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 usually under the eth. In this paper we are interested only in k being the optimum value of some instance. Our question is: What properties make a good Fixed Parameter Inapproximability proof? We claim that Fixed Parameter Inapproximability should be done, whenever possible, with parameter opt. We show simple examples so that k is far from opt for which fixed parameter inapproximability in k is not possible, while fixed parameter tractability in k is trivial to prove. However, these results are meaningless. To reduce with parameter opt, we need opt to be known. The way to achieve that is make opt the value of a yes instance in gap reduction. 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)n (for maximization problems any solution has to be super constant). Our main claim is that the art of Fixed Parameter inapproximability is the art of reducing opt. Reducing the optimum means given some instance I of a certain problem resulting from gap reduction, finding instance I ′ of the same problem with much smaller opt, and roughly the same gap. As r, t get larger and larger, opt has to become smaller and smaller for proofs to work. We develop a systematic way to reduce the optimum, and apply it on three completely different problems. This technique to reduce opt, is of independent interest. Even though we claim that reducing the value of the optimum is the essence of Fixed Parameter Inapproximability, this method was not used before this paper. As an evidence that reducing opt is important consider the problem of finding a Minimum Size Independent Set, that is also Maximal (mmis). In [6] it is shown mmis admits no r(k) approximation for any r. However, their k equals k = g(n) for g an increasing function in n. Thus, there exists a t so that t(k) = t(g(n)) ≥ 2 ·n. Clearly, for such t, no (r, t)-hardness is possible. We significantly improves [6], by proving (r, t)-hardness for any two functions r and t. This is achieved by reducing the value of the optimum. Fellows [9] conjectured that setcover and clique are (r, t)-FPT-hard for any pair of nondecreasing functions r, t and input parameter k. By decreasing the optimum we give a host of (r, t)-hardness results for setcover and clique so that t is much larger than superexponential in opt (and thus in k). All previous inapproximability results for these two problem [3], had t strictly sub exponential in opt. For example, for clique we show that for some constant there is no 1/(1 − ) approximation for clique in time t(opt)n for an arbitrary t (however huge). This substantially improves the main result of [8] by Feige and Killian. We show that in many cases, if subexponential time in opt is allowed, then inapproximability result translates without any changes to a Fixed parameter inapproximability result.