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

In Fixed Parameter Tractability (FPT) theory, we are given a problem with a special parameter k. In this paper we are only interested in k equal the size of the optimum. A FPT algorithm for a problem is an exact algorithm that runs in time h(k) · nO(1) for a function h that may be arbitrarily large. In FPT approximation we seek a g(k) ratio that runs in time h(k) · nO(1), so that h,g are two increasing functions. For such results we want to minimize g and h. FPT inapproximability is the oposite of FPT approximability. Albeit, its not hard to see that for minimization problems we need the inequality k ≥ opt and for maximization we need k ≤ opt (with opt the optimum of some concrete instance). A more strict notion of FPT inapproximability assumes that k = opt with opt the optimum value of some concrete instance. Clearly, if we allow k ≥ opt (for minimization problems) it may be easier to prove inapproximability than when restricting k to opt. In this paper we adopt the latter definition for inapproximability. As it will become clear later, opt will always be the value of a yes instance in some gap reduction from 3-SAT. Thus a FPT inapproximability in opt is defined as follows. We given a problem and an instance I of the problem, we show that the problem can not be approximated within g(opt) in time h(opt) ·nO(1) for some increasing functions h,g, with opt the value of the solution for I. An inapproximability result would like to get h,g that are as large as possible. We study FPT inapproximability in opt for Clique and Set Cover and of the Minimum size Maximal Independent Set Problem. We restrict h(opt) to time super exponential in opt. Otherwise, some hardness results may be directly translated to FPT inapproximability results. Fellows [6] conjectured that Clique and Set Cover admit no g(opt) ratio approximation in time h(opt) ·nO(1) for any pair of increasing functions h,g. We prove that under the exponential time hypothesis (ETH) [11] and the projection game conjecture [15], there are two constants f ,τ > 0 so that Set Cover is (logopt)1+τ -inapproximable, even in time exp(Θ(opt(logopt) f )). This running time is significantly higher than exponential time in opt. Under a qualitatively better version of the projection game conjecture, we can show that Set-Cover admits no optd ′ approximation for some constant d′, in time exp(2opt d′ ) which is almost double exponential time in opt. For the Clique problem, we show that under the ETH there exists a constant ε > 0 so that for any increasing function h, Clique admits no 1− ε approximation in time h(opt) ·nO(1). In [5] it was shown that Clique for opt ≤ logn can not be solved in time significantly smaller than nopt . We improve one aspect of [5], namely, we can show inapproximability for opt ≤ logn which is stronger than ruling an exact solution.