DAG Covers: The Steiner Point Effect

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.

Figures (3)

• Figure 1: On top: Illustration of the DAG $D_x$ created using \ref{['lem:vertexGadget']} for the digraph $G$ (on the left) with respect to the designated vertex $x$ ($v_8$), and the order $(v_1,v_2,\dots,v_{10})$. The DAG $D_x$ contains an auxiliary vertex $u_i$ for each $v_i$. We add a $0$ weight edge between any two consecutive $u_i,u_{i+1}$. In addition for every $i$ there is an edge from $u_i$ to $v_i$ of weight $d_G(x,v_i)$, and from $v_i$ to $u_{i+1}$ of weight $d_G(v_i,x)$. For every $i<j$, $d_{D_x}(v_i,v_j)\le d_{D_x}(v_i,u_{i+1})+d_{D_x}(u_{i+1},u_j)+d_{D_x}(u_j,v_j)=d_G(v_i,x)+0+d_G(x,v_j)$. That is, the shortest $v_i$-$v_j$ path that goes through $x$. On bottom: Illustration of the bidirected star $\mathcal{S}_7$ (used in the proof of \ref{['thm:bidirectedLB']}), and its DAG cover. Here we take the order $\sigma=(v_1,\dots,v_7)$ where the root vertex $v_1=\hbox{\rm rt}$ is first. $D_1=D_{\hbox{\rm rt}}$ is created using \ref{['lem:vertexGadget']} with respect to vertex $x$ and order $\sigma$. $D_2$ is created in the same way, but with respect to order $\sigma^{-1}$. As all shortest paths in $\mathcal{S}_7$ go through $\hbox{\rm rt}$, $D_1\& D_2$ preserve all the distances exactly.
• Figure 2: On top: Illustration of the path $P\in\mathcal{P}$ and how it is broken to subpaths by deleting centroid vertices. The hierarchical tree of the subpaths is drawn using a thin gray line. For each vertex $y_i\in P$, $\mathcal{Q}(y_i,P)$ contains all the subpaths containing $y_i$. Consider two vertices $u<_\sigma v$ such that $P\in \mathcal{P}[u]\cap \mathcal{P}[v]$, and there are vertices $u'\in C(u,p)$, $v'\in C(v,p)$ where $d_G(u,u') + d_P(u',v') + d_G(v', v) \le (1+\varepsilon) d_G(u,v)$. In the illustration, $u'=y_7$ and $v'=y_{11}$. The subpath $P_2$ contains both $y_7,y_{11}$, and thus both $X_u$ and $X_v$ will contain $x(P_2)=y_9$, the centroid of $P_2$. On bottom: Illustrated the DAG $D_{y_9}$ over $A_{y_9}$ that contains both $u,v$ (constructed using \ref{['lem:vertexGadget']}). As $u<_\sigma v$, $d_{D_{y_9}}\le d_G(u,y_9)+d_G(y_9,v)\le (1+\varepsilon)\cdot d_{G}(u,v)$.
• Figure :

Theorems & Definitions (11)

• Definition 1: DAG Cover
• Theorem 1: Non-Steiner LB
• Theorem 2
• proof
• Remark 1
• Theorem 3
• proof
• Lemma 1
• proof
• Theorem 4