Geometric Clustering to Minimize the Sum of Cluster Sizes

  • Vittorio Bilò
  • Ioannis Caragiannis
  • Christos Kaklamanis
  • Panagiotis Kanellopoulos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3669)


We study geometric versions of the min-size k -clustering problem, a clustering problem which generalizes clustering to minimize the sum of cluster radii and has important applications. We prove that the problem can be solved in polynomial time when the points to be clustered are located on a line. For Euclidean spaces of higher dimensions, we show that the problem is NP-hard and present polynomial time approximation schemes. The latter result yields an improved approximation algorithm for the related problem of k-clustering to minimize the sum of cluster diameters.


Polynomial Time Cluster Problem Active Line Polynomial Time Approximation Scheme Minimum Vertex Cover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arora, S., Raghavan, P., Rao, S.: Approximation schemes for the Euclidean k-medians and related problems. In: Proc. of the 30th ACM Symposium on Theory of Computing (STOC 1998), pp. 106–113 (1998)Google Scholar
  2. 2.
    Bǎdoiu, M., Har-Peled, S., Indyk, P.: Approximate clustering via core-sets. In: Proc. of the 34th Annual ACM Symposium on Theory of Computing (STOC 2002), pp. 250–257 (2002)Google Scholar
  3. 3.
    Bartal, Y., Charikar, M., Raz, D.: Approximating min-sum k-clustering in metric spaces. In: Proc. of the 33rd Annual ACM Symposium on Theory of computing (STOC 2001), pp.11–20 (2001)Google Scholar
  4. 4.
    Brucker, P.: On the complexity of clustering problems. Optimization and Operations Research, Lecture Notes in Economics and Mathematical Sciences 157, 45–54 (1978)MathSciNetGoogle Scholar
  5. 5.
    Charikar, M., Guha, S., Tardos, E., Shmoys, D.S.: A constant factor approximation algorithm for the k-median problem. Journal of Computer and Systems Sciences 65(1), 129–149 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Charikar, M., Panigrahy, R.: Clustering to minimize the sum of cluster diameters. Journal of Computer and Systems Sciences 68(2), 417–441 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Capoyleas, V., Rote, G., Woeginger, G.J.: Geometric Clusterings. Journal of Algorithms 12(2), 341–356 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Doddi, S.R., Marathe, M.V., Ravi, S.S., Taylor, D.S., Widmayer, P.: Approximation algorithms for clustering to minimize the sum of diameters. Nordic Journal of Computing 7(3), 185–203 (2000)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation schemes for geometric graphs. In: Proc. of the 12th Annual Symposium on Discrete Algorithms (SODA 2001), pp. 671–679 (2001)Google Scholar
  10. 10.
    de la Vega, W.F., Karpinski, M., Kenyon, C., Rabani, Y.: Approximation schemes for clustering problems. In: Proc. of the 35th Annual ACM Symposium on Theory of Computing (STOC 2003), pp. 50–58 (2003)Google Scholar
  11. 11.
    Freund, A., Rawitz, D.: Combinatorial interpretations of dual fitting and primal fitting. In: Solis-Oba, R., Jansen, K. (eds.) WAOA 2003. LNCS, vol. 2909, pp. 137–150. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Hansen, P., Jaumard, B.: Minimum sum of diameters clustering. Journal of Classification 4, 215–226 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and k-median problems using the primal-dual scheme and Lagrangian relaxation. Journal of the ACM 48, 274–296 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lev-Tov, N., Peleg, D.: Polynomial time approximation schemes for base station coverage with minimum total radii. Computer Networks 47, 489–501 (2005)zbMATHCrossRefGoogle Scholar
  15. 15.
    Monma, C.L., Suri, S.: Partitioning points and graphs to minimize the maximum or the sum of diameters. In: Graph Theory, Combinatorics and Applications, pp. 880–912. John Wiley and Sons, Chichester (1991)Google Scholar
  16. 16.
    Ostrovsky, R., Rabani, Y.: Polynomial-time approximation schemes for geometric clustering problems. Journal of the ACM 49(2), 139–156 (2002)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Vittorio Bilò
    • 1
  • Ioannis Caragiannis
    • 2
  • Christos Kaklamanis
    • 2
  • Panagiotis Kanellopoulos
    • 2
  1. 1.Dipartimento di Matematica “Ennio De Giorgi”Università di Lecce, Provinciale Lecce-ArnesanoLecceItaly
  2. 2.Research Academic Computer Technology Institute &, Department of Computer Engineering and InformaticsUniversity of PatrasRioGreece

Personalised recommendations