Complexity of Local Search for Euclidean Clustering Problems

Bodo Manthey; Nils Morawietz; Jesse van Rhijn; Frank Sommer

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arXiv · 2312.14916

Complexity of Local Search for Euclidean Clustering Problems

Bodo Manthey·Nils Morawietz·Jesse van Rhijn·Frank Sommer

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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.

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Bodo Manthey, Nils Morawietz, Jesse van Rhijn, Frank Sommer. 2023-12-22. Complexity of Local Search for Euclidean Clustering Problems. https://arxiv.org/abs/2312.14916

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