A Polylogarithmic Approximation for Directed Steiner Forest in Planar Digraphs

We consider Directed Steiner Forest (DSF), a fundamental problem in network design. The input to DSF is a directed edge-weighted graph $G = (V, E)$ and a collection of vertex pairs $\{(s_i, t_i)\}_{i \in [k]}$. The goal is to find a minimum cost subgraph $H$ of $G$ such that $H$ contains an $s_i$-$t_i$ path for each $i \in [k]$. DSF is NP-Hard and is known to be hard to approximate to a factor of $\Omega(2^{\log^{1 - \epsilon}(n)})$ for any fixed $\epsilon > 0$ [DK'99]. DSF admits approximation ratios of $O(k^{1/2 + \epsilon})$ [CEGS'11] and $O(n^{2/3 + \epsilon})$ [BBMRY'13]. In this work we show that in planar digraphs, an important and useful class of graphs in both theory and practice, DSF is much more tractable. We obtain an $O(\log^6 k)$-approximation algorithm via the junction tree technique. Our main technical contribution is to prove the existence of a low density junction tree in planar digraphs. To find an approximate junction tree we rely on recent results on rooted directed network design problems [FM'23, CJKZZ'24], in particular, on an LP-based algorithm for the Directed Steiner Tree problem [CJKZZ'24]. Our work and several other recent ones on algorithms for planar digraphs [FM'23, KS'21, CJKZZ'24] are built upon structural insights on planar graph reachability and shortest path separators [Thorup'04].