Conditional Hardness for Approximate Coloring

Irit Dinur; Elchanan Mossel; Oded Regev

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Conditional Hardness for Approximate Coloring

Irit Dinur·Elchanan Mossel·Oded Regev

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Abstract

We study the coloring problem: Given a graph G, decide whether $c(G) \leq q$ or $c(G) \ge Q$, where c(G) is the chromatic number of G. We derive conditional hardness for this problem for any constant $3 \le q < Q$. For $q\ge 4$, our result is based on Khot's 2-to-1 conjecture [Khot'02]. For $q=3$, we base our hardness result on a certain `fish shaped' variant of his conjecture. We also prove that the problem almost coloring is hard for any constant $\eps>0$, assuming Khot's Unique Games conjecture. This is the problem of deciding for a given graph, between the case where one can 3-color all but a $\eps$ fraction of the vertices without monochromatic edges, and the case where the graph contains no independent set of relative size at least $\eps$. Our result is based on bounding various generalized noise-stability quantities using the invariance principle of Mossel et al [MOO'05].

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Irit Dinur, Elchanan Mossel, Oded Regev. 2005-04-14. Conditional Hardness for Approximate Coloring. https://arxiv.org/abs/cs/0504062

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