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A PTAS for One Cardinality-Weighted 2-Clustering Problem

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Mathematical Optimization Theory and Operations Research (MOTOR 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11548))

Abstract

We consider one strongly NP-hard problem of clustering a finite set of points in Euclidean space. In this problem, we need to partition a finite set of points into two clusters minimizing the sum over both clusters of the weighted intracluster sums. Each of these sums is the sum of squared distances between the elements of the cluster and their center. The center of the one cluster is unknown and determined as the centroid, while the center of the other one is fixed at the origin. The weight factors for both intracluster sums are the given sizes of the clusters. In this paper, we present an approximation algorithm for the problem and prove that it is a polynomial-time approximation scheme (PTAS).

The study presented in Sects. 2 and 3 was supported by the Russian Foundation for Basic Research, project 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 Avenue, 630090, Novosibirsk, Russia

    Anna Panasenko

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

    Anna Panasenko

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  1. Anna Panasenko
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Correspondence to Anna Panasenko .

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Editors and Affiliations

  1. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Yury Kochetov

  3. University of Florida, Gainesville, FL, USA

    Panos Pardalos

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Panasenko, A. (2019). A PTAS for One Cardinality-Weighted 2-Clustering Problem. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Lecture Notes in Computer Science(), vol 11548. Springer, Cham. https://doi.org/10.1007/978-3-030-22629-9_41

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  • DOI: https://doi.org/10.1007/978-3-030-22629-9_41

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  • Online ISBN: 978-3-030-22629-9

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