Centrality of Trees for Capacitated k-Center

  • Hyung-Chan An
  • Aditya Bhaskara
  • Chandra Chekuri
  • Shalmoli Gupta
  • Vivek Madan
  • Ola Svensson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)


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.


approximation algorithms capacitated network location problems capacitated k-center problem LP-rounding algorithms 


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  1. 1.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bansal, M., Garg, N., Gupta, N.: A 5-approximation for capacitated facility location. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 133–144. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  3. 3.
    Byrka, J.: An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem. In: APPROX-RANDOM, pp. 29–43 (2007)Google Scholar
  4. 4.
    Charikar, M., Guha, S.: Improved combinatorial algorithms for facility location problems. SIAM J. Comput. 34(4), 803–824 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chudak, F.A., Williamson, D.P.: Improved approximation algorithms for capacitated facility location problems. Math. Program. 102(2), 207–222 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Chuzhoy, J., Rabani, Y.: Approximating k-median with non-uniform capacities. In: SODA, pp. 952–958 (2005)Google Scholar
  8. 8.
    Cygan, M., Hajiaghayi, M., Khuller, S.: LP rounding for k-centers with non-uniform hard capacities. In: FOCS, pp. 273–282 (2012)Google Scholar
  9. 9.
    Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci. 38, 293–306 (1985)CrossRefzbMATHGoogle Scholar
  10. 10.
    Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. J. Algorithms 31(1), 228–248 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Hall, P.: On representatives of subsets. Journal of the London Mathematical Society 10, 26–30 (1935)Google Scholar
  12. 12.
    Hochbaum, D.S., Shmoys, D.B.: A best possible heuristic for the k-center problem. Mathematics of Operations Research 10, 180–184 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Jain, K., Mahdian, M., Saberi, A.: A new greedy approach for facility location problems. In: STOC, pp. 731–740 (2002)Google Scholar
  14. 14.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Khuller, S., Sussmann, Y.J.: The capacitated k-center problem. SIAM J. Discrete Math. 13(3), 403–418 (2000)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Korupolu, M.R., Plaxton, C.G., Rajaraman, R.: Analysis of a local search heuristic for facility location problems. J. Algorithms 37(1), 146–188 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Levi, R., Shmoys, D.B., Swamy, C.: LP-based approximation algorithms for capacitated facility location. In: Bienstock, D., Nemhauser, G.L. (eds.) IPCO 2004. LNCS, vol. 3064, pp. 206–218. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Li, S.: A 1.488 approximation algorithm for the uncapacitated facility location problem. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 77–88. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  19. 19.
    Li, S., Svensson, O.: Approximating k-median problem via pseudo-approximation. In: STOC, pp. 901–910 (2013)Google Scholar
  20. 20.
    Pál, M., Tardos, É., Wexler, T.: Facility location with nonuniform hard capacities. In: FOCS, pp. 329–338 (2001)Google Scholar
  21. 21.
    Shmoys, D.B., Tardos, É., Aardal, K.: Approximation algorithms for facility location problems (extended abstract). In: STOC, pp. 265–274 (1997)Google Scholar
  22. 22.
    Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press (2011)Google Scholar
  23. 23.
    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)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Hyung-Chan An
    • 1
  • Aditya Bhaskara
    • 2
  • Chandra Chekuri
    • 3
  • Shalmoli Gupta
    • 3
  • Vivek Madan
    • 3
  • Ola Svensson
    • 1
  1. 1.École Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Google ResearchUSA
  3. 3.University of Illinois at Urbana-ChampaignUSA

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