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].

翻译：我们研究定向斯坦纳森林问题，这是网络设计中的一个基础问题。DSF的输入是一个带权有向图$G = (V, E)$和一个顶点对集合$\{(s_i, t_i)\}_{i \in [k]}$。目标是找到$G$的一个最小代价子图$H$，使得对于每个$i \in [k]$，$H$都包含一条$s_i$-$t_i$路径。DSF是NP难问题，并且已知对于任意固定的$\epsilon > 0$，其近似难度达到$\Omega(2^{\log^{1 - \epsilon}(n)})$因子[DK'99]。DSF已有的近似比包括$O(k^{1/2 + \epsilon})$[CEGS'11]和$O(n^{2/3 + \epsilon})$[BBMRY'13]。在本工作中，我们证明在平面有向图（理论和实践中一类重要且实用的图）中，DSF问题变得更容易处理。我们通过连接树技术获得了一个$O(\log^6 k)$近似算法。我们的主要技术贡献是证明了平面有向图中存在低密度连接树。为了寻找近似连接树，我们依赖于有根定向网络设计问题的最新研究成果[FM'23, CJKZZ'24]，特别是针对定向斯坦纳树问题的基于线性规划的算法[CJKZZ'24]。我们的工作以及近期其他关于平面有向图算法的研究[FM'23, KS'21, CJKZZ'24]，都建立在平面图可达性与最短路径分隔器的结构洞见之上[Thorup'04]。