On the edge capacitated Steiner tree problem

Given a graph G = (V, E) with a root r is an element of V, positive capacities {c(e)vertical bar e is an element of E}, and non-negative lengths {l(e)vertical bar e is an element of E}, the minimum-length (rooted) edge capacitated Steiner tree problem is to find a tree in G of minimum total length, rooted at r, spanning a given subset T subset of V of vertices, and such that, for each e is an element of E, there are at most c(e) paths, linking r to vertices in T, that contain e. We study the complexity and approximability of the problem, considering several relevant parameters such as the number of terminals, the edge lengths and the minimum and maximum edge capacities. For all but one combinations of assumptions regarding these parameters, we settle the question, giving a complete characterization that separates tractable cases from hard ones. The only remaining open case is proved to be equivalent to a long-standing open problem. We also prove close relations between our problem and classic Steiner tree as well as vertex-disjoint paths problems. (C) 2020 Elsevier B.V. All rights reserved.