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Pattern Recognition and Image Analysis

, Volume 28, Issue 1, pp 17–23 | Cite as

Approximation Scheme for a Quadratic Euclidean Weighted 2-Clustering Problem

  • A. V. Kel’manov
  • A. V. Motkova
Mathematical Method in Pattern Recognition

Abstract

We consider the strongly NP-hard problem of partitioning a finite set of Euclidean points into two clusters so as to minimize the sum (over both clusters) of the weighted sums of the squared intra-cluster distances from the elements of the cluster to its center. The weights of the sums are equal to the cardinalities of the clusters. The center of one of the clusters is given as input, while the center of the other cluster is unknown and is determined as the mean value over all points in this cluster, i.e., as the geometric center (centroid). The version of the problem with constrained cardinalities of the clusters is analyzed. We construct an approximation algorithm for the problem and show that it is a fully polynomial-time approximation scheme (FPTAS) if the space dimension is bounded by a constant.

Keywords

data analysis weighted 2-clustering Euclidean space NP-hardness fixed space dimension FPTAS 

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References

  1. 1.
    A. V. Kel’manov and A. V. Motkova, “Exact pseudopolynomial algorithms for a balanced 2-clustering problem,” J. Appl. Indust. Math. 10 (3), 349–355 (2016).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    A. V. Kel’manov and A. V. Pyatkin, “NP-hardness of some quadratic Euclidean 2-clustering problems,” Dokl. Math. 92 (2), 634–637 (2015).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    A. V. Kel’manov and A. V. Pyatkin, “On the complexity of some quadratic Euclidean 2-clustering problems,” Comput. Math. Math. Phys. 56 (3), 491–497 (2015).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    C. C. Aggarwal, Data Mining: The Textbook (Springer International Publishing, 2015).CrossRefMATHGoogle Scholar
  5. 5.
    C. M. Bishop, Pattern Recognition and Machine Learning (Springer Science + Business Media, New York, 2006).MATHGoogle Scholar
  6. 6.
    R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis (John Wiley & Sons, New York, 1973; Mir, Moscow, 1976).MATHGoogle Scholar
  7. 7.
    K. Fukunaga, Introduction to Statistical Pattern Recognition, 2nd ed. (Academic Press, New York, 1990).MATHGoogle Scholar
  8. 8.
    T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd ed. (Springer-Verlag, New York, 2009).CrossRefMATHGoogle Scholar
  9. 9.
    A. K. Jain, “Data clustering: 50 years beyond - means,” Pattern Recogn. Lett. 31 (8), 651–666 (2010).CrossRefGoogle Scholar
  10. 10.
    G. James, D. Witten, T. Hastie, and R. Tibshirani, An Introduction to Statistical Learning: with Application in R (Springer Science + Business Media, New York, 2013).CrossRefMATHGoogle Scholar
  11. 11.
    T.-C. Fu, “A review on time series data mining,” Eng. Appl. Artif. Intell. 24 (1), 164–181 (2011).CrossRefGoogle Scholar
  12. 12.
    J. T. Tou and R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, Reading, MA, 1974; Mir, Moscow, 1978).MATHGoogle Scholar
  13. 13.
    P. Brucker, “On the complexity of clustering problems,” in Optimization and Operations Research: Proc. of the Workshop Held at University Bonn (Bonn, Germany, October 2–8, 1977), Lecture Notes Econom. Math. Syst. 157, 45–54 (1978).CrossRefGoogle Scholar
  14. 14.
    S. Sahni and T. Gonzalez, “P-complete approximation problems,” J. ACM. 23, 555–566 (1976).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    S. Hasegawa, H. Imai, M. Inaba, N. Katoh, and J. Nakano, “Efficient algorithms for variance-based - Clustering,” in Proc. 1st Pacific Conf. on Computer Graphics and Applications Seoul, Korea, August 30–September 2, 1993), Vol. 1 (World Scientific, River Edge, NJ, 1993), pp. 75–89.Google Scholar
  16. 16.
    M. Inaba, N. Katoh, and H. Imai, “Applications of weighted Voronoi diagrams and randomization to variance- based -clustering: (extended abstract),” in Proc. 10th ACM Symposium on Computational Geometry (Stony Brook, New York, USA, June 6–8, 1994), (ACM, New York, 1994), pp. 332–339.Google Scholar
  17. 17.
    F. de la Vega, M. Karpinski, C. Kenyon, and Y. Rabani, “Polynomial time approximation schemes for metric min-sum clustering,” Electronic Colloquium on Computational Complexity (ECCC), Report No. 25 (2002).Google Scholar
  18. 18.
    F. de la Vega and C. Kenyon, “A randomized approximation scheme for metric max-cut,” J. Comput. Syst. Sci. 63, 531–541 (2001).MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W. H. Freeman and Co., San Francisco, 1979; Mir, Moscow, 1982).MATHGoogle Scholar
  20. 20.
    A. V. Kel’manov and S. M. Romanchenko, “An approximation algorithm for solving a problem of search for a vector subset,” J. Appl. Indust. Math. 6 (1), 90–96 (2012).CrossRefMATHGoogle Scholar
  21. 21.
    A. V. Kel’manov and S. M. Romanchenko, “An FPTAS for a vector subset search problem,” J. Appl. Indust. Math. 8 (3), 329–336 (2014).CrossRefMATHGoogle Scholar
  22. 22.
    A. V. Kel’manov and V. I. Khandeev, “Fully polynomial- time approximation scheme for a special case of a quadratic Euclidean 2-clustering problem,” Comput. Math. Math. Phys. 56 (2), 334–341 (2016).MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    A. V. Kel’manov and A. V. Motkova, “A fully polynomial- time approximation scheme for a special case of a balanced 2-clustering problem,” in Discrete Optimization and Operations Research: Proc. 9th Intern. Conf. DOOR 2016 (Vladivostok, Russia, September 19–23, 2016), Lecture Notes Comp. Sci. 9869, 182–192 (2016).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    N. Wirth, Algorithms + Data Structures = Programs (Prentice Hall, New Jersey, 1976; Mir, Moscow, 1985).MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

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

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