Approximating Capacitated Tree-Routings in Networks

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 M = {Z1, Z2,..., Zl} of M and a set T = {T1, T2,..., Tl} of trees of G such that each Ti spans Zi ∪ {s} and satisfies Σv∈Ziq(v) ≤ κ. A subset of trees in 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 (M, T) that minimizes the installing cost Σe∈E⌈|{T ∈ T | T contains e}|/λw(e). In this paper, we propose a (2+ρST)-approximation algorithm to the CTR, where ρST is any approximation ratio achievable for the Steiner tree problem.