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Exact Algorithms for the Special Cases of Two Hard to Solve Problems of Searching for the Largest Subset

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Analysis of Images, Social Networks and Texts (AIST 2018)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 11179))

Abstract

We consider two problems of searching for the largest subset in a finite set of points in Euclidean space. Both problems have applications, in particular, in data analysis and data approximation. In each problem, an input set and a positive real number are given and it is required to find a subset (i.e., a cluster) of the largest size under a constraint on the value of a quadratic function. The points in the input set which are outside the sought subset are treated as a second cluster. In the first problem, the function in the constraint is the sum over both clusters of the intracluster sums of the squared distances between the elements of the clusters and their centers. The center of the first (i.e., sought) cluster is unknown and determined as the centroid, while the center of the second one is fixed in a given point in Euclidean space (without loss of generality in the origin). In the second problem, the function in the constraint is the sum over both clusters of the weighted intracluster sums of the squared distances between the elements of the clusters and their centers. As in the first problem, the center of the first cluster is unknown and determined as the centroid, while the center of the second one is fixed in the origin. In this paper, we show that both problems are strongly NP-hard. Also, we present exact algorithms for the cases of these problems in which the input points have integer components. If the space dimension is bounded by some constant, the algorithms are pseudopolynomial.

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Acknowledgments

The study of Problem 1 was supported by the Russian Science Foundation, project 16-11-10041. The study of Problem 2 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, Vladimir Khandeev & Anna Panasenko

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

    Alexander Kel’manov, Vladimir Khandeev & Anna Panasenko

Authors
  1. Alexander Kel’manov
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  2. Vladimir Khandeev
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Corresponding author

Correspondence to Vladimir Khandeev .

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Editors and Affiliations

  1. RWTH Aachen University, Aachen, Germany

    Wil M. P. van der Aalst

  2. University of Ljubljana, Ljubljana, Slovenia

    Vladimir Batagelj

  3. University of Mannheim, Mannheim, Germany

    Goran Glavaš

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

    Dmitry I. Ignatov

  5. Institute of Mathematics and Mechanics, Yekaterinburg, Russia

    Michael Khachay

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

    Sergei O. Kuznetsov

  7. National Research University Higher School of Economics , Saint Petersburg, Russia

    Olessia Koltsova

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

    Irina A. Lomazova

  9. Moscow State University, Moscow, Russia

    Natalia Loukachevitch

  10. Loria, Vandoeuvre lès Nancy, France

    Amedeo Napoli

  11. University of Hamburg, Hamburg, Germany

    Alexander Panchenko

  12. University of Florida, Gainesville, FL, USA

    Panos M. Pardalos

  13. Ca Foscari University of Venice, Venice, Italy

    Marcello Pelillo

  14. National Research University Higher School of Economics, Nizhny Novgorod, Russia

    Andrey V. Savchenko

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Kel’manov, A., Khandeev, V., Panasenko, A. (2018). Exact Algorithms for the Special Cases of Two Hard to Solve Problems of Searching for the Largest Subset. In: van der Aalst, W., et al. Analysis of Images, Social Networks and Texts. AIST 2018. Lecture Notes in Computer Science(), vol 11179. Springer, Cham. https://doi.org/10.1007/978-3-030-11027-7_28

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  • DOI: https://doi.org/10.1007/978-3-030-11027-7_28

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