# 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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