Using the PCP to prove fixed parameter hardness

For a minimization (resp, maximization) problem P , and a parameter k, an algorithm is called a (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 certificate that k < opt (resp, k > opt), in time t(k) · poly(n). For maximization problems, the condition k/r(k) = o(k) is also 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. All FPT-hardness results, we are aware off, use the assumptions W [1] 6= FPT or W [2] 6= FPT. This immediately implies that for a minimization (respectively maximization) problems, its not possible to tell between opt = k or opt = k + 1 (respectively, opt = k + 1 and opt = k) in time poly(n) · t(opt) for any function t. Making the hardness stronger requires increasing the gap from k to k + 1 to a large gap r(k). This is usually a very hard thing to do. The difficulty of increasing a gap of k versus k + 1 to a larger one motivates this paper. We suggest a novel way to prove FPT -hardness. Use the standard idea of Gap reductions via the PCP followed by Gap preserving/increasing Reductions that drastically reduces opt. In all our hardness proofs k = opt(I) for some instance I. This does not hold in previous reductions,. In addition the value of k is known, in all our reductions, again a new property. Our proofs are simple, because we overcome the k versus k+1 difficulty. Assuming the eth and the pgc (to be described later) we prove that setcover admits no log opt ratio, for c > 0 for any algorithm with running and time t(opt) = exp ( opt(log f opt) ) · poly(n) for constant f > 0. ∗Maryland University. Supported in part by NSF CAREER award 1053605, NSF grant CCF1161626, ONR YIP award N000141110662, and DARPA/AFOSR grant FA9550-12-1-0423. Email: [email protected]. †KCG holdings Inc., USA Email: [email protected] ‡Computer Science department, Rutgers University, Camden. Email: [email protected] Partially supported by NSF grant number 1218620. Under the eth alone, we prove that setcover admits no √ logopt ratio, in time exp ( opt(log f opt) ) · poly(n) for constant f > 0. Under the eth we prove clique admits no (c, t(opt)), approximation, for any constant c in time exp(exp(optd)) with d a constant that depends on c. We study the subject of increasing/preserving gaps reductions that make opt small. As a toy problem we study Minimum Maximal Independent Set, (mmis) problem. This problem is not that relevant for FPT, as its not monotone. However, the reduction of opt in this case is non trivial and may have future application. For every slowly increasing function f , and for some parameter q, we can reduce from any instance I of mmis to another instance I ′ of mmis, so that the optimum for a yes instance I ′ is f(q) and the gap between the values of the no instance over the value of a yes instance is q/f(q).