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Polynomial Constraint Satisfaction Problems, Graph Bisection, and the Ising Partition Function
 ACM Transactions on Algorithms
, 2009
"... We introduce a problem class we call Polynomial Constraint Satisfaction Problems, or PCSP. Where the usual CSPs from computer science and optimization have realvalued score functions, and partition functions from physics have monomials, PCSP has scores that are arbitrary multivariate formal polynom ..."
Abstract

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We introduce a problem class we call Polynomial Constraint Satisfaction Problems, or PCSP. Where the usual CSPs from computer science and optimization have realvalued score functions, and partition functions from physics have monomials, PCSP has scores that are arbitrary multivariate formal polynomials, or indeed take values in an arbitrary ring. Although PCSP is much more general than CSP, remarkably, all (exact, exponentialtime) algorithms we know of for 2CSP (where each score depends on at most 2 variables) extend to 2PCSP, at the expense of just a polynomial factor in running time. Specifically, we extend the reductionbased algorithm of Scott and Sorkin; the specialization of that approach to sparse random instances, where the algorithm runs in polynomial expected time; dynamicprogramming algorithms based on tree decompositions; and the splitandlist matrixmultiplication algorithm of Williams. This gives the first polynomialspace exact algorithm more efficient than exhaustive enumeration for the wellstudied problems of finding a maximum bisection of a graph, and calculating the partition function of an Ising model. It also yields the most efficient algorithm known for certain instances of counting and/or weighted Maximum Independent Set. Furthermore, PCSP