Exploiting a Theory of Phase Transitions in Three-Satisfiability Problems

In the past few years there have been several empirical discoveries of phase transitions in constraint satisfaction problems (CSPs), and a growth of interest in the area among the arti cial intelligence community. This paper extends a simple analytical theory of phase transitions in three-satis ability (3-SAT) problems in two directions. First, a more accurate, problem-dependent calculation leads to a new polynomial time probabilistic estimate of the satis ability of 3-SAT problems called PE-SAT (Probabilistic Estimate SATis ability algorithm). PE-SAT empirically classi es 3-SAT problems with about 70% accuracy at the hardest region (the so-called crossover point or 50% satis able region) of random 3-SAT space. Furthermore, the estimate has a meaningful magnitude such that extreme estimates are much more likely to be correct. Second, the same estimate is used to improve the running time of a backtracking search for a solution to 3-SAT by pruning unlikely branches of the search. The speed-up is achieved at the expense of accuracy|the search remains sound but is no longer complete. The trade-o between speed-up and accuracy is shown to improve as the size of problems increases.