Complexity of the directed Steiner tree problem on special graph classes I am interested in the complexity of the directed Steiner tree problem: Given a weighted digraph $D=(V,E)$, a root $r\in V$ of $D$, and a set of terminals $T\subseteq V$. The objective is to find a minimum cost arborescence of $D$ rooted at $r$, in which there is a path from $r$ to each node in $T$.

Is a polynomial time algorithm known if the input is restricted to digraphs whose underlying undirected graph is a series-parallel graph (or equivalently is a $K_4$-free graph or is a partial $2$-tree or has treewidth at most $2$)?
This manuscript seems to prove exactly that. (It doesn't; the complexity parameter is $|T|$, not the treewidth.)

In general, most NP-hard optimisation problems have polynomial-time algorithms when the input is restricted to bounded-treewidth graphs. These algorithms use the rather well-known tree-decomposition machinery, which is also used in the linked paper.

EDIT: The undirected Steiner tree is, on the other hand, known to be fixed-parameter tractable with regards to parameter $w$ the treewidth of the underlying graph. My suggestion would be to try and adapt this algorithm to the directed case, which would in particular give a polynomial time algorithm for series-parallel graphs.