Survey propagation: an algorithm for satisfiability

TL;DR

This paper introduces a survey propagation algorithm that efficiently finds satisfying assignments in difficult SAT problems near the phase transition, leveraging message passing over solution clusters.

Contribution

It presents a novel message passing algorithm that generalizes belief propagation to handle clustered solution spaces in satisfiability problems.

Findings

Efficiently finds solutions near the satisfiability threshold

Demonstrates the effectiveness of survey propagation over traditional methods

Provides insights into the structure of solution spaces in SAT problems

Abstract

We study the satisfiability of randomly generated formulas formed by $M$ clauses of exactly $K$ literals over $N$ Boolean variables. For a given value of $N$ the problem is known to be most difficult with $\alpha=M/N$ close to the experimental threshold $\alpha_c$ separating the region where almost all formulas are SAT from the region where all formulas are UNSAT. Recent results from a statistical physics analysis suggest that the difficulty is related to the existence of a clustering phenomenon of the solutions when $\alpha$ is close to (but smaller than) $\alpha_c$. We introduce a new type of message passing algorithm which allows to find efficiently a satisfiable assignment of the variables in the difficult region. This algorithm is iterative and composed of two main parts. The first is a message-passing procedure which generalizes the usual methods like Sum-Product or Belief Propagation: it passes messages that are surveys over clusters of the ordinary messages. The second part uses the detailed probabilistic information obtained from the surveys in order to fix variables and simplify the problem. Eventually, the simplified problem that remains is solved by a conventional heuristic.