Conditional Hardness for Approximate Coloring

TL;DR

This paper establishes conditional hardness results for approximate graph coloring problems based on prominent conjectures, showing the computational difficulty of coloring graphs within certain bounds.

Contribution

It introduces new hardness results for approximate coloring, leveraging Khot's conjectures and advanced invariance principles, extending the understanding of coloring complexity.

Findings

Hardness for deciding if c(G) ≤ q or c(G) ≥ Q for constant q, Q

Almost coloring problem is hard assuming Khot's Unique Games conjecture

Uses invariance principle to bound noise-stability quantities

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].