# Fault tolerant K-center problems

## Abstract

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.

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## References

- 1.J. Bar-Ilan, G. Kortsarz, and D. Peleg. How to allocate network centers.
*Journal of Algorithms*, 15:385–415, 1993.Google Scholar - 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.M. Dyer and A. M. Frieze. A simple heuristic for the
*p*-center problem.*Operations Research Letters*, 3:285–288, 1985.Google Scholar - 4.J. Edmonds and D. R. Fulkerson. Bottleneck extrema.
*Journal of Combinatorial Theory*, 8:299–306, 1970.Google Scholar - 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.T. Gonzalez. Clustering to minimize the maximum inter-cluster distance.
*Theoretical Computer Science*, 38:293–306, 1985.Google Scholar - 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.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.W. L. Hsu and G. L. Nemhauser. Easy and hard bottleneck location problems.
*Discrete Applied Mathematics*, 1:209–216, 1979.Google Scholar - 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 ftp.cs.umd.edu/pub/papers/papers/3652/3652.ps.Z.Google Scholar - 11.S. Khuller and Y. J. Sussmann. The capacitated K-Center problem. In
*Proc. of the*4^{th}*Annual European Symposium on Algorithms*, volume 1136 of*LNCS*, pages 152–166, 1996.Google Scholar - 12.S. O. Krumke. On a generalization of the
*p*-center problem.*Information Processing Letters*, 56:67–71, 1995.Google Scholar - 13.J. Plesnik. A heuristic for the
*p*-center problem in graphs.*Discrete Applied Mathematics*, 17:263–268, 1987.Google Scholar - 14.L. Smith. Volunteers' rescue response rates worsen in Pr. William.
*The Washington Post*, April 17, 1996.Google Scholar - 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