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### NP-Hardness of Coloring 2-Colorable Hypergraph with Poly-Logarithmically Many Colors

Amey Bhangale, Michael Wagner
<span title="2018-06-29">2018</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/pf6v2q6ji5h7birkbrtpdc2uhy" style="color: black;">International Colloquium on Automata, Languages and Programming</a> </i> &nbsp;
Our result is the first to show the NP-hardness of coloring a c-colorable k-uniform hypergraph with poly-logarithmically many colors, for any constants c ≥ 2 and k ≥ 3.  ...  We give very short and simple proofs of the following statements: Given a 2-colorable 4-uniform hypergraph on n vertices, 1. It is NP-hard to color it with log δ n colors for some δ > 0. 2.  ...  Can we go beyond poly(log n) coloring NP-hardness factor for coloring a c-colorable k-uniform hypergraph for some constants c ≥ 2 and k ≥ 3?  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.icalp.2018.15">doi:10.4230/lipics.icalp.2018.15</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/icalp/Bhangale18.html">dblp:conf/icalp/Bhangale18</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mgbf47aqozgf5dbcg6a33wvnba">fatcat:mgbf47aqozgf5dbcg6a33wvnba</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220127101012/https://drops.dagstuhl.de/opus/volltexte/2018/9019/pdf/LIPIcs-ICALP-2018-15.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/3c/b6/3cb61085e1c5349b6b3d059932aa613b841001cf.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.icalp.2018.15"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

### The complexity of finding independent sets in bounded degree (hyper)graphs of low chromatic number [chapter]

Venkatesan Guruswami, Ali Kemal Sinop
<span title="2011-01-23">2011</span> <i title="Society for Industrial and Applied Mathematics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ja7dpjgz2nhjlovlil3ij5m7ei" style="color: black;">Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms</a> </i> &nbsp;
We prove almost tight hardness results under randomized reductions for finding independent sets in bounded degree graphs and hypergraphs that admit a good coloring.  ...  1 r−1 in a 2-colorable r-uniform hypergraph for each fixed r 4.  ...  Acknowledgments We thank Rishi Saket for useful discussions and for pointing us to the reference [3] on the pairwise independent distribution of Theorem 5.1.  ...
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### Approximate coloring of uniform hypergraphs

Michael Krivelevich, Benny Sudakov
<span title="">2003</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/anp7kqfyerdljfxydqb4objj6a" style="color: black;">Journal of Algorithms</a> </i> &nbsp;
Using a simple construction and known results on hardness of graph coloring, we show that for any r 3 it is impossible to approximate in polynomial time the chromatic number of r-uniform hypergraphs on  ...  On the positive side, improving a result of Hofmeister and Lefmann, we present an approximation algorithm for coloring r-uniform hypergraphs on n vertices, whose performance ratio is O(n(log log n) 2 /  ...  Acknowledgments We thank Thomas Hofmeister for bringing paper [15] to our attention.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0196-6774(03)00077-4">doi:10.1016/s0196-6774(03)00077-4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/u5j6a2xjnralxixk73h7xypnqm">fatcat:u5j6a2xjnralxixk73h7xypnqm</a> </span>
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### Strong Colorings of Hypergraphs [chapter]

<span title="">2005</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
A strong vertex coloring of a hypergraph assigns distinct colors to vertices that are contained in a common hyperedge. This captures many previously studied graph coloring problems.  ...  We present nearly tight upper and lower bound on approximating general hypergraphs, both offline and online.  ...  Morris Jr. for many helpful discussions regarding integer programming.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-540-31833-0_21">doi:10.1007/978-3-540-31833-0_21</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wpgt4mjnfbaldfxx26dyyssyoe">fatcat:wpgt4mjnfbaldfxx26dyyssyoe</a> </span>
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### Towards the boundary between easy and hard control problems in multicast Clos networks

