Communities of solutions in single solution clusters of a random K-Satisfiability formula

TL;DR

This paper investigates the internal community structure of solution clusters in random K-SAT problems, revealing early community formation and differences in solution similarity and entropy density, which could inform better heuristic algorithms.

Contribution

It uncovers the community structure within solution clusters of random K-SAT formulas using physics-inspired methods, providing new insights into solution space evolution.

Findings

Community structures form before the clustering transition point.

Solutions within the same community are more similar and densely connected.

Entropy density varies across different communities within the same cluster.

Abstract

The solution space of a K-satisfiability (K-SAT) formula is a collection of solution clusters, each of which contains all the solutions that are mutually reachable through a sequence of single-spin flips. Knowledge of the statistical property of solution clusters is valuable for a complete understanding of the solution space structure and the computational complexity of the random K-SAT problem. This paper explores single solution clusters of random 3- and 4-SAT formulas through unbiased and biased random walk processes and the replica-symmetric cavity method of statistical physics. We find that the giant connected component of the solution space has already formed many different communities when the constraint density of the formula is still lower than the solution space clustering transition point. Solutions of the same community are more similar with each other and more densely connected with each other than with the other solutions. The entropy density of a solution community is calculated using belief propagation and is found to be different for different communities of the same cluster. When the constraint density is beyond the clustering transition point, the same behavior is observed for the solution clusters reached by several stochastic search algorithms. Taking together, the results of this work suggests a refined picture on the evolution of the solution space structure of the random K-SAT problem; they may also be helpful for designing new heuristic algorithms.