Given a weighted digraph $G$, a $(t,g,μ)$-DAG cover is a collection of $g$ dominating DAGs $D_1,\dots,D_g$ such that all distances are approximately preserved: for every pair $(u,v)$ of vertices, $\min_id_{D_i}(u,v)\le t\cdot d_{G}(u,v)$, and the total number of non-$G$ edges is bounded by $|(\cup_i D_i)\setminus G|\le μ$. Assadi, Hoppenworth, and Wein [STOC 25] and Filtser [SODA 26] studied DAG covers for general digraphs. This paper initiates the study of \emph{Steiner} DAG cover, where the DAGs are allowed to contain Steiner points. We obtain Steiner DAG covers on the important classes of planar digraphs and low-treewidth digraphs. Specifically, we show that any digraph with treewidth tw admits a $(1,2,\tilde{O}(n\cdot tw))$-Steiner DAG cover. For planar digraphs we provide a $(1+\varepsilon,2,\tilde{O}_\varepsilon(n))$-Steiner DAG cover. We also demonstrate a stark difference between Steiner and non-Steiner DAG covers. As a lower bound, we show that any non-Steiner DAG cover for graphs with treewidth $1$ with stretch $t<2$ and sub-quadratic number of extra edges requires $Ω(\log n)$ DAGs.

翻译：给定一个加权有向图 $G$，一个 $(t,g,μ)$-有向无环图覆盖是指集合包含 $g$ 个支配性有向无环图 $D_1,\dots,D_g$，使得所有距离近似保持：对于每对顶点 $(u,v)$，有 $\min_i d_{D_i}(u,v)\le t\cdot d_{G}(u,v)$，且非 $G$ 边的总数受限于 $|(\cup_i D_i)\setminus G|\le μ$。Assadi、Hoppenworth 和 Wein [STOC 25] 以及 Filtser [SODA 26] 研究了针对一般有向图的有向无环图覆盖问题。本文首次研究了*斯坦纳*有向无环图覆盖，其中允许有向无环图包含斯坦纳点。我们在平面有向图和低树宽有向图这两类重要图上获得了斯坦纳有向无环图覆盖。具体而言，我们证明任何树宽为 tw 的有向图都允许一个 $(1,2,\tilde{O}(n\cdot tw))$-斯坦纳有向无环图覆盖。对于平面有向图，我们给出了一个 $(1+\varepsilon,2,\tilde{O}_\varepsilon(n))$-斯坦纳有向无环图覆盖。我们还展示了斯坦纳与非斯坦纳有向无环图覆盖之间的显著差异。作为一个下界，我们证明对于树宽为 $1$、拉伸因子 $t<2$ 且额外边数为亚二次的图，任何非斯坦纳有向无环图覆盖都需要 $Ω(\log n)$ 个有向无环图。