We develop new techniques for deriving strong computational lower bounds for a class of well-known NP-hard problems. This class includes WEIGHTED SATISFIABILITY, DOMINATING SET, HITTING SET, SET COVER, CLIQUE, and INDEPENDENT SET. For example, although a trivial enumeration can easily test in time O(n k ) if a given graph of n vertices has a clique of size k, we prove that unless an unlikely collapse occurs in parameterized complexity theory, the problem is not solvable in time f (k)n o(k) for any function f , even if we restrict the parameter values to be bounded by an arbitrarily small function of n. Under the same assumption, we prove that even if we restrict the parameter values k to be of the order Θ(μ(n)) for any reasonable function μ, no algorithm of running time n o(k) can test if a graph of n vertices has a clique of size k. Similar strong lower bounds on the computational complexity are also derived for other NP-hard problems in the above class. Our techniques can be further extended to derive computational lower bounds on polynomial time approximation schemes for NP-hard optimization problems. For example, we prove that the NP-hard DISTINGUISHING SUBSTRING SELECTION problem, for which a polynomial time approximation scheme has been recently developed, has no polynomial time approximation schemes of running time f (1/ )n o(1/ ) for any function f unless an unlikely collapse occurs in parameterized complexity theory. ✩ A preliminary version of this paper "Linear FPT reductions and computational lower bounds" was presented at
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) poly(m), 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 Whierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NP-hard problems, including weighted sat, dominating set, hitting set, set cover, and feature set, cannot be solved in time n o(k) poly(m), 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, and independent 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 poly(m) or O(n k).
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