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New NP-hardness results for 3-Coloring and 2-to-1 Label Cover [article]

Per Austrin, Ryan O'Donnell, Li-Yang Tan, John Wright
2012 arXiv   pre-print
In a related result, we show that given a satisfiable instance of the 2-to-1 Label Cover problem, it is NP-hard to find a (23/24 + )-satisfying assignment.  ...  We show that given a 3-colorable graph, it is NP-hard to find a 3-coloring with (16/17 + ) of the edges bichromatic.  ...  Interestingly, to a certain extent the reverse is also true: it is "folklore" that hardness results for graph 3-Coloring immediately imply hardness results for the 2-to-1 Label Cover problem with label  ... 
arXiv:1210.5648v2 fatcat:qjxagczemfdppbflroqazi6geq

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

Amey Bhangale, Michael Wagner
2018 International Colloquium on Automata, Languages and Programming  
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?  ... 
doi:10.4230/lipics.icalp.2018.15 dblp:conf/icalp/Bhangale18 fatcat:mgbf47aqozgf5dbcg6a33wvnba

Improved Hardness of Approximating Chromatic Number [article]

Sangxia Huang
2013 arXiv   pre-print
We prove that for sufficiently large K, it is NP-hard to color K-colorable graphs with less than 2^K^1/3 colors. This improves the previous result of K versus K^O(log K) in Khot [14].  ...  Acknowledgements The author is grateful to Siu On Chan for pointing out Theorem E.1 in his paper, leading to a gap of K vs. 2 K 1/3 , improving the original gap of K vs. 2 K 1/5 .  ...  It is known that coloring 3-colorable graph with 5 colors is NP-hard, and for general K-colorable graph it is NP-hard to color with K + 2⌈ K 3colors [13, 9] .  ... 
arXiv:1301.5216v3 fatcat:akcrrznhqjhynm3puyq7feoocu

Improved Hardness of Approximating Chromatic Number [chapter]

Sangxia Huang
2013 Lecture Notes in Computer Science  
We prove that for sufficiently large K, it is NP-hard to color K-colorable graphs with less than 2 K 1/3 colors. This improves the previous result of K versus K O(log K) in Khot [14] .  ...  Acknowledgements The author is grateful to Siu On Chan for pointing out Theorem E.1 in his paper, leading to a gap of K vs. 2 K 1/3 , improving the original gap of K vs. 2 K 1/5 .  ...  It is known that coloring 3-colorable graph with 5 colors is NP-hard, and for general K-colorable graph it is NP-hard to color with K + 2⌈ K 3colors [13, 9] .  ... 
doi:10.1007/978-3-642-40328-6_17 fatcat:nqgtmn7hrbbfbisaxzzhpsozjy

d-To-1 Hardness of Coloring 3-Colorable Graphs with O(1) Colors

Venkatesan Guruswami, Sai Sandeep, Emanuela Merelli, Anuj Dawar, Artur Czumaj
2020 International Colloquium on Automata, Languages and Programming  
The d-to-1 conjecture of Khot asserts that it is NP-hard to satisfy an ε fraction of constraints of a satisfiable d-to-1 Label Cover instance, for arbitrarily small ε > 0.  ...  Earlier, the hardness of O(1)-coloring a 4-colorable graphs is known under the 2-to-1 conjecture, which is the strongest in the family of d-to-1 conjectures, and the hardness for 3-colorable graphs is  ...  Since b(n) is increasing and b(n) > n for all n ≥ 4, it follows that a NP-hardness result for O(1)-coloring q-colorable graphs also implies NP-hardness of O(1)-coloring 4-colorable graphs.  ... 
doi:10.4230/lipics.icalp.2020.62 dblp:conf/icalp/GuruswamiS20 fatcat:jlb2a77e25cvpcjwi7tbdklmja

A New Point of NP-Hardness for 2-to-1 Label Cover [chapter]

Per Austrin, Ryan O'Donnell, John Wright
2012 Lecture Notes in Computer Science  
For every integer d ≥ 2 and > 0, there is a label set size q such that it is NP-hard to (1, )-decide the d-to-1 Label Cover problem.  ...  The first result along these lines was due to Dinur, Mossel, and Regev [DMR09] who showed that the 2-to-1 Conjecture implies that it is NP-hard to C-color a 4-colorable graph for any constant C.  ...  Our results In this paper, we focus on proving NP-hardness for the 2-to-1 Label Cover problem.  ... 
doi:10.1007/978-3-642-32512-0_1 fatcat:6qhlb4q4uzdqffgf5dnoj3qkwm

Rainbow Coloring Hardness via Low Sensitivity Polymorphisms

Venkatesan Guruswami, Sai Sandeep
2020 SIAM Journal on Discrete Mathematics  
Given as input such a hypergraph, finding a r-rainbow coloring of it is NP-hard for all k ≥ 3 and r ≥ 2.  ...  In this work we consider the next smaller value of r = k − 1, and prove that in this case it is NP-hard to rainbow color the hypergraph with q := k−2 2 colors.  ...  This is the basis of the hardness results for (2 + )-SAT [4] and 3-coloring graphs that admit a homomorphism to C k for any fixed odd integer k [24] .  ... 
doi:10.1137/19m127731x fatcat:ji3puh626fb43psqamoqrhjjky

Inapproximability Results for Combinatorial Auctions with Submodular Utility Functions

