml"><mn id="S6.F6.40.20.m20.2.2.2.2.2.1.1.2" xref="S6.F6.40.20.m20.2.2.2.2.2.1.1.2.cmml">1</mn><mo id="S6.F6.40.20.m20.2.2.2.2.2.1.1.1" xref="S6.F6.40.20.m20.2.2.2.2.2.1.1.1.cmml">−</mo><mi id="S6.F6.40.20.m20.2.2.2.2.2.1.1.3" xref="S6.F6.40.20.m20.2.2.2.2.2.1.1.3.cmml">p</mi></mrow><mo id="S6.F6.40.20.m20.2.2.2.2.2.1.3" stretchy="false" xref="S6.F6.40.20.m20.2.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S6.F6.40.20.m20.10.10.10.11" xref="S6.F6.40.20.m20.10.10.10.11.cmml">+</mo><mrow id="S6.F6.40.20.m20.3.3.3.3" xref="S6.F6.40.20.m20.3.3.3.3.cmml"><mrow id="S6.F6.40.20.m20.3.3.3.3.3" xref="S6.F6.40.20.m20.3.3.3.3.3.cmml"><mn id="S6.F6.40.20.m20.3.3.3.3.3.2" xref="S6.F6.40.20.m20.3.3.3.3.3.2.cmml">94.185</mn><mo id="S6.F6.40.20.m20.3.3.3.3.3.1" lspace="0.222em" rspace="0.222em" xref="S6.F6.40.20.m20.3.3.3.3.3.1.cmml">⋅</mo><mfrac id="S6.F6.40.20.m20.3.3.3.3.3.3" xref="S6.F6.40.20.m20.3.3.3.3.3.3.cmml"><mn id="S6.F6.40.20.m20.3.3.3.3.3.3.2" xref="S6.F6.40.20.m20.3.3.3.3.3.3.2.cmml">1</mn><mn id="S6.F6.40.20.m20.3.3.3.3.3.3.3" xref="S6.F6.40.20.m20.3.3.3.3.3.3.3.cmml">2</mn></mfrac></mrow><mo id="S6.F6.40.20.m20.3.3.3.3.2" xref="S6.F6.40.20.m20.3.3.3.3.2.cmml">⁢</mo><mrow id="S6.F6.40.20.m20.3.3.3.3.1.1" xref="S6.F6.40.20.m20.3.3.3.3.1.1.1.cmml"><mo id="S6.F6.40.20.m20.3.3.3.3.1.1.2" stretchy="false" xref="S6.F6.40.20.m20.3.3.3.3.1.1.1.cmml">(</mo><mrow id="S6.F6.40.20.m20.3.3.3.3.1.1.1" xref="S6.F6.40.20.m20.3.3.3.3.1.1.1.cmml"><mn id="S6.F6.40.20.m20.3.3.3.3.1.1.1.2" xref="S6.F6.40.20.m20.3.3.3.3.1.1.1.2.cmml">1</mn><mo id="S6.F6.40.20.m20.3.3.3.3.1.1.1.1" xref="S6.F6.40.20.m20.3.3.3.3.1.1.1.1.cmml">−</mo><mi id="S6.F6.40.20.m20.3.3.3.3.1.1.1.3" xref="S6.F6.40.20.m20.3.3.3.3.1.1.1.3.cmml">p</mi></mrow><mo id="S6.F6.40.20.m20.3.3.3.3.1.1.3" stretchy="false" xref="S6.F6.40.20.m20.3.3.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.F6.40.20.m20.10.10.10.11b" xref="S6.F6.40.20.m20.10.10.10.11.cmml">+</mo><mrow id="S6.F6.40.20.m20.10.10.10.12" xref="S6.F6.40.20.m20.10.10.10.12.cmml"><mrow id="S6.F6.40.20.m20.10.10.10.12.2" xref="S6.F6.40.20.m20.10.10.10.12.2.cmml"><mn id="S6.F6.40.20.m20.10.10.10.12.2.2" xref="S6.F6.40.20.m20.10.10.10.12.2.2.cmml">118.215</mn><mo id="S6.F6.40.20.m20.10.10.10.12.2.1" lspace="0.222em" rspace="0.222em" xref="S6.F6.40.20.m20.10.10.10.12.2.1.cmml">⋅</mo><mfrac id="S6.F6.40.20.m20.10.10.10.12.2.3" xref="S6.F6.40.20.m20.10.10.10.12.2.3.cmml"><mn id="S6.F6.40.20.m20.10.10.10.12.2.3.2" xref="S6.F6.40.20.m20.10.10.10.12.2.3.2.cmml">1</mn><mn id="S6.F6.40.20.m20.10.10.10.12.2.3.3" xref="S6.F6.40.20.m20.10.10.10.12.2.3.3.cmml">2</mn></mfrac></mrow><mo id="S6.F6.40.20.m20.10.10.10.12.1" xref="S6.F6.40.20.m20.10.10.10.12.1.cmml">⁢</mo><mi id="S6.F6.40.20.m20.10.10.10.12.3" xref="S6.F6.40.20.m20.10.10.10.12.3.cmml">p</mi></mrow><mo id="S6.F6.40.20.m20.10.10.10.11c" xref="S6.F6.40.20.m20.10.10.10.11.cmml">+</mo><mrow id="S6.F6.40.20.m20.10.10.10.13" xref="S6.F6.40.20.m20.10.10.10.13.cmml"><mn id="S6.F6.40.20.m20.10.10.10.13.2" xref="S6.F6.40.20.m20.10.10.10.13.2.cmml">24.03</mn><mo id="S6.F6.40.20.m20.10.10.10.13.1" lspace="0.222em" rspace="0.222em" xref="S6.F6.40.20.m20.10.10.10.13.1.cmml">⋅</mo><mfrac id="S6.F6.40.20.m20.10.10.10.13.3" xref="S6.F6.40.20.m20.10.10.10.13.3.cmml"><mn id="S6.F6.40.20.m20.10.10.10.13.3.2" xref="S6.F6.40.20.m20.10.10.10.13.3.2.cmml">1</mn><mn id="S6.F6.40.20.m20.10.10.10.13.3.3" xref="S6.F6.40.20.m20.10.10.10.13.3.3.cmml">4</mn></mfrac></mrow><mo id="S6.F6.40.20.m20.10.10.10.11d" xref="S6.F6.40.20.m20.10.10.10.11.cmml">+</mo><mrow id="S6.F6.40.20.m20.4.4.4.4" xref="S6.F6.40.20.m20.4.4.4.4.cmml"><mrow id="S6.F6.40.20.m20.4.4.4.4.3" xref="S6.F6.40.20.m20.4.4.4.4.3.cmml"><mn id="S6.F6.40.20.m20.4.4.4.4.3.2" xref="S6.F6.40.20.m20.4.4.4.4.3.2.cmml">1.035</mn><mo id="S6.F6.40.20.m20.4.4.4.4.3.1" lspace="0.222em" rspace="0.222em" xref="S6.F6.40.20.m20.4.4.4.4.3.1.cmml">⋅</mo><mfrac id="S6.F6.40.20.m20.4.4.4.4.3.3" xref="S6.F6.40.20.m20.4.4.4.4.3.3.cmml"><mn id="S6.F6.40.20.m20.4.4.4.4.3.3.2" xref="S6.F6.40.20.m20.4.4.4.4.3.3.2.cmml">1</mn><mn id="S6.F6.40.20.m20.4.4.4.4.3.3.3" xref="S6.F6.40.20.m20.4.4.4.4.3.3.3.cmml">2</mn></mfrac></mrow><mo id="S6.F6.40.20.m20.4.4.4.4.2" xref="S6.F6.40.20.m20.4.4.4.4.2.cmml">⁢</mo><mrow id="S6.F6.40.20.m20.4.4.4.4.1.1" xref="S6.F6.40.20.m20.4.4.4.4.1.1.1.cmml"><mo id="S6.F6.40.20.m20.4.4.4.4.1.1.2" stretchy="false" xref="S6.F6.40.20.m20.4.4.4.4.1.1.1.cmml">(</mo><mrow id="S6.F6.40.20.m20.4.4.4.4.1.1.1" xref="S6.F6.40.20.m20.4.4.4.4.1.1.1.cmml"><mn id="S6.F6.40.20.m20.4.4.4.4.1.1.1.2" xref="S6.F6.40.20.m20.4.4.4.4.1.1.1.2.cmml">1</mn><mo id="S6.F6.40.20.m20.4.4.4.4.1.1.1.1" xref="S6.F6.40.20.m20.4.4.4.4.1.1.1.1.cmml">−</mo><mi id="S6.F6.40.20.m20.4.4.4.4.1.1.1.3" xref="S6.F6.40.20.m20.4.4.4.4.1.1.1.3.cmml">q</mi></mrow><mo id="S6.F6.40.20.m20.4.4.4.4.1.1.3" stretchy="false" xref="S6.F6.40.20.m20.4.4.4.4.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.F6.40.20.m20.10.10.10.11e" xref="S6.F6.40.20.m20.10.10.10.11.cmml">+</mo><mrow id="S6.F6.40.20.m20.10.10.10.14" xref="S6.F6.40.20.m20.10.10.10.14.cmml"><mrow id="S6.F6.40.20.m20.10.10.10.14.2" xref="S6.F6.40.20.m20.10.10.10.14.2.cmml"><mn id="S6.F6.40.20.m20.10.10.10.14.2.2" xref="S6.F6.40.20.m20.10.10.10.14.2.2.cmml">25.065</mn><mo id="S6.F6.40.20.m20.10.10.10.14.2.1" lspace="0.222em" rspace="0.222em" xref="S6.F6.40.20.m20.10.10.10.14.2.1.cmml">⋅</mo><mfrac id="S6.F6.40.20.m20.10.10.10.14.2.3" xref="S6.F6.40.20.m20.10.10.10.14.2.3.cmml"><mn id="S6.F6.40.20.m20.10.10.10.14.2.3.2" xref="S6.F6.40.20.m20.10.10.10.14.2.3.2.cmml">1</mn><mn id="S6.F6.40.20.m20.10.10.10.14.2.3.3" xref="S6.F6.40.20.m20.10.10.10.14.2.3.3.cmml">2</mn></mfrac></mrow><mo id="S6.F6.40.20.m20.10.10.10.14.1" xref="S6.F6.40.20.m20.10.10.10.14.1.cmml">⁢</mo><mi id="S6.F6.40.20.m20.10.10.10.14.3" xref="S6.F6.40.20.m20.10.10.10.14.3.cmml">q</mi></mrow><mo id="S6.F6.40.20.m20.10.10.10.11f" xref="S6.F6.40.20.m20.10.10.10.11.cmml">+</mo><mrow id="S6.F6.40.20.m20.5.5.5.5" xref="S6.F6.40.20.m20.5.5.5.5.cmml"><mn id="S6.F6.40.20.m20.5.5.5.5.3" xref="S6.F6.40.20.m20.5.5.5.5.3.cmml">22.005</mn><mo id="S6.F6.40.20.m20.5.5.5.5.2" xref="S6.F6.40.20.m20.5.5.5.5.2.cmml">⁢</mo><mi id="S6.F6.40.20.m20.5.5.5.5.4" xref="S6.F6.40.20.m20.5.5.5.5.4.cmml">p</mi><mo id="S6.F6.40.20.m20.5.5.5.5.2b" xref="S6.F6.40.20.m20.5.5.5.5.2.cmml">⁢</mo><mrow id="S6.F6.40.20.m20.5.5.5.5.1.1" xref="S6.F6.40.20.m20.5.5.5.5.1.1.1.cmml"><mo id="S6.F6.40.20.m20.5.5.5.5.1.1.2" stretchy="false" xref="S6.F6.40.20.m20.5.5.5.5.1.1.1.cmml">(</mo><mrow id="S6.F6.40.20.m20.5.5.5.5.1.1.1" xref="S6.F6.40.20.m20.5.5.5.5.1.1.1.cmml"><mn id="S6.F6.40.20.m20.5.5.5.5.1.1.1.2" xref="S6.F6.40.20.m20.5.5.5.5.1.1.1.2.cmml">1</mn><mo id="S6.F6.40.20.m20.5.5.5.5.1.1.1.1" xref="S6.F6.40.20.m20.5.5.5.5.1.1.1.1.cmml">−</mo><mi id="S6.F6.40.20.m20.5.5.5.5.1.1.1.3" xref="S6.F6.40.20.m20.5.5.5.5.1.1.1.3.cmml">q</mi></mrow><mo id="S6.F6.40.20.m20.5.5.5.5.1.1.3" stretchy="false" xref="S6.F6.40.20.m20.5.5.5.5.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.F6.40.20.m20.10.10.10.11g" xref="S6.F6.40.20.m20.10.10.10.11.cmml">+</mo><mrow id="S6.F6.40.20.m20.6.6.6.6" xref="S6.F6.40.20.m20.6.6.6.6.cmml"><mn id="S6.F6.40.20.m20.6.6.6.6.3" xref="S6.F6.40.20.m20.6.6.6.6.3.cmml">13.995</mn><mo id="S6.F6.40.20.m20.6.6.6.6.2" xref="S6.F6.40.20.m20.6.6.6.6.2.cmml">⁢</mo><mi id="S6.F6.40.20.m20.6.6.6.6.4" xref="S6.F6.40.20.m20.6.6.6.6.4.cmml">q</mi><mo id="S6.F6.40.20.m20.6.6.6.6.2b" xref="S6.F6.40.20.m20.6.6.6.6.2.cmml">⁢</mo><mrow id="S6.F6.40.20.m20.6.6.6.6.1.1" xref="S6.F6.40.20.m20.6.6.6.6.1.1.1.cmml"><mo id="S6.F6.40.20.m20.6.6.6.6.1.1.2" stretchy="false" xref="S6.F6.40.20.m20.6.6.6.6.1.1.1.cmml">(</mo><mrow id="S6.F6.40.20.m20.6.6.6.6.1.1.1" xref="S6.F6.40.20.m20.6.6.6.6.1.1.1.cmml"><mn id="S6.F6.40.20.m20.6.6.6.6.1.1.1.2" xref="S6.F6.40.20.m20.6.6.6.6.1.1.1.2.cmml">1</mn><mo id="S6.F6.40.20.m20.6.6.6.6.1.1.1.1" xref="S6.F6.40.20.m20.6.6.6.6.1.1.1.1.cmml">−</mo><mi id="S6.F6.40.20.m20.6.6.6.6.1.1.1.3" xref="S6.F6.40.20.m20.6.6.6.6.1.1.1.3.cmml">p</mi></mrow><mo id="S6.F6.40.20.m20.6.6.6.6.1.1.3" stretchy="false" xref="S6.F6.40.20.m20.6.6.6.6.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.F6.40.20.m20.10.10.10.11h" xref="S6.F6.40.20.m20.10.10.10.11.cmml">+</mo><mrow id="S6.F6.40.20.m20.8.8.8.8" xref="S6.F6.40.20.m20.8.8.8.8.cmml"><mrow id="S6.F6.40.20.m20.7.7.7.7.1.1" xref="S6.F6.40.20.m20.7.7.7.7.1.1.1.cmml"><mo id="S6.F6.40.20.m20.7.7.7.7.1.1.2" stretchy="false" xref="S6.F6.40.20.m20.7.7.7.7.1.1.1.cmml">(</mo><mrow id="S6.F6.40.20.m20.7.7.7.7.1.1.1" xref="S6.F6.40.20.m20.7.7.7.7.1.1.1.cmml"><mn id="S6.F6.40.20.m20.7.7.7.7.1.1.1.2" xref="S6.F6.40.20.m20.7.7.7.7.1.1.1.2.cmml">3.6</mn><mo id="S6.F6.40.20.m20.7.7.7.7.1.1.1.1" xref="S6.F6.40.20.m20.7.7.7.7.1.1.1.1.cmml">+</mo><mn id="S6.F6.40.20.m20.7.7.7.7.1.1.1.3" xref="S6.F6.40.20.m20.7.7.7.7.1.1.1.3.cmml">5.4</mn></mrow><mo id="S6.F6.40.20.m20.7.7.7.7.1.1.3" stretchy="false" xref="S6.F6.40.20.m20.7.7.7.7.1.1.1.cmml">)</mo></mrow><mo id="S6.F6.40.20.m20.8.8.8.8.3" xref="S6.F6.40.20.m20.8.8.8.8.3.cmml">⁢</mo><mi id="S6.F6.40.20.m20.8.8.8.8.4" xref="S6.F6.40.20.m20.8.8.8.8.4.cmml">q</mi><mo id="S6.F6.40.20.m20.8.8.8.8.3b" xref="S6.F6.40.20.m20.8.8.8.8.3.cmml">⁢</mo><mrow id="S6.F6.40.20.m20.8.8.8.8.2.1" xref="S6.F6.40.20.m20.8.8.8.8.2.1.1.cmml"><mo id="S6.F6.40.20.m20.8.8.8.8.2.1.2" stretchy="false" xref="S6.F6.40.20.m20.8.8.8.8.2.1.1.cmml">(</mo><mrow id="S6.F6.40.20.m20.8.8.8.8.2.1.1" xref="S6.F6.40.20.m20.8.8.8.8.2.1.1.cmml"><mn id="S6.F6.40.20.m20.8.8.8.8.2.1.1.2" xref="S6.F6.40.20.m20.8.8.8.8.2.1.1.2.cmml">1</mn><mo id="S6.F6.40.20.m20.8.8.8.8.2.1.1.1" xref="S6.F6.40.20.m20.8.8.8.8.2.1.1.1.cmml">−</mo><mi id="S6.F6.40.20.m20.8.8.8.8.2.1.1.3" xref="S6.F6.40.20.m20.8.8.8.8.2.1.1.3.cmml">q</mi></mrow><mo id="S6.F6.40.20.m20.8.8.8.8.2.1.3" stretchy="false" xref="S6.F6.40.20.m20.8.8.8.8.2.1.1.cmml">)</mo></mrow></mrow><mo id="S6.F6.40.20.m20.10.10.10.11i" xref="S6.F6.40.20.m20.10.10.10.11.cmml">+</mo><mrow id="S6.F6.40.20.m20.10.10.10.10" xref="S6.F6.40.20.m20.10.10.10.10.cmml"><mrow id="S6.F6.40.20.m20.9.9.9.9.1" xref="S6.F6.40.20.m20.9.9.9.9.1.cmml"><mrow id="S6.F6.40.20.m20.9.9.9.9.1.1.1" xref="S6.F6.40.20.m20.9.9.9.9.1.1.1.1.cmml"><mo id="S6.F6.40.20.m20.9.9.9.9.1.1.1.2" stretchy="false" xref="S6.F6.40.20.m20.9.9.9.9.1.1.1.1.cmml">(</mo><mrow id="S6.F6.40.20.m20.9.9.9.9.1.1.1.1" xref="S6.F6.40.20.m20.9.9.9.9.1.1.1.1.cmml"><mn id="S6.F6.40.20.m20.9.9.9.9.1.1.1.1.2" xref="S6.F6.40.20.m20.9.9.9.9.1.1.1.1.2.cmml">12.1275</mn><mo id="S6.F6.40.20.m20.9.9.9.9.1.1.1.1.1" xref="S6.F6.40.20.m20.9.9.9.9.1.1.1.1.1.cmml">+</mo><mn id="S6.F6.40.20.m20.9.9.9.9.1.1.1.1.3" xref="S6.F6.40.20.m20.9.9.9.9.1.1.1.1.3.cmml">9.9225</mn></mrow><mo id="S6.F6.40.20.m20.9.9.9.9.1.1.1.3" rspace="0.055em" stretchy="false" xref="S6.F6.40.20.m20.9.9.9.9.1.1.1.1.cmml">)</mo></mrow><mo id="S6.F6.40.20.m20.9.9.9.9.1.2" rspace="0.222em" xref="S6.F6.40.20.m20.9.9.9.9.1.2.cmml">⋅</mo><mi id="S6.F6.40.20.m20.9.9.9.9.1.3" xref="S6.F6.40.20.m20.9.9.9.9.1.3.cmml">p</mi></mrow><mo id="S6.F6.40.20.m20.10.10.10.10.3" xref="S6.F6.40.20.m20.10.10.10.10.3.cmml">⁢</mo><mrow id="S6.F6.40.20.m20.10.10.10.10.2.1" xref="S6.F6.40.20.m20.10.10.10.10.2.1.1.cmml"><mo id="S6.F6.40.20.m20.10.10.10.10.2.1.2" stretchy="false" xref="S6.F6.40.20.m20.10.10.10.10.2.1.1.cmml">(</mo><mrow id="S6.F6.40.20.m20.10.10.10.10.2.1.1" xref="S6.F6.40.20.m20.10.10.10.10.2.1.1.cmml"><mn id="S6.F6.40.20.m20.10.10.10.10.2.1.1.2" xref="S6.F6.40.20.m20.10.10.10.10.2.1.1.2.cmml">1</mn><mo id="S6.F6.40.20.m20.10.10.10.10.2.1.1.1" xref="S6.F6.40.20.m20.10.10.10.10.2.1.1.1.cmml">−</mo><mi id="S6.F6.40.20.m20.10.10.10.10.2.1.1.3" xref="S6.F6.40.20.m20.10.10.10.10.2.1.1.3.cmml">p</mi></mrow><mo id="S6.F6.40.20.m20.10.10.10.10.2.1.3" stretchy="false" xref="S6.F6.40.20.m20.10.10.10.10.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.F6.40.20.m20.11.11.12" xref="S6.F6.40.20.m20.11.11.12.cmml">=</mo><mrow id="S6.F6.40.20.m20.11.11.11" xref="S6.F6.40.20.m20.11.11.11.cmml"><mrow id="S6.F6.40.20.m20.11.11.11.1" xref="S6.F6.40.20.m20.11.11.11.1.cmml"><mrow id="S6.F6.40.20.m20.11.11.11.1.3" xref="S6.F6.40.20.m20.11.11.11.1.3.cmml"><mn id="S6.F6.40.20.m20.11.11.11.1.3.2" xref="S6.F6.40.20.m20.11.11.11.1.3.2.cmml">53.6175</mn><mo id="S6.F6.40.20.m20.11.11.11.1.3.1" xref="S6.F6.40.20.m20.11.11.11.1.3.1.cmml">−</mo><mrow id="S6.F6.40.20.m20.11.11.11.1.3.3" xref="S6.F6.40.20.m20.11.11.11.1.3.3.cmml"><mn id="S6.F6.40.20.m20.11.11.11.1.3.3.2" xref="S6.F6.40.20.m20.11.11.11.1.3.3.2.cmml">36</mn><mo id="S6.F6.40.20.m20.11.11.11.1.3.3.1" xref="S6.F6.40.20.m20.11.11.11.1.3.3.1.cmml">⁢</mo><msup id="S6.F6.40.20.m20.11.11.11.1.3.3.3" xref="S6.F6.40.20.m20.11.11.11.1.3.3.3.cmml"><mi id="S6.F6.40.20.m20.11.11.11.1.3.3.3.2" xref="S6.F6.40.20.m20.11.11.11.1.3.3.3.2.cmml">p</mi><mn id="S6.F6.40.20.m20.11.11.11.1.3.3.3.3" xref="S6.F6.40.20.m20.11.11.11.1.3.3.3.3.cmml">2</mn></msup></mrow></mrow><mo id="S6.F6.40.20.m20.11.11.11.1.2" xref="S6.F6.40.20.m20.11.11.11.1.2.cmml">+</mo><mrow id="S6.F6.40.20.m20.11.11.11.1.1" xref="S6.F6.40.20.m20.11.11.11.1.1.cmml"><mi id="S6.F6.40.20.m20.11.11.11.1.1.3" xref="S6.F6.40.20.m20.11.11.11.1.1.3.cmml">p</mi><mo id="S6.F6.40.20.m20.11.11.11.1.1.2" xref="S6.F6.40.20.m20.11.11.11.1.1.2.cmml">⁢</mo><mrow id="S6.F6.40.20.m20.11.11.11.1.1.1.1" xref="S6.F6.40.20.m20.11.11.11.1.1.1.1.1.cmml"><mo id="S6.F6.40.20.m20.11.11.11.1.1.1.1.2" stretchy="false" 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id="S6.F6.40.20.m20.11d">(6.2775+7.6725)p(1-p)+94.185\cdot\frac{1}{2}(1-p)+118.215\cdot\frac{1}{2}p+24.% 03\cdot\frac{1}{4}+1.035\cdot\frac{1}{2}(1-q)+25.065\cdot\frac{1}{2}q+22.005p(% 1-q)+13.995q(1-p)+(3.6+5.4)q(1-q)+(12.1275+9.9225)\cdot p(1-p)=53.6175-36p^{2}% +p(70.02-36q)+35.01q-9q^{2}</annotation><annotation encoding="application/x-llamapun" id="S6.F6.40.20.m20.11e">( 6.2775 + 7.6725 ) italic_p ( 1 - italic_p ) + 94.185 ⋅ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( 1 - italic_p ) + 118.215 ⋅ divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_p + 24.03 ⋅ divide start_ARG 1 end_ARG start_ARG 4 end_ARG + 1.035 ⋅ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( 1 - italic_q ) + 25.065 ⋅ divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_q + 22.005 italic_p ( 1 - italic_q ) + 13.995 italic_q ( 1 - italic_p ) + ( 3.6 + 5.4 ) italic_q ( 1 - italic_q ) + ( 12.1275 + 9.9225 ) ⋅ italic_p ( 1 - italic_p ) = 53.6175 - 36 italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_p ( 70.02 - 36 italic_q ) + 35.01 italic_q - 9 italic_q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. </span></figcaption> </figure> <div class="ltx_proof" id="S6.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.1.p1"> <p class="ltx_p" id="S6.1.p1.12">Let <math alttext="G" class="ltx_Math" display="inline" id="S6.1.p1.1.m1.1"><semantics id="S6.1.p1.1.m1.1a"><mi id="S6.1.p1.1.m1.1.1" xref="S6.1.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S6.1.p1.1.m1.1b"><ci id="S6.1.p1.1.m1.1.1.cmml" xref="S6.1.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.1.m1.1d">italic_G</annotation></semantics></math> denote the graph in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S6.F6" title="In 6 Lower bounds for antisymmetric selection functions (Theorem 1.8) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Fig.</span> <span class="ltx_text ltx_ref_tag">6</span></a>.<span class="ltx_note ltx_role_footnote" id="footnote8"><sup class="ltx_note_mark">8</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">8</sup><span class="ltx_tag ltx_tag_note">8</span>We found the graph in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S6.F6" title="In 6 Lower bounds for antisymmetric selection functions (Theorem 1.8) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Fig.</span> <span class="ltx_text ltx_ref_tag">6</span></a> by employing the linear programming methodology described in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S4.SS1" title="4.1 Methodology for finding hard instances ‣ 4 Lower bounds ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.1</span></a>. We considered <math alttext="L=5" class="ltx_Math" display="inline" id="footnote8.m1.1"><semantics id="footnote8.m1.1b"><mrow id="footnote8.m1.1.1" xref="footnote8.m1.1.1.cmml"><mi id="footnote8.m1.1.1.2" xref="footnote8.m1.1.1.2.cmml">L</mi><mo id="footnote8.m1.1.1.1" xref="footnote8.m1.1.1.1.cmml">=</mo><mn id="footnote8.m1.1.1.3" xref="footnote8.m1.1.1.3.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="footnote8.m1.1c"><apply id="footnote8.m1.1.1.cmml" xref="footnote8.m1.1.1"><eq id="footnote8.m1.1.1.1.cmml" xref="footnote8.m1.1.1.1"></eq><ci id="footnote8.m1.1.1.2.cmml" xref="footnote8.m1.1.1.2">𝐿</ci><cn id="footnote8.m1.1.1.3.cmml" type="integer" xref="footnote8.m1.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote8.m1.1d">L=5</annotation><annotation encoding="application/x-llamapun" id="footnote8.m1.1e">italic_L = 5</annotation></semantics></math> possible bias classes, <math alttext="\{\pm b_{1},\pm b_{2},0\}" class="ltx_Math" display="inline" id="footnote8.m2.3"><semantics id="footnote8.m2.3b"><mrow id="footnote8.m2.3.3.2" xref="footnote8.m2.3.3.3.cmml"><mo id="footnote8.m2.3.3.2.3" stretchy="false" xref="footnote8.m2.3.3.3.cmml">{</mo><mrow id="footnote8.m2.2.2.1.1" xref="footnote8.m2.2.2.1.1.cmml"><mo id="footnote8.m2.2.2.1.1b" xref="footnote8.m2.2.2.1.1.cmml">±</mo><msub id="footnote8.m2.2.2.1.1.2" xref="footnote8.m2.2.2.1.1.2.cmml"><mi id="footnote8.m2.2.2.1.1.2.2" xref="footnote8.m2.2.2.1.1.2.2.cmml">b</mi><mn id="footnote8.m2.2.2.1.1.2.3" xref="footnote8.m2.2.2.1.1.2.3.cmml">1</mn></msub></mrow><mo id="footnote8.m2.3.3.2.4" xref="footnote8.m2.3.3.3.cmml">,</mo><mrow id="footnote8.m2.3.3.2.2" xref="footnote8.m2.3.3.2.2.cmml"><mo id="footnote8.m2.3.3.2.2b" xref="footnote8.m2.3.3.2.2.cmml">±</mo><msub id="footnote8.m2.3.3.2.2.2" xref="footnote8.m2.3.3.2.2.2.cmml"><mi id="footnote8.m2.3.3.2.2.2.2" xref="footnote8.m2.3.3.2.2.2.2.cmml">b</mi><mn id="footnote8.m2.3.3.2.2.2.3" xref="footnote8.m2.3.3.2.2.2.3.cmml">2</mn></msub></mrow><mo id="footnote8.m2.3.3.2.5" xref="footnote8.m2.3.3.3.cmml">,</mo><mn id="footnote8.m2.1.1" xref="footnote8.m2.1.1.cmml">0</mn><mo id="footnote8.m2.3.3.2.6" stretchy="false" xref="footnote8.m2.3.3.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="footnote8.m2.3c"><set id="footnote8.m2.3.3.3.cmml" xref="footnote8.m2.3.3.2"><apply id="footnote8.m2.2.2.1.1.cmml" xref="footnote8.m2.2.2.1.1"><csymbol cd="latexml" id="footnote8.m2.2.2.1.1.1.cmml" xref="footnote8.m2.2.2.1.1">plus-or-minus</csymbol><apply id="footnote8.m2.2.2.1.1.2.cmml" xref="footnote8.m2.2.2.1.1.2"><csymbol cd="ambiguous" id="footnote8.m2.2.2.1.1.2.1.cmml" xref="footnote8.m2.2.2.1.1.2">subscript</csymbol><ci id="footnote8.m2.2.2.1.1.2.2.cmml" xref="footnote8.m2.2.2.1.1.2.2">𝑏</ci><cn id="footnote8.m2.2.2.1.1.2.3.cmml" type="integer" xref="footnote8.m2.2.2.1.1.2.3">1</cn></apply></apply><apply id="footnote8.m2.3.3.2.2.cmml" xref="footnote8.m2.3.3.2.2"><csymbol cd="latexml" id="footnote8.m2.3.3.2.2.1.cmml" xref="footnote8.m2.3.3.2.2">plus-or-minus</csymbol><apply id="footnote8.m2.3.3.2.2.2.cmml" xref="footnote8.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="footnote8.m2.3.3.2.2.2.1.cmml" xref="footnote8.m2.3.3.2.2.2">subscript</csymbol><ci id="footnote8.m2.3.3.2.2.2.2.cmml" xref="footnote8.m2.3.3.2.2.2.2">𝑏</ci><cn id="footnote8.m2.3.3.2.2.2.3.cmml" type="integer" xref="footnote8.m2.3.3.2.2.2.3">2</cn></apply></apply><cn id="footnote8.m2.1.1.cmml" type="integer" xref="footnote8.m2.1.1">0</cn></set></annotation-xml><annotation encoding="application/x-tex" id="footnote8.m2.3d">\{\pm b_{1},\pm b_{2},0\}</annotation><annotation encoding="application/x-llamapun" id="footnote8.m2.3e">{ ± italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ± italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , 0 }</annotation></semantics></math>, where <math alttext="b_{1}" class="ltx_Math" display="inline" id="footnote8.m3.1"><semantics id="footnote8.m3.1b"><msub id="footnote8.m3.1.1" xref="footnote8.m3.1.1.cmml"><mi id="footnote8.m3.1.1.2" xref="footnote8.m3.1.1.2.cmml">b</mi><mn id="footnote8.m3.1.1.3" xref="footnote8.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="footnote8.m3.1c"><apply id="footnote8.m3.1.1.cmml" xref="footnote8.m3.1.1"><csymbol cd="ambiguous" id="footnote8.m3.1.1.1.cmml" xref="footnote8.m3.1.1">subscript</csymbol><ci id="footnote8.m3.1.1.2.cmml" xref="footnote8.m3.1.1.2">𝑏</ci><cn id="footnote8.m3.1.1.3.cmml" type="integer" xref="footnote8.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote8.m3.1d">b_{1}</annotation><annotation encoding="application/x-llamapun" id="footnote8.m3.1e">italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="b_{2}" class="ltx_Math" display="inline" id="footnote8.m4.1"><semantics id="footnote8.m4.1b"><msub id="footnote8.m4.1.1" xref="footnote8.m4.1.1.cmml"><mi id="footnote8.m4.1.1.2" xref="footnote8.m4.1.1.2.cmml">b</mi><mn id="footnote8.m4.1.1.3" xref="footnote8.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="footnote8.m4.1c"><apply id="footnote8.m4.1.1.cmml" xref="footnote8.m4.1.1"><csymbol cd="ambiguous" id="footnote8.m4.1.1.1.cmml" xref="footnote8.m4.1.1">subscript</csymbol><ci id="footnote8.m4.1.1.2.cmml" xref="footnote8.m4.1.1.2">𝑏</ci><cn id="footnote8.m4.1.1.3.cmml" type="integer" xref="footnote8.