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

2018
*
International Colloquium on Automata, Languages and Programming
*

Our

doi:10.4230/lipics.icalp.2018.15
dblp:conf/icalp/Bhangale18
fatcat:mgbf47aqozgf5dbcg6a33wvnba
*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? ...##
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The complexity of finding independent sets in bounded degree (hyper)graphs of low chromatic number
[chapter]

2011
*
Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms
*

We prove almost tight

doi:10.1137/1.9781611973082.125
dblp:conf/soda/GuruswamiS11
fatcat:gwwvoafbmvajreh6awqsoz7dva
*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. ...##
###
Approximate coloring of uniform hypergraphs

2003
*
Journal of Algorithms
*

Using a simple construction

doi:10.1016/s0196-6774(03)00077-4
fatcat:u5j6a2xjnralxixk73h7xypnqm
*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. ...##
###
Strong Colorings of Hypergraphs
[chapter]

2005
*
Lecture Notes in Computer Science
*

A strong vertex

doi:10.1007/978-3-540-31833-0_21
fatcat:wpgt4mjnfbaldfxx26dyyssyoe
*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. ...##
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Towards the boundary between easy and hard control problems in multicast Clos networks

2015
*
Bulletin of the Polish Academy of Sciences: Technical Sciences
*

To express our

doi:10.1515/bpasts-2015-0085
fatcat:eu2ff4oafjchllqcdgcukbycme
*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. ...##
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Page 7341 of Mathematical Reviews Vol. , Issue 97M
[page]

1997
*
Mathematical Reviews
*

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 = ...##
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The complexity of generalized graph colorings

1996
*
Discrete Applied Mathematics
*

The techniques rely on

doi:10.1016/0166-218x(96)00096-0
fatcat:rcb3l3banjgmhdbrgtd3nkc2ya
*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. ...##
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Page 6465 of Mathematical Reviews Vol. , Issue 96k
[page]

1996
*
Mathematical Reviews
*

*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. ...

##
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Pure Nash Equilibria: Complete Characterization of Hard and Easy Graphical Games
[article]

2010
*
arXiv
*
pre-print

In this work, we provide such a characterization

arXiv:1002.1363v1
fatcat:5mg2u3eq6za4hgxtejrcqt4jmm
*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] . ...##
###
On Feasible Sets of Mixed Hypergraphs

2004
*
Electronic Journal of Combinatorics
*

This fact has several interesting corollaries, e.g., that deciding whether a feasible set of a mixed

doi:10.37236/1772
fatcat:gavkaumebrbrzdyx6wu5xsw4fi
*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]

2019
*
arXiv
*
pre-print

We then show examples of applying the

arXiv:1910.11302v1
fatcat:zjv764nrmvexhhtzdcxv7usr2m
*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*. ...##
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Approximate Coloring of Uniform Hypergraphs (Extended Abstract)
[chapter]

1998
*
Lecture Notes in Computer Science
*

Using a simple construction

doi:10.1007/3-540-68530-8_40
fatcat:qy3kpwszg5hcvaphb4emftewfm
*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*...##
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Semi-Definite positive Programming Relaxations for Graph Kn-Coloring in Frequency Assignment

2001
*
Reserche operationelle
*

We will propose two relaxations based on Semi-Definite Programming models

doi:10.1051/ro:2001112
fatcat:whc6ezuvt5hohhzawy6evz2dxq
*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. ...##
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The r-coloring and maximum stable set problem in hypergraphs with bounded matching number and edge size
[article]

2022
*
arXiv
*
pre-print

*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]

2003
*
Mathematical Reviews
*

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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