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

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 chromatic number 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−ǫ for arbitrarily small constant ǫ > 0 unless NP=ZPP. In this paper, we aim at getting the best subconstant value of ǫ in Håstad’s result. We prove that clique size is inapproximable within a factor n 2(log n) 1−γ (corresponding to ǫ = 1 (logn)γ ) for some constant γ > 0 unless NP ⊆ ZPTIME(2(logn)O(1)). This improves the previous best inapproximability factor of n 2 n/ √ log log n) (corresponding to ǫ = O( 1 √ log logn ) ) 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−ǫ for any constant ǫ > 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 n 2(log n)1−γ for some constant γ > 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 1 25 (log k) colors. This improves a result of Fürer [11] that for arbitrarily small constant ǫ > 0, for sufficiently large constants k, it is hard to color a k-colorable graph with k3/2−ǫ colors. This work was partially supported by S. Arora’s David and Lucile Packard Fellowship.