CCCP Algorithms to Minimize the Bethe free energy of 3-SAT Problem
This work addresses a specific bottleneck in constraint satisfaction problems for researchers in statistical physics and machine learning, but it is incremental as it builds on prior applications of CCCP to other problems.
The authors tackled the convergence issue of the Belief Propagation (BP) algorithm for the 3-SAT problem at high constraint densities by applying the Concave Convex Procedure (CCCP) algorithm, achieving convergence where BP fails.
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 $α$(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.