P. Obszarski, A. Jastrzębski, M. Kubale
<span title="2015-09-01">2015</span> <i title="Walter de Gruyter GmbH"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/gw7kalohf5dsdag2u3svsyd66y" style="color: black;">Bulletin of the Polish Academy of Sciences: Technical Sciences</a> </i> &nbsp;
To express our results in a formal way we introduce a model of hypergraph edge-coloring. A new class of bipartite hypergraphs corresponding to Clos networks is studied.  ...  This is in sharp contrast to classical unicast Clos networks for which all the control problems are polynomially solvable.  ...  The authors wish to thank anonymous referees for their valuable comments. This article has been partially supported by National Science Centre under the grant DEC-2011/02/A/ST6/00201.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1515/bpasts-2015-0085">doi:10.1515/bpasts-2015-0085</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eu2ff4oafjchllqcdgcukbycme">fatcat:eu2ff4oafjchllqcdgcukbycme</a> </span>
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### Page 7341 of Mathematical Reviews Vol. , Issue 97M [page]

<span title="">1997</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Our first result is an algorithm that colors any 2- colorable hypergraph with 7m vertices and dimension d using O(n'~'/4log'~'/4n) colors.  ...  Previously obtained results for simple hypergraphs and multihypergraphs di- rectly follow from Theorem | in the boundary cases r = | and r =  ...
<span class="external-identifiers"> </span>
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### The complexity of generalized graph colorings

Jason I. Brown
<span title="">1996</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
The techniques rely on new NP-completeness results for hypergraph colorings.  ...  Given a graph property P and positive integer k, a P k-coloring of a graph G is an assignment of one of k colors to each vertex of the graph so that the subgraphs induced by each color class have property  ...  The author wishes to thank Derek Comeil for his insightful comments and suggestions.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0166-218x(96)00096-0">doi:10.1016/0166-218x(96)00096-0</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rcb3l3banjgmhdbrgtd3nkc2ya">fatcat:rcb3l3banjgmhdbrgtd3nkc2ya</a> </span>
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### Page 6465 of Mathematical Reviews Vol. , Issue 96k [page]

<span title="">1996</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
For a general hypergraph, the determination of packing and covering parame- ters is NP-hard.  ...  The second section, “Hypergraphs versus graphs”, focuses on results that extend graph theory. These results include some gen- eralizations about cycles and hypergraph minors.  ...
<span class="external-identifiers"> </span>
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### Pure Nash Equilibria: Complete Characterization of Hard and Easy Graphical Games [article]

<span title="2010-02-06">2010</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this work, we provide such a characterization for the case of bounded in-degree graphs, thereby resolving the gap between existing hardness and tractability results.  ...  We show that the tractable classes of bounded-arity colored hypergraphical games are precisely those whose reduced graphs have bounded treewidth modulo homomorphic equivalence.  ...  On the other hand, our proof for our hardness result is quite unlike the existing NP-hardness proof for graphical games [11] .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1002.1363v1">arXiv:1002.1363v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5mg2u3eq6za4hgxtejrcqt4jmm">fatcat:5mg2u3eq6za4hgxtejrcqt4jmm</a> </span>
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### On Feasible Sets of Mixed Hypergraphs

Daniel Král
<span title="2004-03-04">2004</span> <i title="The Electronic Journal of Combinatorics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v5dyak6ulffqfara7hmuchh24a" style="color: black;">Electronic Journal of Combinatorics</a> </i> &nbsp;
This fact has several interesting corollaries, e.g., that deciding whether a feasible set of a mixed hypergraph is gap-free is both NP-hard and coNP-hard.  ...  We further prove that for any fixed finite sets of positive integers \$A_1\subset A_2\$ (\$1\notin A_2\$), it is NP-hard to decide whether the feasible set of a given mixed hypergraph is equal to \$A_2\$ even  ...  of this paper and the anonymous referee for helpful comments on the presentation of the results and pointing out several related references.  ...
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### Color-critical Graphs and Hereditary Hypergraphs [article]

