Fixed parameter inapproximability for Clique and Set-Cover with super exponential time in k

A minimization (resp., maximization) problem is called fixed parameter (r, t)-hard for two r, t if there does not exist an algorithm that given a problem instance I with optimum value opt and an integer k, either finds a feasible solution of value at most r(k) · k (resp., at least k/r(k)) in time t(k) or finds in time t(k) a certificate that k < opt (resp., k > opt) in time t(k) · |I|O(1) for some function t. In maximization problems like clique only o(k) hardness can be proven. Indeed, returning a single vertex is an opt approximation for the clique problem. Fellows [14] conjectured that clique and setcover have no (r, t) approximation for any functions r and t. We prove fixed parameter hardness for clique and setcover, for super exponential in k time t(k) (later we explain why reductions with sub exponential in k time are undesireable). Our hardness, is proved via gap reductions from 3sat and assuming the Exponential Time Hypothesis (eth) [12]. The value of k equals in all cases to the value of the optimum on a yes instance. We find out that the crux of fixed parameter inapproximability is educing opt while maintaining the gap. While it sems that ∗Supported in part by NSF CAREER award 1053605, NSF grant CCF-1161626, ONR YIP award N000141110662, and DARPA/AFOSR grant FA9550-12-1-0423. Email: [email protected] †Partially supported by NSF award number 1218620.