Abstract
We consider a min-max version of the previously studied r-gathering problem with unit-demands. The problem we consider is a metric facility-location problem, in which each open facility must serve at least r customers, and the maximum of all the facility and connection costs should be minimized (rather than their sum). This problem is motivated by scenarios in which r customers are required for a facility to be worth opening, and the costs represent the time until the facility/connection will be available (i.e., we want to have the complete solution ready as soon as possible).
We present a 3-approximation algorithm for this problem, and prove that it cannot be approximated better (assuming P ≠ NP). Next we consider this problem with the additional natural requirement that each customer will be assigned to a nearest open facility, and present a 9-approximation algorithm. We further consider previously introduced special cases and variants, and obtain improved algorithmic and hardness results.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aggarwal, G., Feder, T., Kenthapadi, K., Khuller, S., Panigrahy, R., Thomas, D., Zhu, A.: Achieving anonymity via clustering. In: Proceedings of the 25th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, pp. 153–162 (2006)
Anshelevich, E., Karagiozova, A.: Terminal backup, 3D matching, and covering cubic graphs. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pp. 391–400 (2007)
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.: An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem. In: Proceedings of the 10th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pp. 29–43 (2007)
Demaine, E.D., Hajiaghayi, M., Mahini, H., Sayedi-Roshkhar, A.S., Oveisgharan, S., Zadimoghaddam, M.: Minimizing movement. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 731–740 (2007)
Drezner, Z.: Facility Location. Springer, Heidelberg (1995)
Ergun, F., Kumar, R., Rubinfeld, R.: Fast approximate PCPs. In: Proceedings of the 31st Annual ACM Symposium on Theory of Computing, pp. 41–50 (1999)
Garey, M.R., Johnson, D.S.: Computers and Intractability – A Guide to the Theory of NP-Completeness, p. 221. Freeman publishing, San Francisco (1979)
Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoretical Comput. Sci. 38, 293–306 (1985)
Gudmundsson, J., van Kreveld, M.J., Narasimhan, G.: Region-restricted clustering for geographic data mining. In: Proceedings of the 14th Annual European Symposium on Algorithms, pp. 399–410 (2006)
Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. J. Algorithms 31(1), 228–248 (1999)
Guha, S., Meyerson, A., Munagala, K.: Hierarchical placement and network design problems. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pp. 603–612 (2000)
Karger, D.R., Minkoff, M.: Building steiner trees with incomplete global knowledge. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pp. 613–623 (2000)
Mahdian, M., Ye, Y., Zhang, J.: Approximation algorithm for the soft-capacitated facility location problem. In: Proceedings of the 6th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pp. 129–140 (2003)
Schneier, B.: Applied Cryptography, pp. 71–73. John Wiley and Sons, Chichester (1996)
Sviridenko, M.: An improved approximation algorithm for the metric uncapacitated facility location problem. In: Cook, W.J., Schulz, A.S. (eds.) IPCO 2002. LNCS, vol. 2337, pp. 240–257. Springer, Heidelberg (2002)
Truta, T.M., Bindu, V.: Privacy protection: p-sensitive k-anonymity property. In: Proceedings of the Workshop on Privacy Data Management, in conjunction with the 22nd IEEE International Conference of Data Engineering (ICDE), pp. 94–103 (2006)
Vygen, J.: Approximation algorithms for facility location problems (2005), citeseer.ist.psu.edu/vygen05approximation.html
Zhang, J., Chen, B., Ye, Y.: Multi-exchange local search algorithm for the capacitated facility location problem. In: Bienstock, D., Nemhauser, G.L. (eds.) IPCO 2004. LNCS, vol. 3064, pp. 219–233. Springer, Heidelberg (2004)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Armon, A. (2008). On Min-Max r-Gatherings. In: Kaklamanis, C., Skutella, M. (eds) Approximation and Online Algorithms. WAOA 2007. Lecture Notes in Computer Science, vol 4927. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77918-6_11
Download citation
DOI: https://doi.org/10.1007/978-3-540-77918-6_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-77917-9
Online ISBN: 978-3-540-77918-6
eBook Packages: Computer ScienceComputer Science (R0)