A Dichotomy Theorem for Polynomial Evaluation

TL;DR

This paper establishes a dichotomy theorem for polynomial evaluation, classifying sets of polynomials into those that are computationally easy (VP) and those that are hard (#VNP-complete), extending complexity classifications.

Contribution

It provides a clear characterization of sets S that lead to easy versus hard polynomial families, and refines complexity results for related counting problems.

Findings

Characterization of sets S leading to VP or VNP-complete polynomial families

Proof that certain #P-complete problems are also #P-complete under many-one reductions

Extension of dichotomy theorems from counting problems to polynomial evaluation

Abstract

A dichotomy theorem for counting problems due to Creignou and Hermann states that or any nite 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.