Abstract
Consideration was given to the problem of seeking a family of disjoint subsets of given cardinalities in a finite set of Euclidean space vectors. The minimal sum of the squared distances from the subset elements to their centers was used as the search criterion. The subset centers are optimizable variables defined as the mean values over the elements of the required subsets. The problem was shown to be NP-hard in the strong sense. To solve it, a 2-approximate algorithm was proposed which is polynomial for a fixed number of the desired subsets.
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References
Hastie, T., Tibshirani, R., and Friedman, J., The Elements of Statistical Learning: Data Mining, Inference, and Prediction, New York: Springer-Verlag, 2009.
Anil, K. and Jain, K., Data Clustering: 50 Years Beyond k-Means, Pattern Recogn. Lett., 2010, vol. 31, pp. 651–666.
Edwards, A.W.F. and Cavalli-Sforza, L.L., A Method for Cluster Analysis, Biometrics, 1965, vol. 21, pp. 362–375.
MacQueen, J.B., Some Methods for Classification and Analysis of Multivariate Observations, in Proc. 5th Berkeley Symp. Math. Statist. Probability, 1967, vol. 1, pp. 281–297.
Rao, M., Cluster Analysis and Mathematical Programming, J. Am. Stat. Assoc., 1971, vol. 66, pp. 622–626.
Aloise, D. and Hansen, P., On the Complexity of Minimum Sum-of-Squares Clustering, Les Cahiers du GERAD, 2007, vol. G-2007-50.
Aloise, D., Deshpande, A., Hansen, P., et al., NP-Hardness of Euclidean Sum-of-Squares Clustering, Les Cahiers du GERAD, 2008, vol. G-2008-33.
Mahajan, M., Nimbhorkar, P., and Varadarajan, K., The Planar k-means Problem is NP-Hard, Lect. Notes Comput. Sci., 2009, vol. 5431, pp. 284–285.
Dolgushev, A.V. and Kel’manov, A.V., On the Algorithmic Complexity of a Problem in Cluster Analysis, J. Appl. Ind. Math., 2011, vol. 5, no. 2, pp. 191–194.
Kel’manov, A.V. and Pyatkin, A.V., NP-Completeness of Some Problems of Choosing a Vector Subset, J. Appl. Ind. Math., 2011, vol. 5, no. 3, pp. 352–357.
Garey, M.L. and Johnson, D.S., Computers and Intractability: A Guide to the Theory of NP-Completeness, San Francisco: Freeman, 1979. Translated under the title Vychislitel’nye machiny i trudnoreshaemye zadachi, Moscow: Mir, 1982.
Kel’manov, A.V. and Romanchenko, S.M., An Approximation Algorithm for Solving a Problem of Search for a Vector Subset, J. Appl. Ind. Math., 2012, vol. 6, no. 1, pp. 90–96.
Kel’manov, A.V. and Romanchenko, S.M., Pseudopolynomial Algorithms for Certain Computationally Hard Vector Subset and Cluster Analysis Problems, Autom. Remote Control, 2012, vol. 73, no. 2, pp. 349–354.
Shenmaier, V.V., An Approximation Scheme for a Problem of Search for a Vector Subset, J. Appl. Ind. Math., 2012, vol. 6, no. 3, pp. 381–386.
Kel’manov, A.V. and Romanchenko, S.M., A Fully Polynomial Time Approximation Scheme for a Problem of Choosing a Vector Subset, in Proc. IV Int. Conf. “Optimization and Applications” (OPTIMA-2013), Petrovac, Montenegro, 2013, p. 89.
Papadimitrou, Ch. and Steiglitz, K., Combinatorial Optimization: Algorithms and Complexity, Englewood Cliffs: Prentice Hall, 1982. Translated under the title Kombinatornaya optimizatsiya: Algoritmy i slozhnost’, Moscow: Mir, 1985.
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Original Russian Text © A.E. Galashov, A.V. Kel’manov, 2014, published in Avtomatika i Telemekhanika, 2014, No. 4, pp. 5–19.
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Galashov, A.E., Kel’manov, A.V. A 2-approximate algorithm to solve one problem of the family of disjoint vector subsets. Autom Remote Control 75, 595–606 (2014). https://doi.org/10.1134/S0005117914040018
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DOI: https://doi.org/10.1134/S0005117914040018