Abstract
Given an undirected edge-capacitated graph and a collection of subsets of vertices, we consider the problem of selecting a maximum (weighted) set of Steiner trees, each spanning a given subset of vertices without violating the capacity constraints. We give an integer linear programming (ILP) formulation, and observe that its linear programming (LP-) relaxation is a fractional packing problem with exponentially many variables and with a block (sub-)problem that cannot be solved in polynomial time. To this end, we take an r-approximate block solver to develop a (1 − ε)/r approximation algorithm for the LP-relaxation. The algorithm has a polynomial coordination complexity for any ε ∈ (0,1). To the best of our knowledge, this is the first approximation result for fractional packing problems with only approximate block solvers and a coordination complexity that is polynomial in the input size and ε − 1. This leads to an approximation algorithm for the underlying tree packing problem. Finally, we extend our results to an important multicast routing and wavelength assignment problem in optical networks, where each Steiner tree is also to be assigned one of a limited set of given wavelengths, so that trees crossing the same fiber are assigned different wavelengths.
Research supported by a MITACS grant for all the authors, an NSERC post doctoral fellowship for the first author, the NSERC Discovery Grant #5-48923 for the second and fourth author, the NSERC Grant #15296 for the third author, and the Canada Research Chair Program for the second author.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Anstreicher, K.M.: Towards a practical volumetric cutting plane method for convex programming. SIAM J. Optimization 9, 190–206 (1999)
Baltz, A., Srivastav, A.: Fast approximation of minimum multicast congestion- implementation versus theory. RAIRO Oper. Res. 38, 319–344 (2004)
Bern, M., Plassmann, P.: The Steiner problem with edge lengths 1 and 2. Inf. Process. Lett. 32, 171–176 (1989)
Cai, M., Deng, X., Wang, L.: Minimum k arborescences with bandwidth constraints. Algorithmica 38, 529–537 (2004)
Carr, R., Vempala, S.: Randomized meta-rounding. In: Proceedings of STOC 2000, pp. 58–62 (2000)
Chen, S., Günlük, O., Yener, B.: The multicast packing problem. IEEE/ACM Trans. Networking 8(3), 311–318 (2000)
Garg, N., Könemann, J.: Fast and simpler algorithms for multicommodity flow and other fractional packing problems. In: Proceedings of FOCS 1998, pp. 300–309 (1998)
Grigoriadis, M.D., Khachiyan, L.G.: Coordination complexity of parallel price-directive decomposition. Math. Oper. Res. 2, 321–340 (1996)
Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)
Jain, K., Mahdian, M., Salavatipour, M.R.: Packing Steiner trees. In: Proceedings of SODA 2003, pp. 266–274 (2003)
Jansen, K., Zhang, H.: Approximation algorithms for general packing problems with modified logarithmic potential function. In: Proceedings of TCS 2002, pp. 255–266 (2002)
Jansen, K., Zhang, H.: An approximation algorithm for the multicast congestion problem via minimum Steiner trees. In: Proceedings of ARACNE 2002, pp. 77–90 (2002)
Jia, X., Wang, L.: A group multicast routing algorithm by using multiple minimum Steiner trees. Comput. Commun. 20, 750–758 (1997)
Lau, L.C.: An approximate max-Steiner-tree-packing min-Steiner-cut theorem. In: Proceedings of STOC 2004, pp. 61–70 (2004)
Lau, L.C.: Packing Steiner forests. In: Jünger, M., Kaibel, V. (eds.) IPCO 2005. LNCS, vol. 3509, pp. 362–376. Springer, Heidelberg (2005)
Lu, Q., Zhang, H.: Implementation of approximation algorithms for the multicast congestion problem. In: Nikoletseas, S.E. (ed.) WEA 2005. LNCS, vol. 3503, pp. 152–164. Springer, Heidelberg (2005)
Plotkin, S.A., Shmoys, D.B., Tardos, E.: Fast approximation algorithms for fractional packing and covering problems. Math. Oper. Res. 2, 257–301 (1995)
Raghavan, P.: Probabilistic construction of deterministic algorithms: approximating packing integer programs. J. Comput. Syst. Sci. 37, 130–143 (1988)
Raghavan, P., Thompson, C.D.: Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica 7(4), 365–374 (1987)
Saad, M., Terlaky, T., Vannelli, A., Zhang, H.: A provably good global routing algorithm in multilayer IC and MCM layout designs. Technical Report, AdvOL #2005-15, Advanced Optimization Lab., McMaster University, Hamilton, ON, Canada, http://www.cas.mcmaster.ca/~oplab/research.htm
Saad, M., Terlaky, T., Vannelli, A., Zhang, H.: Packing Trees in Communication Networks, Technical Report, AdvOL #2005-14, Advanced Optimization Lab., McMaster University, Hamilton, ON, Canada, http://www.cas.mcmaster.ca/~oplab/research.htm
Terlaky, T., Vannelli, A., Zhang, H.: On routing in VLSI design and communication networks. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 1051–1060. Springer, Heidelberg (2005)
Vempala, S., Vöcking, B.: Approximating multicast congestion. In: Aggarwal, A.K., Pandu Rangan, C. (eds.) ISAAC 1999. LNCS, vol. 1741, pp. 367–372. Springer, Heidelberg (1999)
Villavicencio, J., Grigoriadis, M.D.: Approximate Lagrangian decomposition with a modified Karmarkar logarithmic potential. In: Pardalos, P., Hearn, D.W., Hager, W.W. (eds.) Network Optimization. LNEMS, vol. 450, pp. 471–485. Springer, Berlin (1997)
Young, N.E.: Randomized rounding without solving the linear program. In: Proceedings of SODA, pp. 170–178 (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Saad, M., Terlaky, T., Vannelli, A., Zhang, H. (2005). Packing Trees in Communication Networks. In: Deng, X., Ye, Y. (eds) Internet and Network Economics. WINE 2005. Lecture Notes in Computer Science, vol 3828. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11600930_69
Download citation
DOI: https://doi.org/10.1007/11600930_69
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30900-0
Online ISBN: 978-3-540-32293-1
eBook Packages: Computer ScienceComputer Science (R0)