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

  • Alexander Kel’manov
  • Vladimir Khandeev
  • Anna Panasenko
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

Keywords

Euclidean space 2-clustering Largest subset NP-hardness Exact algorithm Pseudopolynomial-time solvability 

Notes

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.

References

  1. 1.
    MacQueen, J.B.: Some methods for classification and analysis of multivariate observations. In: Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, vol. 1, pp. 281–297 (1967)Google Scholar
  2. 2.
    Rao, M.: Cluster analysis and mathematical programming. J. Am. Stat. Assoc. 66, 622–626 (1971)CrossRefGoogle Scholar
  3. 3.
    Hansen, P., Jaumard, B., Mladenovich, N.: Minimum sum of squares clustering in a low dimensional space. J. Classif. 15, 37–55 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hansen, P., Jaumard, B.: Cluster analysis and mathematical programming. Math. Program. 79, 191–215 (1997)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Fisher, R.A.: Statistical Methods and Scientific Inference. Hafner, New York (1956)zbMATHGoogle Scholar
  6. 6.
    Jain, A.K.: Data clustering: 50 years beyond \(k\)-means. Pattern Recogn. Lett. 31(8), 651–666 (2010)CrossRefGoogle Scholar
  7. 7.
    Aloise, D., Deshpande, A., Hansen, P., Popat, P.: NP-hardness of Euclidean sum-of-squares clustering. Mach. Learn. 75(2), 245–248 (2009)CrossRefGoogle Scholar
  8. 8.
    Drineas, P., Frieze, A., Kannan, R., Vempala, S., Vinay, V.: Clustering large graphs via the singular value decomposition. Mach. Learn. 56, 9–33 (2004)CrossRefGoogle Scholar
  9. 9.
    Dolgushev, A.V., Kel’manov, A.V.: On the algorithmic complexity of a problem in cluster analysis. J. Appl. Ind. Math. 5(2), 191–194 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mahajan, M., Nimbhorkar, P., Varadarajan, K.: The planar k-means problem is NP-hard. Theoret. Comput. Sci. 442, 13–21 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kel’manov, A.V., Pyatkin, A.V.: On the complexity of a search for a subset of “similar” vectors. Doklady Math. 78(1), 574–575 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kel’manov, A.V., Pyatkin, A.V.: On a version of the problem of choosing a vector subset. J. Appl. Ind. Math. 3(4), 447–455 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Brucker, P.: On the complexity of clustering problems. In: Henn, R., Korte, B., Oettli, W. (eds.) Optimization and Operations Research. LNEMS, vol. 157, pp. 45–54. Springer, Heidelberg (1978).  https://doi.org/10.1007/978-3-642-95322-4_5CrossRefGoogle Scholar
  14. 14.
    Bern, M., Eppstein, D.: Approximation algorithms for geometric problems. In: Approximation Algorithms for NP-Hard Problems, pp. 296–345. PWS Publishing, Boston (1997)Google Scholar
  15. 15.
    Indyk, P.: A sublinear time approximation scheme for clustering in metric space. In: Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 154–159 (1999)Google Scholar
  16. 16.
    de la Vega, F., Kenyon, C.: A randomized approximation scheme for metric max-cut. J. Comput. Syst. Sci. 63, 531–541 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    de la Vega, F., Karpinski, M., Kenyon, C., Rabani, Y.: Polynomial time approximation schemes for metric min-sum clustering. Electronic Colloquium on Computational Complexity (ECCC). Report No. 25 (2002)Google Scholar
  18. 18.
    Hasegawa, S., Imai, H., Inaba, M., Katoh, N., Nakano, J.: Efficient algorithms for variance-based \(k\)-clustering. In: Proceedings of the 1st Pacific Conference on Computer Graphics and Applications, Pacific Graphics 1993, Seoul, Korea, vol. 1. pp. 75–89. World Scientific, River Edge (1993)Google Scholar
  19. 19.
    Inaba, M., Katoh, N., Imai, H.: Applications of weighted Voronoi diagrams and randomization to variance-based \(k\)-clustering: (extended abstract), 6–8 June 1994, Stony Brook, NY, USA, pp. 332–339. ACM, New York (1994)Google Scholar
  20. 20.
    Sahni, S., Gonzalez, T.: P-complete approximation problems. J. ACM 23, 555–566 (1976)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ageev, A.A., Kel’manov, A.V., Pyatkin, A.V.: NP-hardness of the Euclidean maxcut problem. Doklady Math. 89(3), 343–345 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ageev, A.A., Kel’manov, A.V., Pyatkin, A.V.: Complexity of the weighted max-cut in Euclidean space. J. Appl. Ind. Math. 8(4), 453–457 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kel’manov, A.V., Pyatkin, A.V.: NP-hardness of some quadratic Euclidean 2-clustering problems. Doklady Math. 92(2), 634–637 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kel’manov, A.V., Pyatkin, A.V.: On the complexity of some quadratic Euclidean 2-clustering problems. Comput. Math. Math. Phys. 56(3), 491–497 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Bishop, C.M.: Pattern Recognition and Machine Learning. Springer Science+Business Media, LLC, New York (2006)zbMATHGoogle Scholar
  26. 26.
