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

TL;DR

This paper proves that the single-swap heuristic for weighted metric uncapacitated facility location and weighted discrete K-means problems is computationally hard, with exponential worst-case running time, despite its practical effectiveness.

Contribution

It establishes the PLS-completeness of single-swap heuristics for these problems, showing their inherent computational complexity.

Findings

Single-swap heuristic is PLS-complete for weighted problems.

Exponential worst-case running time for the heuristic.

Maintains known approximation guarantees despite complexity.

Abstract

Metric facility location and $K$-means are well-known problems of 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. In this paper, we show that single-swap applied to the weighted metric uncapacitated facility location and weighted discrete $K$-means problem is tightly PLS-complete and hence has exponential worst-case running time.