In the Directed Steiner Network problem we are given an arc-weighted digraph
G, a set of terminals T⊆V(G), and an (unweighted) directed
request graph R with V(R)=T. Our task is to output a subgraph G′⊆G of the minimum cost such that there is a directed path from s to t in
G′ for all st∈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)⋅∣V(G)∣O(cg⋅∣T∣), where cg 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)⋅∣V(G)∣o(∣T∣2/log∣T∣) algorithm for
any function f for the problem on general graphs, unless ETH fails