Improved Inapproximability Results for Maximum k-Colorable Subgraph

TL;DR

This paper establishes tighter NP-hardness bounds for approximating the maximum k-colorable subgraph problem, matching the asymptotic dependence on k, and explores the limits of semidefinite programming approaches under the 2-to-1 conjecture.

Contribution

It improves the inapproximability bounds for maximum k-colorable subgraph, confirming the asymptotic hardness factor and analyzing the limits of SDP-based algorithms.

Findings

NP-hard to color more than ~1-O(1/k) fraction of edges in k-colorable graphs

Approximation within 32/33 for maximum 3-colorable subgraph is NP-hard

Under the 2-to-1 conjecture, it is hard to surpass the fraction 1-1/k + O(ln k/ k^2]

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 1-1/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-O(1/k} 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. 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^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 1-1/k + O(ln k/ k^2) of edges of a k-colorable graph.