The Complexity of Weighted Boolean #CSP with Mixed Signs

TL;DR

This paper establishes a complexity classification for computing the partition function of weighted Boolean CSPs with mixed signs, extending previous non-negative weight results and identifying cases of polynomial-time solvability versus #P-hardness.

Contribution

It provides a dichotomy theorem for weighted Boolean CSPs with signed weights, generalizing prior work restricted to non-negative weights.

Findings

Polynomial-time solvable when weights are pure affine with quadratic sign polynomial.

Polynomial-time solvable when weights are product type with linear sign polynomial.

Otherwise, computing the partition function is FP^#P-complete.

Abstract

We give a complexity dichotomy for the problem of computing the partition function of a weighted Boolean constraint satisfaction problem. Such a problem is parameterized by a set of rational-valued functions, which generalize constraints. Each function assigns a weight to every assignment to a set of Boolean variables. Our dichotomy extends previous work in which the weight functions were restricted to being non-negative. We represent a weight function as a product of the form (-1)^s g, where the polynomial s determines the sign of the weight and the non-negative function g determines its magnitude. We show that the problem of computing the partition function (the sum of the weights of all possible variable assignments) is in polynomial time if either every weight function can be defined by a "pure affine" magnitude with a quadratic sign polynomial or every function can be defined by a magnitude of "product type" with a linear sign polynomial. In all other cases, computing the partition function is FP^#P-complete.