  We study a variant of Min Cost Flow in which the flow needs to be connected.
Specifically, in the Connected Flow problem one is given a directed graph $G$,
along with a set of demand vertices $D \subseteq V(G)$ with demands
$\mathsf{dem}: D \rightarrow \mathbb{N}$, and costs and capacities for each
edge. The goal is to find a minimum cost flow that satisfies the demands,
respects the capacities and induces a (strongly) connected subgraph. This
generalizes previously studied problems like the (Many Visits) TSP.
  We study the parameterized complexity of Connected Flow parameterized by
$|D|$, the treewidth $tw$ and by vertex cover size $k$ of $G$ and provide:
  (i) $\mathsf{NP}$-completeness already for the case $|D|=2$ with only unit
demands and capacities and no edge costs, and fixed-parameter tractability if
there are no capacities,
  (ii) a fixed-parameter tractable $\mathcal{O}^{\star}(k^{\mathcal{O}(k)})$
time algorithm for the general case, and a kernel of size polynomial in $k$ for
the special case of Many Visits TSP,
  (iii) a $|V(G)|^{\mathcal{O}(tw)}$ time algorithm and a matching
$|V(G)|^{o(tw)}$ time conditional lower bound conditioned on the Exponential
Time Hypothesis.
  To achieve some of our results, we significantly extend an approach by
Kowalik et al.~[ESA'20].
