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

166 Downloads (Pure)

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

Complexity of Local Search for Euclidean Clustering Problems
Manthey, B., Morawietz, N., van Rhijn, J. & Sommer, F., 4 Dec 2024, 35th International Symposium on Algorithms and Computation, ISAAC 2024. Mestre, J. & Wirth, A. (eds.). Dagstuhl, (Leibniz International Proceedings in Informatics, LIPIcs; vol. 322).
Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

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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 \$\textbackslash{}mathcal\{X\} \textbackslash{}subseteq \textbackslash{}mathbb\{R\}\textasciicircum{}d\$; a problem which is equivalent to Min Sum 2-Clustering. ",

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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.

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KW - cs.DS

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BT - Complexity of Local Search for Euclidean Clustering Problems

PB - ArXiv.org

ER -

Complexity of Local Search for Euclidean Clustering Problems

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Complexity of Local Search for Euclidean Clustering Problems

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