Fault tolerant K-center problems

Extended abstract
  • Samir Khuller
  • Robert Pless
  • Yoram J. Sussmann
Regular Presentations
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1203)


The basic K-center problem is a fundamental facility location problem, where we are asked to locate K facilities in a graph, and to assign vertices to facilities, so as to minimize the maximum distance from a vertex to the facility to which it is assigned. This problem is known to be NP-hard, and several optimal approximation algorithms that achieve a factor of 2 have been developed for it.

We focus our attention on a generalization of this problem, where each vertex is required to have a set of α (α≤K) centers close to it. In particular, we study two different versions of this problem. In the first version, each vertex is required to have at least α centers close to it. In the second version, each vertex that does not have a center placed on it is required to have at least α centers close to it. For both these versions we are able to provide polynomial time approximation algorithms that achieve constant approximation factors for any α. For the first version we give an algorithm that achieves an approximation factor of 3 for any α, and achieves an approximation factor of 2 for α<4. For the second version, we provide algorithms with approximation factors of 2 for any α. The best possible approximation factor for even the basic K-center problem is 2. In addition, we give a polynomial time approximation algorithm for a generalization of the K-supplier problem where a subset of at most K supplier nodes must be selected as centers so that every demand node has at least α centers close to it. We also provide polynomial time approximation algorithms for all the above problems for generalizations when cost and weight functions are defined on the set of vertices.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Bar-Ilan, G. Kortsarz, and D. Peleg. How to allocate network centers. Journal of Algorithms, 15:385–415, 1993.Google Scholar
  2. 2.
    S. Chaudhuri, N. Garg, and R. Ravi. Best possible approximation algorithms for generalized k-Center problems. Technical Report MPI-I-96-1-021, Max-Planck-Institut für Informatik, 66123 Saarbrücken, Germany, 1996.Google Scholar
  3. 3.
    M. Dyer and A. M. Frieze. A simple heuristic for the p-center problem. Operations Research Letters, 3:285–288, 1985.Google Scholar
  4. 4.
    J. Edmonds and D. R. Fulkerson. Bottleneck extrema. Journal of Combinatorial Theory, 8:299–306, 1970.Google Scholar
  5. 5.
    M. R. Garey and D. S. Johnson. Computers and Intractibility: A guide to the theory of NP-completeness. Freeman, San Francisco, 1978.Google Scholar
  6. 6.
    T. Gonzalez. Clustering to minimize the maximum inter-cluster distance. Theoretical Computer Science, 38:293–306, 1985.Google Scholar
  7. 7.
    D. Hochbaum and D. B. Shmoys. A best possible heuristic for the k-center problem. Mathematics of Operations Research, 10:180–184, 1985.Google Scholar
  8. 8.
    D. Hochbaum and D. B. Shmoys. A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM, 33(3):533–550, 1986.Google Scholar
  9. 9.
    W. L. Hsu and G. L. Nemhauser. Easy and hard bottleneck location problems. Discrete Applied Mathematics, 1:209–216, 1979.Google Scholar
  10. 10.
    S. Khuller, R. Pless, and Y. J. Sussmann. Fault tolerant K-Center problems. Technical Report CS-TR-3652, University of Maryland, College Park, 1996. Available by ftp at Scholar
  11. 11.
    S. Khuller and Y. J. Sussmann. The capacitated K-Center problem. In Proc. of the 4th Annual European Symposium on Algorithms, volume 1136 of LNCS, pages 152–166, 1996.Google Scholar
  12. 12.
    S. O. Krumke. On a generalization of the p-center problem. Information Processing Letters, 56:67–71, 1995.Google Scholar
  13. 13.
    J. Plesnik. A heuristic for the p-center problem in graphs. Discrete Applied Mathematics, 17:263–268, 1987.Google Scholar
  14. 14.
    L. Smith. Volunteers' rescue response rates worsen in Pr. William. The Washington Post, April 17, 1996.Google Scholar
  15. 15.
    Q. Wang and K. H. Cheng. A heuristic algorithm for the k-center problem with cost and usage weights. Technical Report UH-CS-90-15, University of Houston, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Samir Khuller
    • 1
  • Robert Pless
    • 2
  • Yoram J. Sussmann
    • 2
  1. 1.Dept. of Computer Science and UMIACSUniversity of MarylandCollege Park
  2. 2.Dept. of Computer ScienceUniversity of MarylandCollege Park

Personalised recommendations