Solution space heterogeneity of the random K-satisfiability problem: Theory and simulations

TL;DR

This paper investigates the detailed structure of the solution space in random K-SAT problems, identifying a heterogeneity transition and its effects on solution clustering and algorithm performance.

Contribution

It introduces the concept of solution space heterogeneity transition and analyzes its implications for solution clustering and algorithm dynamics in random K-SAT.

Findings

Heterogeneity transition occurs at a critical constraint density alpha_cm.

Solution communities emerge exponentially after the heterogeneity transition.

Solution space remains ergodic until a larger threshold alpha_d where communities disconnect.

Abstract

The random K-satisfiability (K-SAT) problem is an important problem for studying typical-case complexity of NP-complete combinatorial satisfaction; it is also a representative model of finite-connectivity spin-glasses. In this paper we review our recent efforts on the solution space fine structures of the random K-SAT problem. A heterogeneity transition is predicted to occur in the solution space as the constraint density alpha reaches a critical value alpha_cm. This transition marks the emergency of exponentially many solution communities in the solution space. After the heterogeneity transition the solution space is still ergodic until alpha reaches a larger threshold value alpha_d, at which the solution communities disconnect from each other to become different solution clusters (ergodicity-breaking). The existence of solution communities in the solution space is confirmed by numerical simulations of solution space random walking, and the effect of solution space heterogeneity on a stochastic local search algorithm SEQSAT, which performs a random walk of single-spin flips, is investigated. The relevance of this work to glassy dynamics studies is briefly mentioned.