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Complexity of Local Search for Euclidean Clustering Problems

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

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Author Details

Bodo Manthey
  • Faculty of Electrical Engineering, Mathematics, and Computer Science, University of Twente, The Netherlands
Nils Morawietz
  • Institute of Computer Science, Friedrich Schiller University Jena, Germany
Jesse van Rhijn
  • Faculty of Electrical Engineering, Mathematics, and Computer Science, University of Twente, The Netherlands
Frank Sommer
  • Institute of Logic and Computation, TU Wien, Austria

Funding

  • van Rhijn, Jesse: Supported by NWO grant OCENW.KLEIN.176.
  • Sommer, Frank: Supported by the Alexander von Humboldt Foundation.

Cite As Get BibTex

Bodo Manthey, Nils Morawietz, Jesse van Rhijn, and Frank Sommer. Complexity of Local Search for Euclidean Clustering Problems. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 48:1-48:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.48

Abstract

We show that the simplest local search heuristics for two natural Euclidean clustering problems are PLS-hard. First, we show that the Hartigan-Wong method, which is essentially the Flip heuristic, for k-Means clustering is PLS-hard, 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 𝒳 ⊆ R^d; a problem which is equivalent to Min Sum 2-Clustering.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Discrete optimization
  • Theory of computation → Facility location and clustering
Keywords
  • Local search
  • PLS-complete
  • max cut
  • k-means
  • partitioning problem
  • flip-neighborhood

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