On Directed Steiner Trees with Multiple Roots

We introduce a new Steiner-type problem for directed graphs named q-Root Steiner Tree. Here one is given a directed graph G = (V,A) and two subsets of its vertices, R of size q and T , and the task is to find a minimum size subgraph of G that contains a path from each vertex of R to each vertex of T . The special case of this problem with q = 1 is the well known Directed Steiner Tree problem, while the special case with T = R is the Strongly Connected Steiner Subgraph problem. We first show that the problem is W[1]-hard with respect to |T | for any q ≥ 2. Then we restrict ourselves to instances with R ⊆ T . Generalizing the methods of Feldman and Ruhl [SIAM J. Comput. 2006], we present an algorithm for this restriction with running time O(22q+4|T | ·n), i.e., this restriction is FPT with respect to |T | for any constant q. We further show that we can, without significantly affecting the achievable running time, loosen the restriction to only requiring that in the solution there are a vertex v and a path from each vertex of R to v and from v to