Using the PCP for proving Fixed parameter inapproximability

For a minimization (resp, maximization) problem P , and a parameter k, an algorithm is called an (r(k), t(k))-FPT-approximation algorithm, if for an instance I of size n with optimum opt, the algorithm either computes a feasible solution for I, with value at most k · r(k) (resp, at least k/r(k)) or it computes a certi cate that k < opt (resp, k > opt), in time t(k) · poly(n). For maximization problems k/r(k) = o(k) is required. A problem P is (r, t)-FPT-inapproximable (or, (r, t)-FPT-hard) if the problem does not admit an (r, t)-FPT-approximation algorithm. We show how to use the PCP theorem to prove xed parameter hardness. The way all Fixed Parameter hardness were proved before our paper, used the assumptions W [1] 6= FPT or W [2] 6= FPT. This implies that for any function t, and any W [1], or a W [2] hard problem P , we can not tell between the the values k and k+1, in time t(k). Thus all W [1] and W [2] problems were treated in the same way. In addition it is usually very hard to extend a k versus k + 1 gap to a large one. In contrast, Gap reductions that use the PCP may give very large gaps between a yes and a no instance. We use a new idea: Gap preserving/increasing Reductions that drastically reduces opt. In all our hardness proofs k = opt(I) for some instance I. In addition the value of k is known. Our new techniques makes proving FPT-hardness much easier, as the k versus k + 1 di culty is removed, opening the door for countless FPT-hardness proofs via our method. We give the rst super exponential time inapproximability for setcover: the problem admits no log opt ratio, for c > 1 for any algorithm with running in time exp ( opt (log opt) ) · poly(n) for constant f > 0. Recently [2] gave a log k inapproximability for setcover for any time t(k) · poly(n). Thus we give a stronger hardness and [2] gives better running time in k. This technique [2] does not give, so far, hardness for clique. We give a constant inapproximability for clique for any constant, and for any algorithm that runs in time doubly exponential in opt. Using the PCP, leads to much simpler proof than the highly complex proof of [2]. Finally, we study the Minimum Maximal Independent Set (mmis) problem and show is hard to approximate in opt for any two functions r, t. This hardness is already known in k [4]. However k 6= opt(I) in [4] hence our reduction is slightly better. Nevertheless, the mmis problem is not particularly interesting in FPT as the problem is non monotone. This result is included in order to initiate the subject of designing gap increasing reduction that makes opt small. For the mmis problem such a reduction is non trivial and should nd future applications. 1998 ACM Subject Classi cation Classi cation F.2.2 Non numerical Algorithms and Problems