# A Technical Report on PLS-Completeness of Single-Swap for Unweighted Metric Facility Location and $K$-Means

## TL;DR

This paper proves that the single-swap heuristic for unweighted metric facility location and K-means is tightly PLS-complete, strengthening previous results for weighted versions.

## Contribution

It provides a stronger reduction demonstrating tight PLS-completeness for the unweighted versions of these problems.

## Key findings

- Single-swap heuristic is tightly PLS-complete for unweighted problems.
- Strengthens previous weighted problem results.
- Implications for local search algorithm complexity.

## Abstract

Recently, [Bra17] showed that the single-swap heuristic for weighted metric uncapacitated facility location and $K$-Means is tightly PLS-complete. We build upon this work and present a stronger reduction, which proves tight PLS-completeness for the unweighted version of both problems.