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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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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Venkatesan Guruswami
    • 1
  • Ali Kemal Sinop
    • 1
  1. 1.Computer Science Department, School of Computer ScienceCarnegie Mellon UniversityPittsburghUSA

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