Improved Inapproximability Results for Maximum k-Colorable Subgraph

摘要

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 fc-coloring that properly colors more than a fraction ≈ 1- 1/33k of edges. Previously, only a hardness factor of 1- O(1/k~2) 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.rnUsing 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~2 of edges in polynomial time. We show that, assuming the 2-to-l conjecture, it is hard to properly color (using k colors) more than a fraction 1 - 1/k + O (ln k/k~2) of edges of a k-colorable graph.