Abstract
The buy-at-bulk network design problem has been extensively studied in the general graph model. In this paper we consider the geometric version of the problem, where all points in a Euclidean space are candidates for network nodes. We present the first general approach for geometric versions of basic variants of the buy-at-bulk network design problem. It enables us to obtain quasi-polynomial-time approximation schemes for basic variants of the buy-at-bulk geometric network design problem with polynomial total demand. Then, for instances with few sinks and low capacity links, we design very fast polynomial-time low-constant approximations algorithms.
Research supported in part by VR grant 621-2005-4085, the Royal Society IJP - 2006/R2, the Centre for Discrete Mathematics and its Applications, the Special Coordination Funds for Promoting Science and Technology (Japan), and the Visby Programme Scholarship 01224/2007.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM 45(5), 753–782 (1998)
Arora, S., Raghavan, P., Rao, S.: Approximation schemes for Euclidean k-medians and related problems. In: Proc. 30th ACM STOC, pp. 106–113 (1998)
Awerbuch, B., Azar, Y.: Buy-at-bulk network design. In: Proc 38th IEEE FOCS, pp. 542–547 (1997)
Bienstock, D., Chopra, S., Günlük, O., Tsai, C.-Y.: Minimum cost capacity installation for multicommodity network flows. Mathematical Programming 81(2), 177–199 (1998)
Chekuri, C., Hajiaghayi, M.T., Kortsarz, G., Salavatipour, M.R.: Polylogarithmic approximation algorithms for non-uniform multicommodity buy-at-bulk network design. In: Proc. 47th IEEE FOCS, pp. 677–686 (2006)
Chekuri, C., Hajiaghayi, M.T., Kortsarz, G., Salavatipour, M.R.: Approximation algorithms for node-weighted buy-at-bulk network design. In: SODA, pp. 1265–1274 (2007)
Chopra, S., Gilboa, I., Sastry, S.T.: Source sink flows with capacity installation in batches. Discrete Applied Mathematics 85(3), 165–192 (1998)
Czumaj, A., Lingas, A.: On approximability of the minimum-cost k-connected spanning subgraph problem. In: Proc. 10th IEEE-SIAM SODA, pp. 281–290 (1999)
Czumaj, A., Lingas, A.: Fast approximation schemes for euclidean multi-connectivity problems. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 856–868. Springer, Heidelberg (2000)
Gupta, A., Kumar, A., Roughgarden, T.: Simpler and better approximation algorithms for network design. In: Proc. 35th ACM STOC, pp. 365–372 (2003)
Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-completeness. W.H. Freeman and Company, New York (1979)
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: Proc. 8th IPCO, pp. 170–184 (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)
Guha, S., Meyerson, A., Munagala, K.: A constant factor approximation for the single sink edge installation problem. In: Proc. 33rd ACM STOC, pp. 383–399 (2001)
Hassin, R., Ravi, R., Salman, F.S.: Approximation algorithms for a capacitated network design problem. In: Jansen, K., Khuller, S. (eds.) APPROX 2000. LNCS, vol. 1913, pp. 167–176. Springer, Heidelberg (2000)
Hwang, F.K.: An O(nlogn) algorithm for rectilinear minimal spanning trees. JACM 26(2), 177–182 (1979)
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)
Mansour, Y., Peleg, D.: An approximation algorithm for minimum-cost network design. Technical report, The Weizman Institute of Science, Revohot, Israel, CS94-22 (1994)
Meyerson, A., Munagala, K., Plotkin, S.: COST-DISTANCE: Two metric network design. In: Proc. 41st IEEE FOCS, pp. 624–630 (2000)
Minoux, M.: Discrete cost multicommodity network optimization problems and exact solution methods. Annals of Operations Research 106, 19–46 (2001)
Preparata, F., Shamos, M.: Computational Geometry. Springer, New York (1985)
Rao, S.B., Smith, W.D.: Approximating geometrical graphs via “spanners” and “banyans”. In: Proc. ACM STOC, pp. 540–550 (1998); Full version appeared as TR, NEC (1998)
Salman, F.S., Cheriyan, J., Ravi, R., Subramanian, S.: Approximating the single-sink link-installation problem in network design. SIAM J. Optimization 11(3), 595–610 (2000)
Talwar, K.: Single-sink buy-at-bulk LP has constant integrality gap. In: Cook, W.J., Schulz, A.S. (eds.) IPCO 2002. LNCS, vol. 2337. Springer, Heidelberg (2002)
Zachariasen, M.: A catalog of Hanan grid problems. Networks 38(2), 76–83 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Czumaj, A., Czyzowicz, J., Gąsieniec, L., Jansson, J., Lingas, A., Zylinski, P. (2009). Approximation Algorithms for Buy-at-Bulk Geometric Network Design. In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-03367-4_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03366-7
Online ISBN: 978-3-642-03367-4
eBook Packages: Computer ScienceComputer Science (R0)