Polynomial Constraint Satisfaction, Graph Bisection, and the Ising Partition Function

TL;DR

This paper introduces Polynomial Constraint Satisfaction Problems (PCSP), a broad generalization of CSPs with polynomial score functions, and extends existing algorithms to efficiently solve and analyze these problems, impacting graph bisection, Ising models, and sampling methods.

Contribution

The paper extends known algorithms for 2-CSP to 2-PCSP, enabling more efficient solutions for complex problems like graph bisection and Ising partition functions.

Findings

First polynomial-space exact algorithm surpassing exhaustive search for graph bisection.

Efficient algorithms for computing Ising model partition functions.

Enhanced methods for sampling optimal solutions and Gibbs sampling.

Abstract

We introduce a problem class we call Polynomial Constraint Satisfaction Problems, or PCSP. 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; 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. This gives the first polynomial-space exact algorithm more efficient than exhaustive enumeration for the well-studied problems of finding a minimum bisection of a graph, and calculating the partition function of an Ising model, and the most efficient algorithm known for certain instances of 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.