Abstract
In the classical facility location problem we are given a set of facilities, with associated opening costs, and a set of clients. The goal is to open a subset of facilities, and to connect each client to the closest open facility, so that the total connection and opening cost is minimized. In some applications, however, open facilities need to be connected via an infrastructure. Furthermore, connecting two facilities among them is typically more expensive than connecting a client to a facility (for a given path length). This scenario motivated the study of the connected facility location problem (CFL). Here we are also given a parameter M ≥ 1. A feasible solution consists of a subset of open facilities and a Steiner tree connecting them. The cost of the solution is now the opening cost, plus the connection cost, plus M times the cost of the Steiner tree.
In this paper we investigate the approximability of CFL and related problems. More precisely, we achieve the following results:
-
We present a new, simple 3.19 approximation algorithm for CFL. The previous best approximation factor is 3.92 [Eisenbrand, Grandoni, Rothvoß, Schäfer-’10].
-
We show that SROB, i.e. the special case of CFL where all opening costs are 0, is hard to approximate within 1.28. The previous best lower bound for SROB is 1.01, and derives trivially from Steiner tree inapproximability [Chlebík, Chlebíková-’08]. The same inapproximability result extends to other well-studied problems, such as virtual private network and single-sink buy-at-bulk.
-
We introduce and study a natural multi-commodity generalization MCFL of CFL. In MCFL we are given source-sink pairs (rather than clients) that we wish to connect. A feasible solution consists of a subset of open facilities, and a forest (rather than a tree) spanning them. Source-sink connection paths can use several trees in the forest, but must enter and leave each tree at open facilities. We present the first constant approximation for MCFL.
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
Agrawal, A., Klein, P., Ravi, R.: When trees collide: an approximation algorithm for the generalized Steiner problem on networks. SIAM Journal on Computing 24, 440–456 (1995)
Awerbuch, B., Azar, Y.: Buy-at-bulk network design. In: FOCS, pp. 542–547 (1997)
Becchetti, L., Könemann, J., Leonardi, S., Pál, M.: Sharing the cost more efficiently: improved approximation for multicommodity rent-or-buy. In: SODA, pp. 375–384 (2005)
Byrka, J.: An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) RANDOM 2007 and APPROX 2007. LNCS, vol. 4627, pp. 29–43. Springer, Heidelberg (2007)
Byrka, J., Grandoni, F., Rothvoß, T., Sanità, L.: An improved LP-based approximation for Steiner tree. In: STOC, pp. 583–592 (2010)
Chlebík, M., Chlebíková, J.: The Steiner tree problem on graphs: Inapproximability results. Theoretical Computer Science 406(3), 207–214 (2008)
Eisenbrand, F., Grandoni, F.: An improved approximation algorithm for virtual private network design. In: SODA, pp. 928–932 (2005)
Eisenbrand, F., Grandoni, F., Oriolo, G., Skutella, M.: New approaches for virtual private network design. SIAM Journal on Computing 37(3), 706–721 (2007)
Eisenbrand, F., Grandoni, F., Rothvoß, T., Schäfer, G.: Connected facility location via random facility sampling and core detouring. Journal of Computer and System Sciences 76, 709–726 (2010)
Feige, U.: A Threshold of ln n for Approximating Set Cover. Journal of the ACM 45(4) (1998)
Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. Journal of Computer and System Sciences 69(3), 485–497 (2004)
Fleischer, L., Könemann, J., Leonardi, S., Schäfer, G.: Simple cost sharing schemes for multicommodity rent-or-buy and stochastic steiner tree. In: STOC, pp. 663–670 (2006)
Garg, N., Khandekar, R., Konjevod, G., Ravi, R., Salman, F., Sinha, A.: On the integrality gap of a natural formulation of the single-sink buy-at-bulk network design problem. In: Aardal, K., Gerards, B. (eds.) IPCO 2001. LNCS, vol. 2081, pp. 170–184. Springer, Heidelberg (2001)
Grandoni, F., Italiano, G.F.: Improved approximation for single-sink buy-at-bulk. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 111–120. Springer, Heidelberg (2006)
Grandoni, F., Rothvoß, T.: Network design via core detouring for problems without a core. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 490–502. Springer, Heidelberg (2010)
Guha, S., Khuller, S.: Greedy Strikes Back: Improved Facility Location Algorithms. Journal of Algorithms 31(1), 228–248 (1999)
Guha, S., Meyerson, A., Munagala, K.: A constant factor approximation for the single sink edge installation problem. SIAM Journal on Computing 38(6), 2426–2442 (2009)
Gupta, A., Kleinberg, J., Kumar, A., Rastogi, R., Yener, B.: Provisioning a virtual private network: a network design problem for multicommodity flow. In: STOC, pp. 389–398 (2001)
Gupta, A., Kumar, A.: A constant-factor approximation for stochastic Steiner forest. In: STOC, pp. 659–668 (2009)
Gupta, A., Kumar, A., Pal, M., Roughgarden, T.: Approximation via cost-sharing: simpler and better approximation algorithms for network design. Journal of the ACM 54(3), 11 (2007)
Gupta, A., Srinivasan, A., Tardos, É.: Cost-sharing mechanisms for network design. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 139–150. Springer, Heidelberg (2004)
Jothi, R., Raghavachari, B.: Improved approximation algorithms for the single-sink buy-at-bulk network design problems. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 336–348. Springer, Heidelberg (2004)
Karger, D.R., Minkoff, M.: Building Steiner trees with incomplete global knowledge. In: FOCS, pp. 613–623 (2000)
Kumar, A., Gupta, A., Roughgarden, T.: A constant-factor approximation algorithm for the multicommodity rent-or-buy problem. In: FOCS, pp. 333–342 (2002)
Leonardi, S.: Private communication (2008)
Meyerson, A., Munagala, K., Plotkin, S.: Cost-distance: two metric network design. In: FOCS, pp. 624–630 (2000)
Rothvoß, T., Sanità, L.: On the complexity of the asymmetric VPN problem. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX 2009. LNCS, vol. 5687, pp. 326–338. Springer, Heidelberg (2009)
Swamy, C., Kumar, A.: Primal–dual algorithms for connected facility location problems. Algorithmica 40(4), 245–269 (2004)
Talwar, K.: The single-sink buy-at-bulk LP has constant integrality gap. In: Cook, W.J., Schulz, A.S. (eds.) IPCO 2002. LNCS, vol. 2337, pp. 475–480. Springer, Heidelberg (2002)
Xu, G., Xu, J.: An improved approximation algorithm for uncapacitated facility location problem with penalties. Journal of Combinatorial Optimization 17(4), 424–436 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Grandoni, F., Rothvoß, T. (2011). Approximation Algorithms for Single and Multi-Commodity Connected Facility Location. In: Günlük, O., Woeginger, G.J. (eds) Integer Programming and Combinatoral Optimization. IPCO 2011. Lecture Notes in Computer Science, vol 6655. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20807-2_20
Download citation
DOI: https://doi.org/10.1007/978-3-642-20807-2_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-20806-5
Online ISBN: 978-3-642-20807-2
eBook Packages: Computer ScienceComputer Science (R0)