A Dichotomy Theorem for Polynomial Evaluation

Abstract

A dichotomy theorem for counting problems due to Creignou and Hermann states that or any finite set S of logical relations, the counting problem #SAT(S) is either in FP, or #P-complete. In the present paper we show a dichotomy theorem for polynomial evaluation. That is, we show that for a given set S, either there exists a VNP-complete family of polynomials associated to S, or the associated families of polynomials are all in VP. We give a concise characterization of the sets S that give rise to easy and hard polynomials. We also prove that several problems which were known to be # P-complete under Turing reductions only are in fact # P-complete under many-one reductions.