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote8.m4.1d">b_{2}</annotation><annotation encoding="application/x-llamapun" id="footnote8.m4.1e">italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> were multiples of <math alttext="\frac{1}{10}" class="ltx_Math" display="inline" id="footnote8.m5.1"><semantics id="footnote8.m5.1b"><mfrac id="footnote8.m5.1.1" xref="footnote8.m5.1.1.cmml"><mn id="footnote8.m5.1.1.2" xref="footnote8.m5.1.1.2.cmml">1</mn><mn id="footnote8.m5.1.1.3" xref="footnote8.m5.1.1.3.cmml">10</mn></mfrac><annotation-xml encoding="MathML-Content" id="footnote8.m5.1c"><apply id="footnote8.m5.1.1.cmml" xref="footnote8.m5.1.1"><divide id="footnote8.m5.1.1.1.cmml" xref="footnote8.m5.1.1"></divide><cn id="footnote8.m5.1.1.2.cmml" type="integer" xref="footnote8.m5.1.1.2">1</cn><cn id="footnote8.m5.1.1.3.cmml" type="integer" xref="footnote8.m5.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote8.m5.1d">\frac{1}{10}</annotation><annotation encoding="application/x-llamapun" id="footnote8.m5.1e">divide start_ARG 1 end_ARG start_ARG 10 end_ARG</annotation></semantics></math>, and performed a grid search over these possibilities. To discretize the space of all antisymmetric functions, we considered all functions mapping <math alttext="b_{1}" class="ltx_Math" display="inline" id="footnote8.m6.1"><semantics id="footnote8.m6.1b"><msub id="footnote8.m6.1.1" xref="footnote8.m6.1.1.cmml"><mi id="footnote8.m6.1.1.2" xref="footnote8.m6.1.1.2.cmml">b</mi><mn id="footnote8.m6.1.1.3" xref="footnote8.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="footnote8.m6.1c"><apply id="footnote8.m6.1.1.cmml" xref="footnote8.m6.1.1"><csymbol cd="ambiguous" id="footnote8.m6.1.1.1.cmml" xref="footnote8.m6.1.1">subscript</csymbol><ci id="footnote8.m6.1.1.2.cmml" xref="footnote8.m6.1.1.2">𝑏</ci><cn id="footnote8.m6.1.1.3.cmml" type="integer" xref="footnote8.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote8.m6.1d">b_{1}</annotation><annotation encoding="application/x-llamapun" id="footnote8.m6.1e">italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="b_{2}" class="ltx_Math" display="inline" id="footnote8.m7.1"><semantics id="footnote8.m7.1b"><msub id="footnote8.m7.1.1" xref="footnote8.m7.1.1.cmml"><mi id="footnote8.m7.1.1.2" xref="footnote8.m7.1.1.2.cmml">b</mi><mn id="footnote8.m7.1.1.3" xref="footnote8.m7.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="footnote8.m7.1c"><apply id="footnote8.m7.1.1.cmml" xref="footnote8.m7.1.1"><csymbol cd="ambiguous" id="footnote8.m7.1.1.1.cmml" xref="footnote8.m7.1.1">subscript</csymbol><ci id="footnote8.m7.1.1.2.cmml" xref="footnote8.m7.1.1.2">𝑏</ci><cn id="footnote8.m7.1.1.3.cmml" type="integer" xref="footnote8.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote8.m7.1d">b_{2}</annotation><annotation encoding="application/x-llamapun" id="footnote8.m7.1e">italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> to multiples of <math alttext="\frac{1}{100}" class="ltx_Math" display="inline" id="footnote8.m8.1"><semantics id="footnote8.m8.1b"><mfrac id="footnote8.m8.1.1" xref="footnote8.m8.1.1.cmml"><mn id="footnote8.m8.1.1.2" xref="footnote8.m8.1.1.2.cmml">1</mn><mn id="footnote8.m8.1.1.3" xref="footnote8.m8.1.1.3.cmml">100</mn></mfrac><annotation-xml encoding="MathML-Content" id="footnote8.m8.1c"><apply id="footnote8.m8.1.1.cmml" xref="footnote8.m8.1.1"><divide id="footnote8.m8.1.1.1.cmml" xref="footnote8.m8.1.1"></divide><cn id="footnote8.m8.1.1.2.cmml" type="integer" xref="footnote8.m8.1.1.2">1</cn><cn id="footnote8.m8.1.1.3.cmml" type="integer" xref="footnote8.m8.1.1.3">100</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote8.m8.1d">\frac{1}{100}</annotation><annotation encoding="application/x-llamapun" id="footnote8.m8.1e">divide start_ARG 1 end_ARG start_ARG 100 end_ARG</annotation></semantics></math> between <math alttext="\frac{1}{2}" class="ltx_Math" display="inline" id="footnote8.m9.1"><semantics id="footnote8.m9.1b"><mfrac id="footnote8.m9.1.1" xref="footnote8.m9.1.1.cmml"><mn id="footnote8.m9.1.1.2" xref="footnote8.m9.1.1.2.cmml">1</mn><mn id="footnote8.m9.1.1.3" xref="footnote8.m9.1.1.3.cmml">2</mn></mfrac><annotation-xml encoding="MathML-Content" id="footnote8.m9.1c"><apply id="footnote8.m9.1.1.cmml" xref="footnote8.m9.1.1"><divide id="footnote8.m9.1.1.1.cmml" xref="footnote8.m9.1.1"></divide><cn id="footnote8.m9.1.1.2.cmml" type="integer" xref="footnote8.m9.1.1.2">1</cn><cn id="footnote8.m9.1.1.3.cmml" type="integer" xref="footnote8.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote8.m9.1d">\frac{1}{2}</annotation><annotation encoding="application/x-llamapun" id="footnote8.m9.1e">divide start_ARG 1 end_ARG start_ARG 2 end_ARG</annotation></semantics></math> and <math alttext="1" class="ltx_Math" display="inline" id="footnote8.m10.1"><semantics id="footnote8.m10.1b"><mn id="footnote8.m10.1.1" xref="footnote8.m10.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="footnote8.m10.1c"><cn id="footnote8.m10.1.1.cmml" type="integer" xref="footnote8.m10.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="footnote8.m10.1d">1</annotation><annotation encoding="application/x-llamapun" id="footnote8.m10.1e">1</annotation></semantics></math>.</span></span></span> As described in the figure’s caption, this graph has a cut of weight <math alttext="179.28" class="ltx_Math" display="inline" id="S6.1.p1.2.m2.1"><semantics id="S6.1.p1.2.m2.1a"><mn id="S6.1.p1.2.m2.1.1" xref="S6.1.p1.2.m2.1.1.cmml">179.28</mn><annotation-xml encoding="MathML-Content" id="S6.1.p1.2.m2.1b"><cn id="S6.1.p1.2.m2.1.1.cmml" type="float" xref="S6.1.p1.2.m2.1.1">179.28</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.2.m2.1c">179.28</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.2.m2.1d">179.28</annotation></semantics></math>, while any oblivious cut has value <math alttext="v(p,q)=53.6175-36p^{2}+p(70.02-36q)+35.01q-9q^{2}" class="ltx_Math" display="inline" id="S6.1.p1.3.m3.3"><semantics id="S6.1.p1.3.m3.3a"><mrow id="S6.1.p1.3.m3.3.3" xref="S6.1.p1.3.m3.3.3.cmml"><mrow id="S6.1.p1.3.m3.3.3.3" xref="S6.1.p1.3.m3.3.3.3.cmml"><mi id="S6.1.p1.3.m3.3.3.3.2" xref="S6.1.p1.3.m3.3.3.3.2.cmml">v</mi><mo id="S6.1.p1.3.m3.3.3.3.1" xref="S6.1.p1.3.m3.3.3.3.1.cmml">⁢</mo><mrow id="S6.1.p1.3.m3.3.3.3.3.2" xref="S6.1.p1.3.m3.3.3.3.3.1.cmml"><mo id="S6.1.p1.3.m3.3.3.3.3.2.1" stretchy="false" xref="S6.1.p1.3.m3.3.3.3.3.1.cmml">(</mo><mi id="S6.1.p1.3.m3.1.1" xref="S6.1.p1.3.m3.1.1.cmml">p</mi><mo id="S6.1.p1.3.m3.3.3.3.3.2.2" xref="S6.1.p1.3.m3.3.3.3.3.1.cmml">,</mo><mi id="S6.1.p1.3.m3.2.2" xref="S6.1.p1.3.m3.2.2.cmml">q</mi><mo id="S6.1.p1.3.m3.3.3.3.3.2.3" stretchy="false" xref="S6.1.p1.3.m3.3.3.3.3.1.cmml">)</mo></mrow></mrow><mo id="S6.1.p1.3.m3.3.3.2" xref="S6.1.p1.3.m3.3.3.2.cmml">=</mo><mrow id="S6.1.p1.3.m3.3.3.1" xref="S6.1.p1.3.m3.3.3.1.cmml"><mrow id="S6.1.p1.3.m3.3.3.1.1" xref="S6.1.p1.3.m3.3.3.1.1.cmml"><mrow id="S6.1.p1.3.m3.3.3.1.1.3" xref="S6.1.p1.3.m3.3.3.1.1.3.cmml"><mn id="S6.1.p1.3.m3.3.3.1.1.3.2" xref="S6.1.p1.3.m3.3.3.1.1.3.2.cmml">53.6175</mn><mo id="S6.1.p1.3.m3.3.3.1.1.3.1" xref="S6.1.p1.3.m3.3.3.1.1.3.1.cmml">−</mo><mrow id="S6.1.p1.3.m3.3.3.1.1.3.3" xref="S6.1.p1.3.m3.3.3.1.1.3.3.cmml"><mn id="S6.1.p1.3.m3.3.3.1.1.3.3.2" xref="S6.1.p1.3.m3.3.3.1.1.3.3.2.cmml">36</mn><mo id="S6.1.p1.3.m3.3.3.1.1.3.3.1" xref="S6.1.p1.3.m3.3.3.1.1.3.3.1.cmml">⁢</mo><msup id="S6.1.p1.3.m3.3.3.1.1.3.3.3" xref="S6.1.p1.3.m3.3.3.1.1.3.3.3.cmml"><mi id="S6.1.p1.3.m3.3.3.1.1.3.3.3.2" xref="S6.1.p1.3.m3.3.3.1.1.3.3.3.2.cmml">p</mi><mn id="S6.1.p1.3.m3.3.3.1.1.3.3.3.3" xref="S6.1.p1.3.m3.3.3.1.1.3.3.3.3.cmml">2</mn></msup></mrow></mrow><mo id="S6.1.p1.3.m3.3.3.1.1.2" xref="S6.1.p1.3.m3.3.3.1.1.2.cmml">+</mo><mrow id="S6.1.p1.3.m3.3.3.1.1.1" xref="S6.1.p1.3.m3.3.3.1.1.1.cmml"><mi id="S6.1.p1.3.m3.3.3.1.1.1.3" xref="S6.1.p1.3.m3.3.3.1.1.1.3.cmml">p</mi><mo id="S6.1.p1.3.m3.3.3.1.1.1.2" xref="S6.1.p1.3.m3.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S6.1.p1.3.m3.3.3.1.1.1.1.1" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.cmml"><mo id="S6.1.p1.3.m3.3.3.1.1.1.1.1.2" stretchy="false" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.1.p1.3.m3.3.3.1.1.1.1.1.1" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.cmml"><mn id="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.2" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.2.cmml">70.02</mn><mo id="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.1" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3.cmml"><mn id="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3.2" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3.2.cmml">36</mn><mo id="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3.1" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3.3" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3.3.cmml">q</mi></mrow></mrow><mo id="S6.1.p1.3.m3.3.3.1.1.1.1.1.3" stretchy="false" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.1.p1.3.m3.3.3.1.1.2a" xref="S6.1.p1.3.m3.3.3.1.1.2.cmml">+</mo><mrow id="S6.1.p1.3.m3.3.3.1.1.4" xref="S6.1.p1.3.m3.3.3.1.1.4.cmml"><mn id="S6.1.p1.3.m3.3.3.1.1.4.2" xref="S6.1.p1.3.m3.3.3.1.1.4.2.cmml">35.01</mn><mo id="S6.1.p1.3.m3.3.3.1.1.4.1" xref="S6.1.p1.3.m3.3.3.1.1.4.1.cmml">⁢</mo><mi id="S6.1.p1.3.m3.3.3.1.1.4.3" xref="S6.1.p1.3.m3.3.3.1.1.4.3.cmml">q</mi></mrow></mrow><mo id="S6.1.p1.3.m3.3.3.1.2" xref="S6.1.p1.3.m3.3.3.1.2.cmml">−</mo><mrow id="S6.1.p1.3.m3.3.3.1.3" xref="S6.1.p1.3.m3.3.3.1.3.cmml"><mn id="S6.1.p1.3.m3.3.3.1.3.2" xref="S6.1.p1.3.m3.3.3.1.3.2.cmml">9</mn><mo id="S6.1.p1.3.m3.3.3.1.3.1" xref="S6.1.p1.3.m3.3.3.1.3.1.cmml">⁢</mo><msup id="S6.1.p1.3.m3.3.3.1.3.3" xref="S6.1.p1.3.m3.3.3.1.3.3.cmml"><mi id="S6.1.p1.3.m3.3.3.1.3.3.2" xref="S6.1.p1.3.m3.3.3.1.3.3.2.cmml">q</mi><mn id="S6.1.p1.3.m3.3.3.1.3.3.3" xref="S6.1.p1.3.m3.3.3.1.3.3.3.cmml">2</mn></msup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.3.m3.3b"><apply id="S6.1.p1.3.m3.3.3.cmml" xref="S6.1.p1.3.m3.3.3"><eq id="S6.1.p1.3.m3.3.3.2.cmml" xref="S6.1.p1.3.m3.3.3.2"></eq><apply id="S6.1.p1.3.m3.3.3.3.cmml" xref="S6.1.p1.3.m3.3.3.3"><times id="S6.1.p1.3.m3.3.3.3.1.cmml" xref="S6.1.p1.3.m3.3.3.3.1"></times><ci id="S6.1.p1.3.m3.3.3.3.2.cmml" xref="S6.1.p1.3.m3.3.3.3.2">𝑣</ci><interval closure="open" id="S6.1.p1.3.m3.3.3.3.3.1.cmml" xref="S6.1.p1.3.m3.3.3.3.3.2"><ci id="S6.1.p1.3.m3.1.1.cmml" xref="S6.1.p1.3.m3.1.1">𝑝</ci><ci id="S6.1.p1.3.m3.2.2.cmml" xref="S6.1.p1.3.m3.2.2">𝑞</ci></interval></apply><apply id="S6.1.p1.3.m3.3.3.1.cmml" xref="S6.1.p1.3.m3.3.3.1"><minus id="S6.1.p1.3.m3.3.3.1.2.cmml" xref="S6.1.p1.3.m3.3.3.1.2"></minus><apply id="S6.1.p1.3.m3.3.3.1.1.cmml" xref="S6.1.p1.3.m3.3.3.1.1"><plus id="S6.1.p1.3.m3.3.3.1.1.2.cmml" xref="S6.1.p1.3.m3.3.3.1.1.2"></plus><apply id="S6.1.p1.3.m3.3.3.1.1.3.cmml" xref="S6.1.p1.3.m3.3.3.1.1.3"><minus id="S6.1.p1.3.m3.3.3.1.1.3.1.cmml" xref="S6.1.p1.3.m3.3.3.1.1.3.1"></minus><cn id="S6.1.p1.3.m3.3.3.1.1.3.2.cmml" type="float" xref="S6.1.p1.3.m3.3.3.1.1.3.2">53.6175</cn><apply id="S6.1.p1.3.m3.3.3.1.1.3.3.cmml" xref="S6.1.p1.3.m3.3.3.1.1.3.3"><times id="S6.1.p1.3.m3.3.3.1.1.3.3.1.cmml" xref="S6.1.p1.3.m3.3.3.1.1.3.3.1"></times><cn id="S6.1.p1.3.m3.3.3.1.1.3.3.2.cmml" type="integer" xref="S6.1.p1.3.m3.3.3.1.1.3.3.2">36</cn><apply id="S6.1.p1.3.m3.3.3.1.1.3.3.3.cmml" xref="S6.1.p1.3.m3.3.3.1.1.3.3.3"><csymbol cd="ambiguous" id="S6.1.p1.3.m3.3.3.1.1.3.3.3.1.cmml" xref="S6.1.p1.3.m3.3.3.1.1.3.3.3">superscript</csymbol><ci id="S6.1.p1.3.m3.3.3.1.1.3.3.3.2.cmml" xref="S6.1.p1.3.m3.3.3.1.1.3.3.3.2">𝑝</ci><cn id="S6.1.p1.3.m3.3.3.1.1.3.3.3.3.cmml" type="integer" xref="S6.1.p1.3.m3.3.3.1.1.3.3.3.3">2</cn></apply></apply></apply><apply id="S6.1.p1.3.m3.3.3.1.1.1.cmml" xref="S6.1.p1.3.m3.3.3.1.1.1"><times id="S6.1.p1.3.m3.3.3.1.1.1.2.cmml" xref="S6.1.p1.3.m3.3.3.1.1.1.2"></times><ci id="S6.1.p1.3.m3.3.3.1.1.1.3.cmml" xref="S6.1.p1.3.m3.3.3.1.1.1.3">𝑝</ci><apply id="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.cmml" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1"><minus id="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.1.cmml" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.1"></minus><cn id="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.2.cmml" type="float" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.2">70.02</cn><apply id="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3.cmml" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3"><times id="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3.1.cmml" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3.1"></times><cn id="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3.2">36</cn><ci id="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3.3.cmml" xref="S6.1.p1.3.m3.3.3.1.1.1.1.1.1.3.3">𝑞</ci></apply></apply></apply><apply id="S6.1.p1.3.m3.3.3.1.1.4.cmml" xref="S6.1.p1.3.m3.3.3.1.1.4"><times id="S6.1.p1.3.m3.3.3.1.1.4.1.cmml" xref="S6.1.p1.3.m3.3.3.1.1.4.1"></times><cn id="S6.1.p1.3.m3.3.3.1.1.4.2.cmml" type="float" xref="S6.1.p1.3.m3.3.3.1.1.4.2">35.01</cn><ci id="S6.1.p1.3.m3.3.3.1.1.4.3.cmml" xref="S6.1.p1.3.m3.3.3.1.1.4.3">𝑞</ci></apply></apply><apply id="S6.1.p1.3.m3.3.3.1.3.cmml" xref="S6.1.p1.3.m3.3.3.1.3"><times id="S6.1.p1.3.m3.3.3.1.3.1.cmml" xref="S6.1.p1.3.m3.3.3.1.3.1"></times><cn id="S6.1.p1.3.m3.3.3.1.3.2.cmml" type="integer" xref="S6.1.p1.3.m3.3.3.1.3.2">9</cn><apply id="S6.1.p1.3.m3.3.3.1.3.3.cmml" xref="S6.1.p1.3.m3.3.3.1.3.3"><csymbol cd="ambiguous" id="S6.1.p1.3.m3.3.3.1.3.3.1.cmml" xref="S6.1.p1.3.m3.3.3.1.3.3">superscript</csymbol><ci id="S6.1.p1.3.m3.3.3.1.3.3.2.cmml" xref="S6.1.p1.3.m3.3.3.1.3.3.2">𝑞</ci><cn id="S6.1.p1.3.m3.3.3.1.3.3.3.cmml" type="integer" xref="S6.1.p1.3.m3.3.3.1.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.3.m3.3c">v(p,q)=53.6175-36p^{2}+p(70.02-36q)+35.01q-9q^{2}</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.3.m3.3d">italic_v ( italic_p , italic_q ) = 53.6175 - 36 italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_p ( 70.02 - 36 italic_q ) + 35.01 italic_q - 9 italic_q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> where <math alttext="0\leq p,q\leq 1" class="ltx_Math" display="inline" id="S6.1.p1.4.m4.2"><semantics id="S6.1.p1.4.m4.2a"><mrow id="S6.1.p1.4.m4.2.2.2" xref="S6.1.p1.4.m4.2.2.3.cmml"><mrow id="S6.1.p1.4.m4.1.1.1.1" xref="S6.1.p1.4.m4.1.1.1.1.cmml"><mn id="S6.1.p1.4.m4.1.1.1.1.2" xref="S6.1.p1.4.m4.1.1.1.1.2.cmml">0</mn><mo id="S6.1.p1.4.m4.1.1.1.1.1" xref="S6.1.p1.4.m4.1.1.1.1.1.cmml">≤</mo><mi id="S6.1.p1.4.m4.1.1.1.1.3" xref="S6.1.p1.4.m4.1.1.1.1.3.cmml">p</mi></mrow><mo id="S6.1.p1.4.m4.2.2.2.3" xref="S6.1.p1.4.m4.2.2.3a.cmml">,</mo><mrow id="S6.1.p1.4.m4.2.2.2.2" xref="S6.1.p1.4.m4.2.2.2.2.cmml"><mi id="S6.1.p1.4.m4.2.2.2.2.2" xref="S6.1.p1.4.m4.2.2.2.2.2.cmml">q</mi><mo id="S6.1.p1.4.m4.2.2.2.2.1" xref="S6.1.p1.4.m4.2.2.2.2.1.cmml">≤</mo><mn id="S6.1.p1.4.m4.2.2.2.2.3" xref="S6.1.p1.4.m4.2.2.2.2.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.4.m4.2b"><apply id="S6.1.p1.4.m4.2.2.3.cmml" xref="S6.1.p1.4.m4.2.2.2"><csymbol cd="ambiguous" id="S6.1.p1.4.m4.2.2.3a.cmml" xref="S6.1.p1.4.m4.2.2.2.3">formulae-sequence</csymbol><apply id="S6.1.p1.4.m4.1.1.1.1.cmml" xref="S6.1.p1.4.m4.1.1.1.1"><leq id="S6.1.p1.4.m4.1.1.1.1.1.cmml" xref="S6.1.p1.4.m4.1.1.1.1.1"></leq><cn id="S6.1.p1.4.m4.1.1.1.1.2.cmml" type="integer" xref="S6.1.p1.4.m4.1.1.1.1.2">0</cn><ci id="S6.1.p1.4.m4.1.1.1.1.3.cmml" xref="S6.1.p1.4.m4.1.1.1.1.3">𝑝</ci></apply><apply id="S6.1.p1.4.m4.2.2.2.2.cmml" xref="S6.1.p1.4.m4.2.2.2.2"><leq id="S6.1.p1.4.m4.2.2.2.2.1.cmml" xref="S6.1.p1.4.m4.2.2.2.2.1"></leq><ci id="S6.1.p1.4.m4.2.2.2.2.2.cmml" xref="S6.1.p1.4.m4.2.2.2.2.2">𝑞</ci><cn id="S6.1.p1.4.m4.2.2.2.2.3.cmml" type="integer" xref="S6.1.p1.4.m4.2.2.2.2.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.4.m4.2c">0\leq p,q\leq 1</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.4.m4.2d">0 ≤ italic_p , italic_q ≤ 1</annotation></semantics></math> are assignment probabilities for the two bias classes. Since <math alttext="\frac{\partial v}{\partial p}=70.02-72p-36q" class="ltx_Math" display="inline" id="S6.1.p1.5.m5.1"><semantics id="S6.1.p1.5.m5.1a"><mrow id="S6.1.p1.5.m5.1.1" xref="S6.1.p1.5.m5.1.1.cmml"><mfrac id="S6.1.p1.5.m5.1.1.2" xref="S6.1.p1.5.m5.1.1.2.cmml"><mrow id="S6.1.p1.5.m5.1.1.2.2" xref="S6.1.p1.5.m5.1.1.2.2.cmml"><mo id="S6.1.p1.5.m5.1.1.2.2.1" rspace="0em" xref="S6.1.p1.5.m5.1.1.2.2.1.cmml">∂</mo><mi id="S6.1.p1.5.m5.1.1.2.2.2" xref="S6.1.p1.5.m5.1.1.2.2.2.cmml">v</mi></mrow><mrow id="S6.1.p1.5.m5.1.1.2.3" xref="S6.1.p1.5.m5.1.1.2.3.cmml"><mo id="S6.1.p1.5.m5.1.1.2.3.1" rspace="0em" xref="S6.1.p1.5.m5.1.1.2.3.1.cmml">∂</mo><mi id="S6.1.p1.5.m5.1.1.2.3.2" xref="S6.1.p1.5.m5.1.1.2.3.2.cmml">p</mi></mrow></mfrac><mo id="S6.1.p1.5.m5.1.1.1" xref="S6.1.p1.5.m5.1.1.1.cmml">=</mo><mrow id="S6.1.p1.5.m5.1.1.3" xref="S6.1.p1.5.m5.1.1.3.cmml"><mn id="S6.1.p1.5.m5.1.1.3.2" xref="S6.1.p1.5.m5.1.1.3.2.cmml">70.02</mn><mo id="S6.1.p1.5.m5.1.1.3.1" xref="S6.1.p1.5.m5.1.1.3.1.cmml">−</mo><mrow id="S6.1.p1.5.m5.1.1.3.3" xref="S6.1.p1.5.m5.1.1.3.3.cmml"><mn id="S6.1.p1.5.m5.1.1.3.3.2" xref="S6.1.p1.5.m5.1.1.3.3.2.cmml">72</mn><mo id="S6.1.p1.5.m5.1.1.3.3.1" xref="S6.1.p1.5.m5.1.1.3.3.1.cmml">⁢</mo><mi id="S6.1.p1.5.m5.1.1.3.3.3" xref="S6.1.p1.5.m5.1.1.3.3.3.cmml">p</mi></mrow><mo id="S6.1.p1.5.m5.1.1.3.1a" xref="S6.1.p1.5.m5.1.1.3.1.cmml">−</mo><mrow id="S6.1.p1.5.m5.1.1.3.4" xref="S6.1.p1.5.m5.1.1.3.4.cmml"><mn id="S6.1.p1.5.m5.1.1.3.4.2" xref="S6.1.p1.5.m5.1.1.3.4.2.cmml">36</mn><mo id="S6.1.p1.5.m5.1.1.3.4.1" xref="S6.1.p1.5.m5.1.1.3.4.1.cmml">⁢</mo><mi id="S6.1.p1.5.m5.1.1.3.4.3" xref="S6.1.p1.5.m5.1.1.3.4.3.cmml">q</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.5.m5.1b"><apply id="S6.1.p1.5.m5.1.1.cmml" xref="S6.1.p1.5.m5.1.1"><eq id="S6.1.p1.5.m5.1.1.1.cmml" xref="S6.1.p1.5.m5.1.1.1"></eq><apply id="S6.1.p1.5.m5.1.1.2.cmml" xref="S6.1.p1.5.m5.1.1.2"><divide id="S6.1.p1.5.m5.1.1.2.1.cmml" xref="S6.1.p1.5.m5.1.1.2"></divide><apply id="S6.1.p1.5.m5.1.1.2.2.cmml" xref="S6.1.p1.5.m5.1.1.2.2"><partialdiff id="S6.1.p1.5.m5.1.1.2.2.1.cmml" xref="S6.1.p1.5.m5.1.1.2.2.1"></partialdiff><ci id="S6.1.p1.5.m5.1.1.2.2.2.cmml" xref="S6.1.p1.5.m5.1.1.2.2.2">𝑣</ci></apply><apply id="S6.1.p1.5.m5.1.1.2.3.cmml" xref="S6.1.p1.5.m5.1.1.2.3"><partialdiff id="S6.1.p1.5.m5.1.1.2.3.1.cmml" xref="S6.1.p1.5.m5.1.1.2.3.1"></partialdiff><ci id="S6.1.p1.5.m5.1.1.2.3.2.cmml" xref="S6.1.p1.5.m5.1.1.2.3.2">𝑝</ci></apply></apply><apply id="S6.1.p1.5.m5.1.1.3.cmml" xref="S6.1.p1.5.m5.1.1.3"><minus id="S6.1.p1.5.m5.1.1.3.1.cmml" xref="S6.1.p1.5.m5.1.1.3.1"></minus><cn id="S6.1.p1.5.m5.1.1.3.2.cmml" type="float" xref="S6.1.p1.5.m5.1.1.3.2">70.02</cn><apply id="S6.1.p1.5.m5.1.1.3.3.cmml" xref="S6.1.p1.5.m5.1.1.3.3"><times id="S6.1.p1.5.m5.1.1.3.3.1.cmml" xref="S6.1.p1.5.m5.1.1.3.3.1"></times><cn id="S6.1.p1.5.m5.1.1.3.3.2.cmml" type="integer" xref="S6.1.p1.5.m5.1.1.3.3.2">72</cn><ci id="S6.1.p1.5.m5.1.1.3.3.3.cmml" xref="S6.1.p1.5.m5.1.1.3.3.3">𝑝</ci></apply><apply id="S6.1.p1.5.m5.1.1.3.4.cmml" xref="S6.1.p1.5.m5.1.1.3.4"><times id="S6.1.p1.5.m5.1.1.3.4.1.cmml" xref="S6.1.p1.5.m5.1.1.3.4.1"></times><cn id="S6.1.p1.5.m5.1.1.3.4.2.cmml" type="integer" xref="S6.1.p1.5.m5.1.1.3.4.2">36</cn><ci id="S6.1.p1.5.m5.1.1.3.4.3.cmml" xref="S6.1.p1.5.m5.1.1.3.4.3">𝑞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.5.m5.1c">\frac{\partial v}{\partial p}=70.02-72p-36q</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.5.m5.1d">divide start_ARG ∂ italic_v end_ARG start_ARG ∂ italic_p end_ARG = 70.02 - 72 italic_p - 36 italic_q</annotation></semantics></math> and <math alttext="\frac{\partial v}{\partial q}=35.01-36p-18q" class="ltx_Math" display="inline" id="S6.1.p1.6.m6.1"><semantics id="S6.1.p1.6.m6.1a"><mrow id="S6.1.p1.6.m6.1.1" xref="S6.1.p1.6.m6.1.1.cmml"><mfrac id="S6.1.p1.6.m6.1.1.2" xref="S6.1.p1.6.m6.1.1.2.cmml"><mrow id="S6.1.p1.6.m6.1.1.2.2" xref="S6.1.p1.6.m6.1.1.2.2.cmml"><mo id="S6.1.p1.6.m6.1.1.2.2.1" rspace="0em" xref="S6.1.p1.6.m6.1.1.2.2.1.cmml">∂</mo><mi id="S6.1.p1.6.m6.1.1.2.2.2" xref="S6.1.p1.6.m6.1.1.2.2.2.cmml">v</mi></mrow><mrow id="S6.1.p1.6.m6.1.1.2.3" xref="S6.1.p1.6.m6.1.1.2.3.cmml"><mo id="S6.1.p1.6.m6.1.1.2.3.1" rspace="0em" xref="S6.1.p1.6.m6.1.1.2.3.1.cmml">∂</mo><mi id="S6.1.p1.6.m6.1.1.2.3.2" xref="S6.1.p1.6.m6.1.1.2.3.2.cmml">q</mi></mrow></mfrac><mo id="S6.1.p1.6.m6.1.1.1" xref="S6.1.p1.6.m6.1.1.1.cmml">=</mo><mrow id="S6.1.p1.6.m6.1.1.3" xref="S6.1.p1.6.m6.1.1.3.cmml"><mn id="S6.1.p1.6.m6.1.1.3.2" xref="S6.1.p1.6.m6.1.1.3.2.cmml">35.01</mn><mo id="S6.1.p1.6.m6.1.1.3.1" xref="S6.1.p1.6.m6.1.1.3.1.cmml">−</mo><mrow id="S6.1.p1.6.m6.1.1.3.3" xref="S6.1.p1.6.m6.1.1.3.3.cmml"><mn id="S6.1.p1.6.m6.1.1.3.3.2" xref="S6.1.p1.6.m6.1.1.3.3.2.cmml">36</mn><mo id="S6.1.p1.6.m6.1.1.3.3.1" xref="S6.1.p1.6.m6.1.1.3.3.1.cmml">⁢</mo><mi id="S6.1.p1.6.m6.1.1.3.3.3" xref="S6.1.p1.6.m6.1.1.3.3.3.cmml">p</mi></mrow><mo id="S6.1.p1.6.m6.1.1.3.1a" xref="S6.1.p1.6.m6.1.1.3.1.cmml">−</mo><mrow id="S6.1.p1.6.m6.1.1.3.4" xref="S6.1.p1.6.m6.1.1.3.4.cmml"><mn id="S6.1.p1.6.m6.1.1.3.4.2" xref="S6.1.p1.6.m6.1.1.3.4.2.cmml">18</mn><mo id="S6.1.p1.6.m6.1.1.3.4.1" xref="S6.1.p1.6.m6.1.1.3.4.1.cmml">⁢</mo><mi id="S6.1.p1.6.m6.1.1.3.4.3" xref="S6.1.p1.6.m6.1.1.3.4.3.cmml">q</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.6.m6.1b"><apply id="S6.1.p1.6.m6.1.1.cmml" xref="S6.1.p1.6.m6.1.1"><eq id="S6.1.p1.6.m6.1.1.1.cmml" xref="S6.1.p1.6.m6.1.1.1"></eq><apply id="S6.1.p1.6.m6.1.1.2.cmml" xref="S6.1.p1.6.m6.1.1.2"><divide id="S6.1.p1.6.m6.1.1.2.1.cmml" xref="S6.1.p1.6.m6.1.1.2"></divide><apply id="S6.1.p1.6.m6.1.1.2.2.cmml" xref="S6.1.p1.6.m6.1.1.2.2"><partialdiff id="S6.1.p1.6.m6.1.1.2.2.1.cmml" xref="S6.1.p1.6.m6.1.1.2.2.1"></partialdiff><ci id="S6.1.p1.6.m6.1.1.2.2.2.cmml" xref="S6.1.p1.6.m6.1.1.2.2.2">𝑣</ci></apply><apply id="S6.1.p1.6.m6.1.1.2.3.cmml" xref="S6.1.p1.6.m6.1.1.2.3"><partialdiff id="S6.1.p1.6.m6.1.1.2.3.1.cmml" xref="S6.1.p1.6.m6.1.1.2.3.1"></partialdiff><ci id="S6.1.p1.6.m6.1.1.2.3.2.cmml" xref="S6.1.p1.6.m6.1.1.2.3.2">𝑞</ci></apply></apply><apply id="S6.1.p1.6.m6.1.1.3.cmml" xref="S6.1.p1.6.m6.1.1.3"><minus id="S6.1.p1.6.m6.1.1.3.1.cmml" xref="S6.1.p1.6.m6.1.1.3.1"></minus><cn id="S6.1.p1.6.m6.1.1.3.2.cmml" type="float" xref="S6.1.p1.6.m6.1.1.3.2">35.01</cn><apply id="S6.1.p1.6.m6.1.1.3.3.cmml" xref="S6.1.p1.6.m6.1.1.3.3"><times id="S6.1.p1.6.m6.1.1.3.3.1.cmml" xref="S6.1.p1.6.m6.1.1.3.3.1"></times><cn id="S6.1.p1.6.m6.1.1.3.3.2.cmml" type="integer" xref="S6.1.p1.6.m6.1.1.3.3.2">36</cn><ci id="S6.1.p1.6.m6.1.1.3.3.3.cmml" xref="S6.1.p1.6.m6.1.1.3.3.3">𝑝</ci></apply><apply id="S6.1.p1.6.m6.1.1.3.4.cmml" xref="S6.1.p1.6.m6.1.1.3.4"><times id="S6.1.p1.6.m6.1.1.3.4.1.cmml" xref="S6.1.p1.6.m6.1.1.3.4.1"></times><cn id="S6.1.p1.6.m6.1.1.3.4.2.cmml" type="integer" xref="S6.1.p1.6.m6.1.1.3.4.2">18</cn><ci id="S6.1.p1.6.m6.1.1.3.4.3.cmml" xref="S6.1.p1.6.m6.1.1.3.4.3">𝑞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.6.m6.1c">\frac{\partial v}{\partial q}=35.01-36p-18q</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.6.m6.1d">divide start_ARG ∂ italic_v end_ARG start_ARG ∂ italic_q end_ARG = 35.01 - 36 italic_p - 18 italic_q</annotation></semantics></math>, both derivatives vanish when <math alttext="q=1.945-2p" class="ltx_Math" display="inline" id="S6.1.p1.7.m7.1"><semantics id="S6.1.p1.7.m7.1a"><mrow id="S6.1.p1.7.m7.1.1" xref="S6.1.p1.7.m7.1.1.cmml"><mi id="S6.1.p1.7.m7.1.1.2" xref="S6.1.p1.7.m7.1.1.2.cmml">q</mi><mo id="S6.1.p1.7.m7.1.1.1" xref="S6.1.p1.7.m7.1.1.1.cmml">=</mo><mrow id="S6.1.p1.7.m7.1.1.3" xref="S6.1.p1.7.m7.1.1.3.cmml"><mn id="S6.1.p1.7.m7.1.1.3.2" xref="S6.1.p1.7.m7.1.1.3.2.cmml">1.945</mn><mo id="S6.1.p1.7.m7.1.1.3.1" xref="S6.1.p1.7.m7.1.1.3.1.cmml">−</mo><mrow id="S6.1.p1.7.m7.1.1.3.3" xref="S6.1.p1.7.m7.1.1.3.3.cmml"><mn id="S6.1.p1.7.m7.1.1.3.3.2" xref="S6.1.p1.7.m7.1.1.3.3.2.cmml">2</mn><mo id="S6.1.p1.7.m7.1.1.3.3.1" xref="S6.1.p1.7.m7.1.1.3.3.1.cmml">⁢</mo><mi id="S6.1.p1.7.m7.1.1.3.3.3" xref="S6.1.p1.7.m7.1.1.3.3.3.cmml">p</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.7.m7.1b"><apply id="S6.1.p1.7.m7.1.1.cmml" xref="S6.1.p1.7.m7.1.1"><eq id="S6.1.p1.7.m7.1.1.1.cmml" xref="S6.1.p1.7.m7.1.1.1"></eq><ci id="S6.1.p1.7.m7.1.1.2.cmml" xref="S6.1.p1.7.m7.1.1.2">𝑞</ci><apply id="S6.1.p1.7.m7.1.1.3.cmml" xref="S6.1.p1.7.m7.1.1.3"><minus id="S6.1.p1.7.m7.1.1.3.1.cmml" xref="S6.1.p1.7.m7.1.1.3.1"></minus><cn id="S6.1.p1.7.m7.1.1.3.2.cmml" type="float" xref="S6.1.p1.7.m7.1.1.3.2">1.945</cn><apply id="S6.1.p1.7.m7.1.1.3.3.cmml" xref="S6.1.p1.7.m7.1.1.3.3"><times id="S6.1.p1.7.m7.1.1.3.3.1.cmml" xref="S6.1.p1.7.m7.1.1.3.3.1"></times><cn id="S6.1.p1.7.m7.1.1.3.3.2.cmml" type="integer" xref="S6.1.p1.7.m7.1.1.3.3.2">2</cn><ci id="S6.1.p1.7.m7.1.1.3.3.3.cmml" xref="S6.1.p1.7.m7.