On a logical approach to estimating computational complexity of potentially intractable problems

Abstract

In the paper we present a purely logical approach to estimating computational complexity of potentially intractable problems. The approach is based on descriptive complexity and second-order quantifier elimination techniques. We illustrate the approach on the case of the transversal hypergraph problem, TRANSHYP, which has attracted a great deal of attention. The complexity of the problem remains unsolved for over twenty years. Given two hypergraphs, G and H, TRANSHYP depends on checking whether G=Hd, where Hd is the transversal hypergraph of H. In the paper we provide a logical characterization of minimal transversals of a given hypergraph and prove that checking whether G ⊆ Hd is tractable. For the opposite inclusion the problem still remains open. However, we interpret the resulting quantifier sequences in terms of determinism and bounded nondeterminism. The results give better upper bounds than those known from the literature, e.g., in the case when hypergraph H has a sub-logarithmic number of hyperedges and (for the deterministic case) all hyperedges have the cardinality bounded by a function sub-linear wrt maximum of sizes of G and H. © Springer-Verlag Berlin Heidelberg 2003.