Conditional Hardness for Approximate Coloring

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&#39;s 2-to-1 conjecture [Khot&#39;02]. For $q=3$, we base our hardness result on a certain `fish shaped&#39; variant of his conjecture. We also prove that the problem almost coloring is hard for any constant $\eps>0$, assuming Khot&#39;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&#39;05].