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.7.m7.1c">q=1.945-2p</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.7.m7.1d">italic_q = 1.945 - 2 italic_p</annotation></semantics></math>. Substituting back, we see that <math alttext="v(p,1.945-2p)=87.6647" class="ltx_Math" display="inline" id="S6.1.p1.8.m8.2"><semantics id="S6.1.p1.8.m8.2a"><mrow id="S6.1.p1.8.m8.2.2" xref="S6.1.p1.8.m8.2.2.cmml"><mrow id="S6.1.p1.8.m8.2.2.1" xref="S6.1.p1.8.m8.2.2.1.cmml"><mi id="S6.1.p1.8.m8.2.2.1.3" xref="S6.1.p1.8.m8.2.2.1.3.cmml">v</mi><mo id="S6.1.p1.8.m8.2.2.1.2" xref="S6.1.p1.8.m8.2.2.1.2.cmml">⁢</mo><mrow id="S6.1.p1.8.m8.2.2.1.1.1" xref="S6.1.p1.8.m8.2.2.1.1.2.cmml"><mo id="S6.1.p1.8.m8.2.2.1.1.1.2" stretchy="false" xref="S6.1.p1.8.m8.2.2.1.1.2.cmml">(</mo><mi id="S6.1.p1.8.m8.1.1" xref="S6.1.p1.8.m8.1.1.cmml">p</mi><mo id="S6.1.p1.8.m8.2.2.1.1.1.3" xref="S6.1.p1.8.m8.2.2.1.1.2.cmml">,</mo><mrow id="S6.1.p1.8.m8.2.2.1.1.1.1" xref="S6.1.p1.8.m8.2.2.1.1.1.1.cmml"><mn id="S6.1.p1.8.m8.2.2.1.1.1.1.2" xref="S6.1.p1.8.m8.2.2.1.1.1.1.2.cmml">1.945</mn><mo id="S6.1.p1.8.m8.2.2.1.1.1.1.1" xref="S6.1.p1.8.m8.2.2.1.1.1.1.1.cmml">−</mo><mrow id="S6.1.p1.8.m8.2.2.1.1.1.1.3" xref="S6.1.p1.8.m8.2.2.1.1.1.1.3.cmml"><mn id="S6.1.p1.8.m8.2.2.1.1.1.1.3.2" xref="S6.1.p1.8.m8.2.2.1.1.1.1.3.2.cmml">2</mn><mo id="S6.1.p1.8.m8.2.2.1.1.1.1.3.1" xref="S6.1.p1.8.m8.2.2.1.1.1.1.3.1.cmml">⁢</mo><mi id="S6.1.p1.8.m8.2.2.1.1.1.1.3.3" xref="S6.1.p1.8.m8.2.2.1.1.1.1.3.3.cmml">p</mi></mrow></mrow><mo id="S6.1.p1.8.m8.2.2.1.1.1.4" stretchy="false" xref="S6.1.p1.8.m8.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.1.p1.8.m8.2.2.2" xref="S6.1.p1.8.m8.2.2.2.cmml">=</mo><mn id="S6.1.p1.8.m8.2.2.3" xref="S6.1.p1.8.m8.2.2.3.cmml">87.6647</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.8.m8.2b"><apply id="S6.1.p1.8.m8.2.2.cmml" xref="S6.1.p1.8.m8.2.2"><eq id="S6.1.p1.8.m8.2.2.2.cmml" xref="S6.1.p1.8.m8.2.2.2"></eq><apply id="S6.1.p1.8.m8.2.2.1.cmml" xref="S6.1.p1.8.m8.2.2.1"><times id="S6.1.p1.8.m8.2.2.1.2.cmml" xref="S6.1.p1.8.m8.2.2.1.2"></times><ci id="S6.1.p1.8.m8.2.2.1.3.cmml" xref="S6.1.p1.8.m8.2.2.1.3">𝑣</ci><interval closure="open" id="S6.1.p1.8.m8.2.2.1.1.2.cmml" xref="S6.1.p1.8.m8.2.2.1.1.1"><ci id="S6.1.p1.8.m8.1.1.cmml" xref="S6.1.p1.8.m8.1.1">𝑝</ci><apply id="S6.1.p1.8.m8.2.2.1.1.1.1.cmml" xref="S6.1.p1.8.m8.2.2.1.1.1.1"><minus id="S6.1.p1.8.m8.2.2.1.1.1.1.1.cmml" xref="S6.1.p1.8.m8.2.2.1.1.1.1.1"></minus><cn id="S6.1.p1.8.m8.2.2.1.1.1.1.2.cmml" type="float" xref="S6.1.p1.8.m8.2.2.1.1.1.1.2">1.945</cn><apply id="S6.1.p1.8.m8.2.2.1.1.1.1.3.cmml" xref="S6.1.p1.8.m8.2.2.1.1.1.1.3"><times id="S6.1.p1.8.m8.2.2.1.1.1.1.3.1.cmml" xref="S6.1.p1.8.m8.2.2.1.1.1.1.3.1"></times><cn id="S6.1.p1.8.m8.2.2.1.1.1.1.3.2.cmml" type="integer" xref="S6.1.p1.8.m8.2.2.1.1.1.1.3.2">2</cn><ci id="S6.1.p1.8.m8.2.2.1.1.1.1.3.3.cmml" xref="S6.1.p1.8.m8.2.2.1.1.1.1.3.3">𝑝</ci></apply></apply></interval></apply><cn id="S6.1.p1.8.m8.2.2.3.cmml" type="float" xref="S6.1.p1.8.m8.2.2.3">87.6647</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.8.m8.2c">v(p,1.945-2p)=87.6647</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.8.m8.2d">italic_v ( italic_p , 1.945 - 2 italic_p ) = 87.6647</annotation></semantics></math>; this is the minimum value of <math alttext="v(p,q)" class="ltx_Math" display="inline" id="S6.1.p1.9.m9.2"><semantics id="S6.1.p1.9.m9.2a"><mrow id="S6.1.p1.9.m9.2.3" xref="S6.1.p1.9.m9.2.3.cmml"><mi id="S6.1.p1.9.m9.2.3.2" xref="S6.1.p1.9.m9.2.3.2.cmml">v</mi><mo id="S6.1.p1.9.m9.2.3.1" xref="S6.1.p1.9.m9.2.3.1.cmml">⁢</mo><mrow id="S6.1.p1.9.m9.2.3.3.2" xref="S6.1.p1.9.m9.2.3.3.1.cmml"><mo id="S6.1.p1.9.m9.2.3.3.2.1" stretchy="false" xref="S6.1.p1.9.m9.2.3.3.1.cmml">(</mo><mi id="S6.1.p1.9.m9.1.1" xref="S6.1.p1.9.m9.1.1.cmml">p</mi><mo id="S6.1.p1.9.m9.2.3.3.2.2" xref="S6.1.p1.9.m9.2.3.3.1.cmml">,</mo><mi id="S6.1.p1.9.m9.2.2" xref="S6.1.p1.9.m9.2.2.cmml">q</mi><mo id="S6.1.p1.9.m9.2.3.3.2.3" stretchy="false" xref="S6.1.p1.9.m9.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.9.m9.2b"><apply id="S6.1.p1.9.m9.2.3.cmml" xref="S6.1.p1.9.m9.2.3"><times id="S6.1.p1.9.m9.2.3.1.cmml" xref="S6.1.p1.9.m9.2.3.1"></times><ci id="S6.1.p1.9.m9.2.3.2.cmml" xref="S6.1.p1.9.m9.2.3.2">𝑣</ci><interval closure="open" id="S6.1.p1.9.m9.2.3.3.1.cmml" xref="S6.1.p1.9.m9.2.3.3.2"><ci id="S6.1.p1.9.m9.1.1.cmml" xref="S6.1.p1.9.m9.1.1">𝑝</ci><ci id="S6.1.p1.9.m9.2.2.cmml" xref="S6.1.p1.9.m9.2.2">𝑞</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.9.m9.2c">v(p,q)</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.9.m9.2d">italic_v ( italic_p , italic_q )</annotation></semantics></math> over <math alttext="\mathbb{R}^{2}" class="ltx_Math" display="inline" id="S6.1.p1.10.m10.1"><semantics id="S6.1.p1.10.m10.1a"><msup id="S6.1.p1.10.m10.1.1" xref="S6.1.p1.10.m10.1.1.cmml"><mi id="S6.1.p1.10.m10.1.1.2" xref="S6.1.p1.10.m10.1.1.2.cmml">ℝ</mi><mn id="S6.1.p1.10.m10.1.1.3" xref="S6.1.p1.10.m10.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S6.1.p1.10.m10.1b"><apply id="S6.1.p1.10.m10.1.1.cmml" xref="S6.1.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S6.1.p1.10.m10.1.1.1.cmml" xref="S6.1.p1.10.m10.1.1">superscript</csymbol><ci id="S6.1.p1.10.m10.1.1.2.cmml" xref="S6.1.p1.10.m10.1.1.2">ℝ</ci><cn id="S6.1.p1.10.m10.1.1.3.cmml" type="integer" xref="S6.1.p1.10.m10.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.10.m10.1c">\mathbb{R}^{2}</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.10.m10.1d">blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> (and it is achieved at e.g. <math alttext="(p,q)=(0.6,0.745)\in[0,1]^{2}" class="ltx_Math" display="inline" id="S6.1.p1.11.m11.6"><semantics id="S6.1.p1.11.m11.6a"><mrow id="S6.1.p1.11.m11.6.7" xref="S6.1.p1.11.m11.6.7.cmml"><mrow id="S6.1.p1.11.m11.6.7.2.2" xref="S6.1.p1.11.m11.6.7.2.1.cmml"><mo id="S6.1.p1.11.m11.6.7.2.2.1" stretchy="false" xref="S6.1.p1.11.m11.6.7.2.1.cmml">(</mo><mi id="S6.1.p1.11.m11.1.1" xref="S6.1.p1.11.m11.1.1.cmml">p</mi><mo id="S6.1.p1.11.m11.6.7.2.2.2" xref="S6.1.p1.11.m11.6.7.2.1.cmml">,</mo><mi id="S6.1.p1.11.m11.2.2" xref="S6.1.p1.11.m11.2.2.cmml">q</mi><mo id="S6.1.p1.11.m11.6.7.2.2.3" stretchy="false" xref="S6.1.p1.11.m11.6.7.2.1.cmml">)</mo></mrow><mo id="S6.1.p1.11.m11.6.7.3" xref="S6.1.p1.11.m11.6.7.3.cmml">=</mo><mrow id="S6.1.p1.11.m11.6.7.4.2" xref="S6.1.p1.11.m11.6.7.4.1.cmml"><mo id="S6.1.p1.11.m11.6.7.4.2.1" stretchy="false" xref="S6.1.p1.11.m11.6.7.4.1.cmml">(</mo><mn id="S6.1.p1.11.m11.3.3" xref="S6.1.p1.11.m11.3.3.cmml">0.6</mn><mo id="S6.1.p1.11.m11.6.7.4.2.2" xref="S6.1.p1.11.m11.6.7.4.1.cmml">,</mo><mn id="S6.1.p1.11.m11.4.4" xref="S6.1.p1.11.m11.4.4.cmml">0.745</mn><mo id="S6.1.p1.11.m11.6.7.4.2.3" stretchy="false" xref="S6.1.p1.11.m11.6.7.4.1.cmml">)</mo></mrow><mo id="S6.1.p1.11.m11.6.7.5" xref="S6.1.p1.11.m11.6.7.5.cmml">∈</mo><msup id="S6.1.p1.11.m11.6.7.6" xref="S6.1.p1.11.m11.6.7.6.cmml"><mrow id="S6.1.p1.11.m11.6.7.6.2.2" xref="S6.1.p1.11.m11.6.7.6.2.1.cmml"><mo id="S6.1.p1.11.m11.6.7.6.2.2.1" stretchy="false" xref="S6.1.p1.11.m11.6.7.6.2.1.cmml">[</mo><mn id="S6.1.p1.11.m11.5.5" xref="S6.1.p1.11.m11.5.5.cmml">0</mn><mo id="S6.1.p1.11.m11.6.7.6.2.2.2" xref="S6.1.p1.11.m11.6.7.6.2.1.cmml">,</mo><mn id="S6.1.p1.11.m11.6.6" xref="S6.1.p1.11.m11.6.6.cmml">1</mn><mo id="S6.1.p1.11.m11.6.7.6.2.2.3" stretchy="false" xref="S6.1.p1.11.m11.6.7.6.2.1.cmml">]</mo></mrow><mn id="S6.1.p1.11.m11.6.7.6.3" xref="S6.1.p1.11.m11.6.7.6.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.11.m11.6b"><apply id="S6.1.p1.11.m11.6.7.cmml" xref="S6.1.p1.11.m11.6.7"><and id="S6.1.p1.11.m11.6.7a.cmml" xref="S6.1.p1.11.m11.6.7"></and><apply id="S6.1.p1.11.m11.6.7b.cmml" xref="S6.1.p1.11.m11.6.7"><eq id="S6.1.p1.11.m11.6.7.3.cmml" xref="S6.1.p1.11.m11.6.7.3"></eq><interval closure="open" id="S6.1.p1.11.m11.6.7.2.1.cmml" xref="S6.1.p1.11.m11.6.7.2.2"><ci id="S6.1.p1.11.m11.1.1.cmml" xref="S6.1.p1.11.m11.1.1">𝑝</ci><ci id="S6.1.p1.11.m11.2.2.cmml" xref="S6.1.p1.11.m11.2.2">𝑞</ci></interval><interval closure="open" id="S6.1.p1.11.m11.6.7.4.1.cmml" xref="S6.1.p1.11.m11.6.7.4.2"><cn id="S6.1.p1.11.m11.3.3.cmml" type="float" xref="S6.1.p1.11.m11.3.3">0.6</cn><cn id="S6.1.p1.11.m11.4.4.cmml" type="float" xref="S6.1.p1.11.m11.4.4">0.745</cn></interval></apply><apply id="S6.1.p1.11.m11.6.7c.cmml" xref="S6.1.p1.11.m11.6.7"><in id="S6.1.p1.11.m11.6.7.5.cmml" xref="S6.1.p1.11.m11.6.7.5"></in><share href="https://arxiv.org/html/2411.12976v1#S6.1.p1.11.m11.6.7.4.cmml" id="S6.1.p1.11.m11.6.7d.cmml" xref="S6.1.p1.11.m11.6.7"></share><apply id="S6.1.p1.11.m11.6.7.6.cmml" xref="S6.1.p1.11.m11.6.7.6"><csymbol cd="ambiguous" id="S6.1.p1.11.m11.6.7.6.1.cmml" xref="S6.1.p1.11.m11.6.7.6">superscript</csymbol><interval closure="closed" id="S6.1.p1.11.m11.6.7.6.2.1.cmml" xref="S6.1.p1.11.m11.6.7.6.2.2"><cn id="S6.1.p1.11.m11.5.5.cmml" type="integer" xref="S6.1.p1.11.m11.5.5">0</cn><cn id="S6.1.p1.11.m11.6.6.cmml" type="integer" xref="S6.1.p1.11.m11.6.6">1</cn></interval><cn id="S6.1.p1.11.m11.6.7.6.3.cmml" type="integer" xref="S6.1.p1.11.m11.6.7.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.11.m11.6c">(p,q)=(0.6,0.745)\in[0,1]^{2}</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.11.m11.6d">( italic_p , italic_q ) = ( 0.6 , 0.745 ) ∈ [ 0 , 1 ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>). Finally, <math alttext="\frac{87.6647}{179.28}\approx 0.48898" class="ltx_Math" display="inline" id="S6.1.p1.12.m12.1"><semantics id="S6.1.p1.12.m12.1a"><mrow id="S6.1.p1.12.m12.1.1" xref="S6.1.p1.12.m12.1.1.cmml"><mfrac id="S6.1.p1.12.m12.1.1.2" xref="S6.1.p1.12.m12.1.1.2.cmml"><mn id="S6.1.p1.12.m12.1.1.2.2" xref="S6.1.p1.12.m12.1.1.2.2.cmml">87.6647</mn><mn id="S6.1.p1.12.m12.1.1.2.3" xref="S6.1.p1.12.m12.1.1.2.3.cmml">179.28</mn></mfrac><mo id="S6.1.p1.12.m12.1.1.1" xref="S6.1.p1.12.m12.1.1.1.cmml">≈</mo><mn id="S6.1.p1.12.m12.1.1.3" xref="S6.1.p1.12.m12.1.1.3.cmml">0.48898</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.12.m12.1b"><apply id="S6.1.p1.12.m12.1.1.cmml" xref="S6.1.p1.12.m12.1.1"><approx id="S6.1.p1.12.m12.1.1.1.cmml" xref="S6.1.p1.12.m12.1.1.1"></approx><apply id="S6.1.p1.12.m12.1.1.2.cmml" xref="S6.1.p1.12.m12.1.1.2"><divide id="S6.1.p1.12.m12.1.1.2.1.cmml" xref="S6.1.p1.12.m12.1.1.2"></divide><cn id="S6.1.p1.12.m12.1.1.2.2.cmml" type="float" xref="S6.1.p1.12.m12.1.1.2.2">87.6647</cn><cn id="S6.1.p1.12.m12.1.1.2.3.cmml" type="float" xref="S6.1.p1.12.m12.1.1.2.3">179.28</cn></apply><cn id="S6.1.p1.12.m12.1.1.3.cmml" type="float" xref="S6.1.p1.12.m12.1.1.3">0.48898</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.12.m12.1c">\frac{87.6647}{179.28}\approx 0.48898</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.12.m12.1d">divide start_ARG 87.6647 end_ARG start_ARG 179.28 end_ARG ≈ 0.48898</annotation></semantics></math>, as desired. ∎</p> </div> </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span>Recap: The prior lower bound of <span class="ltx_ERROR undefined" id="A1.2.1">\textcite</span>FJ15 (<a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem2" title="Theorem 1.2 (Prior lower bound for antisymmetric selection, [FJ15, Thm. 1.4]). ‣ 1.1 Background: Directed cuts, bias, and oblivious algorithms ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.2</span></a>)</h2> <div class="ltx_para" id="A1.p1"> <p class="ltx_p" id="A1.p1.1">In this appendix, we include a brief description and analysis of the graph used to prove the lower bound of <span class="ltx_ERROR undefined" id="A1.p1.1.1">\textcite</span>FJ15 (<a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem2" title="Theorem 1.2 (Prior lower bound for antisymmetric selection, [FJ15, Thm. 1.4]). ‣ 1.1 Background: Directed cuts, bias, and oblivious algorithms ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.2</span></a>):</p> </div> <div class="ltx_para" id="A1.p2"> <p class="ltx_p" id="A1.p2.1">See <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem2" title="Theorem 1.2 (Prior lower bound for antisymmetric selection, [FJ15, Thm. 1.4]). ‣ 1.1 Background: Directed cuts, bias, and oblivious algorithms ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">1.2</span></a></p> </div> <div class="ltx_para" id="A1.p3"> <p class="ltx_p" id="A1.p3.1">To prove this theorem, <span class="ltx_ERROR undefined" id="A1.p3.1.1">\textcite</span>FJ15 used the pair of graphs appearing in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#A1.F7" title="In Appendix A Recap: The prior lower bound of \textciteFJ15 (Theorem 1.2) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Fig.</span> <span class="ltx_text ltx_ref_tag">7</span></a> below:</p> </div> <figure 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id="A1.F7.sf1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" mathsize="90%" xref="A1.F7.sf1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="A1.F7.sf1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="A1.F7.sf1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="A1.F7.sf1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_c</annotation></semantics></math></foreignobject></g></g></g></svg> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A1.F7.sf1.30.13.1" style="font-size:90%;">(a)</span> </span><em class="ltx_emph ltx_font_italic" id="A1.F7.sf1.31.14" style="font-size:90%;">Parameter:</em><span class="ltx_text" id="A1.F7.sf1.24.12" style="font-size:90%;"> <math alttext="c>1" class="ltx_Math" display="inline" id="A1.F7.sf1.13.1.m1.1"><semantics id="A1.F7.sf1.13.1.m1.1b"><mrow id="A1.F7.sf1.13.1.m1.1.1" xref="A1.F7.sf1.13.1.m1.1.1.cmml"><mi id="A1.F7.sf1.13.1.m1.1.1.2" xref="A1.F7.sf1.13.1.m1.1.1.2.cmml">c</mi><mo id="A1.F7.sf1.13.1.m1.1.1.1" xref="A1.F7.sf1.13.1.m1.1.1.1.cmml">></mo><mn id="A1.F7.sf1.13.1.m1.1.1.3" xref="A1.F7.sf1.13.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf1.13.1.m1.1c"><apply id="A1.F7.sf1.13.1.m1.1.1.cmml" xref="A1.F7.sf1.13.1.m1.1.1"><gt id="A1.F7.sf1.13.1.m1.1.1.1.cmml" xref="A1.F7.sf1.13.1.m1.1.1.1"></gt><ci id="A1.F7.sf1.13.1.m1.1.1.2.cmml" xref="A1.F7.sf1.13.1.m1.1.1.2">𝑐</ci><cn id="A1.F7.sf1.13.1.m1.1.1.3.cmml" type="integer" xref="A1.F7.sf1.13.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf1.13.1.m1.1d">c>1</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf1.13.1.m1.1e">italic_c > 1</annotation></semantics></math> (“<math alttext="G_{1}" class="ltx_Math" display="inline" id="A1.F7.sf1.14.2.m2.1"><semantics id="A1.F7.sf1.14.2.m2.1b"><msub id="A1.F7.sf1.14.2.m2.1.1" xref="A1.F7.sf1.14.2.m2.1.1.cmml"><mi id="A1.F7.sf1.14.2.m2.1.1.2" xref="A1.F7.sf1.14.2.m2.1.1.2.cmml">G</mi><mn id="A1.F7.sf1.14.2.m2.1.1.3" xref="A1.F7.sf1.14.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="A1.F7.sf1.14.2.m2.1c"><apply id="A1.F7.sf1.14.2.m2.1.1.cmml" xref="A1.F7.sf1.14.2.m2.1.1"><csymbol cd="ambiguous" id="A1.F7.sf1.14.2.m2.1.1.1.cmml" xref="A1.F7.sf1.14.2.m2.1.1">subscript</csymbol><ci id="A1.F7.sf1.14.2.m2.1.1.2.cmml" xref="A1.F7.sf1.14.2.m2.1.1.2">𝐺</ci><cn id="A1.F7.sf1.14.2.m2.1.1.3.cmml" type="integer" xref="A1.F7.sf1.14.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf1.14.2.m2.1d">G_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf1.14.2.m2.1e">italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>” from <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">FJ15</span>]</cite>). The <span class="ltx_text" id="A1.F7.sf1.24.12.1" style="background-color:#ABDEE6;">LIGHT BLUE</span> vertices (<math alttext="\{1,2\}" class="ltx_Math" display="inline" id="A1.F7.sf1.15.3.m3.2"><semantics id="A1.F7.sf1.15.3.m3.2b"><mrow id="A1.F7.sf1.15.3.m3.2.3.2" xref="A1.F7.sf1.15.3.m3.2.3.1.cmml"><mo id="A1.F7.sf1.15.3.m3.2.3.2.1" stretchy="false" xref="A1.F7.sf1.15.3.m3.2.3.1.cmml">{</mo><mn id="A1.F7.sf1.15.3.m3.1.1" xref="A1.F7.sf1.15.3.m3.1.1.cmml">1</mn><mo id="A1.F7.sf1.15.3.m3.2.3.2.2" xref="A1.F7.sf1.15.3.m3.2.3.1.cmml">,</mo><mn id="A1.F7.sf1.15.3.m3.2.2" xref="A1.F7.sf1.15.3.m3.2.2.cmml">2</mn><mo id="A1.F7.sf1.15.3.m3.2.3.2.3" stretchy="false" xref="A1.F7.sf1.15.3.m3.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf1.15.3.m3.2c"><set id="A1.F7.sf1.15.3.m3.2.3.1.cmml" xref="A1.F7.sf1.15.3.m3.2.3.2"><cn id="A1.F7.sf1.15.3.m3.1.1.cmml" type="integer" xref="A1.F7.sf1.15.3.m3.1.1">1</cn><cn id="A1.F7.sf1.15.3.m3.2.2.cmml" type="integer" xref="A1.F7.sf1.15.3.m3.2.2">2</cn></set></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf1.15.3.m3.2d">\{1,2\}</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf1.15.3.m3.2e">{ 1 , 2 }</annotation></semantics></math>) have bias <math alttext="+(c-1)/(c+1)" class="ltx_Math" display="inline" id="A1.F7.sf1.16.4.m4.2"><semantics id="A1.F7.sf1.16.4.m4.2b"><mrow id="A1.F7.sf1.16.4.m4.2.2" xref="A1.F7.sf1.16.4.m4.2.2.cmml"><mo id="A1.F7.sf1.16.4.m4.2.2b" xref="A1.F7.sf1.16.4.m4.2.2.cmml">+</mo><mrow id="A1.F7.sf1.16.4.m4.2.2.2" xref="A1.F7.sf1.16.4.m4.2.2.2.cmml"><mrow id="A1.F7.sf1.16.4.m4.1.1.1.1.1" xref="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.cmml"><mo id="A1.F7.sf1.16.4.m4.1.1.1.1.1.2" stretchy="false" xref="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.F7.sf1.16.4.m4.1.1.1.1.1.1" xref="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.cmml"><mi id="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.2" xref="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.2.cmml">c</mi><mo id="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.1" xref="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.1.cmml">−</mo><mn id="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.3" xref="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="A1.F7.sf1.16.4.m4.1.1.1.1.1.3" stretchy="false" xref="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A1.F7.sf1.16.4.m4.2.2.2.3" xref="A1.F7.sf1.16.4.m4.2.2.2.3.cmml">/</mo><mrow id="A1.F7.sf1.16.4.m4.2.2.2.2.1" xref="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.cmml"><mo id="A1.F7.sf1.16.4.m4.2.2.2.2.1.2" stretchy="false" xref="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.cmml">(</mo><mrow id="A1.F7.sf1.16.4.m4.2.2.2.2.1.1" xref="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.cmml"><mi id="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.2" xref="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.2.cmml">c</mi><mo id="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.1" xref="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.1.cmml">+</mo><mn id="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.3" xref="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.3.cmml">1</mn></mrow><mo id="A1.F7.sf1.16.4.m4.2.2.2.2.1.3" stretchy="false" xref="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf1.16.4.m4.2c"><apply id="A1.F7.sf1.16.4.m4.2.2.cmml" xref="A1.F7.sf1.16.4.m4.2.2"><plus id="A1.F7.sf1.16.4.m4.2.2.3.cmml" xref="A1.F7.sf1.16.4.m4.2.2"></plus><apply id="A1.F7.sf1.16.4.m4.2.2.2.cmml" xref="A1.F7.sf1.16.4.m4.2.2.2"><divide id="A1.F7.sf1.16.4.m4.2.2.2.3.cmml" xref="A1.F7.sf1.16.4.m4.2.2.2.3"></divide><apply id="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.cmml" xref="A1.F7.sf1.16.4.m4.1.1.1.1.1"><minus id="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.1.cmml" xref="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.1"></minus><ci id="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.2.cmml" xref="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.2">𝑐</ci><cn id="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="A1.F7.sf1.16.4.m4.1.1.1.1.1.1.3">1</cn></apply><apply id="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.cmml" xref="A1.F7.sf1.16.4.m4.2.2.2.2.1"><plus id="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.1.cmml" xref="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.1"></plus><ci id="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.2.cmml" xref="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.2">𝑐</ci><cn id="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.3.cmml" type="integer" xref="A1.F7.sf1.16.4.m4.2.2.2.2.