Criticality and parallelism in combinatorial optimization.

Local search methods constitute one of the most successful approaches to solving large-scale combinatorial optimization problems. As these methods are increasingly parallelized, optimization performance initially improves but then abruptly degrades to no better than that of random search beyond a certain point. The existence of this transition is demonstrated for a family of generalized spin-glass models and the traveling salesman problem. Finite-size scaling is used to characterize size-dependent effects near the transition, and analytical insight is obtained through a mean-field approximation.