Reducing the optimum value: FPT inapproximability, for Set Cover and Clique, in time super-exponential in opt

Fixed parameter ρ(k) inapproximability in minimization problems, is given some instance I of a problem with optimum opt, find some k ≥ opt, prove that it is not possible to compute a solution of value ρ(k) · k, usually, under the Exponential Time Hypothesis (eth). If opt is known, inapproximability in terms of opt implies inapproximability in terms of k. An (r, t)-fpt-hardness (in opt) for two functions r, t, is showing that the problem admits no r(opt) approximation that runs in time t(opt)nO(1) (for maximization problems any solution has to be super constant). In this paper we are only interested in t(opt) that is super exponential in opt. Fellows [9] conjectured that setcover and clique are (r, t)-fpt-hard for any pair of non-decreasing functions r, t and input parameter k. We give the first inapproximability for these problems that runs in time super exponential in opt. Our paper is also the first to introduce systematic techniques to reduce the value of the optimum. These technique work for 3 totally different problems. We prove that under eth [14] and the projection game conjecture [19], setcover is (r, t)-fpt-hard for r(opt) = (logopt)1+f and t(opt) = exp(exp(log opt)) = exp ( opt(log f opt) ) for a constant f > 0. Note that t(opt) is significantly larger than super-exponential in opt. Under eth alone we can get c √ logopt/polylog(opt) inapproximability for some constant c with the same running time in opt as above. Under a qualitatively stronger version of the pgc, we can improve this hardness to r(opt) = optδ and t(opt) = exp(exp(optδ)) for some constant 0 < δ ≤ 1. For clique we prove that for any non-decreasing function t(opt), there is a nondecreasing function r(opt) = ω(1) so that clique is (r, t)-fpt-hard. We also prove r(opt) inapproximability in 2o(n) time where n denotes the size of the clique instance. Feige et al. [8] show that if instances of clique with opt ≤ log n can be solved exactly in time significantly smaller than nlogn, then np admits sub-exponential simulation. We improve [8] in two ways. We prove r(opt) inapproximability for clique which may be much stronger than ruling out an exact solution, for such small values of opt. However, our main improvement is that this inapproximability holds even for time 2o(n), an almost exponentially larger running time than the running time of Feige et al [8]. The Minimum Maximal Independent Set (mmis) problem, is to find the maximal independent set containing minimum number of vertices, in a given graph. In [6] it is shown that under eth, for any function r, mmis is (r, t′)-fpt-hard, so that t′ is the largest function that obeys t′(k) · nO(1) = 2o(m), We show, under eth, that mmis is (r, t)-fpt-hard for any non-decreasing functions r(opt) and t(opt). This significantly improves [6]. In addition, our proof is considerably simpler than the one in [6].