2 + p-SAT: Relation of typical-case complexity to the nature of the phase transition

Heuristic methods for solution of problems in the NP-complete class of decision problems often reach exact solutions, but fail badly at "phase boundaries," across which the decision to be reached changes from almost always having one value to almost always having a different value. We report an analytic solution and experimental investigations of the phase transition that occurs in the limit of very large problems in K-SAT. Studying a model which interpolates K-SAT between K = 2 and K = 3, we find a change from a continuous to a discontinuous phase transition when K, the average number of inputs per clause, exceeds 0.4. The cost of finding solutions also increases dramatically above this changeover. The nature of its "random first-order" phase transition, seen at values of K large enough to make the computational cost of solving typical instances increase exponentially with problem size, suggests a mechanism for the cost increase. There has been evidence for features like the "backbone" of frozen inputs which characterizes the UNSAT phase in K-SAT in the study of models of disordered materials, but this feature and this transition are uniquely accessible to analysis in K-SAT. The random first-order transition combines properties of the first-order (discontinuous onset of order) and second-order (with power law scaling, e.g., of the width of the critical region in a finite system) transitions known in the physics of pure solids. Such transitions should occur in other combinatoric problems in the large N limit. Finally, improved search heuristics may be developed when a "backbone" is known to exist.