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf1.16.4.m4.2d">+(c-1)/(c+1)</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf1.16.4.m4.2e">+ ( italic_c - 1 ) / ( italic_c + 1 )</annotation></semantics></math>. The <span class="ltx_text" id="A1.F7.sf1.24.12.2" style="background-color:#CBAACB;">PURPLE</span> vertices (<math alttext="\{3,4\}" class="ltx_Math" display="inline" id="A1.F7.sf1.17.5.m5.2"><semantics id="A1.F7.sf1.17.5.m5.2b"><mrow id="A1.F7.sf1.17.5.m5.2.3.2" xref="A1.F7.sf1.17.5.m5.2.3.1.cmml"><mo id="A1.F7.sf1.17.5.m5.2.3.2.1" stretchy="false" xref="A1.F7.sf1.17.5.m5.2.3.1.cmml">{</mo><mn id="A1.F7.sf1.17.5.m5.1.1" xref="A1.F7.sf1.17.5.m5.1.1.cmml">3</mn><mo id="A1.F7.sf1.17.5.m5.2.3.2.2" xref="A1.F7.sf1.17.5.m5.2.3.1.cmml">,</mo><mn id="A1.F7.sf1.17.5.m5.2.2" xref="A1.F7.sf1.17.5.m5.2.2.cmml">4</mn><mo id="A1.F7.sf1.17.5.m5.2.3.2.3" stretchy="false" xref="A1.F7.sf1.17.5.m5.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf1.17.5.m5.2c"><set id="A1.F7.sf1.17.5.m5.2.3.1.cmml" xref="A1.F7.sf1.17.5.m5.2.3.2"><cn id="A1.F7.sf1.17.5.m5.1.1.cmml" type="integer" xref="A1.F7.sf1.17.5.m5.1.1">3</cn><cn id="A1.F7.sf1.17.5.m5.2.2.cmml" type="integer" xref="A1.F7.sf1.17.5.m5.2.2">4</cn></set></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf1.17.5.m5.2d">\{3,4\}</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf1.17.5.m5.2e">{ 3 , 4 }</annotation></semantics></math>) have bias <math alttext="0" class="ltx_Math" display="inline" id="A1.F7.sf1.18.6.m6.1"><semantics id="A1.F7.sf1.18.6.m6.1b"><mn id="A1.F7.sf1.18.6.m6.1.1" xref="A1.F7.sf1.18.6.m6.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A1.F7.sf1.18.6.m6.1c"><cn id="A1.F7.sf1.18.6.m6.1.1.cmml" type="integer" xref="A1.F7.sf1.18.6.m6.1.1">0</cn></annotation-xml></semantics></math>. The <span class="ltx_text" id="A1.F7.sf1.24.12.3" style="background-color:#FEE1E8;">PINK</span> vertices (<math alttext="\{5,6\}" class="ltx_Math" display="inline" id="A1.F7.sf1.19.7.m7.2"><semantics id="A1.F7.sf1.19.7.m7.2b"><mrow id="A1.F7.sf1.19.7.m7.2.3.2" xref="A1.F7.sf1.19.7.m7.2.3.1.cmml"><mo id="A1.F7.sf1.19.7.m7.2.3.2.1" stretchy="false" xref="A1.F7.sf1.19.7.m7.2.3.1.cmml">{</mo><mn id="A1.F7.sf1.19.7.m7.1.1" xref="A1.F7.sf1.19.7.m7.1.1.cmml">5</mn><mo id="A1.F7.sf1.19.7.m7.2.3.2.2" xref="A1.F7.sf1.19.7.m7.2.3.1.cmml">,</mo><mn id="A1.F7.sf1.19.7.m7.2.2" xref="A1.F7.sf1.19.7.m7.2.2.cmml">6</mn><mo id="A1.F7.sf1.19.7.m7.2.3.2.3" stretchy="false" xref="A1.F7.sf1.19.7.m7.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf1.19.7.m7.2c"><set id="A1.F7.sf1.19.7.m7.2.3.1.cmml" xref="A1.F7.sf1.19.7.m7.2.3.2"><cn id="A1.F7.sf1.19.7.m7.1.1.cmml" type="integer" xref="A1.F7.sf1.19.7.m7.1.1">5</cn><cn id="A1.F7.sf1.19.7.m7.2.2.cmml" type="integer" xref="A1.F7.sf1.19.7.m7.2.2">6</cn></set></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf1.19.7.m7.2d">\{5,6\}</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf1.19.7.m7.2e">{ 5 , 6 }</annotation></semantics></math>) have bias <math alttext="-(c-1)/(c+1)" class="ltx_Math" display="inline" id="A1.F7.sf1.20.8.m8.2"><semantics id="A1.F7.sf1.20.8.m8.2b"><mrow id="A1.F7.sf1.20.8.m8.2.2" xref="A1.F7.sf1.20.8.m8.2.2.cmml"><mo id="A1.F7.sf1.20.8.m8.2.2b" xref="A1.F7.sf1.20.8.m8.2.2.cmml">−</mo><mrow id="A1.F7.sf1.20.8.m8.2.2.2" xref="A1.F7.sf1.20.8.m8.2.2.2.cmml"><mrow id="A1.F7.sf1.20.8.m8.1.1.1.1.1" xref="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.cmml"><mo id="A1.F7.sf1.20.8.m8.1.1.1.1.1.2" stretchy="false" xref="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.F7.sf1.20.8.m8.1.1.1.1.1.1" xref="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.cmml"><mi id="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.2" xref="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.2.cmml">c</mi><mo id="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.1" xref="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.1.cmml">−</mo><mn id="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.3" xref="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="A1.F7.sf1.20.8.m8.1.1.1.1.1.3" stretchy="false" xref="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A1.F7.sf1.20.8.m8.2.2.2.3" xref="A1.F7.sf1.20.8.m8.2.2.2.3.cmml">/</mo><mrow id="A1.F7.sf1.20.8.m8.2.2.2.2.1" xref="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.cmml"><mo id="A1.F7.sf1.20.8.m8.2.2.2.2.1.2" stretchy="false" xref="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.cmml">(</mo><mrow id="A1.F7.sf1.20.8.m8.2.2.2.2.1.1" xref="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.cmml"><mi id="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.2" xref="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.2.cmml">c</mi><mo id="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.1" xref="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.1.cmml">+</mo><mn id="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.3" xref="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.3.cmml">1</mn></mrow><mo id="A1.F7.sf1.20.8.m8.2.2.2.2.1.3" stretchy="false" xref="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf1.20.8.m8.2c"><apply id="A1.F7.sf1.20.8.m8.2.2.cmml" xref="A1.F7.sf1.20.8.m8.2.2"><minus id="A1.F7.sf1.20.8.m8.2.2.3.cmml" xref="A1.F7.sf1.20.8.m8.2.2"></minus><apply id="A1.F7.sf1.20.8.m8.2.2.2.cmml" xref="A1.F7.sf1.20.8.m8.2.2.2"><divide id="A1.F7.sf1.20.8.m8.2.2.2.3.cmml" xref="A1.F7.sf1.20.8.m8.2.2.2.3"></divide><apply id="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.cmml" xref="A1.F7.sf1.20.8.m8.1.1.1.1.1"><minus id="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.1.cmml" xref="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.1"></minus><ci id="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.2.cmml" xref="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.2">𝑐</ci><cn id="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.3.cmml" type="integer" xref="A1.F7.sf1.20.8.m8.1.1.1.1.1.1.3">1</cn></apply><apply id="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.cmml" xref="A1.F7.sf1.20.8.m8.2.2.2.2.1"><plus id="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.1.cmml" xref="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.1"></plus><ci id="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.2.cmml" xref="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.2">𝑐</ci><cn id="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.3.cmml" type="integer" xref="A1.F7.sf1.20.8.m8.2.2.2.2.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf1.20.8.m8.2d">-(c-1)/(c+1)</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf1.20.8.m8.2e">- ( italic_c - 1 ) / ( italic_c + 1 )</annotation></semantics></math>. The assignment <math alttext="\{1,3,5\}\to 1,\{2,4,6\}\to 0" class="ltx_Math" display="inline" id="A1.F7.sf1.21.9.m9.8"><semantics id="A1.F7.sf1.21.9.m9.8b"><mrow id="A1.F7.sf1.21.9.m9.8.8.2" xref="A1.F7.sf1.21.9.m9.8.8.3.cmml"><mrow id="A1.F7.sf1.21.9.m9.7.7.1.1" xref="A1.F7.sf1.21.9.m9.7.7.1.1.cmml"><mrow id="A1.F7.sf1.21.9.m9.7.7.1.1.2.2" xref="A1.F7.sf1.21.9.m9.7.7.1.1.2.1.cmml"><mo id="A1.F7.sf1.21.9.m9.7.7.1.1.2.2.1" stretchy="false" xref="A1.F7.sf1.21.9.m9.7.7.1.1.2.1.cmml">{</mo><mn id="A1.F7.sf1.21.9.m9.1.1" xref="A1.F7.sf1.21.9.m9.1.1.cmml">1</mn><mo id="A1.F7.sf1.21.9.m9.7.7.1.1.2.2.2" xref="A1.F7.sf1.21.9.m9.7.7.1.1.2.1.cmml">,</mo><mn id="A1.F7.sf1.21.9.m9.2.2" xref="A1.F7.sf1.21.9.m9.2.2.cmml">3</mn><mo id="A1.F7.sf1.21.9.m9.7.7.1.1.2.2.3" xref="A1.F7.sf1.21.9.m9.7.7.1.1.2.1.cmml">,</mo><mn id="A1.F7.sf1.21.9.m9.3.3" xref="A1.F7.sf1.21.9.m9.3.3.cmml">5</mn><mo id="A1.F7.sf1.21.9.m9.7.7.1.1.2.2.4" stretchy="false" xref="A1.F7.sf1.21.9.m9.7.7.1.1.2.1.cmml">}</mo></mrow><mo id="A1.F7.sf1.21.9.m9.7.7.1.1.1" stretchy="false" xref="A1.F7.sf1.21.9.m9.7.7.1.1.1.cmml">→</mo><mn id="A1.F7.sf1.21.9.m9.7.7.1.1.3" xref="A1.F7.sf1.21.9.m9.7.7.1.1.3.cmml">1</mn></mrow><mo id="A1.F7.sf1.21.9.m9.8.8.2.3" xref="A1.F7.sf1.21.9.m9.8.8.3a.cmml">,</mo><mrow id="A1.F7.sf1.21.9.m9.8.8.2.2" xref="A1.F7.sf1.21.9.m9.8.8.2.2.cmml"><mrow id="A1.F7.sf1.21.9.m9.8.8.2.2.2.2" xref="A1.F7.sf1.21.9.m9.8.8.2.2.2.1.cmml"><mo id="A1.F7.sf1.21.9.m9.8.8.2.2.2.2.1" stretchy="false" xref="A1.F7.sf1.21.9.m9.8.8.2.2.2.1.cmml">{</mo><mn id="A1.F7.sf1.21.9.m9.4.4" xref="A1.F7.sf1.21.9.m9.4.4.cmml">2</mn><mo id="A1.F7.sf1.21.9.m9.8.8.2.2.2.2.2" xref="A1.F7.sf1.21.9.m9.8.8.2.2.2.1.cmml">,</mo><mn id="A1.F7.sf1.21.9.m9.5.5" xref="A1.F7.sf1.21.9.m9.5.5.cmml">4</mn><mo id="A1.F7.sf1.21.9.m9.8.8.2.2.2.2.3" xref="A1.F7.sf1.21.9.m9.8.8.2.2.2.1.cmml">,</mo><mn id="A1.F7.sf1.21.9.m9.6.6" xref="A1.F7.sf1.21.9.m9.6.6.cmml">6</mn><mo id="A1.F7.sf1.21.9.m9.8.8.2.2.2.2.4" stretchy="false" xref="A1.F7.sf1.21.9.m9.8.8.2.2.2.1.cmml">}</mo></mrow><mo id="A1.F7.sf1.21.9.m9.8.8.2.2.1" stretchy="false" xref="A1.F7.sf1.21.9.m9.8.8.2.2.1.cmml">→</mo><mn id="A1.F7.sf1.21.9.m9.8.8.2.2.3" xref="A1.F7.sf1.21.9.m9.8.8.2.2.3.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf1.21.9.m9.8c"><apply id="A1.F7.sf1.21.9.m9.8.8.3.cmml" xref="A1.F7.sf1.21.9.m9.8.8.2"><csymbol cd="ambiguous" id="A1.F7.sf1.21.9.m9.8.8.3a.cmml" xref="A1.F7.sf1.21.9.m9.8.8.2.3">formulae-sequence</csymbol><apply id="A1.F7.sf1.21.9.m9.7.7.1.1.cmml" xref="A1.F7.sf1.21.9.m9.7.7.1.1"><ci id="A1.F7.sf1.21.9.m9.7.7.1.1.1.cmml" xref="A1.F7.sf1.21.9.m9.7.7.1.1.1">→</ci><set id="A1.F7.sf1.21.9.m9.7.7.1.1.2.1.cmml" xref="A1.F7.sf1.21.9.m9.7.7.1.1.2.2"><cn id="A1.F7.sf1.21.9.m9.1.1.cmml" type="integer" xref="A1.F7.sf1.21.9.m9.1.1">1</cn><cn id="A1.F7.sf1.21.9.m9.2.2.cmml" type="integer" xref="A1.F7.sf1.21.9.m9.2.2">3</cn><cn id="A1.F7.sf1.21.9.m9.3.3.cmml" type="integer" xref="A1.F7.sf1.21.9.m9.3.3">5</cn></set><cn id="A1.F7.sf1.21.9.m9.7.7.1.1.3.cmml" type="integer" xref="A1.F7.sf1.21.9.m9.7.7.1.1.3">1</cn></apply><apply id="A1.F7.sf1.21.9.m9.8.8.2.2.cmml" xref="A1.F7.sf1.21.9.m9.8.8.2.2"><ci id="A1.F7.sf1.21.9.m9.8.8.2.2.1.cmml" xref="A1.F7.sf1.21.9.m9.8.8.2.2.1">→</ci><set id="A1.F7.sf1.21.9.m9.8.8.2.2.2.1.cmml" xref="A1.F7.sf1.21.9.m9.8.8.2.2.2.2"><cn id="A1.F7.sf1.21.9.m9.4.4.cmml" type="integer" xref="A1.F7.sf1.21.9.m9.4.4">2</cn><cn id="A1.F7.sf1.21.9.m9.5.5.cmml" type="integer" xref="A1.F7.sf1.21.9.m9.5.5">4</cn><cn id="A1.F7.sf1.21.9.m9.6.6.cmml" type="integer" xref="A1.F7.sf1.21.9.m9.6.6">6</cn></set><cn id="A1.F7.sf1.21.9.m9.8.8.2.2.3.cmml" type="integer" xref="A1.F7.sf1.21.9.m9.8.8.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf1.21.9.m9.8d">\{1,3,5\}\to 1,\{2,4,6\}\to 0</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf1.21.9.m9.8e">{ 1 , 3 , 5 } → 1 , { 2 , 4 , 6 } → 0</annotation></semantics></math> satisfies weight <math alttext="2\cdot(c^{2}-1)+2\cdot 1=2c^{2}" class="ltx_Math" display="inline" id="A1.F7.sf1.22.10.m10.1"><semantics id="A1.F7.sf1.22.10.m10.1b"><mrow id="A1.F7.sf1.22.10.m10.1.1" xref="A1.F7.sf1.22.10.m10.1.1.cmml"><mrow id="A1.F7.sf1.22.10.m10.1.1.1" xref="A1.F7.sf1.22.10.m10.1.1.1.cmml"><mrow id="A1.F7.sf1.22.10.m10.1.1.1.1" xref="A1.F7.sf1.22.10.m10.1.1.1.1.cmml"><mn id="A1.F7.sf1.22.10.m10.1.1.1.1.3" xref="A1.F7.sf1.22.10.m10.1.1.1.1.3.cmml">2</mn><mo id="A1.F7.sf1.22.10.m10.1.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="A1.F7.sf1.22.10.m10.1.1.1.1.2.cmml">⋅</mo><mrow id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.cmml"><mo id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.2" stretchy="false" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.cmml"><msup id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.2" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.2.cmml"><mi id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.2.2" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.2.2.cmml">c</mi><mn id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.2.3" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.2.3.cmml">2</mn></msup><mo id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.1" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.3" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.3" stretchy="false" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.F7.sf1.22.10.m10.1.1.1.2" xref="A1.F7.sf1.22.10.m10.1.1.1.2.cmml">+</mo><mrow id="A1.F7.sf1.22.10.m10.1.1.1.3" xref="A1.F7.sf1.22.10.m10.1.1.1.3.cmml"><mn id="A1.F7.sf1.22.10.m10.1.1.1.3.2" xref="A1.F7.sf1.22.10.m10.1.1.1.3.2.cmml">2</mn><mo id="A1.F7.sf1.22.10.m10.1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="A1.F7.sf1.22.10.m10.1.1.1.3.1.cmml">⋅</mo><mn id="A1.F7.sf1.22.10.m10.1.1.1.3.3" xref="A1.F7.sf1.22.10.m10.1.1.1.3.3.cmml">1</mn></mrow></mrow><mo id="A1.F7.sf1.22.10.m10.1.1.2" xref="A1.F7.sf1.22.10.m10.1.1.2.cmml">=</mo><mrow id="A1.F7.sf1.22.10.m10.1.1.3" xref="A1.F7.sf1.22.10.m10.1.1.3.cmml"><mn id="A1.F7.sf1.22.10.m10.1.1.3.2" xref="A1.F7.sf1.22.10.m10.1.1.3.2.cmml">2</mn><mo id="A1.F7.sf1.22.10.m10.1.1.3.1" xref="A1.F7.sf1.22.10.m10.1.1.3.1.cmml">⁢</mo><msup id="A1.F7.sf1.22.10.m10.1.1.3.3" xref="A1.F7.sf1.22.10.m10.1.1.3.3.cmml"><mi id="A1.F7.sf1.22.10.m10.1.1.3.3.2" xref="A1.F7.sf1.22.10.m10.1.1.3.3.2.cmml">c</mi><mn id="A1.F7.sf1.22.10.m10.1.1.3.3.3" xref="A1.F7.sf1.22.10.m10.1.1.3.3.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf1.22.10.m10.1c"><apply id="A1.F7.sf1.22.10.m10.1.1.cmml" xref="A1.F7.sf1.22.10.m10.1.1"><eq id="A1.F7.sf1.22.10.m10.1.1.2.cmml" xref="A1.F7.sf1.22.10.m10.1.1.2"></eq><apply id="A1.F7.sf1.22.10.m10.1.1.1.cmml" xref="A1.F7.sf1.22.10.m10.1.1.1"><plus id="A1.F7.sf1.22.10.m10.1.1.1.2.cmml" xref="A1.F7.sf1.22.10.m10.1.1.1.2"></plus><apply id="A1.F7.sf1.22.10.m10.1.1.1.1.cmml" xref="A1.F7.sf1.22.10.m10.1.1.1.1"><ci id="A1.F7.sf1.22.10.m10.1.1.1.1.2.cmml" xref="A1.F7.sf1.22.10.m10.1.1.1.1.2">⋅</ci><cn id="A1.F7.sf1.22.10.m10.1.1.1.1.3.cmml" type="integer" xref="A1.F7.sf1.22.10.m10.1.1.1.1.3">2</cn><apply id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.cmml" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1"><minus id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.1.cmml" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.1"></minus><apply id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.2.cmml" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.2.1.cmml" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.2.2.cmml" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.2.2">𝑐</ci><cn id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.2.3">2</cn></apply><cn id="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.3.cmml" type="integer" xref="A1.F7.sf1.22.10.m10.1.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="A1.F7.sf1.22.10.m10.1.1.1.3.cmml" xref="A1.F7.sf1.22.10.m10.1.1.1.3"><ci id="A1.F7.sf1.22.10.m10.1.1.1.3.1.cmml" xref="A1.F7.sf1.22.10.m10.1.1.1.3.1">⋅</ci><cn id="A1.F7.sf1.22.10.m10.1.1.1.3.2.cmml" type="integer" xref="A1.F7.sf1.22.10.m10.1.1.1.3.2">2</cn><cn id="A1.F7.sf1.22.10.m10.1.1.1.3.3.cmml" type="integer" xref="A1.F7.sf1.22.10.m10.1.1.1.3.3">1</cn></apply></apply><apply id="A1.F7.sf1.22.10.m10.1.1.3.cmml" xref="A1.F7.sf1.22.10.m10.1.1.3"><times id="A1.F7.sf1.22.10.m10.1.1.3.1.cmml" xref="A1.F7.sf1.22.10.m10.1.1.3.1"></times><cn id="A1.F7.sf1.22.10.m10.1.1.3.2.cmml" type="integer" xref="A1.F7.sf1.22.10.m10.1.1.3.2">2</cn><apply id="A1.F7.sf1.22.10.m10.1.1.3.3.cmml" xref="A1.F7.sf1.22.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="A1.F7.sf1.22.10.m10.1.1.3.3.1.cmml" xref="A1.F7.sf1.22.10.m10.1.1.3.3">superscript</csymbol><ci id="A1.F7.sf1.22.10.m10.1.1.3.3.2.cmml" xref="A1.F7.sf1.22.10.m10.1.1.3.3.2">𝑐</ci><cn id="A1.F7.sf1.22.10.m10.1.1.3.3.3.cmml" type="integer" xref="A1.F7.sf1.22.10.m10.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf1.22.10.m10.1d">2\cdot(c^{2}-1)+2\cdot 1=2c^{2}</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf1.22.10.m10.1e">2 ⋅ ( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) + 2 ⋅ 1 = 2 italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. An oblivious assignment <math alttext="\{1,2\}\to p,\{3,4\}\to q,\{5,6\}\to r" class="ltx_Math" display="inline" id="A1.F7.sf1.23.11.m11.8"><semantics id="A1.F7.sf1.23.11.m11.8b"><mrow id="A1.F7.sf1.23.11.m11.8.8.2" xref="A1.F7.sf1.23.11.m11.8.8.3.cmml"><mrow id="A1.F7.sf1.23.11.m11.7.7.1.1" xref="A1.F7.sf1.23.11.m11.7.7.1.1.cmml"><mrow id="A1.F7.sf1.23.11.m11.7.7.1.1.2.2" xref="A1.F7.sf1.23.11.m11.7.7.1.1.2.1.cmml"><mo id="A1.F7.sf1.23.11.m11.7.7.1.1.2.2.1" stretchy="false" xref="A1.F7.sf1.23.11.m11.7.7.1.1.2.1.cmml">{</mo><mn id="A1.F7.sf1.23.11.m11.1.1" xref="A1.F7.sf1.23.11.m11.1.1.cmml">1</mn><mo id="A1.F7.sf1.23.11.m11.7.7.1.1.2.2.2" xref="A1.F7.sf1.23.11.m11.7.7.1.1.2.1.cmml">,</mo><mn id="A1.F7.sf1.23.11.m11.2.2" xref="A1.F7.sf1.23.11.m11.2.2.cmml">2</mn><mo id="A1.F7.sf1.23.11.m11.7.7.1.1.2.2.3" stretchy="false" xref="A1.F7.sf1.23.11.m11.7.7.1.1.2.1.cmml">}</mo></mrow><mo id="A1.F7.sf1.23.11.m11.7.7.1.1.1" stretchy="false" xref="A1.F7.sf1.23.11.m11.7.7.1.1.1.cmml">→</mo><mi id="A1.F7.sf1.23.11.m11.7.7.1.1.3" xref="A1.F7.sf1.23.11.m11.7.7.1.1.3.cmml">p</mi></mrow><mo id="A1.F7.sf1.23.11.m11.8.8.2.3" xref="A1.F7.sf1.23.11.m11.8.8.3a.cmml">,</mo><mrow id="A1.F7.sf1.23.11.m11.8.8.2.2.2" xref="A1.F7.sf1.23.11.m11.8.8.2.2.3.cmml"><mrow id="A1.F7.sf1.23.11.m11.8.8.2.2.1.1" xref="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.cmml"><mrow id="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.2.2" xref="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.2.1.cmml"><mo id="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.2.2.1" stretchy="false" xref="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.2.1.cmml">{</mo><mn id="A1.F7.sf1.23.11.m11.3.3" xref="A1.F7.sf1.23.11.m11.3.3.cmml">3</mn><mo id="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.2.2.2" xref="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.2.1.cmml">,</mo><mn id="A1.F7.sf1.23.11.m11.4.4" xref="A1.F7.sf1.23.11.m11.4.4.cmml">4</mn><mo id="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.2.2.3" stretchy="false" xref="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.2.1.cmml">}</mo></mrow><mo id="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.1" stretchy="false" xref="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.1.cmml">→</mo><mi id="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.3" xref="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.3.cmml">q</mi></mrow><mo id="A1.F7.sf1.23.11.m11.8.8.2.2.2.3" xref="A1.F7.sf1.23.11.m11.8.8.2.2.3a.cmml">,</mo><mrow id="A1.F7.sf1.23.11.m11.8.8.2.2.2.2" xref="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.cmml"><mrow id="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.2.2" xref="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.2.1.cmml"><mo id="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.2.2.1" stretchy="false" xref="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.2.1.cmml">{</mo><mn id="A1.F7.sf1.23.11.m11.5.5" xref="A1.F7.sf1.23.11.m11.5.5.cmml">5</mn><mo id="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.2.2.2" xref="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.2.1.cmml">,</mo><mn id="A1.F7.sf1.23.11.m11.6.6" xref="A1.F7.sf1.23.11.m11.6.6.cmml">6</mn><mo id="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.2.2.3" stretchy="false" xref="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.2.1.cmml">}</mo></mrow><mo id="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.1" stretchy="false" xref="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.1.cmml">→</mo><mi id="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.3" xref="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.3.cmml">r</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf1.23.11.m11.8c"><apply id="A1.F7.sf1.23.11.m11.8.8.3.cmml" xref="A1.F7.sf1.23.11.m11.8.8.2"><csymbol cd="ambiguous" id="A1.F7.sf1.23.11.m11.8.8.3a.cmml" xref="A1.F7.sf1.23.11.m11.8.8.2.3">formulae-sequence</csymbol><apply id="A1.F7.sf1.23.11.m11.7.7.1.1.cmml" xref="A1.F7.sf1.23.11.m11.7.7.1.1"><ci id="A1.F7.sf1.23.11.m11.7.7.1.1.1.cmml" xref="A1.F7.sf1.23.11.m11.7.7.1.1.1">→</ci><set id="A1.F7.sf1.23.11.m11.7.7.1.1.2.1.cmml" xref="A1.F7.sf1.23.11.m11.7.7.1.1.2.2"><cn id="A1.F7.sf1.23.11.m11.1.1.cmml" type="integer" xref="A1.F7.sf1.23.11.m11.1.1">1</cn><cn id="A1.F7.sf1.23.11.m11.2.2.cmml" type="integer" xref="A1.F7.sf1.23.11.m11.2.2">2</cn></set><ci id="A1.F7.sf1.23.11.m11.7.7.1.1.3.cmml" xref="A1.F7.sf1.23.11.m11.7.7.1.1.3">𝑝</ci></apply><apply id="A1.F7.sf1.23.11.m11.8.8.2.2.3.cmml" xref="A1.F7.sf1.23.11.m11.8.8.2.2.2"><csymbol cd="ambiguous" id="A1.F7.sf1.23.11.m11.8.8.2.2.3a.cmml" xref="A1.F7.sf1.23.11.m11.8.8.2.2.2.3">formulae-sequence</csymbol><apply id="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.cmml" xref="A1.F7.sf1.23.11.m11.8.8.2.2.1.1"><ci id="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.1.cmml" xref="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.1">→</ci><set id="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.2.1.cmml" xref="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.2.2"><cn id="A1.F7.sf1.23.11.m11.3.3.cmml" type="integer" xref="A1.F7.sf1.23.11.m11.3.3">3</cn><cn id="A1.F7.sf1.23.11.m11.4.4.cmml" type="integer" xref="A1.F7.sf1.23.11.m11.4.4">4</cn></set><ci id="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.3.cmml" xref="A1.F7.sf1.23.11.m11.8.8.2.2.1.1.3">𝑞</ci></apply><apply id="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.cmml" xref="A1.F7.sf1.23.11.m11.8.8.2.2.2.2"><ci id="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.1.cmml" xref="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.1">→</ci><set id="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.2.1.cmml" xref="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.2.2"><cn id="A1.F7.sf1.23.11.m11.5.5.cmml" type="integer" xref="A1.F7.sf1.23.11.m11.5.5">5</cn><cn id="A1.F7.sf1.23.11.m11.6.6.cmml" type="integer" xref="A1.F7.sf1.23.11.m11.6.6">6</cn></set><ci id="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.3.cmml" xref="A1.F7.sf1.23.11.m11.8.8.2.2.2.2.