Improved Inapproximability Results for Maximum k-Colorable Subgraph

We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a k-colorable graph with k colors so that a maximum fraction of edges are properly colored (i.e. their endpoints receive different colors). A random k-coloring properly colors an expected fraction 1 − 1 k of edges. We prove that given a graph promised to be k-colorable, it is NP-hard to find a k-coloring that properly colors more than a fraction ≈ 1− 1 33k of edges. Previously, only a hardness factor of 1−O ( 1 k ) was known. Our result pins down the correct asymptotic dependence of the approximation factor on k. Along the way, we prove that approximating the Maximum 3-colorable subgraph problem within a factor greater than 32 33 is NP-hard. Using semidefinite programming, it is known that one can do better than a random coloring and properly color a fraction 1− 1 k + 2 ln k k of edges in polynomial time. We show that, assuming the d-to-1 conjecture, it is hard to properly color (using k colors) more than a fraction 1− 1 k + O ( d 3/2 ln k k ) of edges of a k-colorable graph. Research supported in part by a Packard Fellowship. Email: [email protected], [email protected] ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 99 (2009)