Approximating Capacitated Tree-Routings in Networks

Abstract

The capacitated tree-routing problem (CTR) in a graph G = (V,E) consists of an edge weight function w:E→R ⁺, a sink s ∈ V, a terminal set M ⊆ V with a demand function q:M→R ⁺, a routing capacity κ> 0, and an integer edge capacity λ ≥ 1. The CTR asks to find a partition ${\cal M}=\{Z_{1},Z_{2},\ldots,Z_{\ell}\}$ of M and a set ${\cal T}=\{T_{1},T_{2},\ldots,T_{\ell}\}$ of trees of G such that each T _i spans Z _i  ∪ {s} and satisfies $\sum_{v\in Z_{i}}q(v)\leq \kappa$. A subset of trees in ${\cal T}$ can pass through a single copy of an edge e ∈ E as long as the number of these trees does not exceed the edge capacity λ; 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 $({\cal M}, {\cal T})$ that minimizes the installing cost $\sum_{e\in E} \lceil |\{T\in {\cal T}\mid T \mbox{ contains }e\}| /\lambda \rceil w(e)$. In this paper, we propose a $(2+\rho_{\mbox{\tiny{\sc ST}}})$-approximation algorithm to the CTR, where $\rho_{\mbox{\tiny{\sc ST}}}$ is any approximation ratio achievable for the Steiner tree problem.