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf1.23.11.m11.8d">\{1,2\}\to p,\{3,4\}\to q,\{5,6\}\to r</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf1.23.11.m11.8e">{ 1 , 2 } → italic_p , { 3 , 4 } → italic_q , { 5 , 6 } → italic_r</annotation></semantics></math> satisfies weight <math alttext="p(1-p)(c+1)+p(1-r)(c^{2}-1)+r(1-r)(c^{2}-1)+r(1-q)(c^{2}-1)+q(1-q)(c+1)" class="ltx_Math" display="inline" id="A1.F7.sf1.24.12.m12.10"><semantics id="A1.F7.sf1.24.12.m12.10b"><mrow id="A1.F7.sf1.24.12.m12.10.10" xref="A1.F7.sf1.24.12.m12.10.10.cmml"><mrow id="A1.F7.sf1.24.12.m12.2.2.2" xref="A1.F7.sf1.24.12.m12.2.2.2.cmml"><mi id="A1.F7.sf1.24.12.m12.2.2.2.4" xref="A1.F7.sf1.24.12.m12.2.2.2.4.cmml">p</mi><mo id="A1.F7.sf1.24.12.m12.2.2.2.3" xref="A1.F7.sf1.24.12.m12.2.2.2.3.cmml">⁢</mo><mrow id="A1.F7.sf1.24.12.m12.1.1.1.1.1" xref="A1.F7.sf1.24.12.m12.1.1.1.1.1.1.cmml"><mo id="A1.F7.sf1.24.12.m12.1.1.1.1.1.2" stretchy="false" xref="A1.F7.sf1.24.12.m12.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.F7.sf1.24.12.m12.1.1.1.1.1.1" xref="A1.F7.sf1.24.12.m12.1.1.1.1.1.1.cmml"><mn id="A1.F7.sf1.24.12.m12.1.1.1.1.1.1.2" xref="A1.F7.sf1.24.12.m12.1.1.1.1.1.1.2.cmml">1</mn><mo id="A1.F7.sf1.24.12.m12.1.1.1.1.1.1.1" xref="A1.F7.sf1.24.12.m12.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.F7.sf1.24.12.m12.1.1.1.1.1.1.3" xref="A1.F7.sf1.24.12.m12.1.1.1.1.1.1.3.cmml">p</mi></mrow><mo id="A1.F7.sf1.24.12.m12.1.1.1.1.1.3" stretchy="false" xref="A1.F7.sf1.24.12.m12.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A1.F7.sf1.24.12.m12.2.2.2.3b" 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style="fill:none"></path><g stroke-dasharray="none" stroke-dashoffset="0.0pt" stroke-linejoin="miter" transform="matrix(-0.42262 0.90631 -0.90631 -0.42262 481.05 99.65)"><path d="M 6.4 0 C 5.61 0.28 2.16 1.87 0 3.6 L 0 -3.6 C 2.16 -1.87 5.61 -0.28 6.4 0 Z"></path></g><g fill="#E6E6E6" stroke="#000000"><path d="M 482.2 51.76 h 14.61 v 14.59 h -14.61 Z"></path></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 486.81 56.37)"><foreignobject height="5.36" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="5.39"><math alttext="c" class="ltx_Math" display="inline" id="A1.F7.sf2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="A1.F7.sf2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="A1.F7.sf2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" mathsize="90%" xref="A1.F7.sf2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">c</mi><annotation-xml 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stroke="#000000"><path d="M 397.82 50.43 h 31.14 v 17.25 h -31.14 Z"></path></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 402.43 55.04)"><foreignobject height="8.03" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="21.92"><math alttext="c-1" class="ltx_Math" display="inline" id="A1.F7.sf2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="A1.F7.sf2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mrow id="A1.F7.sf2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="A1.F7.sf2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="A1.F7.sf2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" mathsize="90%" xref="A1.F7.sf2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">c</mi><mo id="A1.F7.sf2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1" mathsize="90%" 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id="A1.F7.sf2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" type="integer" xref="A1.F7.sf2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">c-1</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_c - 1</annotation></semantics></math></foreignobject></g></g></g></svg> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A1.F7.sf2.30.13.1" style="font-size:90%;">(b)</span> </span><em class="ltx_emph ltx_font_italic" id="A1.F7.sf2.31.14" style="font-size:90%;">Parameter:</em><span class="ltx_text" id="A1.F7.sf2.24.12" style="font-size:90%;"> <math alttext="c>1" class="ltx_Math" display="inline" id="A1.F7.sf2.13.1.m1.1"><semantics id="A1.F7.sf2.13.1.m1.1b"><mrow id="A1.F7.sf2.13.1.m1.1.1" xref="A1.F7.sf2.13.1.m1.1.1.cmml"><mi id="A1.F7.sf2.13.1.m1.1.1.2" xref="A1.F7.sf2.13.1.m1.1.1.2.cmml">c</mi><mo id="A1.F7.sf2.13.1.m1.1.1.1" xref="A1.F7.sf2.13.1.m1.1.1.1.cmml">></mo><mn id="A1.F7.sf2.13.1.m1.1.1.3" xref="A1.F7.sf2.13.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf2.13.1.m1.1c"><apply id="A1.F7.sf2.13.1.m1.1.1.cmml" xref="A1.F7.sf2.13.1.m1.1.1"><gt id="A1.F7.sf2.13.1.m1.1.1.1.cmml" xref="A1.F7.sf2.13.1.m1.1.1.1"></gt><ci id="A1.F7.sf2.13.1.m1.1.1.2.cmml" xref="A1.F7.sf2.13.1.m1.1.1.2">𝑐</ci><cn id="A1.F7.sf2.13.1.m1.1.1.3.cmml" type="integer" xref="A1.F7.sf2.13.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf2.13.1.m1.1d">c>1</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf2.13.1.m1.1e">italic_c > 1</annotation></semantics></math> (“<math alttext="G_{2}" class="ltx_Math" display="inline" id="A1.F7.sf2.14.2.m2.1"><semantics id="A1.F7.sf2.14.2.m2.1b"><msub id="A1.F7.sf2.14.2.m2.1.1" xref="A1.F7.sf2.14.2.m2.1.1.cmml"><mi id="A1.F7.sf2.14.2.m2.1.1.2" xref="A1.F7.sf2.14.2.m2.1.1.2.cmml">G</mi><mn id="A1.F7.sf2.14.2.m2.1.1.3" xref="A1.F7.sf2.14.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A1.F7.sf2.14.2.m2.1c"><apply id="A1.F7.sf2.14.2.m2.1.1.cmml" xref="A1.F7.sf2.14.2.m2.1.1"><csymbol cd="ambiguous" id="A1.F7.sf2.14.2.m2.1.1.1.cmml" xref="A1.F7.sf2.14.2.m2.1.1">subscript</csymbol><ci id="A1.F7.sf2.14.2.m2.1.1.2.cmml" xref="A1.F7.sf2.14.2.m2.1.1.2">𝐺</ci><cn id="A1.F7.sf2.14.2.m2.1.1.3.cmml" type="integer" xref="A1.F7.sf2.14.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf2.14.2.m2.1d">G_{2}</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf2.14.2.m2.1e">italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>” from <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">FJ15</span>]</cite>). The <span class="ltx_text" id="A1.F7.sf2.24.12.1" style="background-color:#ABDEE6;">LIGHT BLUE</span> vertex (<math alttext="2" class="ltx_Math" display="inline" id="A1.F7.sf2.15.3.m3.1"><semantics id="A1.F7.sf2.15.3.m3.1b"><mn id="A1.F7.sf2.15.3.m3.1.1" xref="A1.F7.sf2.15.3.m3.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="A1.F7.sf2.15.3.m3.1c"><cn id="A1.F7.sf2.15.3.m3.1.1.cmml" type="integer" xref="A1.F7.sf2.15.3.m3.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf2.15.3.m3.1d">2</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf2.15.3.m3.1e">2</annotation></semantics></math>) has bias <math alttext="+(c-1)/(c+1)" class="ltx_Math" display="inline" id="A1.F7.sf2.16.4.m4.2"><semantics id="A1.F7.sf2.16.4.m4.2b"><mrow id="A1.F7.sf2.16.4.m4.2.2" xref="A1.F7.sf2.16.4.m4.2.2.cmml"><mo id="A1.F7.sf2.16.4.m4.2.2b" xref="A1.F7.sf2.16.4.m4.2.2.cmml">+</mo><mrow id="A1.F7.sf2.16.4.m4.2.2.2" xref="A1.F7.sf2.16.4.m4.2.2.2.cmml"><mrow id="A1.F7.sf2.16.4.m4.1.1.1.1.1" xref="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.cmml"><mo id="A1.F7.sf2.16.4.m4.1.1.1.1.1.2" stretchy="false" xref="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.F7.sf2.16.4.m4.1.1.1.1.1.1" xref="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.cmml"><mi id="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.2" xref="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.2.cmml">c</mi><mo id="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.1" xref="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.1.cmml">−</mo><mn id="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.3" xref="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="A1.F7.sf2.16.4.m4.1.1.1.1.1.3" stretchy="false" xref="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A1.F7.sf2.16.4.m4.2.2.2.3" xref="A1.F7.sf2.16.4.m4.2.2.2.3.cmml">/</mo><mrow id="A1.F7.sf2.16.4.m4.2.2.2.2.1" xref="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.cmml"><mo id="A1.F7.sf2.16.4.m4.2.2.2.2.1.2" stretchy="false" xref="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.cmml">(</mo><mrow id="A1.F7.sf2.16.4.m4.2.2.2.2.1.1" xref="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.cmml"><mi id="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.2" xref="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.2.cmml">c</mi><mo id="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.1" xref="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.1.cmml">+</mo><mn id="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.3" xref="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.3.cmml">1</mn></mrow><mo id="A1.F7.sf2.16.4.m4.2.2.2.2.1.3" stretchy="false" xref="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf2.16.4.m4.2c"><apply id="A1.F7.sf2.16.4.m4.2.2.cmml" xref="A1.F7.sf2.16.4.m4.2.2"><plus id="A1.F7.sf2.16.4.m4.2.2.3.cmml" xref="A1.F7.sf2.16.4.m4.2.2"></plus><apply id="A1.F7.sf2.16.4.m4.2.2.2.cmml" xref="A1.F7.sf2.16.4.m4.2.2.2"><divide id="A1.F7.sf2.16.4.m4.2.2.2.3.cmml" xref="A1.F7.sf2.16.4.m4.2.2.2.3"></divide><apply id="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.cmml" xref="A1.F7.sf2.16.4.m4.1.1.1.1.1"><minus id="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.1.cmml" xref="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.1"></minus><ci id="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.2.cmml" xref="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.2">𝑐</ci><cn id="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="A1.F7.sf2.16.4.m4.1.1.1.1.1.1.3">1</cn></apply><apply id="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.cmml" xref="A1.F7.sf2.16.4.m4.2.2.2.2.1"><plus id="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.1.cmml" xref="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.1"></plus><ci id="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.2.cmml" xref="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.2">𝑐</ci><cn id="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.3.cmml" type="integer" xref="A1.F7.sf2.16.4.m4.2.2.2.2.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf2.16.4.m4.2d">+(c-1)/(c+1)</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf2.16.4.m4.2e">+ ( italic_c - 1 ) / ( italic_c + 1 )</annotation></semantics></math>. The <span class="ltx_text" id="A1.F7.sf2.24.12.2" style="background-color:#CBAACB;">PURPLE</span> vertices (<math alttext="\{1,4\}" class="ltx_Math" display="inline" id="A1.F7.sf2.17.5.m5.2"><semantics id="A1.F7.sf2.17.5.m5.2b"><mrow id="A1.F7.sf2.17.5.m5.2.3.2" xref="A1.F7.sf2.17.5.m5.2.3.1.cmml"><mo id="A1.F7.sf2.17.5.m5.2.3.2.1" stretchy="false" xref="A1.F7.sf2.17.5.m5.2.3.1.cmml">{</mo><mn id="A1.F7.sf2.17.5.m5.1.1" xref="A1.F7.sf2.17.5.m5.1.1.cmml">1</mn><mo id="A1.F7.sf2.17.5.m5.2.3.2.2" xref="A1.F7.sf2.17.5.m5.2.3.1.cmml">,</mo><mn id="A1.F7.sf2.17.5.m5.2.2" xref="A1.F7.sf2.17.5.m5.2.2.cmml">4</mn><mo id="A1.F7.sf2.17.5.m5.2.3.2.3" stretchy="false" xref="A1.F7.sf2.17.5.m5.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf2.17.5.m5.2c"><set id="A1.F7.sf2.17.5.m5.2.3.1.cmml" xref="A1.F7.sf2.17.5.m5.2.3.2"><cn id="A1.F7.sf2.17.5.m5.1.1.cmml" type="integer" xref="A1.F7.sf2.17.5.m5.1.1">1</cn><cn id="A1.F7.sf2.17.5.m5.2.2.cmml" type="integer" xref="A1.F7.sf2.17.5.m5.2.2">4</cn></set></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf2.17.5.m5.2d">\{1,4\}</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf2.17.5.m5.2e">{ 1 , 4 }</annotation></semantics></math>) have bias <math alttext="0" class="ltx_Math" display="inline" id="A1.F7.sf2.18.6.m6.1"><semantics id="A1.F7.sf2.18.6.m6.1b"><mn id="A1.F7.sf2.18.6.m6.1.1" xref="A1.F7.sf2.18.6.m6.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A1.F7.sf2.18.6.m6.1c"><cn id="A1.F7.sf2.18.6.m6.1.1.cmml" type="integer" xref="A1.F7.sf2.18.6.m6.1.1">0</cn></annotation-xml></semantics></math>. The <span class="ltx_text" id="A1.F7.sf2.24.12.3" style="background-color:#FEE1E8;">PINK</span> vertex (<math alttext="3" class="ltx_Math" display="inline" id="A1.F7.sf2.19.7.m7.1"><semantics id="A1.F7.sf2.19.7.m7.1b"><mn id="A1.F7.sf2.19.7.m7.1.1" xref="A1.F7.sf2.19.7.m7.1.1.cmml">3</mn><annotation-xml encoding="MathML-Content" id="A1.F7.sf2.19.7.m7.1c"><cn id="A1.F7.sf2.19.7.m7.1.1.cmml" type="integer" xref="A1.F7.sf2.19.7.m7.1.1">3</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf2.19.7.m7.1d">3</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf2.19.7.m7.1e">3</annotation></semantics></math>) has bias <math alttext="-(c-1)/(c+1)" class="ltx_Math" display="inline" id="A1.F7.sf2.20.8.m8.2"><semantics id="A1.F7.sf2.20.8.m8.2b"><mrow id="A1.F7.sf2.20.8.m8.2.2" xref="A1.F7.sf2.20.8.m8.2.2.cmml"><mo id="A1.F7.sf2.20.8.m8.2.2b" xref="A1.F7.sf2.20.8.m8.2.2.cmml">−</mo><mrow id="A1.F7.sf2.20.8.m8.2.2.2" xref="A1.F7.sf2.20.8.m8.2.2.2.cmml"><mrow id="A1.F7.sf2.20.8.m8.1.1.1.1.1" xref="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.cmml"><mo id="A1.F7.sf2.20.8.m8.1.1.1.1.1.2" stretchy="false" xref="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.F7.sf2.20.8.m8.1.1.1.1.1.1" xref="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.cmml"><mi id="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.2" xref="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.2.cmml">c</mi><mo id="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.1" xref="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.1.cmml">−</mo><mn id="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.3" xref="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="A1.F7.sf2.20.8.m8.1.1.1.1.1.3" stretchy="false" xref="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A1.F7.sf2.20.8.m8.2.2.2.3" xref="A1.F7.sf2.20.8.m8.2.2.2.3.cmml">/</mo><mrow id="A1.F7.sf2.20.8.m8.2.2.2.2.1" xref="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.cmml"><mo id="A1.F7.sf2.20.8.m8.2.2.2.2.1.2" stretchy="false" xref="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.cmml">(</mo><mrow id="A1.F7.sf2.20.8.m8.2.2.2.2.1.1" xref="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.cmml"><mi id="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.2" xref="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.2.cmml">c</mi><mo id="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.1" xref="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.1.cmml">+</mo><mn id="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.3" xref="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.3.cmml">1</mn></mrow><mo id="A1.F7.sf2.20.8.m8.2.2.2.2.1.3" stretchy="false" xref="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf2.20.8.m8.2c"><apply id="A1.F7.sf2.20.8.m8.2.2.cmml" xref="A1.F7.sf2.20.8.m8.2.2"><minus id="A1.F7.sf2.20.8.m8.2.2.3.cmml" xref="A1.F7.sf2.20.8.m8.2.2"></minus><apply id="A1.F7.sf2.20.8.m8.2.2.2.cmml" xref="A1.F7.sf2.20.8.m8.2.2.2"><divide id="A1.F7.sf2.20.8.m8.2.2.2.3.cmml" xref="A1.F7.sf2.20.8.m8.2.2.2.3"></divide><apply id="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.cmml" xref="A1.F7.sf2.20.8.m8.1.1.1.1.1"><minus id="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.1.cmml" xref="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.1"></minus><ci id="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.2.cmml" xref="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.2">𝑐</ci><cn id="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.3.cmml" type="integer" xref="A1.F7.sf2.20.8.m8.1.1.1.1.1.1.3">1</cn></apply><apply id="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.cmml" xref="A1.F7.sf2.20.8.m8.2.2.2.2.1"><plus id="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.1.cmml" xref="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.1"></plus><ci id="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.2.cmml" xref="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.2">𝑐</ci><cn id="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.3.cmml" type="integer" xref="A1.F7.sf2.20.8.m8.2.2.2.2.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf2.20.8.m8.2d">-(c-1)/(c+1)</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf2.20.8.m8.2e">- ( italic_c - 1 ) / ( italic_c + 1 )</annotation></semantics></math>. The assignment <math alttext="\{1,3\}\to 0,\{2,4\}\to 1" class="ltx_Math" display="inline" id="A1.F7.sf2.21.9.m9.6"><semantics id="A1.F7.sf2.21.9.m9.6b"><mrow id="A1.F7.sf2.21.9.m9.6.6.2" xref="A1.F7.sf2.21.9.m9.6.6.3.cmml"><mrow id="A1.F7.sf2.21.9.m9.5.5.1.1" xref="A1.F7.sf2.21.9.m9.5.5.1.1.cmml"><mrow id="A1.F7.sf2.21.9.m9.5.5.1.1.2.2" xref="A1.F7.sf2.21.9.m9.5.5.1.1.2.1.cmml"><mo id="A1.F7.sf2.21.9.m9.5.5.1.1.2.2.1" stretchy="false" xref="A1.F7.sf2.21.9.m9.5.5.1.1.2.1.cmml">{</mo><mn id="A1.F7.sf2.21.9.m9.1.1" xref="A1.F7.sf2.21.9.m9.1.1.cmml">1</mn><mo id="A1.F7.sf2.21.9.m9.5.5.1.1.2.2.2" xref="A1.F7.sf2.21.9.m9.5.5.1.1.2.1.cmml">,</mo><mn id="A1.F7.sf2.21.9.m9.2.2" xref="A1.F7.sf2.21.9.m9.2.2.cmml">3</mn><mo id="A1.F7.sf2.21.9.m9.5.5.1.1.2.2.3" stretchy="false" xref="A1.F7.sf2.21.9.m9.5.5.1.1.2.1.cmml">}</mo></mrow><mo id="A1.F7.sf2.21.9.m9.5.5.1.1.1" stretchy="false" xref="A1.F7.sf2.21.9.m9.5.5.1.1.1.cmml">→</mo><mn id="A1.F7.sf2.21.9.m9.5.5.1.1.3" xref="A1.F7.sf2.21.9.m9.5.5.1.1.3.cmml">0</mn></mrow><mo id="A1.F7.sf2.21.9.m9.6.6.2.3" xref="A1.F7.sf2.21.9.m9.6.6.3a.cmml">,</mo><mrow id="A1.F7.sf2.21.9.m9.6.6.2.2" xref="A1.F7.sf2.21.9.m9.6.6.2.2.cmml"><mrow id="A1.F7.sf2.21.9.m9.6.6.2.2.2.2" xref="A1.F7.sf2.21.9.m9.6.6.2.2.2.1.cmml"><mo id="A1.F7.sf2.21.9.m9.6.6.2.2.2.2.1" stretchy="false" xref="A1.F7.sf2.21.9.m9.6.6.2.2.2.1.cmml">{</mo><mn id="A1.F7.sf2.21.9.m9.3.3" xref="A1.F7.sf2.21.9.m9.3.3.cmml">2</mn><mo id="A1.F7.sf2.21.9.m9.6.6.2.2.2.2.2" xref="A1.F7.sf2.21.9.m9.6.6.2.2.2.1.cmml">,</mo><mn id="A1.F7.sf2.21.9.m9.4.4" xref="A1.F7.sf2.21.9.m9.4.4.cmml">4</mn><mo id="A1.F7.sf2.21.9.m9.6.6.2.2.2.2.3" stretchy="false" xref="A1.F7.sf2.21.9.m9.6.6.2.2.2.1.cmml">}</mo></mrow><mo id="A1.F7.sf2.21.9.m9.6.6.2.2.1" stretchy="false" xref="A1.F7.sf2.21.9.m9.6.6.2.2.1.cmml">→</mo><mn id="A1.F7.sf2.21.9.m9.6.6.2.2.3" xref="A1.F7.sf2.21.9.m9.6.6.2.2.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf2.21.9.m9.6c"><apply id="A1.F7.sf2.21.9.m9.6.6.3.cmml" xref="A1.F7.sf2.21.9.m9.6.6.2"><csymbol cd="ambiguous" id="A1.F7.sf2.21.9.m9.6.6.3a.cmml" xref="A1.F7.sf2.21.9.m9.6.6.2.3">formulae-sequence</csymbol><apply id="A1.F7.sf2.21.9.m9.5.5.1.1.cmml" xref="A1.F7.sf2.21.9.m9.5.5.1.1"><ci id="A1.F7.sf2.21.9.m9.5.5.1.1.1.cmml" xref="A1.F7.sf2.21.9.m9.5.5.1.1.1">→</ci><set id="A1.F7.sf2.21.9.m9.5.5.1.1.2.1.cmml" xref="A1.F7.sf2.21.9.m9.5.5.1.1.2.2"><cn id="A1.F7.sf2.21.9.m9.1.1.cmml" type="integer" xref="A1.F7.sf2.21.9.m9.1.1">1</cn><cn id="A1.F7.sf2.21.9.m9.2.2.cmml" type="integer" xref="A1.F7.sf2.21.9.m9.2.2">3</cn></set><cn id="A1.F7.sf2.21.9.m9.5.5.1.1.3.cmml" type="integer" xref="A1.F7.sf2.21.9.m9.5.5.1.1.3">0</cn></apply><apply id="A1.F7.sf2.21.9.m9.6.6.2.2.cmml" xref="A1.F7.sf2.21.9.m9.6.6.2.2"><ci id="A1.F7.sf2.21.9.m9.6.6.2.2.1.cmml" xref="A1.F7.sf2.21.9.m9.6.6.2.2.1">→</ci><set id="A1.F7.sf2.21.9.m9.6.6.2.2.2.1.cmml" xref="A1.F7.sf2.21.9.m9.6.6.2.2.2.2"><cn id="A1.F7.sf2.21.9.m9.3.3.cmml" type="integer" xref="A1.F7.sf2.21.9.m9.3.3">2</cn><cn id="A1.F7.sf2.21.9.m9.4.4.cmml" type="integer" xref="A1.F7.sf2.21.9.m9.4.4">4</cn></set><cn id="A1.F7.sf2.21.9.m9.6.6.2.2.3.cmml" type="integer" xref="A1.F7.sf2.21.9.m9.6.6.2.2.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf2.21.9.m9.6d">\{1,3\}\to 0,\{2,4\}\to 1</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf2.21.9.m9.6e">{ 1 , 3 } → 0 , { 2 , 4 } → 1</annotation></semantics></math> satisfies weight <math alttext="2c" class="ltx_Math" display="inline" id="A1.F7.sf2.22.10.m10.1"><semantics id="A1.F7.sf2.22.10.m10.1b"><mrow id="A1.F7.sf2.22.10.m10.1.1" xref="A1.F7.sf2.22.10.m10.1.1.cmml"><mn id="A1.F7.sf2.22.10.m10.1.1.2" xref="A1.F7.sf2.22.10.m10.1.1.2.cmml">2</mn><mo id="A1.F7.sf2.22.10.m10.1.1.1" xref="A1.F7.sf2.22.10.m10.1.1.1.cmml">⁢</mo><mi id="A1.F7.sf2.22.10.m10.1.1.3" xref="A1.F7.sf2.22.10.m10.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf2.22.10.m10.1c"><apply id="A1.F7.sf2.22.10.m10.1.1.cmml" xref="A1.F7.sf2.22.10.m10.1.1"><times id="A1.F7.sf2.22.10.m10.1.1.1.cmml" xref="A1.F7.sf2.22.10.m10.1.1.1"></times><cn id="A1.F7.sf2.22.10.m10.1.1.2.cmml" type="integer" xref="A1.F7.sf2.22.10.m10.1.1.2">2</cn><ci id="A1.F7.sf2.22.10.m10.1.1.3.cmml" xref="A1.F7.sf2.22.10.m10.