Skip to main content
SpringerLink
Log in

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

  • MATHEMATICS
  • Published:
Doklady Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. D. Aloise, A. Deshpande, P. Hansen, and P. Popat, Mach. Learn. 75 (2), 245–248 (2009).

    Article  Google Scholar 

  2. C. H. Papadimitriou, SIAM J. Comput. 10 (3), 542–557 (1981).

    Article  MathSciNet  Google Scholar 

  3. S. Masuyama, T. Ibaraki, and T. Hasegawa, IEEE Trans. IECE Jpn. 64 (2), 57–64 (1981).

    Google Scholar 

  4. H. Aggarwal, N. Imai, N. Katoh, and S. Suri, J. Algorithms 12 (1), 38–56 (1991).

    Article  MathSciNet  Google Scholar 

  5. G. W. Snedecor and W. G. Cochran, Statistical Methods, 8th ed. (Iowa State University Press, 1989).

    MATH  Google Scholar 

  6. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979).

    MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    A. V. Kel’manov, A. V. Pyatkin & V. I. Khandeev

  2. Novosibirsk State University, 630090, Novosibirsk, Russia

    A. V. Kel’manov, A. V. Pyatkin & V. I. Khandeev

Authors
  1. A. V. Kel’manov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Pyatkin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. I. Khandeev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. V. Kel’manov, A. V. Pyatkin or V. I. Khandeev.

Additional information

Translated by I. Ruzanova

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kel’manov, A.V., Pyatkin, A.V. & Khandeev, V.I. NP-Completeness of Some Problems of Partitioning a Finite Set of Points in Euclidean Space into Balanced Clusters. Dokl. Math. 100, 416–419 (2019). https://doi.org/10.1134/S1064562419050028

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064562419050028

Search

Navigation