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name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> <input id="query" name="query" type="text" value="Duraj, L"> <ul id="abstracts"><li><input checked id="abstracts-0" name="abstracts" type="radio" value="show"> <label for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" name="abstracts" type="radio" value="hide"> <label for="abstracts-1">Hide abstracts</label></li></ul> </div> <div class="box field is-grouped is-grouped-multiline level-item"> <div class="control"> <span class="select is-small"> <select id="size" name="size"><option value="25">25</option><option selected value="50">50</option><option value="100">100</option><option value="200">200</option></select> </span> <label for="size">results per page</label>. </div> <div class="control"> <label for="order">Sort results by</label> <span class="select is-small"> <select id="order" name="order"><option selected value="-announced_date_first">Announcement date (newest first)</option><option value="announced_date_first">Announcement date (oldest first)</option><option value="-submitted_date">Submission date (newest first)</option><option value="submitted_date">Submission date (oldest first)</option><option value="">Relevance</option></select> </span> </div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.06072">arXiv:2309.06072</a> <span> [<a href="https://arxiv.org/pdf/2309.06072">pdf</a>, <a href="https://arxiv.org/ps/2309.06072">ps</a>, <a href="https://arxiv.org/format/2309.06072">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computational Geometry">cs.CG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> </div> </div> <p class="title is-5 mathjax"> The $蠂$-binding function of $d$-directional segment graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Duraj%2C+L">Lech Duraj</a>, <a href="/search/cs?searchtype=author&query=Kang%2C+R+J">Ross J. Kang</a>, <a href="/search/cs?searchtype=author&query=La%2C+H">Hoang La</a>, <a href="/search/cs?searchtype=author&query=Narboni%2C+J">Jonathan Narboni</a>, <a href="/search/cs?searchtype=author&query=Pokr%C3%BDvka%2C+F">Filip Pokr媒vka</a>, <a href="/search/cs?searchtype=author&query=Rambaud%2C+C">Cl茅ment Rambaud</a>, <a href="/search/cs?searchtype=author&query=Reinald%2C+A">Amadeus Reinald</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2309.06072v2-abstract-short" style="display: inline;"> Given a positive integer $d$, the class $d$-DIR is defined as all those intersection graphs formed from a finite collection of line segments in ${\mathbb R}^2$ having at most $d$ slopes. Since each slope induces an interval graph, it easily follows for every $G$ in $d$-DIR with clique number at most $蠅$ that the chromatic number $蠂(G)$ of $G$ is at most $d蠅$. We show for every even value of $蠅$ ho… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.06072v2-abstract-full').style.display = 'inline'; document.getElementById('2309.06072v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.06072v2-abstract-full" style="display: none;"> Given a positive integer $d$, the class $d$-DIR is defined as all those intersection graphs formed from a finite collection of line segments in ${\mathbb R}^2$ having at most $d$ slopes. Since each slope induces an interval graph, it easily follows for every $G$ in $d$-DIR with clique number at most $蠅$ that the chromatic number $蠂(G)$ of $G$ is at most $d蠅$. We show for every even value of $蠅$ how to construct a graph in $d$-DIR that meets this bound exactly. This partially confirms a conjecture of Bhattacharya, Dvo艡谩k and Noorizadeh. Furthermore, we show that the $蠂$-binding function of $d$-DIR is $蠅\mapsto d蠅$ for $蠅$ even and $蠅\mapsto d(蠅-1)+1$ for $蠅$ odd. This extends an earlier result by Kostochka and Ne拧et艡il, which treated the special case $d=2$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.06072v2-abstract-full').style.display = 'none'; document.getElementById('2309.06072v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">11 pages, 3 figures; v2 includes corrections for referee comments</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C15; 05C62; 05C17 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2307.08162">arXiv:2307.08162</a> <span> [<a href="https://arxiv.org/pdf/2307.08162">pdf</a>, <a href="https://arxiv.org/format/2307.08162">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computational Geometry">cs.CG</span> </div> </div> <p class="title is-5 mathjax"> Better Diameter Algorithms for Bounded VC-dimension Graphs and Geometric Intersection Graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Duraj%2C+L">Lech Duraj</a>, <a href="/search/cs?searchtype=author&query=Konieczny%2C+F">Filip Konieczny</a>, <a href="/search/cs?searchtype=author&query=Pot%C4%99pa%2C+K">Krzysztof Pot臋pa</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2307.08162v2-abstract-short" style="display: inline;"> We develop a framework for algorithms finding the diameter in graphs of bounded distance Vapnik-Chervonenkis dimension, in (parameterized) subquadratic time complexity. The class of bounded distance VC-dimension graphs is wide, including, e.g. all minor-free graphs. We build on the work of Ducoffe et al. [SODA'20, SIGCOMP'22], improving their technique. With our approach the algorithms become si… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.08162v2-abstract-full').style.display = 'inline'; document.getElementById('2307.08162v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.08162v2-abstract-full" style="display: none;"> We develop a framework for algorithms finding the diameter in graphs of bounded distance Vapnik-Chervonenkis dimension, in (parameterized) subquadratic time complexity. The class of bounded distance VC-dimension graphs is wide, including, e.g. all minor-free graphs. We build on the work of Ducoffe et al. [SODA'20, SIGCOMP'22], improving their technique. With our approach the algorithms become simpler and faster, working in $\mathcal{O}(k \cdot n^{1-1/d} \cdot m \cdot \mathrm{polylog}(n))$ time complexity for the graph on $n$ vertices and $m$ edges, where $k$ is the diameter and $d$ is the distance VC-dimension of the graph. Furthermore, it allows us to use the improved technique in more general setting. In particular, we use this framework for geometric intersection graphs, i.e. graphs where vertices are identical geometric objects on a plane and the adjacency is defined by intersection. Applying our approach for these graphs, we partially answer a question posed by Bringmann et al. [SoCG'22], finding an $\mathcal{O}(n^{7/4} \cdot \mathrm{polylog}(n))$ parameterized diameter algorithm for unit square intersection graph of size $n$, as well as a more general algorithm for convex polygon intersection graphs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.08162v2-abstract-full').style.display = 'none'; document.getElementById('2307.08162v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">36 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2102.12968">arXiv:2102.12968</a> <span> [<a href="https://arxiv.org/pdf/2102.12968">pdf</a>, <a href="https://arxiv.org/ps/2102.12968">ps</a>, <a href="https://arxiv.org/format/2102.12968">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.ejc.2020.103205">10.1016/j.ejc.2020.103205 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Random hypergraphs and property B </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Duraj%2C+L">Lech Duraj</a>, <a href="/search/cs?searchtype=author&query=Kozik%2C+J">Jakub Kozik</a>, <a href="/search/cs?searchtype=author&query=Shabanov%2C+D">Dmitry Shabanov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2102.12968v1-abstract-short" style="display: inline;"> In 1964 Erd艖s proved that $(1+\oh{1})) \frac{\eul \ln(2)}{4} k^2 2^{k}$ edges are sufficient to build a $k$-graph which is not two colorable. To this day, it is not known whether there exist such $k$-graphs with smaller number of edges. Erd艖s' bound is consequence of the fact that a hypergraph with $k^2/2$ vertices and $M(k)=(1+\oh{1}) \frac{\eul \ln(2)}{4} k^2 2^{k}$ randomly chosen edges of size… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.12968v1-abstract-full').style.display = 'inline'; document.getElementById('2102.12968v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2102.12968v1-abstract-full" style="display: none;"> In 1964 Erd艖s proved that $(1+\oh{1})) \frac{\eul \ln(2)}{4} k^2 2^{k}$ edges are sufficient to build a $k$-graph which is not two colorable. To this day, it is not known whether there exist such $k$-graphs with smaller number of edges. Erd艖s' bound is consequence of the fact that a hypergraph with $k^2/2$ vertices and $M(k)=(1+\oh{1}) \frac{\eul \ln(2)}{4} k^2 2^{k}$ randomly chosen edges of size $k$ is asymptotically almost surely not two colorable. Our first main result implies that for any $\varepsilon > 0$, any $k$-graph with $(1-\varepsilon) M(k)$ randomly and uniformly chosen edges is a.a.s. two colorable. The presented proof is an adaptation of the second moment method analogous to the developments of Achlioptas and Moore from 2002 who considered the problem with fixed size of edges and number of vertices tending to infinity. In the second part of the paper we consider the problem of algorithmic coloring of random $k$-graphs. We show that quite simple, and somewhat greedy procedure, a.a.s. finds a proper two coloring for random $k$-graphs on $k^2/2$ vertices, with at most $\Oh{k\ln k\cdot 2^k}$ edges. That is of the same asymptotic order as the analogue of the \emph{algorithmic barrier} defined by Achlioptas and Coja-Oghlan in 2008, for the case of fixed $k$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.12968v1-abstract-full').style.display = 'none'; document.getElementById('2102.12968v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C65; 05C80; 05D40 <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> F.2.2; G.2.2; G.3 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> European Journal of Combinatorics, Volume 91, January 2021, 103205 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1908.11819">arXiv:1908.11819</a> <span> [<a href="https://arxiv.org/pdf/1908.11819">pdf</a>, <a href="https://arxiv.org/ps/1908.11819">ps</a>, <a href="https://arxiv.org/format/1908.11819">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computational Complexity">cs.CC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1137/1.9781611975994.3">10.1137/1.9781611975994.3 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Equivalences between triangle and range query problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Duraj%2C+L">Lech Duraj</a>, <a href="/search/cs?searchtype=author&query=Kleiner%2C+K">Krzysztof Kleiner</a>, <a