# Improved Inapproximability Results for Maximum k-Colorable Subgraph

• Venkatesan Guruswami
• Ali Kemal Sinop
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5687)

## 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.

## Keywords

Edge Weight Constraint Satisfaction Problem Noise Operator Hardness Result Markov Operator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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