Exact and Heuristic Computation of the Scanwidth of Directed Acyclic Graphs

TL;DR

The first algorithm that efficiently computes the exact scanwidth of general DAGs is presented, proving that the scanwidth is bounded from below by the treewidth of the underlying undirected graph, and experiments suggest that for networks the parameters are close in practice.

Abstract

To measure the tree-likeness of a directed acyclic graph (DAG), a new width parameter that considers the directions of the arcs was recently introduced: scanwidth. We present the first algorithm that efficiently computes the exact scanwidth of general DAGs. For DAGs with one root and scanwidth $k$ it runs in $O(k \cdot n^k \cdot m)$ time. The algorithm also functions as an FPT algorithm with complexity $O(2^{4 \ell - 1} \cdot \ell \cdot n + n^2)$ for phylogenetic networks of level-$\ell$, a type of DAG used to depict evolutionary relationships among species. Our algorithm performs well in practice, being able to compute the scanwidth of synthetic networks up to 30 reticulations and 100 leaves within 500 seconds. Furthermore, we propose a heuristic that obtains an average practical approximation ratio of 1.5 on these networks. While we prove that the scanwidth is bounded from below by the treewidth of the underlying undirected graph, experiments suggest that for networks the parameters are close in practice.

Figures (17)

• Figure 1: A DAG with a single root (a), and a tree extension of the DAG (b), functioning as a route for the scanner lines. The tree extension is indicated by the grey edges, while the arcs of the DAG are drawn back in, following the edges of the tree extension. The scanner lines start at the leaves and move up through the tree extension, at each step becoming brighter red. The tree extension is optimal, thus the scanwidth of this DAG is 3: the maximum number of DAG arcs that are cut by one of the scanner lines.
• Figure 2: (a): Weakly connected DAG $G$. (b): An extension $\sigma$ of $G$ with the arcs of $G$ also drawn. (c): A tree extension $\Gamma$ of $G$ indicated by the grey arcs, whose direction is downwards. The arcs of $G$ are also drawn in $\Gamma$ and are made to follow the grey arcs. (d): A tree $H$ on $V(G)$ that is not a tree extension, because $z$ and $c$ are not comparable in $H$, while they are adjacent in $G$. Visually this means that the corresponding thick red arc $zc$ of $G$ 'crosses' two branches of the tree.
• Figure 3: (a): Weakly connected DAG $G$. (b): An optimal extension $\sigma$ of $G$ with cutwidth 4, attained at the red cut. (c): A non-optimal extension $\pi$ of $G$ with cutwidth 5, attained at the red cut.
• Figure 4: (a): Weakly connected, rooted DAG $G$. (b): Optimal canonical tree extension $\Gamma^\sigma$ with scanwidth 3, attained at the vertex $v$. (c): Non-canonical tree extension $\Gamma'$ with scanwidth 4, attained at the vertex $z$. (d): Optimal extension $\sigma$ with scanwidth 3, attained at the vertex $v$. For each $i\leq 8$, the outermost grey shaded areas containing only vertices belonging to $\sigma [1 \ldots i]$ depict the weakly connected components of $G[1 \ldots i]$. For $i \geq 8$, $G[1 \ldots i]$ is weakly connected and therefore consists of just one component.
• Figure 5: (a): The ladder-graph$L_n$ (with $n \geq 3$), which is a rooted binary network with level $n-1$ and $2n$ vertices. (b): An optimal tree extension $\Gamma_1$ of $L_n$ with scanwidth 3. (c): The worst-case tree extension $\Gamma_2$ of $L_n$ with scanwidth $n$.

Theorems & Definitions (57)

• Definition 2.1: Cutwidth
• Definition 2.2: Scanwidth
• Definition 2.3: Scanwidth
• Definition 2.4: Canonical tree extension
• Proposition 2.4
• Definition 2.5: Treewidth
• Lemma 2.5
• Lemma 2.5
• Definition 3.1: S-block
• Theorem 3.2