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On Complexity of Searching a Subset of Vectors with Shortest Average Under a Cardinality Restriction

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Analysis of Images, Social Networks and Texts (AIST 2016)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 661))

Abstract

In this paper, we study the computational complexity of the following subset search problem in a set of vectors. Given a set of N Euclidean q-dimensional vectors and an integer M, choose a subset of at least M vectors minimizing the Euclidean norm of the arithmetic mean of chosen vectors. This problem is induced, in particular, by a problem of clustering a set of points into two clusters where one of the clusters consists of points with a mean close to a given point. Without loss of generality the given point may be assumed to be the origin.

We show that the considered problem is NP-hard in the strong sense and it does not admit any approximation algorithm with guaranteed performance, unless P = NP. An exact algorithm with pseudo-polynomial time complexity is proposed for the special case of the problem, where the dimension q of the space is bounded from above by a constant and the input data are integer.

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Acknowledgements

This research is supported by RFBR, projects 15-01-00462, 16-01-00740 and 15-01-00976.

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Authors and Affiliations

  1. Omsk Branch of Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Sciences, Omsk, Russia

    Anton V. Eremeev

  2. Omsk State University n.a. F.M. Dostoevsky, Omsk, Russia

    Anton V. Eremeev

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

    Alexander V. Kel’manov & Artem V. Pyatkin

  4. Novosibirsk State University, Novosibirsk, Russia

    Alexander V. Kel’manov & Artem V. Pyatkin

Authors
  1. Anton V. Eremeev
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  2. Alexander V. Kel’manov
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  3. Artem V. Pyatkin
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Corresponding author

Correspondence to Anton V. Eremeev .

Editor information

Editors and Affiliations

  1. National Research University Higher School of Economics, Moscow, Russia

    Dmitry I. Ignatov

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

    Mikhail Yu. Khachay

  3. Ural Federal University, Yekaterinbug, Russia

    Valeri G. Labunets

  4. Research Computing Center, Lomonosov Moscow State University, Moscow, Russia

    Natalia Loukachevitch

  5. National Research University Higher School of Economics, St. Petersburg, Russia

    Sergey I. Nikolenko

  6. Technische Universität Darmstadt, Darmstadt, Germany

    Alexander Panchenko

  7. Laboratory of Algorithms and Technologies for Networks Analysis, National Research University Higher School of Economics, Nizhny Novgorod, Russia

    Andrey V. Savchenko

  8. Dorodnicyn Computing Centre of Russian Academy of Sciences, Moscow, Russia

    Konstantin Vorontsov

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Eremeev, A.V., Kel’manov, A.V., Pyatkin, A.V. (2017). On Complexity of Searching a Subset of Vectors with Shortest Average Under a Cardinality Restriction. In: Ignatov, D., et al. Analysis of Images, Social Networks and Texts. AIST 2016. Communications in Computer and Information Science, vol 661. Springer, Cham. https://doi.org/10.1007/978-3-319-52920-2_5

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  • DOI: https://doi.org/10.1007/978-3-319-52920-2_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-52919-6

  • Online ISBN: 978-3-319-52920-2

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