Abstract
For several NP-hard network design problems, the best known approximation algorithms are remarkably simple randomized algorithms called Sample-Augment algorithms in [11]. The algorithms draw a random sample from the input, solve a certain subproblem on the random sample, and augment the solution for the subproblem to a solution for the original problem. We give a general framework that allows us to derandomize most Sample-Augment algorithms, i.e. to specify a specific sample for which the cost of the solution created by the Sample-Augment algorithm is at most a constant factor away from optimal. Our approach allows us to give deterministic versions of the Sample-Augment algorithms for the connected facility location problem, in which the open facilities need to be connected by either a tree or a tour, the virtual private network design problem, 2-stage rooted stochastic Steiner tree problem with independent decisions, the a priori traveling salesman problem and the single sink buy-at-bulk problem. This partially answers an open question posed in Gupta et al. [11].
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
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 J. Comput. 37(3), 706–721 (2007)
Eisenbrand, F., Grandoni, F., Rothvoß, T., Schäfer, G.: Approximating connected facility location problems via random facility sampling and core detouring. In: SODA, pp. 1174–1183 (2008)
Erdős, P., Spencer, J.: Probabilistic methods in combinatorics. Academic Press, London (1974)
Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. System Sci. 69(3), 485–497 (2004)
Garg, N., Gupta, A., Leonardi, S., Sankowski, P.: Stochastic analyses for online combinatorial optimization problems. In: SODA, pp. 942–951 (2008)
Garg, N., Khandekar, R., Konjevod, G., Ravi, R., Salman, F.S., 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)
Goemans, M.X., Bertsimas, D.J.: Survivable networks, linear programming relaxations and the parsimonious property. Math. Program. 60(2, Ser. A), 145–166 (1993)
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)
Gupta, A., Kumar, A., Pál, M., Roughgarden, T.: Approximation via cost-sharing: A simple approximation algorithm for the multicommodity rent-or-buy problem. In: FOCS, pp. 606–615 (2003)
Gupta, A., Kumar, A., Pál, M., Roughgarden, T.: Approximation via cost sharing: simpler and better approximation algorithms for network design. J. ACM 54(3), 11 (2007)
Gupta, A., Kumar, A., Roughgarden, T.: Simpler and better approximation algorithms for network design. In: STOC, pp. 365–372 (2003)
Gupta, A., Pál, M., Ravi, R., Sinha, A.: Boosted sampling: approximation algorithms for stochastic optimization. In: STOC, pp. 417–426 (2004)
Gupta, A., Srinivasan, A., Tardos, É.: Cost-sharing mechanisms for network design. Algorithmica 50(1), 98–119 (2008)
Hasan, M.K., Jung, H., Chwa, K.-Y.: Approximation algorithms for connected facility location problems. J. Comb. Optim. (to appear, 2008)
Immorlica, N., Karger, D., Minkoff, M., Mirrokni, V.S.: On the costs and benefits of procrastination: approximation algorithms for stochastic combinatorial optimization problems. In: SODA, pp. 691–700 (2004)
Raghavan, P.: Probabilistic construction of deterministic algorithms: approximating packing integer programs. J. Comput. System Sci. 37(2), 130–143 (1988)
Shmoys, D., Talwar, K.: A constant approximation algorithm for the a priori traveling salesman problem. In: IPCO (2008)
Shmoys, D.B., Williamson, D.P.: Analyzing the Held-Karp TSP bound: A monotonicity property with application. Inf. Process. Lett. 35, 281–285 (1990)
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. 475–486. Springer, Heidelberg (2002)
Williamson, D.P., van Zuylen, A.: A simpler and better derandomization of an approximation algorithm for single source rent-or-buy. Oper. Res. Lett. 35(6), 707–712 (2007)
Wolsey, L.A.: Heuristic analysis, linear programming and branch and bound. Math. Prog. Study 13, 121–134 (1980)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
van Zuylen, A. (2008). Deterministic Sampling Algorithms for Network Design. In: Halperin, D., Mehlhorn, K. (eds) Algorithms - ESA 2008. ESA 2008. Lecture Notes in Computer Science, vol 5193. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87744-8_69
Download citation
DOI: https://doi.org/10.1007/978-3-540-87744-8_69
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87743-1
Online ISBN: 978-3-540-87744-8
eBook Packages: Computer ScienceComputer Science (R0)