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International Conference on Optimization and Applications

OPTIMA 2018: Optimization and Applications pp 131–143Cite as

An Exact Algorithm of Searching for the Largest Size Cluster in an Integer Sequence 2-Clustering Problem

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Part of the book series: Communications in Computer and Information Science ((CCIS,volume 974))

Abstract

A problem of partitioning a finite sequence of points in Euclidean space into two subsequences (clusters) maximizing the size of the first cluster subject to two constraints is considered. The first constraint deals with every two consecutive indices of elements of the first cluster: the difference between them is bounded from above and below by some constants. The second one restricts the value of a quadratic clustering function that is the sum of the intracluster sums over both clusters. The intracluster sum is the sum of squared distances between cluster elements and the cluster center. The center of the first cluster is unknown and determined as the centroid (i.e. as the mean value of its elements), while the center of the second one is zero.

The strong NP-hardness of the problem is shown and an exact algorithm is suggested for the case of integer coordinates of input points. If the space dimension is bounded by some constant this algorithm runs in a pseudopolynomial time.

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Acknowledgments

The study presented in Sects. 3 and 5 was supported by the Russian Science Foundation, project 16-11-10041. The study presented in Sects. 2 and 4 was supported by the Russian Foundation for Basic Research, projects 16-07-00168 and 18-31-00398, by the Russian Academy of Science (the Program of basic research), project 0314-2016-0015, 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, Sergey Khamidullin, Vladimir Khandeev & Artem Pyatkin

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

    Alexander Kel’manov, Vladimir Khandeev & Artem Pyatkin

Authors
  1. Alexander Kel’manov
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  2. Sergey Khamidullin
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  3. Vladimir Khandeev
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  4. Artem Pyatkin
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Corresponding author

Correspondence to Vladimir Khandeev .

Editor information

Editors and Affiliations

  1. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Yury Evtushenko

  2. University of Montenegro, Podgorica, Montenegro

    Milojica Jaćimović

  3. Krasovsky Institute of Mathematics and Mechanics, Yekaterinburg, Russia

    Michael Khachay

  4. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

    Yury Kochetov

  5. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Vlasta Malkova

  6. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Mikhail Posypkin

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Kel’manov, A., Khamidullin, S., Khandeev, V., Pyatkin, A. (2019). An Exact Algorithm of Searching for the Largest Size Cluster in an Integer Sequence 2-Clustering Problem. In: Evtushenko, Y., Jaćimović, M., Khachay, M., Kochetov, Y., Malkova, V., Posypkin, M. (eds) Optimization and Applications. OPTIMA 2018. Communications in Computer and Information Science, vol 974. Springer, Cham. https://doi.org/10.1007/978-3-030-10934-9_10

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  • DOI: https://doi.org/10.1007/978-3-030-10934-9_10

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