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf2.22.10.m10.1d">2c</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf2.22.10.m10.1e">2 italic_c</annotation></semantics></math>. An oblivious assignment <math alttext="2\to p,3\to q,\{1,4\}\to r" class="ltx_Math" display="inline" id="A1.F7.sf2.23.11.m11.4"><semantics id="A1.F7.sf2.23.11.m11.4b"><mrow id="A1.F7.sf2.23.11.m11.4.4.2" xref="A1.F7.sf2.23.11.m11.4.4.3.cmml"><mrow id="A1.F7.sf2.23.11.m11.3.3.1.1" xref="A1.F7.sf2.23.11.m11.3.3.1.1.cmml"><mn id="A1.F7.sf2.23.11.m11.3.3.1.1.2" xref="A1.F7.sf2.23.11.m11.3.3.1.1.2.cmml">2</mn><mo id="A1.F7.sf2.23.11.m11.3.3.1.1.1" stretchy="false" xref="A1.F7.sf2.23.11.m11.3.3.1.1.1.cmml">→</mo><mi id="A1.F7.sf2.23.11.m11.3.3.1.1.3" xref="A1.F7.sf2.23.11.m11.3.3.1.1.3.cmml">p</mi></mrow><mo id="A1.F7.sf2.23.11.m11.4.4.2.3" xref="A1.F7.sf2.23.11.m11.4.4.3a.cmml">,</mo><mrow id="A1.F7.sf2.23.11.m11.4.4.2.2.2" xref="A1.F7.sf2.23.11.m11.4.4.2.2.3.cmml"><mrow id="A1.F7.sf2.23.11.m11.4.4.2.2.1.1" xref="A1.F7.sf2.23.11.m11.4.4.2.2.1.1.cmml"><mn id="A1.F7.sf2.23.11.m11.4.4.2.2.1.1.2" xref="A1.F7.sf2.23.11.m11.4.4.2.2.1.1.2.cmml">3</mn><mo id="A1.F7.sf2.23.11.m11.4.4.2.2.1.1.1" stretchy="false" xref="A1.F7.sf2.23.11.m11.4.4.2.2.1.1.1.cmml">→</mo><mi id="A1.F7.sf2.23.11.m11.4.4.2.2.1.1.3" xref="A1.F7.sf2.23.11.m11.4.4.2.2.1.1.3.cmml">q</mi></mrow><mo id="A1.F7.sf2.23.11.m11.4.4.2.2.2.3" xref="A1.F7.sf2.23.11.m11.4.4.2.2.3a.cmml">,</mo><mrow id="A1.F7.sf2.23.11.m11.4.4.2.2.2.2" xref="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.cmml"><mrow id="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.2.2" xref="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.2.1.cmml"><mo id="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.2.2.1" stretchy="false" xref="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.2.1.cmml">{</mo><mn id="A1.F7.sf2.23.11.m11.1.1" xref="A1.F7.sf2.23.11.m11.1.1.cmml">1</mn><mo id="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.2.2.2" xref="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.2.1.cmml">,</mo><mn id="A1.F7.sf2.23.11.m11.2.2" xref="A1.F7.sf2.23.11.m11.2.2.cmml">4</mn><mo id="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.2.2.3" stretchy="false" xref="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.2.1.cmml">}</mo></mrow><mo id="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.1" stretchy="false" xref="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.1.cmml">→</mo><mi id="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.3" xref="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.3.cmml">r</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf2.23.11.m11.4c"><apply id="A1.F7.sf2.23.11.m11.4.4.3.cmml" xref="A1.F7.sf2.23.11.m11.4.4.2"><csymbol cd="ambiguous" id="A1.F7.sf2.23.11.m11.4.4.3a.cmml" xref="A1.F7.sf2.23.11.m11.4.4.2.3">formulae-sequence</csymbol><apply id="A1.F7.sf2.23.11.m11.3.3.1.1.cmml" xref="A1.F7.sf2.23.11.m11.3.3.1.1"><ci id="A1.F7.sf2.23.11.m11.3.3.1.1.1.cmml" xref="A1.F7.sf2.23.11.m11.3.3.1.1.1">→</ci><cn id="A1.F7.sf2.23.11.m11.3.3.1.1.2.cmml" type="integer" xref="A1.F7.sf2.23.11.m11.3.3.1.1.2">2</cn><ci id="A1.F7.sf2.23.11.m11.3.3.1.1.3.cmml" xref="A1.F7.sf2.23.11.m11.3.3.1.1.3">𝑝</ci></apply><apply id="A1.F7.sf2.23.11.m11.4.4.2.2.3.cmml" xref="A1.F7.sf2.23.11.m11.4.4.2.2.2"><csymbol cd="ambiguous" id="A1.F7.sf2.23.11.m11.4.4.2.2.3a.cmml" xref="A1.F7.sf2.23.11.m11.4.4.2.2.2.3">formulae-sequence</csymbol><apply id="A1.F7.sf2.23.11.m11.4.4.2.2.1.1.cmml" xref="A1.F7.sf2.23.11.m11.4.4.2.2.1.1"><ci id="A1.F7.sf2.23.11.m11.4.4.2.2.1.1.1.cmml" xref="A1.F7.sf2.23.11.m11.4.4.2.2.1.1.1">→</ci><cn id="A1.F7.sf2.23.11.m11.4.4.2.2.1.1.2.cmml" type="integer" xref="A1.F7.sf2.23.11.m11.4.4.2.2.1.1.2">3</cn><ci id="A1.F7.sf2.23.11.m11.4.4.2.2.1.1.3.cmml" xref="A1.F7.sf2.23.11.m11.4.4.2.2.1.1.3">𝑞</ci></apply><apply id="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.cmml" xref="A1.F7.sf2.23.11.m11.4.4.2.2.2.2"><ci id="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.1.cmml" xref="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.1">→</ci><set id="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.2.1.cmml" xref="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.2.2"><cn id="A1.F7.sf2.23.11.m11.1.1.cmml" type="integer" xref="A1.F7.sf2.23.11.m11.1.1">1</cn><cn id="A1.F7.sf2.23.11.m11.2.2.cmml" type="integer" xref="A1.F7.sf2.23.11.m11.2.2">4</cn></set><ci id="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.3.cmml" xref="A1.F7.sf2.23.11.m11.4.4.2.2.2.2.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf2.23.11.m11.4d">2\to p,3\to q,\{1,4\}\to r</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf2.23.11.m11.4e">2 → italic_p , 3 → italic_q , { 1 , 4 } → italic_r</annotation></semantics></math> satisfies weight <math alttext="p(1-r)(c)+r(1-p)+r(1-r)(c-1)+r(1-q)(c)+q(1-r)" class="ltx_Math" display="inline" id="A1.F7.sf2.24.12.m12.8"><semantics id="A1.F7.sf2.24.12.m12.8b"><mrow id="A1.F7.sf2.24.12.m12.8.8" xref="A1.F7.sf2.24.12.m12.8.8.cmml"><mrow id="A1.F7.sf2.24.12.m12.3.3.1" xref="A1.F7.sf2.24.12.m12.3.3.1.cmml"><mi id="A1.F7.sf2.24.12.m12.3.3.1.3" xref="A1.F7.sf2.24.12.m12.3.3.1.3.cmml">p</mi><mo id="A1.F7.sf2.24.12.m12.3.3.1.2" xref="A1.F7.sf2.24.12.m12.3.3.1.2.cmml">⁢</mo><mrow id="A1.F7.sf2.24.12.m12.3.3.1.1.1" xref="A1.F7.sf2.24.12.m12.3.3.1.1.1.1.cmml"><mo id="A1.F7.sf2.24.12.m12.3.3.1.1.1.2" stretchy="false" xref="A1.F7.sf2.24.12.m12.3.3.1.1.1.1.cmml">(</mo><mrow id="A1.F7.sf2.24.12.m12.3.3.1.1.1.1" xref="A1.F7.sf2.24.12.m12.3.3.1.1.1.1.cmml"><mn id="A1.F7.sf2.24.12.m12.3.3.1.1.1.1.2" xref="A1.F7.sf2.24.12.m12.3.3.1.1.1.1.2.cmml">1</mn><mo id="A1.F7.sf2.24.12.m12.3.3.1.1.1.1.1" xref="A1.F7.sf2.24.12.m12.3.3.1.1.1.1.1.cmml">−</mo><mi id="A1.F7.sf2.24.12.m12.3.3.1.1.1.1.3" xref="A1.F7.sf2.24.12.m12.3.3.1.1.1.1.3.cmml">r</mi></mrow><mo id="A1.F7.sf2.24.12.m12.3.3.1.1.1.3" stretchy="false" xref="A1.F7.sf2.24.12.m12.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="A1.F7.sf2.24.12.m12.3.3.1.2b" xref="A1.F7.sf2.24.12.m12.3.3.1.2.cmml">⁢</mo><mrow id="A1.F7.sf2.24.12.m12.3.3.1.4.2" xref="A1.F7.sf2.24.12.m12.3.3.1.cmml"><mo id="A1.F7.sf2.24.12.m12.3.3.1.4.2.1" stretchy="false" xref="A1.F7.sf2.24.12.m12.3.3.1.cmml">(</mo><mi id="A1.F7.sf2.24.12.m12.1.1" xref="A1.F7.sf2.24.12.m12.1.1.cmml">c</mi><mo id="A1.F7.sf2.24.12.m12.3.3.1.4.2.2" stretchy="false" xref="A1.F7.sf2.24.12.m12.3.3.1.cmml">)</mo></mrow></mrow><mo id="A1.F7.sf2.24.12.m12.8.8.7" xref="A1.F7.sf2.24.12.m12.8.8.7.cmml">+</mo><mrow id="A1.F7.sf2.24.12.m12.4.4.2" xref="A1.F7.sf2.24.12.m12.4.4.2.cmml"><mi id="A1.F7.sf2.24.12.m12.4.4.2.3" xref="A1.F7.sf2.24.12.m12.4.4.2.3.cmml">r</mi><mo id="A1.F7.sf2.24.12.m12.4.4.2.2" xref="A1.F7.sf2.24.12.m12.4.4.2.2.cmml">⁢</mo><mrow id="A1.F7.sf2.24.12.m12.4.4.2.1.1" xref="A1.F7.sf2.24.12.m12.4.4.2.1.1.1.cmml"><mo id="A1.F7.sf2.24.12.m12.4.4.2.1.1.2" stretchy="false" xref="A1.F7.sf2.24.12.m12.4.4.2.1.1.1.cmml">(</mo><mrow id="A1.F7.sf2.24.12.m12.4.4.2.1.1.1" xref="A1.F7.sf2.24.12.m12.4.4.2.1.1.1.cmml"><mn id="A1.F7.sf2.24.12.m12.4.4.2.1.1.1.2" xref="A1.F7.sf2.24.12.m12.4.4.2.1.1.1.2.cmml">1</mn><mo id="A1.F7.sf2.24.12.m12.4.4.2.1.1.1.1" xref="A1.F7.sf2.24.12.m12.4.4.2.1.1.1.1.cmml">−</mo><mi id="A1.F7.sf2.24.12.m12.4.4.2.1.1.1.3" xref="A1.F7.sf2.24.12.m12.4.4.2.1.1.1.3.cmml">p</mi></mrow><mo id="A1.F7.sf2.24.12.m12.4.4.2.1.1.3" stretchy="false" xref="A1.F7.sf2.24.12.m12.4.4.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.F7.sf2.24.12.m12.8.8.7b" xref="A1.F7.sf2.24.12.m12.8.8.7.cmml">+</mo><mrow id="A1.F7.sf2.24.12.m12.6.6.4" xref="A1.F7.sf2.24.12.m12.6.6.4.cmml"><mi id="A1.F7.sf2.24.12.m12.6.6.4.4" xref="A1.F7.sf2.24.12.m12.6.6.4.4.cmml">r</mi><mo id="A1.F7.sf2.24.12.m12.6.6.4.3" xref="A1.F7.sf2.24.12.m12.6.6.4.3.cmml">⁢</mo><mrow id="A1.F7.sf2.24.12.m12.5.5.3.1.1" xref="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.cmml"><mo id="A1.F7.sf2.24.12.m12.5.5.3.1.1.2" stretchy="false" xref="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.cmml">(</mo><mrow id="A1.F7.sf2.24.12.m12.5.5.3.1.1.1" xref="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.cmml"><mn id="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.2" xref="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.2.cmml">1</mn><mo id="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.1" xref="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.1.cmml">−</mo><mi id="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.3" xref="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.3.cmml">r</mi></mrow><mo id="A1.F7.sf2.24.12.m12.5.5.3.1.1.3" stretchy="false" xref="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.cmml">)</mo></mrow><mo id="A1.F7.sf2.24.12.m12.6.6.4.3b" xref="A1.F7.sf2.24.12.m12.6.6.4.3.cmml">⁢</mo><mrow id="A1.F7.sf2.24.12.m12.6.6.4.2.1" xref="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.cmml"><mo id="A1.F7.sf2.24.12.m12.6.6.4.2.1.2" stretchy="false" xref="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.cmml">(</mo><mrow id="A1.F7.sf2.24.12.m12.6.6.4.2.1.1" xref="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.cmml"><mi id="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.2" xref="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.2.cmml">c</mi><mo id="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.1" xref="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.1.cmml">−</mo><mn id="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.3" xref="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.3.cmml">1</mn></mrow><mo id="A1.F7.sf2.24.12.m12.6.6.4.2.1.3" stretchy="false" xref="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.cmml">)</mo></mrow></mrow><mo id="A1.F7.sf2.24.12.m12.8.8.7c" xref="A1.F7.sf2.24.12.m12.8.8.7.cmml">+</mo><mrow id="A1.F7.sf2.24.12.m12.7.7.5" xref="A1.F7.sf2.24.12.m12.7.7.5.cmml"><mi id="A1.F7.sf2.24.12.m12.7.7.5.3" xref="A1.F7.sf2.24.12.m12.7.7.5.3.cmml">r</mi><mo id="A1.F7.sf2.24.12.m12.7.7.5.2" xref="A1.F7.sf2.24.12.m12.7.7.5.2.cmml">⁢</mo><mrow id="A1.F7.sf2.24.12.m12.7.7.5.1.1" xref="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.cmml"><mo id="A1.F7.sf2.24.12.m12.7.7.5.1.1.2" stretchy="false" xref="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.cmml">(</mo><mrow id="A1.F7.sf2.24.12.m12.7.7.5.1.1.1" xref="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.cmml"><mn id="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.2" xref="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.2.cmml">1</mn><mo id="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.1" xref="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.1.cmml">−</mo><mi id="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.3" xref="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.3.cmml">q</mi></mrow><mo id="A1.F7.sf2.24.12.m12.7.7.5.1.1.3" stretchy="false" xref="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.cmml">)</mo></mrow><mo id="A1.F7.sf2.24.12.m12.7.7.5.2b" xref="A1.F7.sf2.24.12.m12.7.7.5.2.cmml">⁢</mo><mrow id="A1.F7.sf2.24.12.m12.7.7.5.4.2" xref="A1.F7.sf2.24.12.m12.7.7.5.cmml"><mo id="A1.F7.sf2.24.12.m12.7.7.5.4.2.1" stretchy="false" xref="A1.F7.sf2.24.12.m12.7.7.5.cmml">(</mo><mi id="A1.F7.sf2.24.12.m12.2.2" xref="A1.F7.sf2.24.12.m12.2.2.cmml">c</mi><mo id="A1.F7.sf2.24.12.m12.7.7.5.4.2.2" stretchy="false" xref="A1.F7.sf2.24.12.m12.7.7.5.cmml">)</mo></mrow></mrow><mo id="A1.F7.sf2.24.12.m12.8.8.7d" xref="A1.F7.sf2.24.12.m12.8.8.7.cmml">+</mo><mrow id="A1.F7.sf2.24.12.m12.8.8.6" xref="A1.F7.sf2.24.12.m12.8.8.6.cmml"><mi id="A1.F7.sf2.24.12.m12.8.8.6.3" xref="A1.F7.sf2.24.12.m12.8.8.6.3.cmml">q</mi><mo id="A1.F7.sf2.24.12.m12.8.8.6.2" xref="A1.F7.sf2.24.12.m12.8.8.6.2.cmml">⁢</mo><mrow id="A1.F7.sf2.24.12.m12.8.8.6.1.1" xref="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.cmml"><mo id="A1.F7.sf2.24.12.m12.8.8.6.1.1.2" stretchy="false" xref="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.cmml">(</mo><mrow id="A1.F7.sf2.24.12.m12.8.8.6.1.1.1" xref="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.cmml"><mn id="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.2" xref="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.2.cmml">1</mn><mo id="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.1" xref="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.1.cmml">−</mo><mi id="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.3" xref="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.3.cmml">r</mi></mrow><mo id="A1.F7.sf2.24.12.m12.8.8.6.1.1.3" stretchy="false" xref="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.F7.sf2.24.12.m12.8c"><apply id="A1.F7.sf2.24.12.m12.8.8.cmml" xref="A1.F7.sf2.24.12.m12.8.8"><plus id="A1.F7.sf2.24.12.m12.8.8.7.cmml" xref="A1.F7.sf2.24.12.m12.8.8.7"></plus><apply id="A1.F7.sf2.24.12.m12.3.3.1.cmml" xref="A1.F7.sf2.24.12.m12.3.3.1"><times id="A1.F7.sf2.24.12.m12.3.3.1.2.cmml" xref="A1.F7.sf2.24.12.m12.3.3.1.2"></times><ci id="A1.F7.sf2.24.12.m12.3.3.1.3.cmml" xref="A1.F7.sf2.24.12.m12.3.3.1.3">𝑝</ci><apply id="A1.F7.sf2.24.12.m12.3.3.1.1.1.1.cmml" 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id="A1.F7.sf2.24.12.m12.4.4.2.1.1.1.3.cmml" xref="A1.F7.sf2.24.12.m12.4.4.2.1.1.1.3">𝑝</ci></apply></apply><apply id="A1.F7.sf2.24.12.m12.6.6.4.cmml" xref="A1.F7.sf2.24.12.m12.6.6.4"><times id="A1.F7.sf2.24.12.m12.6.6.4.3.cmml" xref="A1.F7.sf2.24.12.m12.6.6.4.3"></times><ci id="A1.F7.sf2.24.12.m12.6.6.4.4.cmml" xref="A1.F7.sf2.24.12.m12.6.6.4.4">𝑟</ci><apply id="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.cmml" xref="A1.F7.sf2.24.12.m12.5.5.3.1.1"><minus id="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.1.cmml" xref="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.1"></minus><cn id="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.2.cmml" type="integer" xref="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.2">1</cn><ci id="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.3.cmml" xref="A1.F7.sf2.24.12.m12.5.5.3.1.1.1.3">𝑟</ci></apply><apply id="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.cmml" xref="A1.F7.sf2.24.12.m12.6.6.4.2.1"><minus id="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.1.cmml" xref="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.1"></minus><ci id="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.2.cmml" xref="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.2">𝑐</ci><cn id="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.3.cmml" type="integer" xref="A1.F7.sf2.24.12.m12.6.6.4.2.1.1.3">1</cn></apply></apply><apply id="A1.F7.sf2.24.12.m12.7.7.5.cmml" xref="A1.F7.sf2.24.12.m12.7.7.5"><times id="A1.F7.sf2.24.12.m12.7.7.5.2.cmml" xref="A1.F7.sf2.24.12.m12.7.7.5.2"></times><ci id="A1.F7.sf2.24.12.m12.7.7.5.3.cmml" xref="A1.F7.sf2.24.12.m12.7.7.5.3">𝑟</ci><apply id="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.cmml" xref="A1.F7.sf2.24.12.m12.7.7.5.1.1"><minus id="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.1.cmml" xref="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.1"></minus><cn id="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.2.cmml" type="integer" xref="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.2">1</cn><ci id="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.3.cmml" xref="A1.F7.sf2.24.12.m12.7.7.5.1.1.1.3">𝑞</ci></apply><ci id="A1.F7.sf2.24.12.m12.2.2.cmml" xref="A1.F7.sf2.24.12.m12.2.2">𝑐</ci></apply><apply id="A1.F7.sf2.24.12.m12.8.8.6.cmml" xref="A1.F7.sf2.24.12.m12.8.8.6"><times id="A1.F7.sf2.24.12.m12.8.8.6.2.cmml" xref="A1.F7.sf2.24.12.m12.8.8.6.2"></times><ci id="A1.F7.sf2.24.12.m12.8.8.6.3.cmml" xref="A1.F7.sf2.24.12.m12.8.8.6.3">𝑞</ci><apply id="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.cmml" xref="A1.F7.sf2.24.12.m12.8.8.6.1.1"><minus id="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.1.cmml" xref="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.1"></minus><cn id="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.2.cmml" type="integer" xref="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.2">1</cn><ci id="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.3.cmml" xref="A1.F7.sf2.24.12.m12.8.8.6.1.1.1.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F7.sf2.24.12.m12.8d">p(1-r)(c)+r(1-p)+r(1-r)(c-1)+r(1-q)(c)+q(1-r)</annotation><annotation encoding="application/x-llamapun" id="A1.F7.sf2.24.12.m12.8e">italic_p ( 1 - italic_r ) ( italic_c ) + italic_r ( 1 - italic_p ) + italic_r ( 1 - italic_r ) ( italic_c - 1 ) + italic_r ( 1 - italic_q ) ( italic_c ) + italic_q ( 1 - italic_r )</annotation></semantics></math>.</span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A1.F7.3.1.1" style="font-size:90%;">Figure 7</span>: </span><span class="ltx_text" id="A1.F7.4.2" style="font-size:90%;">The pair of graphs used to prove a lower bound against antisymmetric selection functions by <span class="ltx_ERROR undefined" id="A1.F7.4.2.1">\textcite</span>FJ15.</span></figcaption> </figure> <div class="ltx_proof" id="A1.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A1.1.p1"> <p class="ltx_p" id="A1.1.p1.6">Let <math alttext="0\leq\lambda\leq 1" class="ltx_Math" display="inline" id="A1.1.p1.1.m1.1"><semantics id="A1.1.p1.1.m1.1a"><mrow id="A1.1.p1.1.m1.1.1" xref="A1.1.p1.1.m1.1.1.cmml"><mn id="A1.1.p1.1.m1.1.1.2" xref="A1.1.p1.1.m1.1.1.2.cmml">0</mn><mo id="A1.1.p1.1.m1.1.1.3" xref="A1.1.p1.1.m1.1.1.3.cmml">≤</mo><mi id="A1.1.p1.1.m1.1.1.4" xref="A1.1.p1.1.m1.1.1.4.cmml">λ</mi><mo id="A1.1.p1.1.m1.1.1.5" xref="A1.1.p1.1.m1.1.1.5.cmml">≤</mo><mn id="A1.1.p1.1.m1.1.1.6" xref="A1.1.p1.1.m1.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.1.m1.1b"><apply id="A1.1.p1.1.m1.1.1.cmml" xref="A1.1.p1.1.m1.1.1"><and id="A1.1.p1.1.m1.1.1a.cmml" xref="A1.1.p1.1.m1.1.1"></and><apply id="A1.1.p1.1.m1.1.1b.cmml" xref="A1.1.p1.1.m1.1.1"><leq id="A1.1.p1.1.m1.1.1.3.cmml" xref="A1.1.p1.1.m1.1.1.3"></leq><cn id="A1.1.p1.1.m1.1.1.2.cmml" type="integer" xref="A1.1.p1.1.m1.1.1.2">0</cn><ci id="A1.1.p1.1.m1.1.1.4.cmml" xref="A1.1.p1.1.m1.1.1.4">𝜆</ci></apply><apply id="A1.1.p1.1.m1.1.1c.cmml" xref="A1.1.p1.1.m1.1.1"><leq id="A1.1.p1.1.m1.1.1.5.cmml" xref="A1.1.p1.1.m1.1.1.5"></leq><share href="https://arxiv.org/html/2411.12976v1#A1.1.p1.1.m1.1.1.4.cmml" id="A1.1.p1.1.m1.1.1d.cmml" xref="A1.1.p1.1.m1.1.1"></share><cn id="A1.1.p1.1.m1.1.1.6.cmml" type="integer" xref="A1.1.p1.1.m1.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.1.m1.1c">0\leq\lambda\leq 1</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.1.m1.1d">0 ≤ italic_λ ≤ 1</annotation></semantics></math> and <math alttext="c>1" class="ltx_Math" display="inline" id="A1.1.p1.2.m2.1"><semantics id="A1.1.p1.2.m2.1a"><mrow id="A1.1.p1.2.m2.1.1" xref="A1.1.p1.2.m2.1.1.cmml"><mi id="A1.1.p1.2.m2.1.1.2" xref="A1.1.p1.2.m2.1.1.2.cmml">c</mi><mo id="A1.1.p1.2.m2.1.1.1" xref="A1.1.p1.2.m2.1.1.1.cmml">></mo><mn id="A1.1.p1.2.m2.1.1.3" xref="A1.1.p1.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.2.m2.1b"><apply id="A1.1.p1.2.m2.1.1.cmml" xref="A1.1.p1.2.m2.1.1"><gt id="A1.1.p1.2.m2.1.1.1.cmml" xref="A1.1.p1.2.m2.1.1.1"></gt><ci id="A1.1.p1.2.m2.1.1.2.cmml" xref="A1.1.p1.2.m2.1.1.2">𝑐</ci><cn id="A1.1.p1.2.m2.1.1.3.cmml" type="integer" xref="A1.1.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.2.m2.1c">c>1</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.2.m2.1d">italic_c > 1</annotation></semantics></math> be two TBD constants. Let <math alttext="G" class="ltx_Math" display="inline" id="A1.1.p1.3.m3.1"><semantics id="A1.1.p1.3.m3.1a"><mi id="A1.1.p1.3.m3.1.1" xref="A1.1.p1.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A1.1.p1.3.m3.1b"><ci id="A1.1.p1.3.m3.1.1.cmml" xref="A1.1.p1.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.3.m3.1d">italic_G</annotation></semantics></math> denote a weighted disjoint union of the graphs in <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#A1.F7.sf1" title="In Figure 7 ‣ Appendix A Recap: The prior lower bound of \textciteFJ15 (Theorem 1.2) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Figs.</span> <span class="ltx_text ltx_ref_tag">7(a)</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#A1.F7.sf2" title="Figure 7(b) ‣ Figure 7 ‣ Appendix A Recap: The prior lower bound of \textciteFJ15 (Theorem 1.2) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">7(b)</span></a>, weighted by <math alttext="\lambda" class="ltx_Math" display="inline" id="A1.1.p1.4.m4.1"><semantics id="A1.1.p1.4.m4.1a"><mi id="A1.1.p1.4.m4.1.1" xref="A1.1.p1.4.m4.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="A1.1.p1.4.m4.1b"><ci id="A1.1.p1.4.m4.1.1.cmml" xref="A1.1.p1.4.m4.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.4.m4.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.4.m4.1d">italic_λ</annotation></semantics></math> and <math alttext="1-\lambda" class="ltx_Math" display="inline" id="A1.1.p1.5.m5.1"><semantics id="A1.1.p1.5.m5.1a"><mrow id="A1.1.p1.5.m5.1.1" xref="A1.1.p1.5.m5.1.1.cmml"><mn id="A1.1.p1.5.m5.1.1.2" xref="A1.1.p1.5.m5.1.1.2.cmml">1</mn><mo id="A1.1.p1.5.m5.1.1.1" xref="A1.1.p1.5.m5.1.1.1.cmml">−</mo><mi id="A1.1.p1.5.m5.1.1.3" xref="A1.1.p1.5.m5.1.1.3.cmml">λ</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.5.m5.1b"><apply id="A1.1.p1.5.m5.1.1.cmml" xref="A1.1.p1.5.m5.1.1"><minus id="A1.1.p1.5.m5.1.1.1.cmml" xref="A1.1.p1.5.m5.1.1.1"></minus><cn id="A1.1.p1.5.m5.1.1.2.cmml" type="integer" xref="A1.1.p1.5.m5.1.1.2">1</cn><ci id="A1.1.p1.5.m5.1.1.3.cmml" xref="A1.1.p1.5.m5.1.1.3">𝜆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.5.m5.1c">1-\lambda</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.5.m5.1d">1 - italic_λ</annotation></semantics></math>, respectively. As in the figures’ caption, <math alttext="G" class="ltx_Math" display="inline" id="A1.1.p1.6.m6.1"><semantics id="A1.1.p1.6.m6.1a"><mi id="A1.1.p1.6.m6.1.1" xref="A1.1.p1.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A1.1.p1.6.m6.1b"><ci id="A1.1.p1.6.m6.1.1.cmml" xref="A1.1.p1.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.6.m6.1d">italic_G</annotation></semantics></math> has a cut satisfying weight</p> <table class="ltx_equation ltx_eqn_table" id="A1.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="\lambda(2c^{2})+(1-\lambda)(2c)." class="ltx_Math" display="block" id="A1.Ex1.m1.1"><semantics id="A1.Ex1.m1.1a"><mrow id="A1.Ex1.m1.1.1.1" xref="A1.Ex1.m1.1.1.1.1.cmml"><mrow id="A1.Ex1.m1.1.1.1.1" xref="A1.Ex1.m1.1.1.1.1.cmml"><mrow id="A1.Ex1.m1.1.1.1.1.1" xref="A1.Ex1.m1.1.1.1.1.1.cmml"><mi id="A1.Ex1.m1.1.1.1.1.1.3" xref="A1.Ex1.m1.1.1.1.1.1.3.cmml">λ</mi><mo id="A1.Ex1.m1.1.1.1.1.1.2" xref="A1.Ex1.