Reducing the optimum: superexponential in opt time, Fixed parameter inapproximability for clique and setcover

An (r, t)-FPT-hardness in opt is finding some instance I whose optimum value opt is known and proving that the problem admits no r(opt) approximation that runs in time t(opt)|I|O(1). The usual definition uses r(k)-inapproximability in time t(k) · |I|O(1), for some k ≥ opt in minimization problems, and opt ≤ k for maximization problems. If opt is known then inapproximability in opt implies inapproximability in k (but not vice versa) thus inapproximability in opt is stronger. We restrict our tools to gap reductions from 3-SAT and always assuming the Exponential Time Hypothesis (eth). In all our reductions opt is known as its the ”yes” instance in a gap reduction from 3-SAT. In is obvious that as we want to prove (r, s)-FPT-hardness for larger and larger r, t it is required to change the optimum into smaller and smaller. However, no paper previous to ours proved Fixed Parameter Inapproximability by systematically reducing the optimum value. We develop a general technique to reduce the value of the optimum, which is of independent interest. The technique is robust and we apply it to three completely different problems. By reducing the value of the optimum, we give the first host of (r, t)-FPThardness results for clique and setcover so that t is much larger than super exponential in opt. All previous FPT-hardness [4] had time strictly subexponential in opt. In fact, allowing subexponential time in opt can make some FPT-hardness proofs trivial. One of our results for clique is proving that there is no 1/(1− ) approximation for clique, for some > 0 that runs in time t(opt) · nO(1) time, for any t, however huge. The time can also be set to 2o(n) for any o(n) term. This result improves in two ways the main result in a paper by Feige and Killian [11]. We also show that the The Minimum Maximal Independent Set (mmis) problem is (r(opt, t(opt))-FPT-hard for any r, s however huge. This was already known in terms of k. While hardness in opt is stronger, we mainly present this proof because the reduction in the optimum in this case, is very instructive and its ideas may find further applications. ∗Department of Computer Science , University of Maryland at College Park, USA. 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]. †KCG holdings Inc., USA, [email protected]. ‡Department of Computer Science, Rutgers University-Camden, USA. Supported in part by NSF grant 1218620. [email protected].