Exact Algorithms for the Max-Min Dispersion Problem

  • Toshihiro Akagi
  • Tetsuya Araki
  • Takashi Horiyama
  • Shin-ichi NakanoEmail author
  • Yoshio Okamoto
  • Yota Otachi
  • Toshiki Saitoh
  • Ryuhei Uehara
  • Takeaki Uno
  • Kunihiro Wasa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10823)


Given a set P of n elements, and a function d that assigns a non-negative real number d(pq) for each pair of elements \(p,q \in P\), we want to find a subset \(S\subseteq P\) with \(|S|=k\) such that \(\mathop {\mathrm {cost}}(S)= \min \{ d(p,q) \mid p,q \in S\}\) is maximized. This is the max-min k-dispersion problem. In this paper, exact algorithms for the max-min k-dispersion problem are studied. We first show the max-min k-dispersion problem can be solved in \(O(n^{\omega k/3} \log n)\) time. Then, we show two special cases in which we can solve the problem quickly. Namely, we study the cases where a set of n points lie on a line and where a set of n points lie on a circle (and the distance is measured by the shortest arc length on the circle). We obtain O(n)-time algorithms after sorting.


Dispersion problem Algorithm 


  1. 1.
    Agarwal, P., Sharir, M.: Efficient algorithms for geometric optimization. Comput. Surv. 30, 412–458 (1998)CrossRefGoogle Scholar
  2. 2.
    Akagi, T., Nakano, S.: Dispersion on the line, IPSJ SIG Technical reports, 2016-AL-158-3 (2016)Google Scholar
  3. 3.
    Baur, C., Fekete, S.P.: Approximation of geometric dispersion problems. In: Jansen, K., Rolim, J. (eds.) APPROX 1998. LNCS, vol. 1444, pp. 63–75. Springer, Heidelberg (1998). CrossRefGoogle Scholar
  4. 4.
    Birnbaum, B., Goldman, K.J.: An improved analysis for a greedy remote-clique algorithm using factor-revealing LPs. Algorithmica 50, 42–59 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cevallos, A., Eisenbrand, F., Zenklusen, R.: Max-sum diversity via convex programming. In: Proceedings of SoCG 2016, pp. 26:1–26:14 (2016)Google Scholar
  6. 6.
    Cevallos, A., Eisenbrand, F., Zenklusen, R.: Local search for max-sum diversification. In: Proceedings of SODA 2017, pp. 130–142 (2017)Google Scholar
  7. 7.
    Chandra, B., Halldorsson, M.M.: Approximation algorithms for dispersion problems. J. Algorithms 38, 438–465 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Drezner, Z.: Facility Location: A Survey of Applications and Methods. Springer, New York (1995)CrossRefGoogle Scholar
  9. 9.
    Drezner, Z., Hamacher, H.W.: Facility Location: Applications and Theory. Springer, Heidelberg (2004)zbMATHGoogle Scholar
  10. 10.
    Erkut, E.: The discrete \(p\)-dispersion problem. Eur. J. Oper. Res. 46, 48–60 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fekete, S.P., Meijer, H.: Maximum dispersion and geometric maximum weight cliques. Algorithmica 38, 501–511 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006). zbMATHGoogle Scholar
  13. 13.
    Frederickson, G.: Optimal algorithms for tree partitioning. In: Proceedings of SODA 1991, pp. 168–177 (1991)Google Scholar
  14. 14.
    Hassin, R., Rubinstein, S., Tamir, A.: Approximation algorithms for maximum dispersion. Oper. Res. Lett. 21, 133–137 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kuby, M.J.: Programming models for facility dispersion: the \(p\)-dispersion and maxisum dispersion problems. Geogr. Anal. 19, 315–329 (1987)CrossRefGoogle Scholar
  16. 16.
    Lei, T.L., Church, R.L.: On the unified dispersion problem: efficient formulations and exact algorithms. Eur. J. Oper. Res. 241, 622–630 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Le Gall, F.: Powers of tensors and fast matrix multiplication. In: Proceedings of ISSAC 2014, pp. 296–303 (2014)Google Scholar
  18. 18.
    Marx, D., Pilipczuk, M.: Optimal parameterized algorithms for planar facility location problems using Voronoi diagrams. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 865–877. Springer, Heidelberg (2015). CrossRefGoogle Scholar
  19. 19.
    Marx, D., Sidiropoulos, A.: The limited blessing of low dimensionality: when \(1-1/d\) is the best possible exponent for \(d\)-dimensional geometric problems. In: Proceedings of SoCG 2014, pp. 67–76 (2014)Google Scholar
  20. 20.
    Nešetřil, J., Poljak, S.: On the complexity of the subgraph problem. Commentationes Mathematicae Universitatic Corolinae 26, 415–419 (1985)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Ravi, S.S., Rosenkrantz, D.J., Tayi, G.K.: Heuristic and special case algorithms for dispersion problems. Oper. Res. 42, 299–310 (1994)CrossRefzbMATHGoogle Scholar
  22. 22.
    Shier, D.R.: A min-max theorem for \(p\)-center problems on a tree. Transp. Sci. 11, 243–252 (1977)CrossRefGoogle Scholar
  23. 23.
    Sydow, M.: Approximation guarantees for max sum and max min facility dispersion with parameterised triangle inequality and applications in result diversification. Math. Appl. 42, 241–257 (2014)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Tsai, K.-H., Wang, D.-W.: Optimal algorithms for circle partitioning. In: Jiang, T., Lee, D.T. (eds.) COCOON 1997. LNCS, vol. 1276, pp. 304–310. Springer, Heidelberg (1997). CrossRefGoogle Scholar
  25. 25.
    Wang, D.W., Kuo, Y.-S.: A study on two geometric location problems. Inf. Process. Lett. 28, 281–286 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Toshihiro Akagi
    • 1
  • Tetsuya Araki
    • 2
  • Takashi Horiyama
    • 3
  • Shin-ichi Nakano
    • 1
    Email author
  • Yoshio Okamoto
    • 4
    • 5
  • Yota Otachi
    • 6
  • Toshiki Saitoh
    • 7
  • Ryuhei Uehara
    • 8
  • Takeaki Uno
    • 9
  • Kunihiro Wasa
    • 9
  1. 1.Gunma UniversityKiryuJapan
  2. 2.Tokyo Metropolitan UniversityHachiojiJapan
  3. 3.Saitama UniversitySaitamaJapan
  4. 4.The University of Electro-CommunicationsChofuJapan
  5. 5.RIKEN Center for Advanced Intelligence ProjectTokyoJapan
  6. 6.Kumamoto UnivesityKumamotoJapan
  7. 7.Kyushu Institute of TechnologyKitakyushuJapan
  8. 8.JAISTNomiJapan
  9. 9.National Institute of InformaticsTokyoJapan

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