  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 $\frac{3}{2}$-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.
