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International Conference on Learning and Intelligent Optimization

LION 2019: Learning and Intelligent Optimization pp 46–52Cite as

On Polynomial Solvability of One Quadratic Euclidean Clustering Problem on a Line

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11968))

Abstract

We consider one problem of partitioning a finite set of points in Euclidean space into clusters so as to minimize the sum over all clusters of the intracluster sums of the squared distances between clusters elements and their centers. The centers of some clusters are given as an input, while the other centers are unknown and defined as centroids (geometrical centers). It is known that the general case of the problem is strongly NP-hard. We show that there exists an exact polynomial algorithm for the one-dimensional case of the problem.

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Acknowledgments

The study presented in Sects. 3 and 4 was supported by the Russian Foundation for Basic Research, projects 19-01-00308 and 18-31-00398. The study presented in the other sections was supported by the Russian Academy of Science (the Program of basic research), project 0314-2019-0015, and by the Russian Ministry of Science and Education under the 5-100 Excellence Programme.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, 4 Koptyug Ave., 630090, Novosibirsk, Russia

    Alexander Kel’manov & Vladimir Khandeev

  2. Novosibirsk State University, 2 Pirogova St., 630090, Novosibirsk, Russia

    Alexander Kel’manov & Vladimir Khandeev

Authors
  1. Alexander Kel’manov
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  2. Vladimir Khandeev
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Corresponding author

Correspondence to Vladimir Khandeev .

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Editors and Affiliations

  1. Technical University of Crete, Chania, Greece

    Nikolaos F. Matsatsinis

  2. Technical University of Crete, Chania, Greece

    Yannis Marinakis

  3. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA

    Panos Pardalos

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Kel’manov, A., Khandeev, V. (2020). On Polynomial Solvability of One Quadratic Euclidean Clustering Problem on a Line. In: Matsatsinis, N., Marinakis, Y., Pardalos, P. (eds) Learning and Intelligent Optimization. LION 2019. Lecture Notes in Computer Science(), vol 11968. Springer, Cham. https://doi.org/10.1007/978-3-030-38629-0_4

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