  The Wiener index of a graph is the sum of all pairwise shortest-path
distances between its vertices. In this paper we study the novel problem of
finding a minimum Wiener connector: given a connected graph $G=(V,E)$ and a set
$Q\subseteq V$ of query vertices, find a subgraph of $G$ that connects all
query vertices and has minimum Wiener index.
  We show that The Minimum Wiener Connector admits a polynomial-time (albeit
impractical) exact algorithm for the special case where the number of query
vertices is bounded. We show that in general the problem is NP-hard, and has no
PTAS unless $\mathbf{P} = \mathbf{NP}$. Our main contribution is a
constant-factor approximation algorithm running in time
$\widetilde{O}(|Q||E|)$.
  A thorough experimentation on a large variety of real-world graphs confirms
that our method returns smaller and denser solutions than other methods, and
does so by adding to the query set $Q$ a small number of important vertices
(i.e., vertices with high centrality).
