Abstract
A systematic technique to bound factor-revealing linear programs is presented. We show how to derive a family of upper bound factor-revealing programs (UPFRP), and that each such program can be solved by a computer to bound the approximation factor. Obtaining an UPFRP is straightforward, and can be used as an alternative to analytical proofs, that are usually very long and tedious. We apply this technique to the Metric Facility Location Problem (MFLP) and to a generalization where the distance function is a squared metric. We call this generalization the Squared Metric Facility Location Problem (SMFLP) and prove that there is no approximation factor better than 2.04, assuming \(\mbox{\rm P} \neq\mbox{\rm NP} \). Then, we analyze the best known algorithms for the MFLP based on primal-dual and LP-rounding techniques when they are applied to the SMFLP. We prove very tight bounds for these algorithms, and show that the LP-rounding algorithm achieves a ratio of 2.04, and therefore has the best factor for the SMFLP. We use UPFRPs in the dual-fitting analysis of the primal-dual algorithms for both the SMFLP and the MFLP, improving some of the previous analysis for the MFLP.
This research was partially supported by CNPq (grant numbers 306860/2010-4, 473867/2010-9, and 309657/2009-1) and FAPESP (grant number 2010/20710-4).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Arora, S., Raghavan, P., Rao, S.: Approximation schemes for Euclidean k-medians and related problems. In: Proc. 30th Annual ACM Symp. on Theory of Computing, pp. 106–113. ACM, New York (1998)
Byrka, J., Aardal, K.: An Optimal Bifactor Approximation Algorithm for the Metric Uncapacitated Facility Location Problem. SIAM J. on Comp. 39(6), 2212–2231 (2010)
Byrka, J., Ghodsi, M., Srinivasan, A.: LP-rounding algorithms for facility-location problems (2010), http://arxiv.org/abs/1007.3611
Chudak, F.A., Shmoys, D.B.: Improved Approximation Algorithms for the Uncapacitated Facility Location Problem. SIAM J. on Comp. 33(1), 1–25 (2003)
Fernandes, C.G., Meira, L.A.A., Miyazawa, F.K., Pedrosa, L.L.C.: Squared Metric Facility Location Problem (2012), http://arxiv.org/abs/1111.1672
Guha, S., Khuller, S.: Greedy Strikes Back: Improved Facility Location Algorithms. Journal of Algorithms 31(1), 228–248 (1999)
Hochbaum, D.S.: Heuristics for the fixed cost median problem. Mathematical Programming 22, 148–162 (1982)
Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.V.: Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. Journal of the ACM 50(6), 795–824 (2003)
Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and k-Median problems using the primal-dual schema and Lagrangian relaxation. Journal of the ACM 48(2), 274–296 (2001)
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)
Mahdian, M., Yan, Q.: Online bipartite matching with random arrivals: an approach based on strongly factor-revealing LPs. In: Proc. 43rd Annual ACM Symp. on Theory of Computing, pp. 597–606. ACM, New York (2011)
Mahdian, M., Ye, Y., Zhang, J.: Approximation Algorithms for Metric Facility Location Problems. SIAM J. on Comp. 36(2), 411–432 (2006)
Shmoys, D.B., Tardos, E., Aardal, K.: Approximation algorithms for facility location problems. In: Proc. 29th Annual ACM Symp. on Theory of Computing, pp. 265–274. ACM, New York (1997)
Vygen, J.: Approximation algorithms for facility location problems (Lecture Notes). Tech. Rep. 05950-OR, Research Institute for Discrete Mathematics, University of Bonn (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fernandes, C.G., Meira, L.A.A., Miyazawa, F.K., Pedrosa, L.L.C. (2012). A Systematic Approach to Bound Factor Revealing LPs and Its Application to the Metric and Squared Metric Facility Location Problems. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2012 2012. Lecture Notes in Computer Science, vol 7408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32512-0_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-32512-0_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32511-3
Online ISBN: 978-3-642-32512-0
eBook Packages: Computer ScienceComputer Science (R0)