Rooted k-connections in digraphs

The problem of computing a minimum cost subgraph D′ = (V,A′) of a directed graph D = (V,A) so that D′ contains k edge-disjoint paths from a specified root r ∈ V to every other node in V is known to be nicely solvable since it can be formulated as a matroid intersection problem. A corresponding problem when openly disjoint paths are requested rather than edge-disjoint was solved in [12] with the help of submodular flows. Here we show that the use of submodular flows is actually avoidable and even a common generalization of the two rooted k-connection problems is a matroid intersection problem. We also provide a polyhedral description using supermodular functions on bi-sets and this approach enables us to handle more general rooted k-connection problems. For example, with the help of a submodular flow algorithm the following restricted version of the generalized Steiner-network problem is solvable in polynomial time: given a digraph D = (V,A) with a root-node r, a terminal set T , and a cost function c : A → R+ so that each edge of positive cost has its head in T , find a subgraph D′ = (V,A′) of D of minimum cost so that there are k openly disjoint paths in D′ from r to every node in T .