DAG Covers: The Steiner Point Effect

TL;DR

This paper introduces Steiner DAG covers, allowing Steiner points, and provides constructions for planar and low-treewidth digraphs, highlighting differences from non-Steiner covers.

Contribution

It initiates the study of Steiner DAG covers, offering new bounds for planar and low-treewidth digraphs, and contrasts Steiner and non-Steiner cover complexities.

Findings

Steiner DAG covers exist for graphs with bounded treewidth.

Planar digraphs admit near-optimal Steiner DAG covers.

Non-Steiner covers require logarithmic number of DAGs for certain cases.

Abstract

Given a weighted digraph $G$, a $(t,g,\mu)$-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 $|(\cup_i D_i)\setminus G|\le \mu$. 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 $\Omega(\log n)$ DAGs.