# Centrality of trees for capacitated \(k\)-center

- 389 Downloads
- 3 Citations

## Abstract

We consider the capacitated \(k\)-center problem. In this problem we are given a finite set of locations in a metric space and each location has an associated non-negative integer capacity. The goal is to choose (open) \(k\) locations (called centers) and assign each location to an open center to minimize the maximum, over all locations, of the distance of the location to its assigned center. The number of locations assigned to a center cannot exceed the center’s capacity. The uncapacitated \(k\)-center problem has a simple tight \(2\)-approximation from the 80’s. In contrast, the first constant factor approximation for the capacitated problem was obtained only recently by Cygan, Hajiaghayi and Khuller who gave an intricate LP-rounding algorithm that achieves an approximation guarantee in the hundreds. In this paper we give a simple algorithm with a clean analysis and prove an approximation guarantee of \(9\). It uses the standard LP relaxation and comes close to settling the integrality gap (after necessary preprocessing), which is narrowed down to either \(7, 8\) or \(9\). The algorithm proceeds by first reducing to special *tree instances*, and then uses our best-possible algorithm to solve such instances. Our concept of tree instances is versatile and applies to natural variants of the capacitated \(k\)-center problem for which we also obtain improved algorithms. Finally, we give evidence to show that more powerful preprocessing could lead to better algorithms, by giving an approximation algorithm that beats the integrality gap for instances where all non-zero capacities are the same.

## Keywords

Approximation algorithms Capacitated network location problems Capacitated \(k\)-center problem LP-rounding algorithms## Mathematics Subject Classification

68W25## Notes

### Acknowledgments

The authors thank the anonymous reviewers of this paper and of its preliminary version [1] for their helpful comments.

## References

- 1.An, H.C., Bhaskara, A., Chekuri, C., Gupta, S., Madan, V., Svensson, O.: Centrality of trees for capacitated \(k\)-center. In: IPCO, pp. 52–63 (2014)Google Scholar
- 2.Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for \(k\)-median and facility location problems. SIAM J. Comput.
**33**(3), 544–562 (2004)MATHMathSciNetCrossRefGoogle Scholar - 3.Bansal, M., Garg, N., Gupta, N.: A 5-approximation for capacitated facility location. In: ESA, pp. 133–144 (2012)Google Scholar
- 4.Bar-Ilan, J., Kortsarz, G., Peleg, D.: How to allocate network centers. J. Algorithm.
**15**(3), 385–415 (1993)MATHMathSciNetCrossRefGoogle Scholar - 5.Byrka, J.: An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem. In: Approx-Random, pp. 29–43 (2007)Google Scholar
- 6.Charikar, M., Guha, S.: Improved combinatorial algorithms for facility location problems. SIAM J. Comput.
**34**(4), 803–824 (2005)MATHMathSciNetCrossRefGoogle Scholar - 7.Charikar, M., Guha, S., Tardos, É., Shmoys, D.B.: A constant-factor approximation algorithm for the k-median problem. J. Comput. Syst. Sci.
**65**(1), 129–149 (2002)MATHMathSciNetCrossRefGoogle Scholar - 8.Chudak, F.A., Williamson, D.P.: Improved approximation algorithms for capacitated facility location problems. Math. Program.
**102**(2), 207–222 (2005)MATHMathSciNetCrossRefGoogle Scholar - 9.Chuzhoy, J., Rabani, Y.: Approximating \(k\)-median with non-uniform capacities. In: SODA, pp. 952–958 (2005)Google Scholar
- 10.Cygan, M., Hajiaghayi, M., Khuller, S.: LP rounding for k-centers with non-uniform hard capacities. In: FOCS, pp. 273–282 (2012)Google Scholar
- 11.Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci.
**38**, 293–306 (1985)MATHCrossRefGoogle Scholar - 12.Guha, S., Khuller, S.: Greedy strikes back: improved facility location algorithms. J. Algorithm.
**31**(1), 228–248 (1999)MATHMathSciNetCrossRefGoogle Scholar - 13.Hall, P.: On representatives of subsets. J. Lond. Math. Soc.
**10**, 26–30 (1935)Google Scholar - 14.Hochbaum, D.S., Shmoys, D.B.: A best possible heuristic for the \(k\)-center problem. Math. Oper. Res.
**10**, 180–184 (1985)MATHMathSciNetCrossRefGoogle Scholar - 15.Jain, K., Mahdian, M., Saberi, A.: A new greedy approach for facility location problems. In: STOC, pp. 731–740 (2002)Google Scholar
- 16.Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and \(k\)-median problems using the primal-dual schema and lagrangian relaxation. J. ACM
**48**(2), 274–296 (2001)MATHMathSciNetCrossRefGoogle Scholar - 17.Khuller, S., Sussmann, Y.J.: The capacitated \(k\)-center problem. SIAM J. Discret. Math.
**13**(3), 403–418 (2000)MathSciNetCrossRefGoogle Scholar - 18.Korupolu, M.R., Plaxton, C.G., Rajaraman, R.: Analysis of a local search heuristic for facility location problems. J. Algorithm.
**37**(1), 146–188 (2000)MATHMathSciNetCrossRefGoogle Scholar - 19.Levi, R., Shmoys, D.B., Swamy, C.: LP-based approximation algorithms for capacitated facility location. In: IPCO, pp. 206–218 (2004)Google Scholar
- 20.Li, S.: A 1.488 approximation algorithm for the uncapacitated facility location problem. In: ICALP (2), pp. 77–88 (2011)Google Scholar
- 21.Li, S., Svensson, O.: Approximating \(k\)-median problem via pseudo-approximation. In: STOC, pp. 901–910 (2013)Google Scholar
- 22.Pál, M., Tardos, É., Wexler, T.: Facility location with nonuniform hard capacities. In: FOCS, pp. 329–338 (2001)Google Scholar
- 23.Shmoys, D.B., Tardos, É., Aardal, K.: Approximation algorithms for facility location problems (extended abstract). In: STOC, pp. 265–274 (1997)Google Scholar
- 24.Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, New York (2011)MATHCrossRefGoogle Scholar
- 25.Zhang, J., Chen, B., Ye, Y.: A multiexchange local search algorithm for the capacitated facility location problem. Math. Oper. Res.
**30**(2), 389–403 (2005)MATHMathSciNetCrossRefGoogle Scholar