  In this paper we reassess the parameterized complexity and approximability of
the well-studied Steiner Forest problem in several graph classes of bounded
width. The problem takes an edge-weighted graph and pairs of vertices as input,
and the aim is to find a minimum cost subgraph in which each given vertex pair
lies in the same connected component. It is known that this problem is APX-hard
in general, and NP-hard on graphs of treewidth 3, treedepth 4, and feedback
vertex set size 2. However, Bateni, Hajiaghayi and Marx [JACM, 2011] gave an
approximation scheme with a runtime of $n^{O(\frac{k^2}{\varepsilon})}$ on
graphs of treewidth $k$. Our main result is a much faster efficient
parameterized approximation scheme (EPAS) with a runtime of
$2^{O(\frac{k^2}{\varepsilon} \log \frac{k^2}{\varepsilon})} \cdot n^{O(1)}$.
If $k$ instead is the vertex cover number of the input graph, we show how to
compute the optimum solution in $2^{O(k \log k)} \cdot n^{O(1)}$ time, and we
also prove that this runtime dependence on $k$ is asymptotically best possible,
under ETH. Furthermore, if $k$ is the size of a feedback edge set, then we
obtain a faster $2^{O(k)} \cdot n^{O(1)}$ time algorithm, which again cannot be
improved under ETH.
