Glassy Behavior and Jamming of a Random Walk Process for Sequentially Satisfying a Constraint Satisfaction Formula
Abstract
Random K-satisfiability (K-SAT) is a model system for studying typical-case complexity of combinatorial optimization. Recent theoretical and simulation work revealed that the solution space of a random K-SAT formula has very rich structures, including the emergence of solution communities within single solution clusters. In this paper we investigate the influence of the solution space landscape to a simple stochastic local search process SEQSAT, which satisfies a K-SAT formula in a sequential manner. Before satisfying each newly added clause, SEQSAT walk randomly by single-spin flips in a solution cluster of the old subformula. This search process is efficient when the constraint density α of the satisfied subformula is less than certain value αcm; however it slows down considerably as α > αcm and finally reaches a jammed state at α ≈ αj. The glassy dynamical behavior of SEQSAT for α ≥ αcm probably is due to the entropic trapping of various communities in the solution cluster of the satisfied subformula. For random 3-SAT, the jamming transition point αj is larger than the solution space clustering transition point αd, and its value can be predicted by a long-range frustration mean-field theory. For random K-SAT with K≥ 4, however, our simulation results indicate that αj = αd. The relevance of this work for understanding the dynamic properties of glassy systems is also discussed.