Complexity of Local Search for Euclidean Clustering Problems

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

doi:10.48550/arXiv.2312.14916

Complexity of Local Search for Euclidean Clustering Problems

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

Mathematics of Operations Research

Research output: Working paper › Preprint › Academic

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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.

Original language	English
Publisher	ArXiv.org
DOIs	https://doi.org/10.48550/arXiv.2312.14916
Publication status	Published - 22 Dec 2023

Keywords

cs.CC
cs.DS

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10.48550/arXiv.2312.14916Licence: CC BY

2312.14916v2Final published version, 917 KBLicence: CC BY

Cite this

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title = "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. ",

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AU - van Rhijn, Jesse

AU - Sommer, Frank

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PY - 2023/12/22

Y1 - 2023/12/22

N2 - 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.

AB - 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.

KW - cs.CC

KW - cs.DS

U2 - 10.48550/arXiv.2312.14916

DO - 10.48550/arXiv.2312.14916

M3 - Preprint

BT - Complexity of Local Search for Euclidean Clustering Problems

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Complexity of Local Search for Euclidean Clustering Problems

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Keywords

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