Approximating Rooted Steiner Networks

Abstract

The Directed Steiner Tree (DST) problem is a cornerstone problem in network design. We focus on the generalization of the problem with higher connectivity requirements. The problem with one root and two sinks is APX-hard. The problem with one root and many sinks is as hard to approximate as the directed Steiner forest problem, and the latter is well known to be as hard to approximate as the label cover problem. Utilizing previous techniques (due to others), we strengthen these results and extend them to undirected graphs. Specifically, we give an Ω(k∊) hardness bound for the rooted k-connectivity problem in undirected graphs; this addresses a recent open question of Khanna. As a consequence, we also obtain the Ω(k∊) hardness of the undirected subset k-connectivity problem. Additionally, we give a result on the integrality ratio of the natural linear programming relaxation of the directed rooted k-connectivity problem.