Fixed parameter inapproximability for Clique and Set-Cover

A minimization (resp., maximization) problem is called fixed parameter (r, t)approximable for two functionsr, t if there exists an algorithm that given an integer k and a problem instance I with optimum value opt, finds either a feasible solution of value at most r(k) · k (resp., at least k/r(k)) or a certificate that k < opt (resp., k > opt), in time t(k) · |I|O(1). A problem is called fixed parameter (r, t)-hard (or (r, t)-FPT-hard) if it is not (r, t)-approximable. Fellows [7] conjectured that clique and setcover are (r, t)-FPT-hard for all functions r and t. We prove the first fixed parameter hardness for clique and setcover, for super exponential functions t. Our results are as follows. 1. Assuming eth and Projection Game Conjecture (pgc), setcover is (r, t)-FPThard for r(k) = (log k)γ and t(k) = exp(exp((log k)γ)) = exp(k(log k) f ) for some constant γ > 1 and f = γ − 1. 2. Under eth and a stronger for of pgc, setcover is (r, t)-FPT-hard for r(k) = optd1 and t(k) = exp(exp(kd2)) for some constants d1, d2 > 0. 3. Under eth alone, setcover is (r, t)-FPT-hard for r(k) = c √ log k and t(k) = exp(k(log k) f ) for some constants c, f > 0. 4. Under eth, for any constant c, clique is (c, t)-FPT-hard for t(k) = exp(exp(kd)) for some constant d that depends on c. It is also (r, t)-FPT-hard for some super constant function r(k) and t(k) = exp(exp(k/q(k))) for an arbitrarily slowly increasing function q(k). We show that the crux of FP-hardness is reducing the optimum and suggest simple but effective ways to do so. Feige and Kilian [9] proved that the (log n)-clique problem, i.e. the problem of finding a clique of size log n in a graph of size n, can not be solved exactly in time much better than nlogn. Our results are slightly better as they imply a constant inapproximability for any constant, for the k-clique problem with k = Ω((log log n)1/O(1)) in time less than doubly exponential in k. Marx [1] asked if clique is (2, t)-approximable for some function t. We prove that for any constant c, achieving a c-approximation for clique requires time roughly double exponential in the parameter. ∗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].