From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs

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 $O(k^ε)$-approximation for any fixed $ε> 0$ 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 $Ω(\log^2 n/\log \log n)$ 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 $Ω(k)$ and $Ω(n^δ)$ for some fixed $δ> 0$. - 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 $R= ω(\log^2 k)$.