#### The Complexity of Weighted Boolean #CSP

##### Martin Dyer, Leslie Ann Goldberg, Mark Jerrum

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 the partition function, i.e. the sum of the weights of all configurations, can be computed in polynomial time if either (1) every function in F is of product type'', or (2) every function in F is pure affine''. For every other fixed set F, computing the partition function is FP^{#P}-complete.

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