The complexity of weighted Boolean #CSP with mixed signs

  • Authors:

  • Andrei Bulatov;Martin Dyer;Leslie Ann Goldberg;Markus Jalsenius;David Richerby

  • Affiliations:

  • School of Computing Science, Simon Fraser University, Burnaby, Canada;School of Computing, University of Leeds, Leeds, LS2 9JT, UK;Department of Computer Science, University of Liverpool, Liverpool, L69 3BX, UK;Department of Computer Science, University of Liverpool, Liverpool, L69 3BX, UK;School of Computing, University of Leeds, Leeds, LS2 9JT, UK

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2009

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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 @C 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)^sg, 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 function in @C 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.