Abstract
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.
B. Rybicki—Research supported by NCN 2012/07/N/ST6/03068 grant.
S. Uniyal—Partially supported by the ERC StG project NEWNET No. 279352.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
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)
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)
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)
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)
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)
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)
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)
Gandhi, R., Khuller, S., Parthasarathy, S., Srinivasan, A.: Dependent rounding and its applications to approximation algorithms. J. ACM 53(3), 324–360 (2006)
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)
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)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Byrka, J., Rybicki, B., Uniyal, S. (2016). An Approximation Algorithm for Uniform Capacitated k-Median Problem with \(1+\epsilon \) Capacity Violation. In: Louveaux, Q., Skutella, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 2016. Lecture Notes in Computer Science(), vol 9682. Springer, Cham. https://doi.org/10.1007/978-3-319-33461-5_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-33461-5_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-33460-8
Online ISBN: 978-3-319-33461-5
eBook Packages: Computer ScienceComputer Science (R0)