Approximating capacitated tree-routings in networks

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.