The Complexity of Weighted Boolean CSP

  • Authors:

  • Martin Dyer;Leslie Ann Goldberg;Mark Jerrum

  • Affiliations:

  • dyer@comp.leeds.ac.uk;l.a.goldberg@liverpool.ac.uk;m.jerrum@qmul.ac.uk

  • Venue:
  • SIAM Journal on Computing
  • Year:
  • 2009

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Abstract

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 parameterized by a finite set $\mathcal{F}$ of nonnegative 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 $\text{{\sf FP}}^{\text{{\sf\#P}}}$-complete unless either (1) every function in $\mathcal{F}$ is of “product type,” or (2) every function in $\mathcal{F}$ is “pure affine.” In the remaining cases, computing the partition function is in P.