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A1.Ex1.m1.1.1.1.1.1.1.1" xref="A1.Ex1.m1.1.1.1.1.1.1.1.1.cmml"><mo id="A1.Ex1.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="A1.Ex1.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.Ex1.m1.1.1.1.1.1.1.1.1" xref="A1.Ex1.m1.1.1.1.1.1.1.1.1.cmml"><mn id="A1.Ex1.m1.1.1.1.1.1.1.1.1.2" xref="A1.Ex1.m1.1.1.1.1.1.1.1.1.2.cmml">2</mn><mo id="A1.Ex1.m1.1.1.1.1.1.1.1.1.1" xref="A1.Ex1.m1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><msup id="A1.Ex1.m1.1.1.1.1.1.1.1.1.3" xref="A1.Ex1.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="A1.Ex1.m1.1.1.1.1.1.1.1.1.3.2" xref="A1.Ex1.m1.1.1.1.1.1.1.1.1.3.2.cmml">c</mi><mn id="A1.Ex1.m1.1.1.1.1.1.1.1.1.3.3" xref="A1.Ex1.m1.1.1.1.1.1.1.1.1.3.3.cmml">2</mn></msup></mrow><mo id="A1.Ex1.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="A1.Ex1.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.Ex1.m1.1.1.1.1.4" xref="A1.Ex1.m1.1.1.1.1.4.cmml">+</mo><mrow id="A1.Ex1.m1.1.1.1.1.3" xref="A1.Ex1.m1.1.1.1.1.3.cmml"><mrow id="A1.Ex1.m1.1.1.1.1.2.1.1" xref="A1.Ex1.m1.1.1.1.1.2.1.1.1.cmml"><mo id="A1.Ex1.m1.1.1.1.1.2.1.1.2" stretchy="false" xref="A1.Ex1.m1.1.1.1.1.2.1.1.1.cmml">(</mo><mrow id="A1.Ex1.m1.1.1.1.1.2.1.1.1" xref="A1.Ex1.m1.1.1.1.1.2.1.1.1.cmml"><mn id="A1.Ex1.m1.1.1.1.1.2.1.1.1.2" xref="A1.Ex1.m1.1.1.1.1.2.1.1.1.2.cmml">1</mn><mo id="A1.Ex1.m1.1.1.1.1.2.1.1.1.1" xref="A1.Ex1.m1.1.1.1.1.2.1.1.1.1.cmml">−</mo><mi id="A1.Ex1.m1.1.1.1.1.2.1.1.1.3" xref="A1.Ex1.m1.1.1.1.1.2.1.1.1.3.cmml">λ</mi></mrow><mo id="A1.Ex1.m1.1.1.1.1.2.1.1.3" stretchy="false" xref="A1.Ex1.m1.1.1.1.1.2.1.1.1.cmml">)</mo></mrow><mo id="A1.Ex1.m1.1.1.1.1.3.3" xref="A1.Ex1.m1.1.1.1.1.3.3.cmml">⁢</mo><mrow id="A1.Ex1.m1.1.1.1.1.3.2.1" xref="A1.Ex1.m1.1.1.1.1.3.2.1.1.cmml"><mo id="A1.Ex1.m1.1.1.1.1.3.2.1.2" stretchy="false" xref="A1.Ex1.m1.1.1.1.1.3.2.1.1.cmml">(</mo><mrow id="A1.Ex1.m1.1.1.1.1.3.2.1.1" xref="A1.Ex1.m1.1.1.1.1.3.2.1.1.cmml"><mn id="A1.Ex1.m1.1.1.1.1.3.2.1.1.2" xref="A1.Ex1.m1.1.1.1.1.3.2.1.1.2.cmml">2</mn><mo id="A1.Ex1.m1.1.1.1.1.3.2.1.1.1" xref="A1.Ex1.m1.1.1.1.1.3.2.1.1.1.cmml">⁢</mo><mi id="A1.Ex1.m1.1.1.1.1.3.2.1.1.3" xref="A1.Ex1.m1.1.1.1.1.3.2.1.1.3.cmml">c</mi></mrow><mo id="A1.Ex1.m1.1.1.1.1.3.2.1.3" stretchy="false" xref="A1.Ex1.m1.1.1.1.1.3.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A1.Ex1.m1.1.1.1.2" lspace="0em" xref="A1.Ex1.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex1.m1.1b"><apply id="A1.Ex1.m1.1.1.1.1.cmml" xref="A1.Ex1.m1.1.1.1"><plus id="A1.Ex1.m1.1.1.1.1.4.cmml" xref="A1.Ex1.m1.1.1.1.1.4"></plus><apply id="A1.Ex1.m1.1.1.1.1.1.cmml" xref="A1.Ex1.m1.1.1.1.1.1"><times id="A1.Ex1.m1.1.1.1.1.1.2.cmml" xref="A1.Ex1.m1.1.1.1.1.1.2"></times><ci id="A1.Ex1.m1.1.1.1.1.1.3.cmml" xref="A1.Ex1.m1.1.1.1.1.1.3">𝜆</ci><apply id="A1.Ex1.m1.1.1.1.1.1.1.1.1.cmml" xref="A1.Ex1.m1.1.1.1.1.1.1.1"><times id="A1.Ex1.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A1.Ex1.m1.1.1.1.1.1.1.1.1.1"></times><cn id="A1.Ex1.m1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A1.Ex1.m1.1.1.1.1.1.1.1.1.2">2</cn><apply id="A1.Ex1.m1.1.1.1.1.1.1.1.1.3.cmml" xref="A1.Ex1.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex1.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="A1.Ex1.m1.1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="A1.Ex1.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex1.m1.1.1.1.1.1.1.1.1.3.2">𝑐</ci><cn id="A1.Ex1.m1.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="A1.Ex1.m1.1.1.1.1.1.1.1.1.3.3">2</cn></apply></apply></apply><apply id="A1.Ex1.m1.1.1.1.1.3.cmml" xref="A1.Ex1.m1.1.1.1.1.3"><times id="A1.Ex1.m1.1.1.1.1.3.3.cmml" xref="A1.Ex1.m1.1.1.1.1.3.3"></times><apply id="A1.Ex1.m1.1.1.1.1.2.1.1.1.cmml" xref="A1.Ex1.m1.1.1.1.1.2.1.1"><minus id="A1.Ex1.m1.1.1.1.1.2.1.1.1.1.cmml" xref="A1.Ex1.m1.1.1.1.1.2.1.1.1.1"></minus><cn id="A1.Ex1.m1.1.1.1.1.2.1.1.1.2.cmml" type="integer" xref="A1.Ex1.m1.1.1.1.1.2.1.1.1.2">1</cn><ci id="A1.Ex1.m1.1.1.1.1.2.1.1.1.3.cmml" xref="A1.Ex1.m1.1.1.1.1.2.1.1.1.3">𝜆</ci></apply><apply id="A1.Ex1.m1.1.1.1.1.3.2.1.1.cmml" xref="A1.Ex1.m1.1.1.1.1.3.2.1"><times id="A1.Ex1.m1.1.1.1.1.3.2.1.1.1.cmml" xref="A1.Ex1.m1.1.1.1.1.3.2.1.1.1"></times><cn id="A1.Ex1.m1.1.1.1.1.3.2.1.1.2.cmml" type="integer" xref="A1.Ex1.m1.1.1.1.1.3.2.1.1.2">2</cn><ci id="A1.Ex1.m1.1.1.1.1.3.2.1.1.3.cmml" xref="A1.Ex1.m1.1.1.1.1.3.2.1.1.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex1.m1.1c">\lambda(2c^{2})+(1-\lambda)(2c).</annotation><annotation encoding="application/x-llamapun" id="A1.Ex1.m1.1d">italic_λ ( 2 italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) + ( 1 - italic_λ ) ( 2 italic_c ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.1.p1.11">Now, consider an antisymmetric selection function <math alttext="\mathsf{S}:[-1,+1]\to[0,1]" class="ltx_Math" display="inline" id="A1.1.p1.7.m1.4"><semantics id="A1.1.p1.7.m1.4a"><mrow id="A1.1.p1.7.m1.4.4" xref="A1.1.p1.7.m1.4.4.cmml"><mi id="A1.1.p1.7.m1.4.4.4" xref="A1.1.p1.7.m1.4.4.4.cmml">𝖲</mi><mo id="A1.1.p1.7.m1.4.4.3" lspace="0.278em" rspace="0.278em" xref="A1.1.p1.7.m1.4.4.3.cmml">:</mo><mrow id="A1.1.p1.7.m1.4.4.2" xref="A1.1.p1.7.m1.4.4.2.cmml"><mrow id="A1.1.p1.7.m1.4.4.2.2.2" xref="A1.1.p1.7.m1.4.4.2.2.3.cmml"><mo id="A1.1.p1.7.m1.4.4.2.2.2.3" stretchy="false" xref="A1.1.p1.7.m1.4.4.2.2.3.cmml">[</mo><mrow id="A1.1.p1.7.m1.3.3.1.1.1.1" xref="A1.1.p1.7.m1.3.3.1.1.1.1.cmml"><mo id="A1.1.p1.7.m1.3.3.1.1.1.1a" xref="A1.1.p1.7.m1.3.3.1.1.1.1.cmml">−</mo><mn id="A1.1.p1.7.m1.3.3.1.1.1.1.2" xref="A1.1.p1.7.m1.3.3.1.1.1.1.2.cmml">1</mn></mrow><mo id="A1.1.p1.7.m1.4.4.2.2.2.4" xref="A1.1.p1.7.m1.4.4.2.2.3.cmml">,</mo><mrow id="A1.1.p1.7.m1.4.4.2.2.2.2" xref="A1.1.p1.7.m1.4.4.2.2.2.2.cmml"><mo id="A1.1.p1.7.m1.4.4.2.2.2.2a" xref="A1.1.p1.7.m1.4.4.2.2.2.2.cmml">+</mo><mn id="A1.1.p1.7.m1.4.4.2.2.2.2.2" xref="A1.1.p1.7.m1.4.4.2.2.2.2.2.cmml">1</mn></mrow><mo id="A1.1.p1.7.m1.4.4.2.2.2.5" stretchy="false" xref="A1.1.p1.7.m1.4.4.2.2.3.cmml">]</mo></mrow><mo id="A1.1.p1.7.m1.4.4.2.3" stretchy="false" xref="A1.1.p1.7.m1.4.4.2.3.cmml">→</mo><mrow id="A1.1.p1.7.m1.4.4.2.4.2" xref="A1.1.p1.7.m1.4.4.2.4.1.cmml"><mo id="A1.1.p1.7.m1.4.4.2.4.2.1" stretchy="false" xref="A1.1.p1.7.m1.4.4.2.4.1.cmml">[</mo><mn id="A1.1.p1.7.m1.1.1" xref="A1.1.p1.7.m1.1.1.cmml">0</mn><mo id="A1.1.p1.7.m1.4.4.2.4.2.2" xref="A1.1.p1.7.m1.4.4.2.4.1.cmml">,</mo><mn id="A1.1.p1.7.m1.2.2" xref="A1.1.p1.7.m1.2.2.cmml">1</mn><mo id="A1.1.p1.7.m1.4.4.2.4.2.3" stretchy="false" xref="A1.1.p1.7.m1.4.4.2.4.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.7.m1.4b"><apply id="A1.1.p1.7.m1.4.4.cmml" xref="A1.1.p1.7.m1.4.4"><ci id="A1.1.p1.7.m1.4.4.3.cmml" xref="A1.1.p1.7.m1.4.4.3">:</ci><ci id="A1.1.p1.7.m1.4.4.4.cmml" xref="A1.1.p1.7.m1.4.4.4">𝖲</ci><apply id="A1.1.p1.7.m1.4.4.2.cmml" xref="A1.1.p1.7.m1.4.4.2"><ci id="A1.1.p1.7.m1.4.4.2.3.cmml" xref="A1.1.p1.7.m1.4.4.2.3">→</ci><interval closure="closed" id="A1.1.p1.7.m1.4.4.2.2.3.cmml" xref="A1.1.p1.7.m1.4.4.2.2.2"><apply id="A1.1.p1.7.m1.3.3.1.1.1.1.cmml" xref="A1.1.p1.7.m1.3.3.1.1.1.1"><minus id="A1.1.p1.7.m1.3.3.1.1.1.1.1.cmml" xref="A1.1.p1.7.m1.3.3.1.1.1.1"></minus><cn id="A1.1.p1.7.m1.3.3.1.1.1.1.2.cmml" type="integer" xref="A1.1.p1.7.m1.3.3.1.1.1.1.2">1</cn></apply><apply id="A1.1.p1.7.m1.4.4.2.2.2.2.cmml" xref="A1.1.p1.7.m1.4.4.2.2.2.2"><plus id="A1.1.p1.7.m1.4.4.2.2.2.2.1.cmml" xref="A1.1.p1.7.m1.4.4.2.2.2.2"></plus><cn id="A1.1.p1.7.m1.4.4.2.2.2.2.2.cmml" type="integer" xref="A1.1.p1.7.m1.4.4.2.2.2.2.2">1</cn></apply></interval><interval closure="closed" id="A1.1.p1.7.m1.4.4.2.4.1.cmml" xref="A1.1.p1.7.m1.4.4.2.4.2"><cn id="A1.1.p1.7.m1.1.1.cmml" type="integer" xref="A1.1.p1.7.m1.1.1">0</cn><cn id="A1.1.p1.7.m1.2.2.cmml" type="integer" xref="A1.1.p1.7.m1.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.7.m1.4c">\mathsf{S}:[-1,+1]\to[0,1]</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.7.m1.4d">sansserif_S : [ - 1 , + 1 ] → [ 0 , 1 ]</annotation></semantics></math>. Suppose <math alttext="\mathsf{S}(+\frac{c-1}{c+1})=p" class="ltx_Math" display="inline" id="A1.1.p1.8.m2.1"><semantics id="A1.1.p1.8.m2.1a"><mrow id="A1.1.p1.8.m2.1.1" xref="A1.1.p1.8.m2.1.1.cmml"><mrow id="A1.1.p1.8.m2.1.1.1" xref="A1.1.p1.8.m2.1.1.1.cmml"><mi id="A1.1.p1.8.m2.1.1.1.3" xref="A1.1.p1.8.m2.1.1.1.3.cmml">𝖲</mi><mo id="A1.1.p1.8.m2.1.1.1.2" xref="A1.1.p1.8.m2.1.1.1.2.cmml">⁢</mo><mrow id="A1.1.p1.8.m2.1.1.1.1.1" xref="A1.1.p1.8.m2.1.1.1.1.1.1.cmml"><mo id="A1.1.p1.8.m2.1.1.1.1.1.2" stretchy="false" xref="A1.1.p1.8.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.1.p1.8.m2.1.1.1.1.1.1" xref="A1.1.p1.8.m2.1.1.1.1.1.1.cmml"><mo id="A1.1.p1.8.m2.1.1.1.1.1.1a" xref="A1.1.p1.8.m2.1.1.1.1.1.1.cmml">+</mo><mfrac id="A1.1.p1.8.m2.1.1.1.1.1.1.2" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.cmml"><mrow id="A1.1.p1.8.m2.1.1.1.1.1.1.2.2" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.2.cmml"><mi id="A1.1.p1.8.m2.1.1.1.1.1.1.2.2.2" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.2.2.cmml">c</mi><mo id="A1.1.p1.8.m2.1.1.1.1.1.1.2.2.1" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.2.1.cmml">−</mo><mn id="A1.1.p1.8.m2.1.1.1.1.1.1.2.2.3" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.2.3.cmml">1</mn></mrow><mrow id="A1.1.p1.8.m2.1.1.1.1.1.1.2.3" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.3.cmml"><mi id="A1.1.p1.8.m2.1.1.1.1.1.1.2.3.2" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.3.2.cmml">c</mi><mo id="A1.1.p1.8.m2.1.1.1.1.1.1.2.3.1" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.3.1.cmml">+</mo><mn id="A1.1.p1.8.m2.1.1.1.1.1.1.2.3.3" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.3.3.cmml">1</mn></mrow></mfrac></mrow><mo id="A1.1.p1.8.m2.1.1.1.1.1.3" stretchy="false" xref="A1.1.p1.8.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.1.p1.8.m2.1.1.2" xref="A1.1.p1.8.m2.1.1.2.cmml">=</mo><mi id="A1.1.p1.8.m2.1.1.3" xref="A1.1.p1.8.m2.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.8.m2.1b"><apply id="A1.1.p1.8.m2.1.1.cmml" xref="A1.1.p1.8.m2.1.1"><eq id="A1.1.p1.8.m2.1.1.2.cmml" xref="A1.1.p1.8.m2.1.1.2"></eq><apply id="A1.1.p1.8.m2.1.1.1.cmml" xref="A1.1.p1.8.m2.1.1.1"><times id="A1.1.p1.8.m2.1.1.1.2.cmml" xref="A1.1.p1.8.m2.1.1.1.2"></times><ci id="A1.1.p1.8.m2.1.1.1.3.cmml" xref="A1.1.p1.8.m2.1.1.1.3">𝖲</ci><apply id="A1.1.p1.8.m2.1.1.1.1.1.1.cmml" xref="A1.1.p1.8.m2.1.1.1.1.1"><plus id="A1.1.p1.8.m2.1.1.1.1.1.1.1.cmml" xref="A1.1.p1.8.m2.1.1.1.1.1"></plus><apply id="A1.1.p1.8.m2.1.1.1.1.1.1.2.cmml" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2"><divide id="A1.1.p1.8.m2.1.1.1.1.1.1.2.1.cmml" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2"></divide><apply id="A1.1.p1.8.m2.1.1.1.1.1.1.2.2.cmml" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.2"><minus id="A1.1.p1.8.m2.1.1.1.1.1.1.2.2.1.cmml" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.2.1"></minus><ci id="A1.1.p1.8.m2.1.1.1.1.1.1.2.2.2.cmml" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.2.2">𝑐</ci><cn id="A1.1.p1.8.m2.1.1.1.1.1.1.2.2.3.cmml" type="integer" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.2.3">1</cn></apply><apply id="A1.1.p1.8.m2.1.1.1.1.1.1.2.3.cmml" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.3"><plus id="A1.1.p1.8.m2.1.1.1.1.1.1.2.3.1.cmml" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.3.1"></plus><ci id="A1.1.p1.8.m2.1.1.1.1.1.1.2.3.2.cmml" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.3.2">𝑐</ci><cn id="A1.1.p1.8.m2.1.1.1.1.1.1.2.3.3.cmml" type="integer" xref="A1.1.p1.8.m2.1.1.1.1.1.1.2.3.3">1</cn></apply></apply></apply></apply><ci id="A1.1.p1.8.m2.1.1.3.cmml" xref="A1.1.p1.8.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.8.m2.1c">\mathsf{S}(+\frac{c-1}{c+1})=p</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.8.m2.1d">sansserif_S ( + divide start_ARG italic_c - 1 end_ARG start_ARG italic_c + 1 end_ARG ) = italic_p</annotation></semantics></math>. By antisymmetry, <math alttext="\mathsf{S}(0)=\frac{1}{2}" class="ltx_Math" display="inline" id="A1.1.p1.9.m3.1"><semantics id="A1.1.p1.9.m3.1a"><mrow id="A1.1.p1.9.m3.1.2" xref="A1.1.p1.9.m3.1.2.cmml"><mrow id="A1.1.p1.9.m3.1.2.2" xref="A1.1.p1.9.m3.1.2.2.cmml"><mi id="A1.1.p1.9.m3.1.2.2.2" xref="A1.1.p1.9.m3.1.2.2.2.cmml">𝖲</mi><mo id="A1.1.p1.9.m3.1.2.2.1" xref="A1.1.p1.9.m3.1.2.2.1.cmml">⁢</mo><mrow id="A1.1.p1.9.m3.1.2.2.3.2" xref="A1.1.p1.9.m3.1.2.2.cmml"><mo id="A1.1.p1.9.m3.1.2.2.3.2.1" stretchy="false" xref="A1.1.p1.9.m3.1.2.2.cmml">(</mo><mn id="A1.1.p1.9.m3.1.1" xref="A1.1.p1.9.m3.1.1.cmml">0</mn><mo id="A1.1.p1.9.m3.1.2.2.3.2.2" stretchy="false" xref="A1.1.p1.9.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="A1.1.p1.9.m3.1.2.1" xref="A1.1.p1.9.m3.1.2.1.cmml">=</mo><mfrac id="A1.1.p1.9.m3.1.2.3" xref="A1.1.p1.9.m3.1.2.3.cmml"><mn id="A1.1.p1.9.m3.1.2.3.2" xref="A1.1.p1.9.m3.1.2.3.2.cmml">1</mn><mn id="A1.1.p1.9.m3.1.2.3.3" xref="A1.1.p1.9.m3.1.2.3.3.cmml">2</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.9.m3.1b"><apply id="A1.1.p1.9.m3.1.2.cmml" xref="A1.1.p1.9.m3.1.2"><eq id="A1.1.p1.9.m3.1.2.1.cmml" xref="A1.1.p1.9.m3.1.2.1"></eq><apply id="A1.1.p1.9.m3.1.2.2.cmml" xref="A1.1.p1.9.m3.1.2.2"><times id="A1.1.p1.9.m3.1.2.2.1.cmml" xref="A1.1.p1.9.m3.1.2.2.1"></times><ci id="A1.1.p1.9.m3.1.2.2.2.cmml" xref="A1.1.p1.9.m3.1.2.2.2">𝖲</ci><cn id="A1.1.p1.9.m3.1.1.cmml" type="integer" xref="A1.1.p1.9.m3.1.1">0</cn></apply><apply id="A1.1.p1.9.m3.1.2.3.cmml" xref="A1.1.p1.9.m3.1.2.3"><divide id="A1.1.p1.9.m3.1.2.3.1.cmml" xref="A1.1.p1.9.m3.1.2.3"></divide><cn id="A1.1.p1.9.m3.1.2.3.2.cmml" type="integer" xref="A1.1.p1.9.m3.1.2.3.2">1</cn><cn id="A1.1.p1.9.m3.1.2.3.3.cmml" type="integer" xref="A1.1.p1.9.m3.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.9.m3.1c">\mathsf{S}(0)=\frac{1}{2}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.9.m3.1d">sansserif_S ( 0 ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG</annotation></semantics></math> and <math alttext="\mathsf{S}(-\frac{c-1}{c+1})=1-p" class="ltx_Math" display="inline" id="A1.1.p1.10.m4.1"><semantics id="A1.1.p1.10.m4.1a"><mrow id="A1.1.p1.10.m4.1.1" xref="A1.1.p1.10.m4.1.1.cmml"><mrow id="A1.1.p1.10.m4.1.1.1" xref="A1.1.p1.10.m4.1.1.1.cmml"><mi id="A1.1.p1.10.m4.1.1.1.3" xref="A1.1.p1.10.m4.1.1.1.3.cmml">𝖲</mi><mo id="A1.1.p1.10.m4.1.1.1.2" xref="A1.1.p1.10.m4.1.1.1.2.cmml">⁢</mo><mrow id="A1.1.p1.10.m4.1.1.1.1.1" xref="A1.1.p1.10.m4.1.1.1.1.1.1.cmml"><mo id="A1.1.p1.10.m4.1.1.1.1.1.2" stretchy="false" xref="A1.1.p1.10.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.1.p1.10.m4.1.1.1.1.1.1" xref="A1.1.p1.10.m4.1.1.1.1.1.1.cmml"><mo id="A1.1.p1.10.m4.1.1.1.1.1.1a" xref="A1.1.p1.10.m4.1.1.1.1.1.1.cmml">−</mo><mfrac id="A1.1.p1.10.m4.1.1.1.1.1.1.2" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.cmml"><mrow id="A1.1.p1.10.m4.1.1.1.1.1.1.2.2" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.2.cmml"><mi id="A1.1.p1.10.m4.1.1.1.1.1.1.2.2.2" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.2.2.cmml">c</mi><mo id="A1.1.p1.10.m4.1.1.1.1.1.1.2.2.1" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.2.1.cmml">−</mo><mn id="A1.1.p1.10.m4.1.1.1.1.1.1.2.2.3" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.2.3.cmml">1</mn></mrow><mrow id="A1.1.p1.10.m4.1.1.1.1.1.1.2.3" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.3.cmml"><mi id="A1.1.p1.10.m4.1.1.1.1.1.1.2.3.2" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.3.2.cmml">c</mi><mo id="A1.1.p1.10.m4.1.1.1.1.1.1.2.3.1" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.3.1.cmml">+</mo><mn id="A1.1.p1.10.m4.1.1.1.1.1.1.2.3.3" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.3.3.cmml">1</mn></mrow></mfrac></mrow><mo id="A1.1.p1.10.m4.1.1.1.1.1.3" stretchy="false" xref="A1.1.p1.10.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.1.p1.10.m4.1.1.2" xref="A1.1.p1.10.m4.1.1.2.cmml">=</mo><mrow id="A1.1.p1.10.m4.1.1.3" xref="A1.1.p1.10.m4.1.1.3.cmml"><mn id="A1.1.p1.10.m4.1.1.3.2" xref="A1.1.p1.10.m4.1.1.3.2.cmml">1</mn><mo id="A1.1.p1.10.m4.1.1.3.1" xref="A1.1.p1.10.m4.1.1.3.1.cmml">−</mo><mi id="A1.1.p1.10.m4.1.1.3.3" xref="A1.1.p1.10.m4.1.1.3.3.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.10.m4.1b"><apply id="A1.1.p1.10.m4.1.1.cmml" xref="A1.1.p1.10.m4.1.1"><eq id="A1.1.p1.10.m4.1.1.2.cmml" xref="A1.1.p1.10.m4.1.1.2"></eq><apply id="A1.1.p1.10.m4.1.1.1.cmml" xref="A1.1.p1.10.m4.1.1.1"><times id="A1.1.p1.10.m4.1.1.1.2.cmml" xref="A1.1.p1.10.m4.1.1.1.2"></times><ci id="A1.1.p1.10.m4.1.1.1.3.cmml" xref="A1.1.p1.10.m4.1.1.1.3">𝖲</ci><apply id="A1.1.p1.10.m4.1.1.1.1.1.1.cmml" xref="A1.1.p1.10.m4.1.1.1.1.1"><minus id="A1.1.p1.10.m4.1.1.1.1.1.1.1.cmml" xref="A1.1.p1.10.m4.1.1.1.1.1"></minus><apply id="A1.1.p1.10.m4.1.1.1.1.1.1.2.cmml" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2"><divide id="A1.1.p1.10.m4.1.1.1.1.1.1.2.1.cmml" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2"></divide><apply id="A1.1.p1.10.m4.1.1.1.1.1.1.2.2.cmml" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.2"><minus id="A1.1.p1.10.m4.1.1.1.1.1.1.2.2.1.cmml" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.2.1"></minus><ci id="A1.1.p1.10.m4.1.1.1.1.1.1.2.2.2.cmml" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.2.2">𝑐</ci><cn id="A1.1.p1.10.m4.1.1.1.1.1.1.2.2.3.cmml" type="integer" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.2.3">1</cn></apply><apply id="A1.1.p1.10.m4.1.1.1.1.1.1.2.3.cmml" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.3"><plus id="A1.1.p1.10.m4.1.1.1.1.1.1.2.3.1.cmml" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.3.1"></plus><ci id="A1.1.p1.10.m4.1.1.1.1.1.1.2.3.2.cmml" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.3.2">𝑐</ci><cn id="A1.1.p1.10.m4.1.1.1.1.1.1.2.3.3.cmml" type="integer" xref="A1.1.p1.10.m4.1.1.1.1.1.1.2.3.3">1</cn></apply></apply></apply></apply><apply id="A1.1.p1.10.m4.1.1.3.cmml" xref="A1.1.p1.10.m4.1.1.3"><minus id="A1.1.p1.10.m4.1.1.3.1.cmml" xref="A1.1.p1.10.m4.1.1.3.1"></minus><cn id="A1.1.p1.10.m4.1.1.3.2.cmml" type="integer" xref="A1.1.p1.10.m4.1.1.3.2">1</cn><ci id="A1.1.p1.10.m4.1.1.3.3.cmml" xref="A1.1.p1.10.m4.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.10.m4.1c">\mathsf{S}(-\frac{c-1}{c+1})=1-p</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.10.m4.1d">sansserif_S ( - divide start_ARG italic_c - 1 end_ARG start_ARG italic_c + 1 end_ARG ) = 1 - italic_p</annotation></semantics></math>. Thus, as in the caption, <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="A1.1.p1.11.m5.1"><semantics id="A1.1.p1.11.m5.1a"><msub id="A1.1.p1.11.m5.1.1" xref="A1.1.p1.11.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.1.p1.11.m5.1.1.2" xref="A1.1.p1.11.m5.1.1.2.cmml">𝒪</mi><mi id="A1.1.p1.11.m5.1.1.3" xref="A1.1.p1.11.m5.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="A1.1.p1.11.m5.1b"><apply id="A1.1.p1.11.m5.1.1.cmml" xref="A1.1.p1.11.m5.1.1"><csymbol cd="ambiguous" id="A1.1.p1.11.m5.1.1.1.cmml" xref="A1.1.p1.11.m5.1.1">subscript</csymbol><ci id="A1.1.p1.11.m5.1.1.2.cmml" xref="A1.1.p1.11.m5.1.1.2">𝒪</ci><ci id="A1.1.p1.11.m5.1.1.3.cmml" xref="A1.1.p1.11.m5.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.11.m5.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.11.m5.