Simple Local Search Problems that are Hard to Solve

Abstract

Many algorithms for NP-hard optimization problems find solutions that are locally optimal, in the sense that the solutions cannot be improved by a polynomially computable perturbation. Very little is known about the complexity of finding locally optimal solutions, either by local search algorithms or using other indirect methods. Johnson, Papadimitriou, and Yannakakis [J. Comput. System Sci., 37 (1988), pp. 79–100] studied this question by defining a complexity class PLS that captures local search problems. It was proved that finding a partition of a graph that is locally optimal into equal parts with respect to the acclaimed Kernighan-Lin algorithm is PLS-complete. It is shown here that several natural, simple local search problems are PLS-complete, and thus just as hard. Two examples are: finding a partition that cannot be improved by a single swap of two vertices, and finding a stable configuration for an undirected connectionist network. When edges or other objects are unweighted, then a local optimum can always be found in polynomial time. It is shown that the unweighted versions of the local search problems studied in this paper are P-complete.