Complexity of Local Search for Euclidean Clustering Problems

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.