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 and prove that it is not possible to compute a solution of value ρ(k) · k, usually, under the Exponential Time Conjecture eth. If opt is known inapproximability in terms of opt implies inapproximability in terms of k. An (r, t)-fpt-hardness 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 [6] conjectured that Set cover 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. We prove that under the exponential time hypothesis (eth) [10] and the projection game conjecture [15], Set cover is (r, t)-fpt-hard for r(opt) = (logopt)1+γ and t(opt) = exp(Θ(opt(logopt f ))) for a constants f, γ > 0. Note that t(opt) is significantly super-exponential in opt. Under eth alone we can get c ln 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 δ > 0. For Clique we prove that for any running time t(opt) (even huge) there is a super constant increasing function x(opt) so that Clique is (r, t)-fpt-hard for r(opt) = x(opt). We also prove x(opt) inapproximability in time 2o(n) time with n the Clique instance size. Feige et al. [5], show that instances of Clique with opt ≤ log n can be solved exactly in time significantly smaller than nlogn, np admits sub-exponential simulation. We improve [5], in two ways. We prove x(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 [5]. The Minimum Maximal Independent Set (mmis) problem, is to find the maximal independent set containing minimum number of vertices, in a given graph. Assuming eth, we prove that for any r, t the problem admits no (r, t)-fpt approximation. In this case we say that we proved the the Fellows Conjecture for the problem.