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Doklady Mathematics

, Volume 100, Issue 2, pp 416–419 | Cite as

NP-Completeness of Some Problems of Partitioning a Finite Set of Points in Euclidean Space into Balanced Clusters

  • A. V. Kel’manovEmail author
  • A. V. PyatkinEmail author
  • V. I. KhandeevEmail author
MATHEMATICS
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Abstract

We consider three related problems of partitioning an \(N\)-element set of points in \(d\)-dimensional Euclidean space into two clusters balancing the value of (1) the intracluster quadratic variance normalized by the cluster size in the first problem; (2) the intracluster quadratic variance in the second problem; and (3) the size-weighted intracluster quadratic variance in the third problem. The NP-completeness of all these problems is proved.

Notes

FUNDING

This work was supported by the Russian Foundation for Basic Research (project nos. 19-01-00308 and 18-31-00398), by Basic Research Programs of the Russian Academy of Sciences (project nos. 0314-2019-0015 and 0314-2019-0014), and by the Top-5-100 Program of the Ministry of Education and Science of the Russian Federation.

REFERENCES

  1. 1.
    D. Aloise, A. Deshpande, P. Hansen, and P. Popat, Mach. Learn. 75 (2), 245–248 (2009).CrossRefGoogle Scholar
  2. 2.
    C. H. Papadimitriou, SIAM J. Comput. 10 (3), 542–557 (1981).MathSciNetCrossRefGoogle Scholar
  3. 3.
    S. Masuyama, T. Ibaraki, and T. Hasegawa, IEEE Trans. IECE Jpn. 64 (2), 57–64 (1981).Google Scholar
  4. 4.
    H. Aggarwal, N. Imai, N. Katoh, and S. Suri, J. Algorithms 12 (1), 38–56 (1991).MathSciNetCrossRefGoogle Scholar
  5. 5.
    G. W. Snedecor and W. G. Cochran, Statistical Methods, 8th ed. (Iowa State University Press, 1989).zbMATHGoogle Scholar
  6. 6.
    M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of SciencesNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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