Single-sink fractionally subadditive network design

We study a generalization of the Steiner tree problem, where we are given a weighted network G together with a collection of k subsets of its vertices and a root r. We wish to construct a minimum cost network such that the network supports one unit of flow to the root from every node in a subset simultaneously. The network constructed does not need to support flows from all the subsets simultaneously. We settle an open question regarding the complexity of this problem for k = 2, and give a 3/2 -Approximation algorithm that improves over a (trivial) known 2-Approximation. Furthermore, we prove some structural results that prevent many well-known techniques from doing better than the known O(log n)-Approximation. Despite these obstacles, we conjecture that this problem should have an O(1)-Approximation. We also give an approximation result for a variant of the problem where the solution is required to be a path.