href="/search/cs?searchtype=author&query=Polak%2C+A">Adam Polak</a>, <a href="/search/cs?searchtype=author&query=Williams%2C+V+V">Virginia Vassilevska Williams</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1908.11819v2-abstract-short" style="display: inline;"> We define a natural class of range query problems, and prove that all problems within this class have the same time complexity (up to polylogarithmic factors). The equivalence is very general, and even applies to online algorithms. This allows us to obtain new improved algorithms for all of the problems in the class. We then focus on the special case of the problems when the queries are offline… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1908.11819v2-abstract-full').style.display = 'inline'; document.getElementById('1908.11819v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1908.11819v2-abstract-full" style="display: none;"> We define a natural class of range query problems, and prove that all problems within this class have the same time complexity (up to polylogarithmic factors). The equivalence is very general, and even applies to online algorithms. This allows us to obtain new improved algorithms for all of the problems in the class. We then focus on the special case of the problems when the queries are offline and the number of queries is linear. We show that our range query problems are runtime-equivalent (up to polylogarithmic factors) to counting for each edge $e$ in an $m$-edge graph the number of triangles through $e$. This natural triangle problem can be solved using the best known triangle counting algorithm, running in $O(m^{2蠅/(蠅+1)}) \leq O(m^{1.41})$ time. Moreover, if $蠅=2$, the $O(m^{2蠅/(蠅+1)})$ running time is known to be tight (within $m^{o(1)}$ factors) under the 3SUM Hypothesis. In this case, our equivalence settles the complexity of the range query problems. Our problems constitute the first equivalence class with this peculiar running time bound. To better understand the complexity of these problems, we also provide a deeper insight into the family of triangle problems, in particular showing black-box reductions between triangle listing and per-edge triangle detection and counting. As a byproduct of our reductions, we obtain a simple triangle listing algorithm matching the state-of-the-art for all regimes of the number of triangles. We also give some not necessarily tight, but still surprising reductions from variants of matrix products, such as the $(\min,\max)$-product. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1908.11819v2-abstract-full').style.display = 'none'; document.getElementById('1908.11819v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 August, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1902.06864">arXiv:1902.06864</a> <span> [<a href="https://arxiv.org/pdf/1902.06864">pdf</a>, <a href="https://arxiv.org/format/1902.06864">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> A sub-quadratic algorithm for the longest common increasing subsequence problem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Duraj%2C+L">Lech Duraj</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1902.06864v3-abstract-short" style="display: inline;"> The Longest Common Increasing Subsequence problem (LCIS) is a natural variant of the celebrated Longest Common Subsequence (LCS) problem. For LCIS, as well as for LCS, there is an $O(n^2)$-time algorithm and a SETH-based conditional lower bound of $O(n^{2-\varepsilon})$. For LCS, there is also the Masek-Paterson $O(n^2 / \log{n})$-time algorithm, which does not seem to adapt to LCIS in any obvious… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1902.06864v3-abstract-full').style.display = 'inline'; document.getElementById('1902.06864v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1902.06864v3-abstract-full" style="display: none;"> The Longest Common Increasing Subsequence problem (LCIS) is a natural variant of the celebrated Longest Common Subsequence (LCS) problem. For LCIS, as well as for LCS, there is an $O(n^2)$-time algorithm and a SETH-based conditional lower bound of $O(n^{2-\varepsilon})$. For LCS, there is also the Masek-Paterson $O(n^2 / \log{n})$-time algorithm, which does not seem to adapt to LCIS in any obvious way. Hence, a natural question arises: does any (slightly) sub-quadratic algorithm exist for the Longest Common Increasing Subsequence problem? We answer this question positively, presenting a $O(n^2 / \log^a{n})$-time algorithm for $a = \frac{1}{6}-o(1)$. The algorithm is not based on memorizing small chunks of data (often used for logarithmic speedups, including the "Four Russians Trick" in LCS), but rather utilizes a new technique, bounding the number of significant symbol matches between the two sequences. