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Metric facility location and $K$-means are well-known problems ofcombinatorial optimization. Both admit a fairly simple heuristic calledsingle-swap, which adds, drops or swaps open facilities until it reaches alocal optimum. For both problems, it is known that this algorithm produces asolution that is at most a constant factor worse than the respective globaloptimum. In this paper, we show that single-swap applied to the weighted metricuncapacitated facility location and weighted discrete $K$-means problem istightly PLS-complete and hence has exponential worst-case running time.

Complexity of Single-Swap Heuristics for Metric Facility Location and Related Problem