An Algorithmic Metatheorem for Directed Treewidth

TL;DR

This paper establishes a new algorithmic metatheorem linking monadic second order logic to directed treewidth, enabling polynomial algorithms for complex digraph problems and introducing novel width measures and formal tools.

Contribution

It provides the first algorithmic metatheorem connecting monadic second order logic to directed treewidth and introduces new tools like tree-zig-zag number and z-saturated tree slice language.

Findings

Polynomial algorithms for counting minimal spanning strong subgraphs of bounded directed treewidth.

Polynomial algorithms for maximal subgraphs with union of k paths satisfying minor closed properties.

Most positive algorithmic results for digraphs of constant directed treewidth can be reformulated using the new metatheorem.

Abstract

The notion of directed treewidth was introduced by Johnson, Robertson, Seymour and Thomas [Journal of Combinatorial Theory, Series B, Vol 82, 2001] as a first step towards an algorithmic metatheory for digraphs. They showed that some NP-complete properties such as Hamiltonicity can be decided in polynomial time on digraphs of constant directed treewidth. Nevertheless, despite more than one decade of intensive research, the list of hard combinatorial problems that are known to be solvable in polynomial time when restricted to digraphs of constant directed treewidth has remained scarce. In this work we enrich this list by providing for the first time an algorithmic metatheorem connecting the monadic second order logic of graphs to directed treewidth. We show that most of the known positive algorithmic results for digraphs of constant directed treewidth can be reformulated in terms of our metatheorem. Additionally, we show how to use our metatheorem to provide polynomial time algorithms for two classes of combinatorial problems that have not yet been studied in the context of directed width measures. More precisely, for each fixed $k,w \in \mathbb{N}$, we show how to count in polynomial time on digraphs of directed treewidth $w$, the number of minimum spanning strong subgraphs that are the union of $k$ directed paths, and the number of maximal subgraphs that are the union of $k$ directed paths and satisfy a given minor closed property. To prove our metatheorem we devise two technical tools which we believe to be of independent interest. First, we introduce the notion of tree-zig-zag number of a digraph, a new directed width measure that is at most a constant times directed treewidth. Second, we introduce the notion of $z$-saturated tree slice language, a new formalism for the specification and manipulation of infinite sets of digraphs.