The Foundation of Fixed Parameter Inapproximability

Given an instance I of a minimization problem with optimum opt, fixed parameter ρ(k) inapproximability is to find a k ≥ opt and prove that it is not possible to compute a solution of value ρ(k) · k. In this paper all proofs are under the eth. In this paper all ratios are larger than 1. 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). 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. In this paper we make the claim that the art of Fixed Parmeter inapproximability is the art of reducing opt. Namely, given some instance I of a ceratain problem we should map it to an instance I ′ of the same problem with smaller op. For example in we reduce from 3-SAT, we need to make sure that t(opt) = o(2|I|) for the eth to fail. Thus as t gets larger, opt has to become smaller. As an evidence that reducing opt is critical, consider the Minimum Maximal Independent Set (mmis) problem. Given a graph G(V,E) it requires is to find the minimum size maximal independent set in the graph. In [6] it is shown that under eth, for any function r, and expression 2o(m), mmis is (r, t′)-FPT-hard, where t′ is the largest function that obeys t′(k) · nO(1) = 2o(m). Under eth, we show 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 that of [6]. All the improvements are because we decrease the optimum. We also prove several inapproximability results for clique and setcover. For setcover amongs others we prove that setcover is (r, t)-FPT-hard for r(opt) = (logopt)1+f and t(opt) = exp(exp(log opt)) for a constant f > 0. For clique mongs other we prove that for some constant > 0, and for r(opt) = 1/(1− ) and t any non-decreasing function (however huge) clique is (r, t)-FPT-hard. The time can be also set to 2o(n), for an arbitrary o(n) term. This implies a considerable improvement of the main result in the paper of Feige et al [8] A second principal we suggest is that when ever possible, inapproximability should be proved in terms of opt as if k is far than opt the inapproximability may not mean much. We also note that if we allow time sub exponential in opt many inapproximability results directly translate into Fixed Parameter inapproximability. We suggest avoiding Fixed Parameter inapproximability results that follow directly from the usual inapproximability of the problem.