Journal of Combinatorial Optimization

, Volume 21, Issue 2, pp 254–267 | Cite as

Approximating capacitated tree-routings in networks

  • Ehab Morsy
  • Hiroshi Nagamochi


Let G=(V,E) be a connected graph such that each edge eE is weighted by a nonnegative real w(e). Let s be a vertex designated as a sink, MV be a set of terminals with a demand function q:MR +, κ>0 be a routing capacity, and λ≥1 be an integer edge capacity. The capacitated tree-routing problem (CTR) asks to find a partition ℳ={Z 1,Z 2,…,Z } of M and a set \({\mathcal{T}}=\{T_{1},T_{2},\ldots,T_{\ell}\}\) of trees of G such that each T i contains Z i ∪{s} and satisfies \(\sum_{v\in Z_{i}}q(v)\leq \kappa\) . A single copy of an edge eE can be shared by at most λ trees in  \({\mathcal{T}}\) ; any integer number of copies of e are allowed to be installed, where the cost of installing a copy of e is w(e). The objective is to find a solution \(({\mathcal{M}},{\mathcal{T}})\) that minimizes the total installing cost. In this paper, we propose a (2+ρ ST )-approximation algorithm to CTR, where ρ ST is any approximation ratio achievable for the Steiner tree problem.


Approximation algorithm Graph algorithm Routing problems Network optimization Tree cover 


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  1. Cai Z, Lin G-H, Xue G (2005) Improved approximation algorithms for the capacitated multicast routing problem. LNCS 3595:136–145 MathSciNetGoogle Scholar
  2. Garey MR, Johnson MR (1977) The rectilinear Steiner tree problem is NP-complete. SIAM J Appl Math 32:826–843 zbMATHCrossRefMathSciNetGoogle Scholar
  3. Hassin R, Ravi R, Salman FS (2004) Approximation algorithms for a capacitated network design problem. Algorithmica 38:417–431 zbMATHCrossRefMathSciNetGoogle Scholar
  4. Hu X-D, Shuai T-P, Jia X, Zhang MH (2004) Multicast routing and wavelength assignment in WDM networks with limited drop-offs. In: Proceedings of IEEE INFOCOM Google Scholar
  5. Jothi R, Raghavachari B (2004) Approximation algorithms for the capacitated minimum spanning tree problem and its variants in network design. In: Proceedings of ICALP 2004. LNCS, vol 3142. Springer, Berlin, pp 805–818 Google Scholar
  6. Khuller S, Raghavachari B, Young NN (1993) Balancing minimum spanning and shortest path trees. Algorithmica 14:305–322 CrossRefMathSciNetGoogle Scholar
  7. Lin G-H (2004) An improved approximation algorithm for multicast k-tree routing. J Comb Optim 9:349–356 CrossRefGoogle Scholar
  8. Mansour Y, Peleg D (1998) An approximation algorithm for minimum-cost network design. Tech Report Cs94-22, The Weizman Institute of Science, Rehovot (1994); also presented at the DIMACS Workshop on Robust Communication Network Google Scholar
  9. Morsy E, Nagamochi H (2008) An improved approximation algorithm for capacitated multicast routings in networks. Theor Comput Sci 390(1):81–91 zbMATHCrossRefMathSciNetGoogle Scholar
  10. Robins G, Zelikovsky AZ (2000) Improved Steiner tree approximation in graphs. In: Proceedings of the 11th annual ACM-SIAM symposium on discrete algorithms, pp 770–779 Google Scholar
  11. Salman FS, Cheriyan J, Ravi R, Subramanian S (2000) Approximating the single-sink link-installation problem in network design. SIAM J Optim 11:595–610 zbMATHCrossRefMathSciNetGoogle Scholar
  12. Zhao L, Yamamoto H (2005) Multisource receiver-driven layered multicast. In: Proceedings of IEEE TENCON 2005, pp 1325–1328 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Physics, Graduate School of InformaticsKyoto UniversityKyotoJapan
  2. 2.Department of Mathematics, Faculty of ScienceSuez Canal UniversityIsmailiaEgypt

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