A Dichotomy Theorem for Polynomial Evaluation

Irénée Briquel (1) , Pascal Koiran (1)

1 LIP - Laboratoire de l'Informatique du Parallélisme

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.

Mots clés: Constraint satisfaction problems; polynomial evaluation; dichotomy theorem; VNP; #P-completeness; permanent; algebraic complexity

Domaines: Complexité [cs.CC]

Irénée Briquel, Pascal Koiran. A Dichotomy Theorem for Polynomial Evaluation. Mathematical Foundations of Computer Science 2009, Aug 2009, Novy Smokovec, Slovakia. pp.187-198, ⟨10.1007/978-3-642-03816-7⟩. ⟨ensl-00360974v2⟩