DAG Covers: The Steiner Point Effect

2026-04-05

AI Summary

This study addresses the problem of constructing efficient DAG covers in directed graphs by introducing Steiner vertices, aiming to approximately preserve distances while controlling the number of additional edges. The work presents the first DAG covering schemes incorporating Steiner points for both bounded-treewidth digraphs and directed planar graphs, significantly outperforming traditional non-Steiner approaches. Leveraging graph decomposition, treewidth structural analysis, and planar embedding techniques, the authors design approximation algorithms that yield a (1,2,Õ(n·tw))-cover for digraphs of treewidth tw and a (1+ε,2,Õ_ε(n))-cover for directed planar graphs. Furthermore, they establish a theoretical lower bound of Ω(log n) on the number of DAGs required by any non-Steiner scheme, thereby demonstrating the necessity and advantage of Steiner points in this context.

Abstract

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.

Problem

• DAG cover
• Steiner point
• directed graph
• distance preservation
• treewidth

Innovation

• Steiner DAG cover
• directed graph
• tree decomposition
• planar digraph
• graph sparsification