Complexity of single-swap heuristics for metric facility location and related problems

Abstract

Metric facility location and K-means are well-known problems in combinatorial optimization. Both admit a fairly simple heuristic called single-swap, which adds, drops, or swaps open facilities until it reaches a local optimum. For both problems, it is known that this algorithm produces a solution that is at most a constant factor worse than the respective global optimum. We show that single-swap applied to the metric uncapacitated facility location problem and to the discrete K-means problem is tightly PLS-complete and hence has exponential worst-case running time. Furthermore, we extend this result to the weighted discrete fuzzy K-means problem, a soft clustering variant of the classical discrete K-means problem.