Parameterized Complexity of Directed Steiner Network with Respect to Shared Vertices and Arcs

Abstract

We consider the directed Steiner network problem, where given a weighted directed graph [Formula: see text] and [Formula: see text] pairs of vertices [Formula: see text], one has to find the minimum weight subgraph [Formula: see text] of [Formula: see text] that contains path from [Formula: see text] to [Formula: see text] for all [Formula: see text]. We search for the main cause of the complexity of the problem and introduce the notions of junction vertex and shared segment in an optimal solution. We study the parameterized complexity of the problem with respect to these parameters and achieve several parameterized hardness results. Also, we present two output-sensitive algorithms for the problem, which are much simpler and easier to implement than the previously best known one; and in addition, when the paths corresponding to the input pairs have few shared vertices or arcs in the optimal solution, these algorithms outperform the previous one.