Approximating the k-splittable capacitated network design problem

We consider the k-Splittable Capacitated Network Design Problem (kSCND) in a graph G=(V,E) with edge weight w(e)≥0, e∈E. We are given a vertex t∈V designated as a sink, a cable capacity λ>0, and a source set S⊆V with demand d(v)≥0, v∈S. For any edge e∈E, we are allowed to install an integer number x(e) of copies of e. The kSCND asks to simultaneously send demand d(v) from each source v∈S along at most k paths to the sink t. A set of such paths can pass through an edge in G as long as the total demand along the paths does not exceed the capacity x(e)λ. The objective is to find a set P of paths of G that minimize the installing cost ∑e∈Ex(e)w(e). In this paper, we propose a ((k+1)/k+ρST)-approximation algorithm to the kSCND, where ρST is any approximation ratio achievable for the Steiner tree problem.