Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring

• Authors: S. Khot
• Venue: FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
• Year: 2001

In this paper, we present improved inapproximability results for three problems : the problem of finding the maximum clique size in a graph, the problem of finding the chromaticnumber of a graph, and the problem of coloring a graph with a small chromatic number with a small number of colors.Håstad's celebrated result [13] shows that the maximum clique size in a graph with n vertices is inapproximable in polynomial time within a factor n1- \varepsilon for arbitrarily small constant \varepsilon 0 unless NP=ZPP. In this paper, we aim at getting the best sub-constant value of \varepsilon in Håstad's result. We prove that clique size is inapproximable within a factor \frac{n}{{2(\log n)^{1 - \gamma } }} (corresponding to \varepsilon= \frac{1}{{(\log n)^\gamma}} for some constant \gamma 0 unless NP \subseteq ZPTIME(2^{(\log n)^{0(1)} } ). This improves the previous best inapproximability factor of \frac{n}{{2^{0({{\log n} \mathord{\left/ {\vphantom {{\log n} {\sqrt {\log \log n} }}} \right. \kern-\nulldelimiterspace} {\sqrt {\log \log n} }}} }} (corresponding to \varepsilon= 0(\frac{1} {{\sqrt {\log \log n} }}) due to Engebretsen and Holmerin [7].A similar result is obtained for the problem of approximating chromatic number of a graph. Feige and Kilian [10] prove that chromatic number is hard to approximate within factor n1- \varepsilon for any constant \varepsilon 0 unless NP=ZPP. We use some of their techniques to give a much simpler proof of the same result and also improve the hardness factor to \frac{n}{{2(\log n)^{1 - \gamma } }} for some constant \gamma 0. The above two results are obtained by constructing a new Hadamard code based PCP inner verifier.We also present a new hardness result for approximate graph coloring. We show that for all sufficiently large constants k, it is NP-hard to color a k-colorable graph with K\frac{1}{{25}}(\log k) colors. This improves a result of Fürer [11] that for arbitrarily small constant \varepsilon 0, for sufficiently large constants k, it is hard to color a k-colorable graph with k3/2-\varepsilon colors.