    James, G., Witten, D., Hastie, T., Tibshirani, R.: An Introduction to Statistical Learning. Springer Science+Business Media, LLC, New York (2013).  https://doi.org/10.1007/978-1-4614-7138-7CrossRefzbMATHGoogle Scholar
  27. 27.
    Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning, 2nd edn. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-0-387-84858-7CrossRefzbMATHGoogle Scholar
  28. 28.
    Aggarwal, C.C.: Data Mining: The Textbook. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-319-14142-8CrossRefzbMATHGoogle Scholar
  29. 29.
    Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. Adaptive Computation and Machine Learning Series. The MIT Press, Cambridge (2017)Google Scholar
  30. 30.
    Shirkhorshidi, A.S., Aghabozorgi, S., Wah, T.Y., Herawan, T.: Big data clustering: a review. In: Murgante, B., et al. (eds.) ICCSA 2014. LNCS, vol. 8583, pp. 707–720. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-09156-3_49CrossRefGoogle Scholar
  31. 31.
    Pach, J., Agarwal, P.K.: Combinatorial Geometry. Wiley, New York (1995)CrossRefGoogle Scholar
  32. 32.
    Kel’manov, A.V., Khandeev, V.I.: A 2-approximation polynomial algorithm for a clustering problem. J. Appl. Ind. Math. 7(4), 515–521 (2013)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Gimadi, E.Kh., Kel’manov, A.V., Kel’manova, M.A., Khamidullin, S.A.: A posteriori detection of a quasi periodic fragment in numerical sequences with given number of recurrences. Siberian J. Ind. Math. 9(1(25)), 55–74 (2006). (in Russian)Google Scholar
  34. 34.
    Gimadi, E.K., Kel’manov, A.V., Kel’manova, M.A., Khamidullin, S.A.: A posteriori detecting a quasiperiodic fragment in a numerical sequence. Pattern Recogn. Image Anal. 18(1), 30–42 (2008)CrossRefGoogle Scholar
  35. 35.
    Baburin, A.E., Gimadi, E.K., Glebov, N.I., Pyatkin, A.V.: The problem of finding a subset of vectors with the maximum total weight. J. Appl. Ind. Math. 2(1), 32–38 (2008)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Dolgushev, A.V., Kel’manov, A.V.: An approximation algorithm for solving a problem of cluster analysis. J. Appl. Ind. Math. 5(4), 551–558 (2011)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Dolgushev, A.V., Kel’manov, A.V., Shenmaier, V.V.: Polynomial-time approximation scheme for a problem of partitioning a finite set into two clusters. Proc. Steklov Inst. Math. 295(Suppl. 1), 47–56 (2016)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Kel’manov, A.V., Khandeev, V.I.: A randomized algorithm for two-cluster partition of a set of vectors. Comput. Math. Math. Phys. 55(2), 330–339 (2015)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Gimadi, E.K., Pyatkin, A.V., Rykov, I.A.: On polynomial solvability of some problems of a vector subset choice in a Euclidean space of fixed dimension. J. Appl. Ind. Math. 4(1), 48–53 (2010)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Shenmaier, V.V.: Solving some vector subset problems by Voronoi diagrams. J. Appl. Ind. Math. 10(4), 560–566 (2016)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Kel’manov, A.V., Khandeev, V.I.: An exact pseudopolynomial algorithm for a problem of the two-cluster partitioning of a set of vectors. J. Appl. Ind. Math. 9(4), 497–502 (2015)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Kel’manov, A.V., Khandeev, V.I.: Fully polynomial-time approximation scheme for a special case of a quadratic euclidean 2-clustering problem. J. Appl. Ind. Math. 56(2), 334–341 (2016)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Kel’manov, A.V., Motkova, A.V.: Polynomial-time approximation algorithm for the problem of cardinality-weighted variance-based 2-clustering with a given center. Comput. Math. Math. Phys. 58(1), 130–136 (2018)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Kel’manov, A.V., Motkova, A.V.: Exact pseudopolynomial algorithms for a balanced 2-clustering problem. J. Appl. Ind. Math. 10(3), 349–355 (2016)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Kel’manov, A., Motkova, A.: A fully polynomial-time approximation scheme for a special case of a balanced 2-clustering problem. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 182–192. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-44914-2_15CrossRefGoogle Scholar
  46. 46.
    Kel’manov, A., Motkova, A., Shenmaier, V.: An approximation scheme for a weighted two-cluster partition problem. In: van der Aalst, W.M.P., et al. (eds.) AIST 2017. LNCS, vol. 10716, pp. 323–333. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-73013-4_30CrossRefGoogle Scholar
  47. 47.
    Kel’manov, A., Khandeev, V., Panasenko, A.: Randomized algorithms for some clustering problems. In: Eremeev, A., Khachay, M., Kochetov, Y., Pardalos, P. (eds.) OPTA 2018. CCIS, vol. 871, pp. 109–119. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-93800-4_9CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Alexander Kel’manov
    • 1
    • 2
  • Vladimir Khandeev
    • 1
    • 2
  • Anna Panasenko
    • 1
    • 2
  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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