Approximate Counting CSP Solutions Using Partition Function

TL;DR

This paper introduces an efficient approximate method for counting CSP solutions by leveraging partition functions and belief propagation to estimate marginal probabilities, enabling quick solution counts for various problems.

Contribution

It presents a novel approach combining belief propagation and partition functions to approximate CSP solution counts, improving efficiency over existing methods.

Findings

Effective on random and structural problems

Accurate approximation of solution counts

Computationally efficient

Abstract

We propose a new approximate method for counting the number of the solutions for constraint satisfaction problem (CSP). The method derives from the partition function based on introducing the free energy and capturing the relationship of probabilities of variables and constraints, which requires the marginal probabilities. It firstly obtains the marginal probabilities using the belief propagation, and then computes the number of solutions according to the partition function. This allows us to directly plug the marginal probabilities into the partition function and efficiently count the number of solutions for CSP. The experimental results show that our method can solve both random problems and structural problems efficiently.