Improved Inapproximability Results for Maximum k-Colorable Subgraph

Abstract

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 $1-\frac{1}{k}$ 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 $\approx 1-\frac{1}{33 k}$ of edges. Previously, only a hardness factor of $1- O\bigl(\frac{1}{k^2}\bigr)$ 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 $\frac{32}{33}$ is NP-hard.

Using semidefinite programming, it is known that one can do better than a random coloring and properly color a fraction $1-\frac{1}{k} +\frac{2 \ln k}{k^2}$ of edges in polynomial time. We show that, assuming the 2-to-1 conjecture, it is hard to properly color (using k colors) more than a fraction $1-\frac{1}{k} + O\left(\frac{\ln k}{k^2}\right)$ of edges of a k-colorable graph.