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