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 k > 0, and an integer edge capacity λ > 1. The CTR asks to find a partition M = {Z1, Z2, ,...,Z ℓ) of M and a set T = [T1, T2,..., T ℓ) of trees of G such that each Ti spans Zi ∪ {s} and satisfies ∑v∈zi q(v) ≤ k. 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 ∑e∈E ⌈|{T ∈ T | T contains e}|/λ⌉w(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. © Springer-Verlag Berlin Heidelberg 2007.