  In the Directed Steiner Tree (DST) problem the input is a directed
edge-weighted graph $G=(V,E)$, a root vertex $r$ and a set $S \subseteq V$ of
$k$ terminals. The goal is to find a min-cost subgraph that connects $r$ to
each of the terminals. DST admits an $O(\log^2 k/\log \log k)$-approximation in
quasi-polynomial time, and an $O(k^{\epsilon})$-approximation for any fixed
$\epsilon > 0$ in polynomial-time. Resolving the existence of a polynomial-time
poly-logarithmic approximation is a major open problem in approximation
algorithms. In a recent work, Friggstad and Mousavi [ICALP 2023] obtained a
simple and elegant polynomial-time $O(\log k)$-approximation for DST in planar
digraphs via Thorup's shortest path separator theorem. We build on their work
and obtain several new results on DST and related problems.
  - We develop a tree embedding technique for rooted problems in planar
digraphs via an interpretation of the recursion in Friggstad and Mousavi [ICALP
2023]. Using this we obtain polynomial-time poly-logarithmic approximations for
Group Steiner Tree, Covering Steiner Tree, and the Polymatroid Steiner Tree
problems in planar digraphs. All these problems are hard to approximate to
within a factor of $\Omega(\log^2 n/\log \log n)$ even in trees.
  - We prove that the natural cut-based LP relaxation for DST has an
integrality gap of $O(\log^2 k)$ in planar graphs. This is in contrast to
general graphs where the integrality gap of this LP is known to be $\Omega(k)$
and $\Omega(n^{\delta})$ for some fixed $\delta > 0$.
  - We combine the preceding results with density based arguments to obtain
poly-logarithmic approximations for the multi-rooted versions of the problems
in planar digraphs. For DST our result improves the $O(R + \log k)$
approximation of Friggstad and Mousavi [ICALP 2023] when $R= \omega(\log^2 k)$.
