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Exact Algorithms for Two Quadratic Euclidean Problems of Searching for the Largest Subset and Longest Subsequence

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Learning and Intelligent Optimization (LION 12 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11353))

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Abstract

The following two strongly NP-hard problems are considered. In the first problem, we need to find in the given finite set of points in Euclidean space the subset of largest size such that the sum of squared distances between the elements of this subset and its unknown centroid (geometrical center) does not exceed a given percentage of the sum of squared distances between the elements of the input set and its centroid. In the second problem, the input is a sequence (not a set) and we have some additional constraints on the indices of the elements of the chosen subsequence under the same restriction on the sum of squared distances as in the first problem. Both problems can be treated as data editing problems aimed to find similar elements and removal of extraneous (dissimilar) elements. We propose exact algorithms for the cases of both problems in which the input points have integer-valued coordinates. If the space dimension is bounded by some constant, our algorithms run in a pseudopolynomial time. Some results of numerical experiments illustrating the performance of the algorithms are presented.

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Acknowledgments

The study presented in Sects. 2, 4 was supported by the Russian Science Foundation, project 16-11-10041. The study presented in Sects. 3, 5 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. University of Trento, Trento, Italy

    Roberto Battiti

  2. University of Trento, Trento, Italy

    Mauro Brunato

  3. Wilfrid Laurier University, Waterloo, ON, Canada

    Ilias Kotsireas

  4. University of Florida, Gainesville, FL, USA

    Panos M. Pardalos

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Kel’manov, A., Khamidullin, S., Khandeev, V., Pyatkin, A. (2019). Exact Algorithms for Two Quadratic Euclidean Problems of Searching for the Largest Subset and Longest Subsequence. In: Battiti, R., Brunato, M., Kotsireas, I., Pardalos, P. (eds) Learning and Intelligent Optimization. LION 12 2018. Lecture Notes in Computer Science(), vol 11353. Springer, Cham. https://doi.org/10.1007/978-3-030-05348-2_28

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  • DOI: https://doi.org/10.1007/978-3-030-05348-2_28

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-05347-5

  • Online ISBN: 978-3-030-05348-2

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