A dichotomy theorem for regular types

A Dichotomy Theorem for Regular Types

Toward a Dichotomy Theorem for Polynomial Evaluation

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 study polynomial evaluation from this dichotomic point of view. We show that the “hard” cases in the Creignou-Hermann theorem give rise to VNP-complete families of polynomials, and we giv...

A dichotomy theorem for circular colouring reconfiguration

Let p and q be positive integers with p/q ≥ 2. The “reconfiguration problem” for circular colourings asks, given two (p, q)-colourings f and g of a graph G, is it possible to transform f into g by changing the colour of one vertex at a time such that every intermediate mapping is a (p, q)-colouring? We show that this problem can be solved in polynomial time for 2 ≤ p/q < 4 and that it is PSPACE...

A Dichotomy Theorem for Polynomial Evaluation

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 fa...

A Dichotomy Theorem for Homomorphism Polynomials

In the present paper we show a dichotomy theorem for the complexity of polynomial evaluation. We associate to each graph H a polynomial that encodes all graphs of a fixed size homomorphic to H . We show that this family is computable by arithmetic circuits in constant depth if H has a loop or no edge and that it is hard otherwise (i.e., complete for VNP, the arithmetic class related to #P ). We...