50 32 v 1 3 1 M ay 2 00 0 Computational Complexity and Phase Transitions

Phase transitions in combinatorial problems have recently been shown [2] to be useful in locating “hard” instances of combinatorial problems. The connection between computational complexity and the existence of phase transitions has been addressed in Statistical Mechanics [2] and Artificial Intelligence [3], but not studied rigorously. We take a first step in this direction by investigating the existence of sharp thresholds for the class of generalized satisfiability problems, defined by Schaefer [4]. In the case when all constraints have a special clausal form we completely characterize the generalized satisfiability problems that have a sharp threshold. While NP-completeness does not imply the sharpness of the threshold, our result suggests that the class of counterexamples is rather limited, as all such counterexamples can be predicted, with constant success probability by a single procedure.