The complexity of weighted Boolean #CSP with mixed signs

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 Γ of rational functions, each of which assigns a weight to each variable assignment. 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)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 computable in polynomial time if either every function in Γ 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-complete.