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 $R= \omega(\log^2 k)$.

翻译：在有向斯坦纳树（DST）问题中，输入为一个带权有向图 $G=(V,E)$、一个根顶点 $r$ 以及一个包含 $k$ 个终端的集合 $S \subseteq V$。目标是找到一个最小成本的子图，使得 $r$ 与每个终端相连。DST 在拟多项式时间内存在 $O(\log^2 k/\log \log k)$ 近似算法，在多项式时间内对任意固定 $\epsilon > 0$ 存在 $O(k^{\epsilon})$ 近似算法。是否存在多项式时间的多对数近似算法是近似算法领域的一个主要开放问题。在最近的工作中，Friggstad 和 Mousavi [ICALP 2023] 通过 Thorup 的最短路径分隔定理，为平面有向图中的 DST 问题提出了一个简洁优雅的多项式时间 $O(\log k)$ 近似算法。我们在其工作基础上，针对 DST 及相关问题获得了若干新结果。- 我们通过解释 Friggstad 和 Mousavi [ICALP 2023] 中的递归过程，开发了一种面向平面有向图中根节点问题的树嵌入技术。利用该技术，我们获得了平面有向图中群斯坦纳树、覆盖斯坦纳树及多拟阵斯坦纳树问题的多项式时间多对数近似算法。即使在树结构中，这些问题也难以在 $\Omega(\log^2 n/\log \log n)$ 因子内近似。- 我们证明 DST 问题基于割的自然线性规划松弛在平面图中具有 $O(\log^2 k)$ 的整数间隙。这与一般图形成对比，在一般图中该线性规划的整数间隙已知为 $\Omega(k)$ 及 $\Omega(n^{\delta})$（其中 $\delta > 0$ 为固定值）。- 我们将上述结果与基于密度的论证相结合，为平面有向图中多根版本的问题获得了多对数近似算法。对于 DST 问题，当 $R= \omega(\log^2 k)$ 时，我们的结果改进了 Friggstad 和 Mousavi [ICALP 2023] 提出的 $O(R + \log k)$ 近似算法。