András Sebő
<span title="2019-10-24">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We then show examples of applying the results to new problems and indicate the way to algorithms and refined complexity results for all these examples at the same time.  ...  A quick proof of Gallai's celebrated theorem on color-critical graphs is given from Gallai's simple, ingenious lemma on factor-critical graphs, in terms of partitioning the vertex-set into a minimum number  ...  Conclusion: We tried to show by the Theorem and multiple examples how results on graph colorings may be extended to covers in hypergraphs.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1910.11302v1">arXiv:1910.11302v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zjv764nrmvexhhtzdcxv7usr2m">fatcat:zjv764nrmvexhhtzdcxv7usr2m</a> </span>
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### Approximate Coloring of Uniform Hypergraphs (Extended Abstract) [chapter]

Michael Krivelevich, Benny Sudakov
<span title="">1998</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Using a simple construction and known results on hardness of graph coloring, we show that for any r 3 it is impossible to approximate in polynomial time the chromatic number of r-uniform hypergraphs on  ...  We also describe an algorithm for coloring 3-uniform 2-colorable hypergraphs on n vertices inÕ(n 9=41 ) colors, thus improving previous results of Chen and Frieze and of Kelsen, Mahajan and Ramesh.  ...  At this point we can use the fact that the graph formed by edges of size two is also 2-colorable and therefore we need at most two new colors for each color class in the coloring of the 3-uniform hypergraph  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-68530-8_40">doi:10.1007/3-540-68530-8_40</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qy3kpwszg5hcvaphb4emftewfm">fatcat:qy3kpwszg5hcvaphb4emftewfm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190228130329/http://pdfs.semanticscholar.org/926f/d11a6b1f2df00264051986f723167aa1bf2b.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/92/6f/926fd11a6b1f2df00264051986f723167aa1bf2b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-68530-8_40"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

### Semi-Definite positive Programming Relaxations for Graph Kn-Coloring in Frequency Assignment

Philippe Meurdesoif, Benoît Rottembourg
<span title="">2001</span> <i title="EDP Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/inei2pvlvnaw7lwfzvf7meb67e" style="color: black;">Reserche operationelle</a> </i> &nbsp;
We will propose two relaxations based on Semi-Definite Programming models for graph and hypergraph coloring, to approximate those (generally) NP-hard problems, as well as a generalization of the works  ...  of Karger et al. for hypergraph coloring, to find good feasible solutions with a probabilistic approach.  ...  We can obtain non-approximability results for K n -Coloring by combining this transformation with hardness of approximation results for Coloring, such as the result of Lund et al.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1051/ro:2001112">doi:10.1051/ro:2001112</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/whc6ezuvt5hohhzawy6evz2dxq">fatcat:whc6ezuvt5hohhzawy6evz2dxq</a> </span>
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### The r-coloring and maximum stable set problem in hypergraphs with bounded matching number and edge size [article]

Yanjia Li, Sophie Spirkl
<span title="2022-05-02">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
hypergraphs G with ν(G) ≤ s; Maximum Weight Stable Set in k-uniform hypergraphs with ν(G) ≤ s; as well as partial results for r-Coloring in k-uniform hypergraphs ν(G) ≤ s.  ...  3-coloring linear 3-uniform hypergraphs G with ν(G) ≤ 532 is NP-hard.  ...  The hypergraph coloring problem is a natural extension of the graph coloring problem; see the survey [3] . The following result shows that the problem is NP-hard, even in uniform hypergraphs.  ...
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### Page 4923 of Mathematical Reviews Vol. , Issue 2003g [page]

<span title="">2003</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
They also study operations that preserve unique colorability and show that recognition of uc mixed hypergraphs is NP-hard. Douglas B.  ...  A short proof is given for a result of P. M. Gibson [Proc. Amer. Math.  ...
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