# Approximating capacitated tree-routings in networks

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## Abstract

Let *G*=(*V*,*E*) be a connected graph such that each edge *e*∈*E* is weighted by a nonnegative real *w*(*e*). Let *s* be a vertex designated as a sink, *M*⊆*V* be a set of terminals with a demand function *q*:*M*→*R* ^{+}, *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* _{ ℓ }} 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 *e*∈*E* 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.

## Keywords

Approximation algorithm Graph algorithm Routing problems Network optimization Tree cover## Preview

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## References

- Cai Z, Lin G-H, Xue G (2005) Improved approximation algorithms for the capacitated multicast routing problem. LNCS 3595:136–145 MathSciNetGoogle Scholar
- Garey MR, Johnson MR (1977) The rectilinear Steiner tree problem is NP-complete. SIAM J Appl Math 32:826–843 zbMATHCrossRefMathSciNetGoogle Scholar
- Hassin R, Ravi R, Salman FS (2004) Approximation algorithms for a capacitated network design problem. Algorithmica 38:417–431 zbMATHCrossRefMathSciNetGoogle Scholar
- 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
- 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
- Khuller S, Raghavachari B, Young NN (1993) Balancing minimum spanning and shortest path trees. Algorithmica 14:305–322 CrossRefMathSciNetGoogle Scholar
- Lin G-H (2004) An improved approximation algorithm for multicast k-tree routing. J Comb Optim 9:349–356 CrossRefGoogle Scholar
- 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
- Morsy E, Nagamochi H (2008) An improved approximation algorithm for capacitated multicast routings in networks. Theor Comput Sci 390(1):81–91 zbMATHCrossRefMathSciNetGoogle Scholar
- 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
- 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
- Zhao L, Yamamoto H (2005) Multisource receiver-driven layered multicast. In: Proceedings of IEEE TENCON 2005, pp 1325–1328 Google Scholar