An Approximation Algorithm for Uniform Capacitated k-Median Problem with \(1+\epsilon \) Capacity Violation

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9682)


We study the Capacitated k-Median problem, for which all the known constant factor approximation algorithms violate either the number of facilities or the capacities. While the standard LP-relaxation can only be used for algorithms violating one of the two by a factor of at least two, Li [10, 11] gave algorithms violating the number of facilities by a factor of \(1+\epsilon \) exploring properties of extended relaxations.

In this paper we develop a constant factor approximation algorithm for hard Uniform Capacitated k-Median violating only the capacities by a factor of \(1\,+\,\epsilon \). The algorithm is based on a configuration LP. Unlike in the algorithms violating the number of facilities, we cannot simply open extra few facilities at selected locations. Instead, our algorithm decides about the facility openings in a carefully designed dependent rounding process.


Integral Solution Child Group Fractional Solution Open Facility Root Group 
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.


  1. 1.
    Aardal, K., van den Berg, P.L., Gijswijt, D., Li, S.: Approximation algorithms for hard capacitated \(k\)-facility location problems. Eur. J. Oper. Res. 242(2), 358–368 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    An, H.-C., Singh, M., Svensson, O.: LP-based algorithms for capacitated facility location. In: IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS), pp. 256–265. IEEE (2014)Google Scholar
  3. 3.
    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)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Byrka, J., Fleszar, K., Rybicki, B., Spoerhase, J.: Bi-factor approximation algorithms for hard capacitated \(k\)-median problems. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 722–736. SIAM (2015)Google Scholar
  5. 5.
    Byrka, J., Pensyl, T., Rybicki, B., Srinivasan, A., Trinh, K.: An improved approximation for \(k\)-median, and positive correlation in budgeted optimization. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 737–756. SIAM (2015)Google Scholar
  6. 6.
    Byrka, J., Rybicki, B., Uniyal, S.: An approximation algorithm for uniform capacitated \(k\)-median problem with \(1+\epsilon \) capacity violation. CoRR abs/1511.07494 (2015)Google Scholar
  7. 7.
    Charikar, M., Guha, S., Tardos, É., Shmoys, D.B.: A constant-factor approximation algorithm for the \(k\)-median problem. In: Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pp. 1–10. ACM (1999)Google Scholar
  8. 8.
    Chuzhoy, J., Rabani, Y.: Approximating \(k\)-median with non-uniform capacities. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 952–958. Society for Industrial and Applied Mathematics (2005)Google Scholar
  9. 9.
    Gandhi, R., Khuller, S., Parthasarathy, S., Srinivasan, A.: Dependent rounding and its applications to approximation algorithms. J. ACM 53(3), 324–360 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Li, S.: On uniform capacitated \(k\)-median beyond the natural LP relaxation. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 696–707. SIAM (2015)Google Scholar
  11. 11.
    Li, S.: Approximating capacitated \(k\)-median with \((1+\epsilon )k\) open facilities. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 786–796. SIAM (2016)Google Scholar
  12. 12.
    Li, S., Svensson, O.: Approximating \(k\)-median via pseudo-approximation. In: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, pp. 901–910. ACM (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Jarosław Byrka
    • 1
  • Bartosz Rybicki
    • 1
  • Sumedha Uniyal
    • 2
  1. 1.Institute of Computer ScienceUniversity of WrocławWrocławPoland
  2. 2.IDSIAUniversity of LuganoLuganoSwitzerland

Personalised recommendations