Maximum Diversity Problem with Squared Euclidean Distance

  • Anton V. EremeevEmail author
  • Alexander V. Kel’manov
  • Mikhail Y. Kovalyov
  • Artem V. Pyatkin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11548)


In this paper we consider the following Maximum Diversity Subset problem. Given a set of points in Euclidean space, find a subset of size M maximizing the squared Euclidean distances between the chosen points. We propose an exact dynamic programming algorithm for the case of integer input data. If the dimension of the Euclidean space is bounded by a constant, the algorithm has a pseudo-polynomial time complexity. Using this algorithm, we develop an FPTAS for the special case where the dimension of the Euclidean space is bounded by a constant. We also propose a new proof of strong NP-hardness of the problem in the general case.


Euclidean space Subset of points Given size Maximum variance Strong NP-hardness Integer instance Exact algorithm Fixed space dimension Pseudo-polynomial time 



The study presented in Sect. 4 was supported by the Russian Foundation for Basic Research project 19-01-00308. The study presented in Sect. 6 was supported by the Russian Academy of Science (the Program of basic research), projects 0314-2019-0015, 0314-2019-0019, and in Sect. 3 by the Russian Ministry of Science and Education under the 5-100 Excellence Programme.


  1. 1.
    Aggarwal, H., Imai, N., Katoh, N., Suri, S.: Finding \(k\) points with minimum diameter and related problems. J. Algorithms 12(1), 38–56 (1991)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aringhieri, R.: Composing medical crews with equity and efficiency. Cent. Eur. J. Oper. Res. 17(3), 343–357 (2009). Scholar
  3. 3.
    Cevallos, A., Eisenbrand, F., Morell, S.: Diversity maximization in doubling metrics. In: Proceedings of 29th International Symposium on Algorithms and Computation (ISAAC 2018). LIPIcs, vol. 123, pp. 33:1–33:12. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl (2016).
  4. 4.
    Cevallos, A., Eisenbrand, F., Zenklusen, R.: Max-sum diversity via convex programming. In: 32nd Annual Symposium on Computational Geometry (SoCG). LIPIcs, vol. 51, pp. 26:1–26:14. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl (2016).
  5. 5.
    Edwards, A.W.F., Cavalli-Sforza, L.L.: A method for cluster analysis. Biometrics 21, 362–375 (1965)CrossRefGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and intractability. A guide to the Theory of NP-Completeness. W.H. Freeman and Company, San Francisco (1979)zbMATHGoogle Scholar
  7. 7.
    Ibarra, O., Kim, C.E.: Fast approximation algorithms for the knapsack and sum of subset problems. J. ACM 22, 463–468 (1975)MathSciNetCrossRefGoogle Scholar
  8. 8.
    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). Scholar
  9. 9.
    Kel’manov, A.V., Pyatkin, A.V.: NP-completeness of some problems of choosing a vector subset. J. Appl. Ind. Math. 5(3), 352–357 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kel’manov, A.V., Romanchenko, S.M.: Pseudopolynomial algorithms for certain computationally hard vector subset and cluster analysis problems. Autom. Remote Control 73(2), 349–354 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kel’manov, A.V., Romanchenko, S.M.: An approximation algorithm for solving a problem of search for a vector subset. J. Appl. Ind. Math. 6(1), 90–96 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kel’manov, A.V., Romanchenko, S.M.: An FPTAS for a vector subset search problem. J. Appl. Ind. Math. 8(3), 329–336 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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). Scholar
  14. 14.
    Kuo, C.C., Glover, F., Dhir, K.S.: Analyzing and modeling the maximum diversity problem by zero-one programming. Decis. Sci. 24(6), 1171–1185 (1993)CrossRefGoogle Scholar
  15. 15.
    McConnell, S.: The new battle over immigration. Fortune 117(10), 89–102 (1988)Google Scholar
  16. 16.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, New York (1994)zbMATHGoogle Scholar
  17. 17.
    Porter, W.M., Eawal, K.M., Rachie, K.O., Wien, H.C., Willians, R.C.: Cowpea germplasm catalog No. 1. International Institute of Tropical Agriculture, Ibadan, Nigeria (1975)Google Scholar
  18. 18.
    Shenmaier, V.V.: An approximation scheme for a problem of search for a vector subset. J. Appl. Ind. Math. 6(3), 381–386 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shenmaier, V.V.: Solving some vector subset problems by Voronoi diagrams. J. Appl. Ind. Math. 10(4), 560–566 (2016)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Dostoevsky Omsk State UniversityOmskRussia
  3. 3.Novosibirsk State UniversityNovosibirskRussia
  4. 4.United Institute of Informatics ProblemsMinskBelarus

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