Complexity of Local Search for Euclidean Clustering Problems

Abstract

We show that the simplest local search heuristics for two natural Euclidean clustering problems are PLS-complete. First, we show that the Hartigan--Wong method for $k$-Means clustering is PLS-complete, even when $k=2$. Second, we show the same result for the Flip heuristic for Max Cut, even when the edge weights are given by the (squared) Euclidean distances between the points in some set $\mathcal{X}\subseteq {\mathbb{R}}^d$; a problem which is equivalent to Min Sum 2-Clustering.

29 pages, 4 figures