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2-Approximation Polynomial-Time Algorithm for a Cardinality-Weighted 2-Partitioning Problem of a Sequence

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Numerical Computations: Theory and Algorithms (NUMTA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11974))

Abstract

We consider a problem of 2-partitioning a finite sequence of points in Euclidean space into clusters of the given sizes with some constraints. The solution criterion is the minimum of the sum of weighted intracluster sums of squared distances between the elements of each cluster and its center. The weight of the intracluster sum is equal to the cluster size. The center of one cluster is given as input (is the origin without loss of generality), while the center of the other one is unknown and is determined as a geometric center. The following constraints hold: the difference between the indices of two subsequent points included in the first cluster is bounded from above and below by some given constants. In this paper, we have shown that the considered problem is the strongly NP-hard one and propose a polynomial-time 2-approximation algorithm for solving 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 19-07-00397, 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 Avenue, 630090, Novosibirsk, Russia

    Alexander Kel’manov, Sergey Khamidullin & Anna Panasenko

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

    Alexander Kel’manov & Anna Panasenko

Authors
  1. Alexander Kel’manov
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  2. Sergey Khamidullin
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  3. Anna Panasenko
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Corresponding author

Correspondence to Anna Panasenko .

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Editors and Affiliations

  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

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Kel’manov, A., Khamidullin, S., Panasenko, A. (2020). 2-Approximation Polynomial-Time Algorithm for a Cardinality-Weighted 2-Partitioning Problem of a Sequence. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11974. Springer, Cham. https://doi.org/10.1007/978-3-030-40616-5_34

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  • DOI: https://doi.org/10.1007/978-3-030-40616-5_34

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  • Online ISBN: 978-3-030-40616-5

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