Tight lower bounds for certain parameterized NP-hard problems

Abstract

Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of well-known NP-hard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on depth- t circuits cannot be solved in time n o( k ) m O(1) , where n is the circuit input length, m is the circuit size, and k is the parameter, unless the ( t − 1)-st level W [ t − 1] of the W -hierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NP-hard problems, including weighted sat , hitting set , set cover , and feature set , cannot be solved in time n o( k ) m O(1) , where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W [1] of the W -hierarchy collapses to FPT. We also prove that another group of parameterized problems which includes weighted q -sat (for any fixed q ⩾ 2), clique , independent set , and dominating set , cannot be solved in time n o( k ) unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either n k m O(1) or O( n k ).