CCCP Algorithms to Minimize the Bethe free energy of 3-SAT Problem

TL;DR

This paper applies the CCCP algorithm to the 3-SAT problem, demonstrating improved convergence over the BP algorithm, and explores the properties and potential applications of CCCP in learning and inference tasks.

Contribution

It introduces a novel application of the CCCP algorithm to the 3-SAT problem, showing better convergence properties than BP and analyzing its unique characteristics.

Findings

CCCP converges where BP does not in 3-SAT.

CCCP shows potential for broader learning and inference applications.

Differences from previous CCCP implementations highlight problem-specific properties.

Abstract

The k-sat problem is a prototypical constraint satisfaction problem. There are many algorithms to study k-sat problem, BP algorithm is famous one of them. But BP algorithm does not converge when $\alpha$(constraint density)is bigger than some threshold value. In this paper we use CCCP (Concave Convex Procedure) algorithm to study 3-sat problem and we get better results than BP algorithm that CCCP algorithm still converges when BP algorithm does not converge. Our work almost builds on recent results by Yuille \cite{Yuille2002} who apply the CCCP algorithm to Bethe and Kikuchi free energies and obtained two algorithms on 2D and 3D spin glasses. Our implementation of CCCP algorithm on 3-sat problem is some different from his implementation and we have some different views about CCCP algorithm's some properties. Some difference of these maybe because of CCCP algorithm have different properties and implementation process on different problem and some others of these are related to the CCCP algorithm itself. Our work indicates that CCCP algorithm has more learning and inference applications.