Subhash Khot, Richard J. Lipton, Evangelos Markakis, Aranyak Mehta
2007 Algorithmica  
Our result is based on a reduction from a multi-prover proof system for MAX-3-COLORING.  ...  − 1/e 0.632, unless P= NP.  ...  Acknowledgements We would like to thank Amin Saberi and Vahab Mirrokni for useful discussions and Shahar Dobzinski for comments. We would also like to thank the anonymous referees for their comments.  ... 
doi:10.1007/s00453-007-9105-7 fatcat:jo7odecozrbb7ikluyqsz2se4u

Inapproximability Results for Combinatorial Auctions with Submodular Utility Functions [chapter]

Subhash Khot, Richard J. Lipton, Evangelos Markakis, Aranyak Mehta
2005 Lecture Notes in Computer Science  
Our result is based on a reduction from a multi-prover proof system for MAX-3-COLORING.  ...  − 1/e 0.632, unless P= NP.  ...  Acknowledgements We would like to thank Amin Saberi and Vahab Mirrokni for useful discussions and Shahar Dobzinski for comments. We would also like to thank the anonymous referees for their comments.  ... 
doi:10.1007/11600930_10 fatcat:67sbsrg5nbglrhalhceiovw4m4

On Complexity and Approximability of the Labeled Maximum/Perfect Matching Problems [chapter]

Jérôme Monnot
2005 Lecture Notes in Computer Science  
In this paper, we deal with both the complexity and the approximability of the labeled perfect matching problem in bipartite graphs.  ...  Given a simple graph G = (V, E) with n vertices with a color (or label) function L : E → {c1, . . . , cq}, the labeled maximum matching problem consists in finding a maximum matching on G that uses a minimum  ...  We conjecture that Labeled M in P M is not O(n ε )-approximable in complete bipartite graphs. Thus, a challenge will be to give better approximate algorithms or to improve the lower bound.  ... 
doi:10.1007/11602613_93 fatcat:2ojn36muwbgetldpkvty35n7te

A new point of NP-hardness for 2-to-1 Label Cover [article]

Per Austrin and Ryan O'Donnell and John Wright
2012 arXiv   pre-print
We show that given a satisfiable instance of the 2-to-1 Label Cover problem, it is NP-hard to find a (23/24 + )-satisfying assignment.  ...  Our results In this paper, we focus on proving NP-hardness for the 2-to-1 Label Cover problem.  ...  The main result of our paper gives an improved hardness result: Theorem 1.2. For all > 0, (1, 23 24 + )-deciding the 2-to-1 Label Cover problem with label set sizes 3 & 6 is NP-hard.  ... 
arXiv:1204.5666v1 fatcat:a5b4kdi6ojc7blmjtrbuqmpdei

The labeled perfect matching in bipartite graphs

Jérôme Monnot
2005 Information Processing Letters  
Given a simple graph G = (V, E) with |V | = 2n vertices such that E contains a perfect matching (of size n), together with a color (or label) function L : E → {c 1 , . . . , c q }, the labeled perfect  ...  In this paper, we deal with both the complexity and the approximability of the labeled perfect matching problem in bipartite graphs.  ...  Thus, we deduce from this result that Labeled M ax P M r is NP-hard for any r ≥ 2. We strengthen this result using a reduction from Max balanced 2-Sat.  ... 
doi:10.1016/j.ipl.2005.06.009 fatcat:ygk54vxvxzeo5hnipt3hbe3vpi

Fixed-parameter complexity of λ-labelings

Jiřı́ Fiala, Ton Kloks, Jan Kratochvı́l
2001 Discrete Applied Mathematics  
We show several hardness results for L(G; p; q) including that for any p ¿ q¿1 there is a = (p; q) such that deciding if L(G; p; q)6 is NP-complete, and that for p¿2q, this decision is NP-complete for  ...  We study the minimum value = (G) such that G admits a -labeling. We show that for every ÿxed value k¿4 it is NP-complete to determine whether (G)6k.  ...  Most of our NP-hardness results are obtained via graph covers. Deÿnition 2.3. Let G be a graph and H be a multigraph.  ... 
doi:10.1016/s0166-218x(00)00387-5 fatcat:5f6hb3oln5bnri66hgf6i5rnza

Almost Optimal Inapproximability of Multidimensional Packing Problems [article]

Sai Sandeep
2021 arXiv   pre-print
We also show that the problem is NP-hard to approximate within (loglog d)^ω(1). 3.  ...  Previously, no hardness results that grow with d were known for Vector Scheduling and Vector Bin Covering when d is part of the input and for Vector Bin Packing when d is a fixed constant.  ...  Acknowledgements I am greatly indebted to Venkatesan Guruswami for helpful discussions, for his detailed feedback on the manuscript which significantly improved the presentation, and for his encouragement  ... 
arXiv:2101.02854v2 fatcat:s3nxsafvurbmxmmtdrdubumoz4

The Hardness of 3-Uniform Hypergraph Coloring

Irit Dinur*, Oded Regev†, Clifford Smyth‡
2005 Combinatorica  
This enables us to prove that for any constant c, it is NP-hard to color a 2-colorable 4-uniform hypergraph using just c colors, and also yields a super-constant inapproximability result under a stronger  ...  The covering complexity of PCP verifiers offers a promising route to getting stronger inapproximability results for some minimization problems, and in particular, (hyper)-graph coloring problems.  ...  No non-trivial hardness results seem to be known, and in fact it was not known prior to our work if 3-coloring a 2-colorable 4-uniform hypergraph is NP-hard.  ... 
doi:10.1007/s00493-005-0032-4 fatcat:455einszcfcgncrd4ld62ukeyi
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