The Complexity of Weighted Boolean #CSP

This paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted Boolean constraint satisfaction problem. The problem is parameterised by a finite set F of non-negative functions that may be used to assign weights to the configurations (feasible solutions) of a problem instance. Classical constraint satisfaction problems correspond to the special case of 0,1-valued functions. We show that computing the partition function, i.e. the sum of the weights of all configurations, is FP-complete unless either (1) every function in F is of “product type”, or (2) every function in F is “pure affine”. In the remaining cases, computing the partition function is in P.