Comparing Beliefs, Surveys and Random Walks

TL;DR

This paper unifies survey propagation, belief propagation, and hybrid methods through probability arguments, and uses WSAT to empirically analyze 3-SAT complexity, revealing insights into solution space structure and search strategies.

Contribution

It provides a common probabilistic derivation of multiple SAT solving methods and demonstrates WSAT's effectiveness in analyzing 3-SAT complexity and structure.

Findings

WSAT's mean cost scales with the number of variables in the easy-SAT regime.

WSAT behavior reflects the solution space structure predicted by replica symmetry-breaking.

Hybrid methods and survey propagation offer promising directions for practical SAT algorithms.

Abstract

Survey propagation is a powerful technique from statistical physics that has been applied to solve the 3-SAT problem both in principle and in practice. We give, using only probability arguments, a common derivation of survey propagation, belief propagation and several interesting hybrid methods. We then present numerical experiments which use WSAT (a widely used random-walk based SAT solver) to quantify the complexity of the 3-SAT formulae as a function of their parameters, both as randomly generated and after simplification, guided by survey propagation. Some properties of WSAT which have not previously been reported make it an ideal tool for this purpose -- its mean cost is proportional to the number of variables in the formula (at a fixed ratio of clauses to variables) in the easy-SAT regime and slightly beyond, and its behavior in the hard-SAT regime appears to reflect the underlying structure of the solution space that has been predicted by replica symmetry-breaking arguments. An analysis of the tradeoffs between the various methods of search for satisfying assignments shows WSAT to be far more powerful that has been appreciated, and suggests some interesting new directions for practical algorithm development.