Towards optimal lower bounds for clique and chromatic number

It was previously known that neither Max Clique nor Min Chromatic Number can be approximated in polynomial time within n1− , for any constant > 0, unless NP = ZPP. In this paper, we extend the reductions used to prove these results and combine the extended reductions with a recent result of Samorodnitsky and Trevisan to show that unless NP ⊆ ZPTIME(2 logn) 3/2)), neither Max Clique nor Min Chromatic Number can be approximated within n1−O(1/ √ log . Since there exists polynomial time algorithms approximating both problems within n1−O(log logn/ , our result shows that the best possible ratio we can hope for is of the form n1−o(1), for some—yet unknown—value of o(1) between O(1/ √ log logn) and Ω(log logn/ logn).