# Polynomial constraint satisfaction problems, graph bisection, and the Ising partition function

## Abstract

We introduce a problem classwecall Polynomial Constraint Satisfaction Problems, orPCSP. Where the usual CSPs from computer science and optimization have real-valued 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, exponential-time) algorithms we know of for 2-CSP (where each score depends on at most 2 variables) extend to 2-PCSP, at the expense of just a polynomial factor in running time. Specifically, we extend the reduction-based algorithm of Scott and Sorkin [2007]; the specialization of that approach to sparse random instances, where the algorithm runs in polynomial expected time; dynamic-programming algorithms based on tree decompositions; and the split-and-list matrix-multiplication algorithm of Williams [2004]. This gives the first polynomial-space exact algorithm more efficient than exhaustive enumeration for the well-studied 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 solves both optimization and counting versions of a wide range of problems, including all CSPs, and thus enables samplers including uniform sampling of optimal solutions and Gibbs sampling of all solutions. © 2009 ACM.