Computing $\vec{\mathcal{S}}$-DAGs and Parity Games

TL;DR

This paper introduces the $oldsymbol{ ext{ extbf{ extit{S}}}}$-DAG-width, a new directed graph width measure, and provides algorithms for computing it and solving parity games on graphs with bounded $oldsymbol{ ext{ extbf{ extit{S}}}}$-DAG-width.

Contribution

It defines the $oldsymbol{ ext{ extbf{ extit{S}}}}$-DAG-width, establishes its structural duality, and presents the first algorithms for computing this measure and solving parity games on such graphs.

Findings

Introduces the $oldsymbol{ ext{ extbf{ extit{S}}}}$-DAG-width as a new directed width measure.

Provides an algorithm to compute $oldsymbol{ ext{ extbf{ extit{S}}}}$-DAG-width of a digraph.

Extends polynomial-time solvability of parity games to graphs with bounded $oldsymbol{ ext{ extbf{ extit{S}}}}$-DAG-width.

Abstract

Treewidth on undirected graphs is known to have many algorithmic applications. When considering directed width-measures there are much less results on their deployment for algorithmic results. In 2022 the first author, Rabinovich and Wiederrecht introduced a new directed width measure, $\vec{\mathcal{S}}$-DAG-width, using directed separations and obtained a structural duality for it. In 2012 Berwanger~et~al.~solved Parity Games in polynomial time on digraphs of bounded DAG-width. With generalising this result to digraphs of bounded $\vec{\mathcal{S}}$-DAG-width and also providing an algorithm to compute the $\vec{\mathcal{S}}$-DAG-width of a given digraphs we give first algorithmical results for this new parameter.