  Given an $n$-vertex non-negatively real-weighted graph $G$, whose vertices
are partitioned into a set of $k$ clusters, a \emph{clustered network design
problem} on $G$ consists of solving a given network design optimization problem
on $G$, subject to some additional constraint on its clusters.
  In particular, we focus on the classic problem of designing a
\emph{single-source shortest-path tree}, and we analyze its computational
hardness when in a feasible solution each cluster is required to form a
subtree. We first study the \emph{unweighted} case, and prove that the problem
is \np-hard. However, on the positive side, we show the existence of an
approximation algorithm whose quality essentially depends on few parameters,
but which remarkably is an $O(1)$-approximation when the largest out of all the
\emph{diameters} of the clusters is either $O(1)$ or $\Theta(n)$. Furthermore,
we also show that the problem is \emph{fixed-parameter tractable} with respect
to $k$ or to the number of vertices that belong to clusters of size at least 2.
Then, we focus on the \emph{weighted} case, and show that the problem can be
approximated within a tight factor of $O(n)$, and that it is fixed-parameter
tractable as well. Finally, we analyze the unweighted \emph{single-pair
shortest path problem}, and we show it is hard to approximate within a (tight)
factor of $n^{1-\epsilon}$, for any $\epsilon>0$.