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math> satisfies weight</p> <table class="ltx_equation ltx_eqn_table" id="A1.1.p1.12"> <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="\lambda\left(p(1-p)(c+1)+\frac{1}{2}p(c^{2}-1)+\frac{1}{4}(c^{2}-1)+\frac{1}{2% }p(c^{2}-1)+p(1-p)(c+1)\right)\\ +(1-\lambda)\left(\frac{1}{2}p(c)+\frac{1}{2}(1-p)+\frac{1}{4}(c-1)+\frac{1}{2% }p(c)+\frac{1}{2}(1-p)\right)\\ =(1-\lambda)\left(1+\left(\frac{1}{4}+p\right)(c-1)\right)+\lambda\left(\left(% \frac{1}{4}+p\right)(c^{2}-1)+2(c+1)(1-p)p)\right)" class="ltx_math_unparsed" display="block" id="A1.1.p1.12.m1.144"><semantics id="A1.1.p1.12.m1.144a"><mtable displaystyle="true" id="A1.1.p1.12.m1.144.144.3" rowspacing="0pt"><mtr id="A1.1.p1.12.m1.144.144.3a"><mtd class="ltx_align_left" columnalign="left" id="A1.1.p1.12.m1.144.144.3b"><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53"><mi id="A1.1.p1.12.m1.1.1.1.1.1.1">λ</mi><mo id="A1.1.p1.12.m1.142.142.1.142.53.53.54">⁢</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1"><mo id="A1.1.p1.12.m1.2.2.2.2.2.2">(</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1"><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.2"><mi id="A1.1.p1.12.m1.3.3.3.3.3.3">p</mi><mo id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.2.3">⁢</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.1.1.1"><mo id="A1.1.p1.12.m1.4.4.4.4.4.4" stretchy="false">(</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.1.1.1.1"><mn id="A1.1.p1.12.m1.5.5.5.5.5.5">1</mn><mo id="A1.1.p1.12.m1.6.6.6.6.6.6">−</mo><mi id="A1.1.p1.12.m1.7.7.7.7.7.7">p</mi></mrow><mo id="A1.1.p1.12.m1.8.8.8.8.8.8" stretchy="false">)</mo></mrow><mo id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.2.3a">⁢</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.2.2.1"><mo id="A1.1.p1.12.m1.9.9.9.9.9.9" stretchy="false">(</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.2.2.1.1"><mi id="A1.1.p1.12.m1.10.10.10.10.10.10">c</mi><mo id="A1.1.p1.12.m1.11.11.11.11.11.11">+</mo><mn id="A1.1.p1.12.m1.12.12.12.12.12.12">1</mn></mrow><mo id="A1.1.p1.12.m1.13.13.13.13.13.13" stretchy="false">)</mo></mrow></mrow><mo id="A1.1.p1.12.m1.14.14.14.14.14.14">+</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.3"><mfrac id="A1.1.p1.12.m1.15.15.15.15.15.15"><mn id="A1.1.p1.12.m1.15.15.15.15.15.15.2">1</mn><mn id="A1.1.p1.12.m1.15.15.15.15.15.15.3">2</mn></mfrac><mo id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.3.2">⁢</mo><mi id="A1.1.p1.12.m1.16.16.16.16.16.16">p</mi><mo id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.3.2a">⁢</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.3.1.1"><mo id="A1.1.p1.12.m1.17.17.17.17.17.17" stretchy="false">(</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.3.1.1.1"><msup id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.3.1.1.1.1"><mi id="A1.1.p1.12.m1.18.18.18.18.18.18">c</mi><mn id="A1.1.p1.12.m1.19.19.19.19.19.19.1">2</mn></msup><mo id="A1.1.p1.12.m1.20.20.20.20.20.20">−</mo><mn id="A1.1.p1.12.m1.21.21.21.21.21.21">1</mn></mrow><mo id="A1.1.p1.12.m1.22.22.22.22.22.22" stretchy="false">)</mo></mrow></mrow><mo id="A1.1.p1.12.m1.14.14.14.14.14.14a">+</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.4"><mfrac id="A1.1.p1.12.m1.24.24.24.24.24.24"><mn id="A1.1.p1.12.m1.24.24.24.24.24.24.2">1</mn><mn id="A1.1.p1.12.m1.24.24.24.24.24.24.3">4</mn></mfrac><mo id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.4.2">⁢</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.4.1.1"><mo id="A1.1.p1.12.m1.25.25.25.25.25.25" stretchy="false">(</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.4.1.1.1"><msup id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.4.1.1.1.1"><mi id="A1.1.p1.12.m1.26.26.26.26.26.26">c</mi><mn id="A1.1.p1.12.m1.27.27.27.27.27.27.1">2</mn></msup><mo id="A1.1.p1.12.m1.28.28.28.28.28.28">−</mo><mn id="A1.1.p1.12.m1.29.29.29.29.29.29">1</mn></mrow><mo id="A1.1.p1.12.m1.30.30.30.30.30.30" stretchy="false">)</mo></mrow></mrow><mo id="A1.1.p1.12.m1.14.14.14.14.14.14b">+</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.5"><mfrac id="A1.1.p1.12.m1.32.32.32.32.32.32"><mn id="A1.1.p1.12.m1.32.32.32.32.32.32.2">1</mn><mn id="A1.1.p1.12.m1.32.32.32.32.32.32.3">2</mn></mfrac><mo id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.5.2">⁢</mo><mi id="A1.1.p1.12.m1.33.33.33.33.33.33">p</mi><mo id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.5.2a">⁢</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.5.1.1"><mo id="A1.1.p1.12.m1.34.34.34.34.34.34" stretchy="false">(</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.5.1.1.1"><msup id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.5.1.1.1.1"><mi id="A1.1.p1.12.m1.35.35.35.35.35.35">c</mi><mn id="A1.1.p1.12.m1.36.36.36.36.36.36.1">2</mn></msup><mo id="A1.1.p1.12.m1.37.37.37.37.37.37">−</mo><mn id="A1.1.p1.12.m1.38.38.38.38.38.38">1</mn></mrow><mo id="A1.1.p1.12.m1.39.39.39.39.39.39" stretchy="false">)</mo></mrow></mrow><mo id="A1.1.p1.12.m1.14.14.14.14.14.14c">+</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.7"><mi id="A1.1.p1.12.m1.41.41.41.41.41.41">p</mi><mo id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.7.3">⁢</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.6.1.1"><mo id="A1.1.p1.12.m1.42.42.42.42.42.42" stretchy="false">(</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.6.1.1.1"><mn id="A1.1.p1.12.m1.43.43.43.43.43.43">1</mn><mo id="A1.1.p1.12.m1.44.44.44.44.44.44">−</mo><mi id="A1.1.p1.12.m1.45.45.45.45.45.45">p</mi></mrow><mo id="A1.1.p1.12.m1.46.46.46.46.46.46" stretchy="false">)</mo></mrow><mo id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.7.3a">⁢</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.7.2.1"><mo id="A1.1.p1.12.m1.47.47.47.47.47.47" stretchy="false">(</mo><mrow id="A1.1.p1.12.m1.142.142.1.142.53.53.53.1.1.7.2.1.1"><mi id="A1.1.p1.12.m1.48.48.48.48.48.48">c</mi><mo id="A1.1.p1.12.m1.49.49.49.49.49.49">+</mo><mn id="A1.1.p1.12.m1.50.50.50.50.50.50">1</mn></mrow><mo id="A1.1.p1.12.m1.51.51.51.51.51.51" stretchy="false">)</mo></mrow></mrow></mrow><mo id="A1.1.p1.12.m1.52.52.52.52.52.52">)</mo></mrow></mrow></mtd></mtr><mtr id="A1.1.p1.12.m1.144.144.3c"><mtd class="ltx_align_right" columnalign="right" id="A1.1.p1.12.m1.144.144.3d"><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42"><mo id="A1.1.p1.12.m1.144.144.3.144.42.42a">+</mo><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42"><mrow id="A1.1.p1.12.m1.143.143.2.143.41.41.41.1.1"><mo id="A1.1.p1.12.m1.54.54.54.2.2.2" stretchy="false">(</mo><mrow id="A1.1.p1.12.m1.143.143.2.143.41.41.41.1.1.1"><mn id="A1.1.p1.12.m1.55.55.55.3.3.3">1</mn><mo id="A1.1.p1.12.m1.56.56.56.4.4.4">−</mo><mi id="A1.1.p1.12.m1.57.57.57.5.5.5">λ</mi></mrow><mo id="A1.1.p1.12.m1.58.58.58.6.6.6" stretchy="false">)</mo></mrow><mo id="A1.1.p1.12.m1.144.144.3.144.42.42.42.3">⁢</mo><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1"><mo id="A1.1.p1.12.m1.59.59.59.7.7.7">(</mo><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1"><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.4"><mfrac id="A1.1.p1.12.m1.60.60.60.8.8.8"><mn id="A1.1.p1.12.m1.60.60.60.8.8.8.2">1</mn><mn id="A1.1.p1.12.m1.60.60.60.8.8.8.3">2</mn></mfrac><mo id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.4.1">⁢</mo><mi id="A1.1.p1.12.m1.61.61.61.9.9.9">p</mi><mo id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.4.1a">⁢</mo><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.4.2"><mo id="A1.1.p1.12.m1.62.62.62.10.10.10" stretchy="false">(</mo><mi id="A1.1.p1.12.m1.63.63.63.11.11.11">c</mi><mo id="A1.1.p1.12.m1.64.64.64.12.12.12" stretchy="false">)</mo></mrow></mrow><mo id="A1.1.p1.12.m1.65.65.65.13.13.13">+</mo><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.1"><mfrac id="A1.1.p1.12.m1.66.66.66.14.14.14"><mn id="A1.1.p1.12.m1.66.66.66.14.14.14.2">1</mn><mn id="A1.1.p1.12.m1.66.66.66.14.14.14.3">2</mn></mfrac><mo id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.1.2">⁢</mo><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.1.1.1"><mo id="A1.1.p1.12.m1.67.67.67.15.15.15" stretchy="false">(</mo><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.1.1.1.1"><mn id="A1.1.p1.12.m1.68.68.68.16.16.16">1</mn><mo id="A1.1.p1.12.m1.69.69.69.17.17.17">−</mo><mi id="A1.1.p1.12.m1.70.70.70.18.18.18">p</mi></mrow><mo id="A1.1.p1.12.m1.71.71.71.19.19.19" stretchy="false">)</mo></mrow></mrow><mo id="A1.1.p1.12.m1.65.65.65.13.13.13a">+</mo><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.2"><mfrac id="A1.1.p1.12.m1.73.73.73.21.21.21"><mn id="A1.1.p1.12.m1.73.73.73.21.21.21.2">1</mn><mn id="A1.1.p1.12.m1.73.73.73.21.21.21.3">4</mn></mfrac><mo id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.2.2">⁢</mo><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.2.1.1"><mo id="A1.1.p1.12.m1.74.74.74.22.22.22" stretchy="false">(</mo><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.2.1.1.1"><mi id="A1.1.p1.12.m1.75.75.75.23.23.23">c</mi><mo id="A1.1.p1.12.m1.76.76.76.24.24.24">−</mo><mn id="A1.1.p1.12.m1.77.77.77.25.25.25">1</mn></mrow><mo id="A1.1.p1.12.m1.78.78.78.26.26.26" stretchy="false">)</mo></mrow></mrow><mo id="A1.1.p1.12.m1.65.65.65.13.13.13b">+</mo><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.5"><mfrac id="A1.1.p1.12.m1.80.80.80.28.28.28"><mn id="A1.1.p1.12.m1.80.80.80.28.28.28.2">1</mn><mn id="A1.1.p1.12.m1.80.80.80.28.28.28.3">2</mn></mfrac><mo id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.5.1">⁢</mo><mi id="A1.1.p1.12.m1.81.81.81.29.29.29">p</mi><mo id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.5.1a">⁢</mo><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.5.2"><mo id="A1.1.p1.12.m1.82.82.82.30.30.30" stretchy="false">(</mo><mi id="A1.1.p1.12.m1.83.83.83.31.31.31">c</mi><mo id="A1.1.p1.12.m1.84.84.84.32.32.32" stretchy="false">)</mo></mrow></mrow><mo id="A1.1.p1.12.m1.65.65.65.13.13.13c">+</mo><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.3"><mfrac id="A1.1.p1.12.m1.86.86.86.34.34.34"><mn id="A1.1.p1.12.m1.86.86.86.34.34.34.2">1</mn><mn id="A1.1.p1.12.m1.86.86.86.34.34.34.3">2</mn></mfrac><mo id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.3.2">⁢</mo><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.3.1.1"><mo id="A1.1.p1.12.m1.87.87.87.35.35.35" stretchy="false">(</mo><mrow id="A1.1.p1.12.m1.144.144.3.144.42.42.42.2.1.1.3.1.1.1"><mn id="A1.1.p1.12.m1.88.88.88.36.36.36">1</mn><mo id="A1.1.p1.12.m1.89.89.89.37.37.37">−</mo><mi id="A1.1.p1.12.m1.90.90.90.38.38.38">p</mi></mrow><mo id="A1.1.p1.12.m1.91.91.91.39.39.39" stretchy="false">)</mo></mrow></mrow></mrow><mo id="A1.1.p1.12.m1.92.92.92.40.40.40">)</mo></mrow></mrow></mrow></mtd></mtr><mtr id="A1.1.p1.12.m1.144.144.3e"><mtd class="ltx_align_right" columnalign="right" id="A1.1.p1.12.m1.144.144.3f"><mrow id="A1.1.p1.12.m1.141.141.141.49.49"><mo id="A1.1.p1.12.m1.93.93.93.1.1.1">=</mo><mrow id="A1.1.p1.12.m1.141.141.141.49.49.50"><mo id="A1.1.p1.12.m1.94.94.94.2.2.2" stretchy="false">(</mo><mn id="A1.1.p1.12.m1.95.95.95.3.3.3">1</mn><mo id="A1.1.p1.12.m1.96.96.96.4.4.4">−</mo><mi id="A1.1.p1.12.m1.97.97.97.5.5.5">λ</mi><mo id="A1.1.p1.12.m1.98.98.98.6.6.6" stretchy="false">)</mo></mrow><mrow id="A1.1.p1.12.m1.141.141.141.49.49.51"><mo id="A1.1.p1.12.m1.99.99.99.7.7.7">(</mo><mn id="A1.1.p1.12.m1.100.100.100.8.8.8">1</mn><mo id="A1.1.p1.12.m1.101.101.101.9.9.9">+</mo><mrow id="A1.1.p1.12.m1.141.141.141.49.49.51.1"><mo id="A1.1.p1.12.m1.102.102.102.10.10.10">(</mo><mfrac id="A1.1.p1.12.m1.103.103.103.11.11.11"><mn id="A1.1.p1.12.m1.103.103.103.11.11.11.2">1</mn><mn id="A1.1.p1.12.m1.103.103.103.11.11.11.3">4</mn></mfrac><mo id="A1.1.p1.12.m1.104.104.104.12.12.12">+</mo><mi id="A1.1.p1.12.m1.105.105.105.13.13.13">p</mi><mo id="A1.1.p1.12.m1.106.106.106.14.14.14">)</mo></mrow><mrow id="A1.1.p1.12.m1.141.141.141.49.49.51.2"><mo id="A1.1.p1.12.m1.107.107.107.15.15.15" stretchy="false">(</mo><mi id="A1.1.p1.12.m1.108.108.108.16.16.16">c</mi><mo id="A1.1.p1.12.m1.109.109.109.17.17.17">−</mo><mn id="A1.1.p1.12.m1.110.110.110.18.18.18">1</mn><mo id="A1.1.p1.12.m1.111.111.111.19.19.19" stretchy="false">)</mo></mrow><mo id="A1.1.p1.12.m1.112.112.112.20.20.20">)</mo></mrow><mo id="A1.1.p1.12.m1.113.113.113.21.21.21">+</mo><mi id="A1.1.p1.12.m1.114.114.114.22.22.22">λ</mi><mrow id="A1.1.p1.12.m1.141.141.141.49.49.52"><mo id="A1.1.p1.12.m1.115.115.115.23.23.23">(</mo><mrow id="A1.1.p1.12.m1.141.141.141.49.49.52.1"><mo id="A1.1.p1.12.m1.116.116.116.24.24.24">(</mo><mfrac id="A1.1.p1.12.m1.117.117.117.25.25.25"><mn id="A1.1.p1.12.m1.117.117.117.25.25.25.2">1</mn><mn id="A1.1.p1.12.m1.117.117.117.25.25.25.3">4</mn></mfrac><mo id="A1.1.p1.12.m1.118.118.118.26.26.26">+</mo><mi id="A1.1.p1.12.m1.119.119.119.27.27.27">p</mi><mo id="A1.1.p1.12.m1.120.120.120.28.28.28">)</mo></mrow><mrow id="A1.1.p1.12.m1.141.141.141.49.49.52.2"><mo id="A1.1.p1.12.m1.121.121.121.29.29.29" stretchy="false">(</mo><msup id="A1.1.p1.12.m1.141.141.141.49.49.52.2.1"><mi id="A1.1.p1.12.m1.122.122.122.30.30.30">c</mi><mn id="A1.1.p1.12.m1.123.123.123.31.31.31.1">2</mn></msup><mo id="A1.1.p1.12.m1.124.124.124.32.32.32">−</mo><mn id="A1.1.p1.12.m1.125.125.125.33.33.33">1</mn><mo id="A1.1.p1.12.m1.126.126.126.34.34.34" stretchy="false">)</mo></mrow><mo id="A1.1.p1.12.m1.127.127.127.35.35.35">+</mo><mn id="A1.1.p1.12.m1.128.128.128.36.36.36">2</mn><mrow id="A1.1.p1.12.m1.141.141.141.49.49.52.3"><mo id="A1.1.p1.12.m1.129.129.129.37.37.37" stretchy="false">(</mo><mi id="A1.1.p1.12.m1.130.130.130.38.38.38">c</mi><mo id="A1.1.p1.12.m1.131.131.131.39.39.39">+</mo><mn id="A1.1.p1.12.m1.132.132.132.40.40.40">1</mn><mo id="A1.1.p1.12.m1.133.133.133.41.41.41" stretchy="false">)</mo></mrow><mrow id="A1.1.p1.12.m1.141.141.141.49.49.52.4"><mo id="A1.1.p1.12.m1.134.134.134.42.42.42" stretchy="false">(</mo><mn id="A1.1.p1.12.m1.135.135.135.43.43.43">1</mn><mo id="A1.1.p1.12.m1.136.136.136.44.44.44">−</mo><mi id="A1.1.p1.12.m1.137.137.137.45.45.45">p</mi><mo id="A1.1.p1.12.m1.138.138.138.46.46.46" stretchy="false">)</mo></mrow><mi id="A1.1.p1.12.m1.139.139.139.47.47.47">p</mi><mo id="A1.1.p1.12.m1.140.140.140.48.48.48" stretchy="false">)</mo></mrow><mo id="A1.1.p1.12.m1.141.141.141.49.49.49">)</mo></mrow></mtd></mtr></mtable><annotation encoding="application/x-tex" id="A1.1.p1.12.m1.144b">\lambda\left(p(1-p)(c+1)+\frac{1}{2}p(c^{2}-1)+\frac{1}{4}(c^{2}-1)+\frac{1}{2% }p(c^{2}-1)+p(1-p)(c+1)\right)\\ +(1-\lambda)\left(\frac{1}{2}p(c)+\frac{1}{2}(1-p)+\frac{1}{4}(c-1)+\frac{1}{2% }p(c)+\frac{1}{2}(1-p)\right)\\ =(1-\lambda)\left(1+\left(\frac{1}{4}+p\right)(c-1)\right)+\lambda\left(\left(% \frac{1}{4}+p\right)(c^{2}-1)+2(c+1)(1-p)p)\right)</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.12.m1.144c">start_ROW start_CELL italic_λ ( italic_p ( 1 - italic_p ) ( italic_c + 1 ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_p ( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) + divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_p ( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) + italic_p ( 1 - italic_p ) ( italic_c + 1 ) ) end_CELL end_ROW start_ROW start_CELL + ( 1 - italic_λ ) ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_p ( italic_c ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( 1 - italic_p ) + divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( italic_c - 1 ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_p ( italic_c ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( 1 - italic_p ) ) end_CELL end_ROW start_ROW start_CELL = ( 1 - italic_λ ) ( 1 + ( divide start_ARG 1 end_ARG start_ARG 4 end_ARG + italic_p ) ( italic_c - 1 ) ) + italic_λ ( ( divide start_ARG 1 end_ARG start_ARG 4 end_ARG + italic_p ) ( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) + 2 ( italic_c + 1 ) ( 1 - italic_p ) italic_p ) ) end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.1.p1.13">Thus, <math alttext="\mathcal{O}_{\mathsf{S}}" class="ltx_Math" display="inline" id="A1.1.p1.13.m1.1"><semantics id="A1.1.p1.13.m1.1a"><msub id="A1.1.p1.13.m1.1.1" xref="A1.1.p1.13.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.1.p1.13.m1.1.1.2" xref="A1.1.p1.13.m1.1.1.2.cmml">𝒪</mi><mi id="A1.1.p1.13.m1.1.1.3" xref="A1.1.p1.13.m1.1.1.3.cmml">𝖲</mi></msub><annotation-xml encoding="MathML-Content" id="A1.1.p1.13.m1.1b"><apply id="A1.1.p1.13.m1.1.1.cmml" xref="A1.1.p1.13.m1.1.1"><csymbol cd="ambiguous" id="A1.1.p1.13.m1.1.1.1.cmml" xref="A1.1.p1.13.m1.1.1">subscript</csymbol><ci id="A1.1.p1.13.m1.1.1.2.cmml" xref="A1.1.p1.13.m1.1.1.2">𝒪</ci><ci id="A1.1.p1.13.m1.1.1.3.cmml" xref="A1.1.p1.13.m1.1.1.3">𝖲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.13.m1.1c">\mathcal{O}_{\mathsf{S}}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.13.m1.1d">caligraphic_O start_POSTSUBSCRIPT sansserif_S end_POSTSUBSCRIPT</annotation></semantics></math> has ratio at most</p> <table class="ltx_equation ltx_eqn_table" id="A1.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="\frac{(1-\lambda)\left(1+\left(\frac{1}{4}+p\right)(c-1)\right)+\lambda\left(% \left(\frac{1}{4}+p\right)(c^{2}-1)+2(c+1)(1-p)p\right)}{2(\lambda c^{2}+(1-% \lambda)c)}." class="ltx_Math" display="block" id="A1.Ex2.m1.4"><semantics id="A1.Ex2.m1.4a"><mrow id="A1.Ex2.m1.4.5.2" xref="A1.Ex2.m1.4.4.cmml"><mfrac id="A1.Ex2.m1.4.4" xref="A1.Ex2.m1.4.4.cmml"><mrow id="A1.Ex2.m1.3.3.3" xref="A1.Ex2.m1.3.3.3.cmml"><mrow id="A1.Ex2.m1.2.2.2.2" xref="A1.Ex2.m1.2.2.2.2.cmml"><mrow id="A1.Ex2.m1.1.1.1.1.1.1" xref="A1.Ex2.m1.1.1.1.1.1.1.1.cmml"><mo id="A1.Ex2.m1.1.1.1.1.1.1.2" stretchy="false" xref="A1.Ex2.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.Ex2.m1.1.1.1.1.1.1.1" xref="A1.Ex2.m1.1.1.1.1.1.1.1.cmml"><mn id="A1.Ex2.m1.1.1.1.1.1.1.1.2" xref="A1.Ex2.m1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="A1.Ex2.m1.1.1.1.1.1.1.1.1" xref="A1.Ex2.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A1.Ex2.m1.1.1.1.1.1.1.1.3" xref="A1.Ex2.m1.1.1.1.1.1.1.1.3.cmml">λ</mi></mrow><mo id="A1.Ex2.m1.1.1.1.1.1.1.3" stretchy="false" xref="A1.Ex2.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A1.Ex2.m1.2.2.2.2.3" xref="A1.Ex2.m1.2.2.2.2.3.cmml">⁢</mo><mrow id="A1.Ex2.m1.2.2.2.2.2.1" xref="A1.Ex2.m1.2.2.2.2.2.1.1.cmml"><mo id="A1.Ex2.m1.2.2.2.2.2.1.2" xref="A1.Ex2.m1.2.2.2.2.2.1.1.cmml">(</mo><mrow id="A1.Ex2.m1.2.2.2.2.2.1.1" xref="A1.Ex2.m1.2.2.2.2.2.1.1.cmml"><mn id="A1.Ex2.m1.2.2.2.2.2.1.1.4" xref="A1.Ex2.m1.2.2.2.2.2.1.1.4.cmml">1</mn><mo id="A1.Ex2.m1.2.2.2.2.2.1.1.3" xref="A1.Ex2.m1.2.2.2.2.2.1.1.3.cmml">+</mo><mrow 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encoding="application/x-llamapun" id="A1.Ex2.m1.4d">divide start_ARG ( 1 - italic_λ ) ( 1 + ( divide start_ARG 1 end_ARG start_ARG 4 end_ARG + italic_p ) ( italic_c - 1 ) ) + italic_λ ( ( divide start_ARG 1 end_ARG start_ARG 4 end_ARG + italic_p ) ( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) + 2 ( italic_c + 1 ) ( 1 - italic_p ) italic_p ) end_ARG start_ARG 2 ( italic_λ italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 1 - italic_λ ) italic_c ) end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.1.p1.16">Evaluating at <math alttext="c=\frac{5}{4}" class="ltx_Math" display="inline" id="A1.1.p1.14.m1.1"><semantics id="A1.1.p1.14.m1.1a"><mrow id="A1.1.p1.14.m1.1.1" xref="A1.1.p1.14.m1.1.1.cmml"><mi id="A1.1.p1.14.m1.1.1.2" xref="A1.1.p1.14.m1.1.1.2.cmml">c</mi><mo id="A1.1.p1.14.m1.1.1.1" xref="A1.1.p1.14.m1.1.1.1.cmml">=</mo><mfrac id="A1.1.p1.14.m1.1.1.3" xref="A1.1.p1.14.m1.1.1.3.cmml"><mn id="A1.1.p1.14.m1.1.1.3.2" xref="A1.1.p1.14.m1.1.1.3.2.cmml">5</mn><mn id="A1.1.p1.14.m1.1.1.3.3" xref="A1.1.p1.14.m1.1.1.3.3.cmml">4</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.14.m1.1b"><apply id="A1.1.p1.14.m1.1.1.cmml" xref="A1.1.p1.14.m1.1.1"><eq id="A1.1.p1.14.m1.1.1.1.cmml" xref="A1.1.p1.14.m1.1.1.1"></eq><ci id="A1.1.p1.14.m1.1.1.2.cmml" xref="A1.1.p1.14.m1.1.1.2">𝑐</ci><apply id="A1.1.p1.14.m1.1.1.3.cmml" xref="A1.1.p1.14.m1.1.1.3"><divide id="A1.1.p1.14.m1.1.1.3.1.cmml" xref="A1.1.p1.14.m1.1.1.3"></divide><cn id="A1.1.p1.14.m1.1.1.3.2.cmml" type="integer" xref="A1.1.p1.14.m1.1.1.3.2">5</cn><cn id="A1.1.p1.14.m1.1.1.3.3.cmml" type="integer" xref="A1.1.p1.14.m1.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.14.m1.1c">c=\frac{5}{4}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.14.m1.1d">italic_c = divide start_ARG 5 end_ARG start_ARG 4 end_ARG</annotation></semantics></math> and <math alttext="\lambda=\frac{3}{4}" class="ltx_Math" display="inline" id="A1.1.p1.15.m2.1"><semantics id="A1.1.p1.15.m2.1a"><mrow id="A1.1.p1.15.m2.1.1" xref="A1.1.p1.15.m2.1.1.cmml"><mi id="A1.1.p1.15.m2.1.1.2" xref="A1.1.p1.15.m2.1.1.2.cmml">λ</mi><mo id="A1.1.p1.15.m2.1.1.1" xref="A1.1.p1.15.m2.1.1.1.cmml">=</mo><mfrac id="A1.1.p1.15.m2.1.1.3" xref="A1.1.p1.15.m2.1.1.3.cmml"><mn id="A1.1.p1.15.m2.1.1.3.2" xref="A1.1.p1.15.m2.1.1.3.2.cmml">3</mn><mn id="A1.1.p1.15.m2.1.1.3.3" xref="A1.1.p1.15.m2.1.1.3.3.cmml">4</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="A1.1.p1.15.m2.1b"><apply id="A1.1.p1.15.m2.1.1.cmml" xref="A1.1.p1.15.m2.1.1"><eq id="A1.1.p1.15.m2.1.1.1.cmml" xref="A1.1.p1.15.m2.1.1.1"></eq><ci id="A1.1.p1.15.m2.1.1.2.cmml" xref="A1.1.p1.15.m2.1.1.2">𝜆</ci><apply id="A1.1.p1.15.m2.1.1.3.cmml" xref="A1.1.p1.15.m2.1.1.3"><divide id="A1.1.p1.15.m2.1.1.3.1.cmml" xref="A1.1.p1.15.m2.1.1.3"></divide><cn id="A1.1.p1.15.m2.1.1.3.2.cmml" type="integer" xref="A1.1.p1.15.m2.1.1.3.2">3</cn><cn id="A1.1.p1.15.m2.1.1.3.3.cmml" type="integer" xref="A1.1.p1.15.m2.1.1.3.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.15.m2.1c">\lambda=\frac{3}{4}</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.15.m2.1d">italic_λ = divide start_ARG 3 end_ARG start_ARG 4 end_ARG</annotation></semantics></math> (found in <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">FJ15</span>]</cite> using computer search) and then maximizing over <math alttext="p" class="ltx_Math" display="inline" id="A1.1.p1.16.m3.1"><semantics id="A1.1.p1.16.m3.1a"><mi id="A1.1.p1.16.m3.1.1" xref="A1.1.p1.16.m3.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A1.1.p1.16.m3.1b"><ci id="A1.1.p1.16.m3.1.1.cmml" xref="A1.1.p1.16.m3.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.1.p1.16.m3.1c">p</annotation><annotation encoding="application/x-llamapun" id="A1.1.p1.16.m3.1d">italic_p</annotation></semantics></math> gives the bound. ∎</p> </div> </div> <div class="ltx_para" id="A1.p4"> <p class="ltx_p" id="A1.p4.1">We note that the graphs we used to prove <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem9" title="Theorem 1.9 (Lower bound for general selection). ‣ 1.2 Results ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.9</span></a>, i.e., <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S4.F3" title="In 4.2 Bounds against 𝖯𝖫𝖲𝗂𝗀𝗆𝗈𝗂𝖽_{1/2} (Theorem 1.6) ‣ 4 Lower bounds ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Figs.</span> <span class="ltx_text ltx_ref_tag">3</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S5.F5" title="Figure 5 ‣ 5 Lower bound for arbitrary selection functions (Theorem 1.9) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">5</span></a>, are simpler than the graphs used in <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">FJ15</span>]</cite> to prove <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#S1.Thmtheorem2" title="Theorem 1.2 (Prior lower bound for antisymmetric selection, [FJ15, Thm. 1.4]). ‣ 1.1 Background: Directed cuts, bias, and oblivious algorithms ‣ 1 Introduction ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.2</span></a>, i.e., <a class="ltx_ref" href="https://arxiv.org/html/2411.12976v1#A1.F7" title="In Appendix A Recap: The prior lower bound of \textciteFJ15 (Theorem 1.2) ‣ Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds"><span class="ltx_text ltx_ref_tag">Fig.</span> <span class="ltx_text ltx_ref_tag">7</span></a>: In particular, there are no vertices of bias zero.</p> </div> <div class="ltx_para" id="A1.p5"> <span class="ltx_ERROR undefined" id="A1.p5.1">\printbibliography</span> </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Wed Nov 20 02:08:00 2024 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" 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