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1902.06864v3-abstract-full').style.display = 'none'; document.getElementById('1902.06864v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 February, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages; STACS 2020 version -- heavily corrected from the previous one, might actually be readable</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1803.03060">arXiv:1803.03060</a> <span> [<a href="https://arxiv.org/pdf/1803.03060">pdf</a>, <a href="https://arxiv.org/ps/1803.03060">ps</a>, <a href="https://arxiv.org/format/1803.03060">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.4230/LIPIcs.ICALP.2018.46">10.4230/LIPIcs.ICALP.2018.46 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A note on two-colorability of nonuniform hypergraphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Duraj%2C+L">Lech Duraj</a>, <a href="/search/cs?searchtype=author&query=Gutowski%2C+G">Grzegorz Gutowski</a>, <a href="/search/cs?searchtype=author&query=Kozik%2C+J">Jakub Kozik</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1803.03060v1-abstract-short" style="display: inline;"> For a hypergraph $H$, let $q(H)$ denote the expected number of monochromatic edges when the color of each vertex in $H$ is sampled uniformly at random from the set of size 2. Let $s_{\min}(H)$ denote the minimum size of an edge in $H$. Erd艖s asked in 1963 whether there exists an unbounded function $g(k)$ such that any hypergraph $H$ with $s_{\min}(H) \geq k$ and $q(H) \leq g(k)$ is two colorable.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.03060v1-abstract-full').style.display = 'inline'; document.getElementById('1803.03060v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1803.03060v1-abstract-full" style="display: none;"> For a hypergraph $H$, let $q(H)$ denote the expected number of monochromatic edges when the color of each vertex in $H$ is sampled uniformly at random from the set of size 2. Let $s_{\min}(H)$ denote the minimum size of an edge in $H$. Erd艖s asked in 1963 whether there exists an unbounded function $g(k)$ such that any hypergraph $H$ with $s_{\min}(H) \geq k$ and $q(H) \leq g(k)$ is two colorable. Beck in 1978 answered this question in the affirmative for a function $g(k) = 螛(\log^* k)$. We improve this result by showing that, for an absolute constant $未>0$, a version of random greedy coloring procedure is likely to find a proper two coloring for any hypergraph $H$ with $s_{\min}(H) \geq k$ and $q(H) \leq 未\cdot \log k$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.03060v1-abstract-full').style.display = 'none'; document.getElementById('1803.03060v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 March, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C35; 05C65 <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> G.2.1; G.2.2; G.3; F.2.2 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1709.10075">arXiv:1709.10075</a> <span> [<a href="https://arxiv.org/pdf/1709.10075">pdf</a>, <a href="https://arxiv.org/format/1709.10075">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computational Complexity">cs.CC</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.4230/LIPIcs.IPEC.2017.15">10.4230/LIPIcs.IPEC.2017.15 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00453-018-0485-7">10.1007/s00453-018-0485-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Tight Conditional Lower Bounds for Longest Common Increasing Subsequence </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Duraj%2C+L">Lech Duraj</a>, <a href="/search/cs?searchtype=author&query=K%C3%BCnnemann%2C+M">Marvin K眉nnemann</a>, <a href="/search/cs?searchtype=author&query=Polak%2C+A">Adam Polak</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1709.10075v1-abstract-short" style="display: inline;"> We consider the canonical generalization of the well-studied Longest Increasing Subsequence problem to multiple sequences, called $k$-LCIS: Given $k$ integer sequences $X_1,\dots,X_k$ of length at most $n$, the task is to determine the length of the longest common subsequence of $X_1,\dots,X_k$ that is also strictly increasing. Especially for the case of $k=2$ (called LCIS for short), several algo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.10075v1-abstract-full').style.display = 'inline'; document.getElementById('1709.10075v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1709.10075v1-abstract-full" style="display: none;"> We consider the canonical generalization of the well-studied Longest Increasing Subsequence problem to multiple sequences, called $k$-LCIS: Given $k$ integer sequences $X_1,\dots,X_k$ of length at most $n$, the task is to determine the length of the longest common subsequence of $X_1,\dots,X_k$ that is also strictly increasing. Especially for the case of $k=2$ (called LCIS for short), several algorithms have been proposed that require quadratic time in the worst case. Assuming the Strong Exponential Time Hypothesis (SETH), we prove a tight lower bound, specifically, that no algorithm solves LCIS in (strongly) subquadratic time. Interestingly, the proof makes no use of normalization tricks common to hardness proofs for similar problems such as LCS. We further strengthen this lower bound (1) to rule out $O((nL)^{1-\varepsilon})$ time algorithms for LCIS, where $L$ denotes the solution size, (2) to rule out $O(n^{k-\varepsilon})$ time algorithms for $k$-LCIS, and (3) to follow already from weaker variants of SETH. We obtain the same conditional lower bounds for the related Longest Common Weakly Increasing Subsequence problem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.10075v1-abstract-full').style.display = 'none'; document.getElementById('1709.10075v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages, full version of IPEC 2017 paper</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Algorithmica 81, 3968-3992 (2019) </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" 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