Reducing the Optimum: Fixed Parameter Inapproximability for Clique and Set Cover in Time Super-exponential in Optimum

A minimization problem is called fixed parameter ρinapproximable, for a function ρ ≥ 1, if there does not exist an algorithm that given a problem instance I with optimum value opt and an integer k, either finds a feasible solution of value at most ρ(k) ·k or finds a certificate that k < opt in time t(k) · |I| for some function t. For maximization problem the definition is similar. We motivate the study of inapproximability in terms of the parameter opt, the optimum value of an instance. A problem is called (r, t)-FPT-hard in parameter opt for functions r and t, if it admits no r(opt) approximation that runs in time t(opt)|I|. To prove hardness, we use gap reductions from 3sat and assume the Exponential Time Hypothesis (eth). If the value of opt is known for the yes instance, inapproximability w.r.t. opt implies inapproximability w.r.t. input integer k but not vise versa. Hence inapproximability in opt is stronger. Previous FPT Hardness results [2] have running sub-exponential in opt. Fellows [6] conjectured that setcover 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 results for these problems with running times super-exponential in opt. Our paper introduces systematic techniques to reduce the value of the optimum. These techniques are robust and work for three quite different problems. In particular one of our results shows that, under eth, clique is (r, t)FPT-hard for r(opt) = 1/(1 − ) with some constant > 0 and any non-decreasing function t. The running time can be also set to 2, for an arbitrary o(n) exponent. This improves the main result of Feige and Kilian [5] in two ways. We prove that the Minimum Maximal Independent Set (mmis) problem is (r, t)-FPT-hard in opt, for arbitrarily fast growing functions r, t. In terms of k an elementary reduction [3] shows that an (r(k), t(k)) approximation for any r(k), t(k) is W [2]-hard. The assumption that W [2] 6= FPT removes the need to reduce the value of k. Our reduction is very significantly harder, and technically complex because we need to drastically reduce opty since we assume the eth. The (f(k), t(k)) hardness can be shown under eth, by combining [3] with several papers showed that the eth implies that W [2] 6= FPT. In these papers, indeed, the value of k is drastically reduces. We expect that our technique to reduce opt for mmis will find future applications. ? Supported in part by NSF CAREER award 1053605, NSF grant CCF-1161626, ONR YIP award N000141110662, and DARPA/AFOSR grant FA9550-12-1-0423. Email: [email protected]. ?? Partially supported by NSF award number 1218620.