Approximating the Generalized Capacitated Tree-Routing Problem

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 = {T1, T2, . . . , T`} 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 Ti, each edge e contained in Ti requires α+βq′ amount of capacity for the total demand q′ that passes through edge e along Ti and prescribed constants α, β ≥ 0, where α means a fixed amount used to separate the inside of the routing Ti from the outside while term βq′ means the net capacity proportional to q′. The objective of GCTR is to find a collection T 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)]/bλ/(α + βκ)c + ρST)-approximable if λ ≥ α + βκ holds, where ρST is any approximation ratio achievable for the Steiner tree problem.