Numerical Solution-Space Analysis of Satisfiability Problems

TL;DR

This paper investigates the structure of the solution space in 3-SAT problems using numerical simulations, introduces a new unbiased sampling method, and analyzes how the solution space transitions from a simple to a clustered phase as the clause-to-variable ratio increases.

Contribution

It introduces the ballistic-networking approach for unbiased sampling of solutions and analyzes the phase transition in solution space structure of 3-SAT.

Findings

Solution space transitions to a clustered phase at alpha_c ~ 3.86.

Unbiased sampling reveals clusters without frozen variables near the SAT-UNSAT transition.

SLS algorithms can solve large instances close to the transition due to the solution space structure.

Abstract

The solution-space structure of the 3-Satisfiability Problem (3-SAT) is studied as a function of the control parameter alpha (ratio of number of clauses to the number of variables) using numerical simulations. For this purpose, one has to sample the solution space with uniform weight. It is shown here that standard stochastic local-search (SLS) algorithms like "ASAT" and "MCMCMC" (also known as "parallel tempering") exhibit a sampling bias. Nevertheless, unbiased samples of solutions can be obtained using the "ballistic-networking approach", which is introduced here. It is a generalization of "ballistic search" methods and yields also a cluster structure of the solution space. As application, solutions of 3-SAT instances are generated using ASAT plus ballistic networking. The numerical results are compatible with a previous analytic prediction of a simple solution-space structure for small values of alpha and a transition to a clustered phase at alpha_c ~ 3.86, where the solution space breaks up into several non-negligible clusters. Furthermore, in the thermodynamic limit there are, for values of alpha close to the SATUNSAT transition alpha_s ~ 4.267, always clusters without any frozen variables. This may explain why some SLS algorithms are able to solve very large 3-SAT instances close to the SAT-UNSAT transition.