Solving Constraint Satisfaction Problems through Belief Propagation-guided decimation

TL;DR

This paper analyzes a belief propagation-guided decimation algorithm for solving random constraint satisfaction problems, proposing a tree model to predict its behavior and confirming the predictions through simulations.

Contribution

It introduces a simple randomized decimation algorithm based on belief propagation and develops a tree model to analyze its asymptotic behavior on large random k-satisfiability instances.

Findings

The tree model accurately predicts the algorithm's behavior in large instances.

Numerical simulations confirm the conjecture that the model provides asymptotically exact predictions.

The approach offers insights into the effectiveness of message passing algorithms in CSPs.

Abstract

Message passing algorithms have proved surprisingly successful in solving hard constraint satisfaction problems on sparse random graphs. In such applications, variables are fixed sequentially to satisfy the constraints. Message passing is run after each step. Its outcome provides an heuristic to make choices at next step. This approach has been referred to as `decimation,' with reference to analogous procedures in statistical physics. The behavior of decimation procedures is poorly understood. Here we consider a simple randomized decimation algorithm based on belief propagation (BP), and analyze its behavior on random k-satisfiability formulae. In particular, we propose a tree model for its analysis and we conjecture that it provides asymptotically exact predictions in the limit of large instances. This conjecture is confirmed by numerical simulations.