  Given a graph G = (V,E) with a root r in V, positive capacities {c(e)|e in
E}, and non-negative lengths {l(e)|e in 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 of vertices, and such that, for
each e in 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 classical Steiner tree as well as vertex-disjoint paths problems.
