THE CONSTANT INAPPROXIMABILITY OF THE PARAMETERIZED DOMINATING SET PROBLEM

A set D of vertices of a graph G is a dominating set if every vertex of G is contained in D or adjacent to some vertex of D. The number of vertices in a smallest dominating set of G is denoted by gamma(G). We prove that, under the assumption FPT not equal W[1] from parameterized complexity, for any constant c is an element of N+ and computable function f : N -> N there is no algorithm which on every input graph G finds a dominating set of size at most c . gamma(G) in f(gamma(G)) . vertical bar G vertical bar(O(1) )time. In other words, any constant approximation of the parameterized dominating set problem is W[1]-hard. Furthermore, assuming the exponential time hypothesis (ETH) [R. Impagliazzo and R. Paturi, T. Comput. System Sci., 62 (2001), pp. 367-375], we can even rule out the existence of a f (gamma(G)). vertical bar G vertical bar((log gamma(G))epsilon/12) -time algorithm which on every input graph G outputs a dominating set of size at most (3+epsilon)root log (gamma(G)).gamma(G) for every 0 <epsilon < 1. Our hardness reduction is built on the second author's recent W[1]-hardness proof of the biclique problem [B. Lin, The parameterized complexity of k-BICLIQUE, in Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 (San Diego, CA), ACM, New York, SIAM, Philadelphia, 2015, pp. 605-615]. This yields, among other things, a proof without the probabilistically checkable proof (PCP) machinery that the classic dominating set problem has no polynomial time constant approximation under ETH.