  In the Directed Steiner Network problem we are given an arc-weighted digraph
$G$, a set of terminals $T \subseteq V(G)$, and an (unweighted) directed
request graph $R$ with $V(R)=T$. Our task is to output a subgraph $G' \subseteq
G$ of the minimum cost such that there is a directed path from $s$ to $t$ in
$G'$ for all $st \in A(R)$.
  It is known that the problem can be solved in time $|V(G)|^{O(|A(R)|)}$
[Feldman&Ruhl, SIAM J. Comput. 2006] and cannot be solved in time
$|V(G)|^{o(|A(R)|)}$ even if $G$ is planar, unless Exponential-Time Hypothesis
(ETH) fails [Chitnis et al., SODA 2014]. However, as this reduction (and other
reductions showing hardness of the problem) only shows that the problem cannot
be solved in time $|V(G)|^{o(|T|)}$ unless ETH fails, there is a significant
gap in the complexity with respect to $|T|$ in the exponent.
  We show that Directed Steiner Network is solvable in time $f(R)\cdot
|V(G)|^{O(c_g \cdot |T|)}$, where $c_g$ is a constant depending solely on the
genus of $G$ and $f$ is a computable function. We complement this result by
showing that there is no $f(R)\cdot |V(G)|^{o(|T|^2/ \log |T|)}$ algorithm for
any function $f$ for the problem on general graphs, unless ETH fails.
