Approximating the generalized capacitated tree-routing problem

Abstract

In this paper, we introduce the generalized capacitated tree-routing problem (GCTR), which is described as follows. Given a connected graph G = (V,E) with a sink s ∈ V and a set M ⊆ V - {s} of terminals with a nonnegative demand q(v), v ∈ M, we wish to find a collection T = {T 1, T2,...,Tl}of trees rooted at s to send all the demands to s, where the total demand collected by each tree Ti is bounded from above by a demand capacity κ> 0. Let λ > 0 denote a bulk capacity of an edge, and each edge e ∈ E has an installation cost w(e) ≥ 0 per bulk capacity; each edge e is allowed to have capacity kλ for any integer k, which installation incurs cost kw(e). To establish a tree routing T i , each edge e contained in T i requires α + βq′ amount of capacity for the total demand q′ that passes through edge e along T i and prescribed constants α,β ≥ 0, where α means a fixed amount used to separate the inside of the routing T i from the outside while term βq′ means the net capacity proportional to q′. The objective of GCTR is to find a collection of trees that minimizes the total installation cost of edges. Then GCTR is a new generalization of the several known multicast problems in networks with edge/demand capacities. In this paper, we prove that GCTR is (2[λ/(α+βκ)]/⌊λ/(α+βκ) ⌋ + ρST)-approximable if λ ≥ α + βκ holds, where ρST is any approximation ratio achievable for the Steiner tree problem. © 2008 Springer-Verlag Berlin Heidelberg.