Parametrised Algorithms for Directed Modular Width

TL;DR

This paper explores directed modular width as a parameter for directed graphs, developing fixed-parameter tractable algorithms for several NP-hard problems and showing its effectiveness in structural graph analysis.

Contribution

It introduces directed modular width as a new parameter and provides FPT algorithms for multiple NP-hard problems on digraphs, extending structural graph theory tools.

Findings

FPT algorithms for directed feedback vertex set and dominating set

Polynomial-time detection of topological minors in bounded directed modular width digraphs

Efficient computation of structural parameters like directed pathwidth and cycle-rank

Abstract

Many well-known NP-hard algorithmic problems on directed graphs resist efficient parametrisations with most known width measures for directed graphs, such as directed treewidth, DAG-width, Kelly-width and many others. While these focus on measuring how close a digraph is to an oriented tree resp. a directed acyclic graph, in this paper, we investigate directed modular width as a parameter, which is closer to the concept of clique-width. We investigate applications of modular decompositions of directed graphs to a wide range of algorithmic problems and derive FPT-algorithms for several well-known digraph-specific NP-hard problems, namely minimum (weight) directed feedback vertex set, minimum (weight) directed dominating set, digraph colouring, directed Hamiltonian path/cycle, partitioning into paths, (capacitated) vertex-disjoint directed paths, and the directed subgraph homeomorphism problem. The latter yields a polynomial-time algorithm for detecting topological minors in digraphs of bounded directed modular width. Finally we illustrate that also other structural digraph parameters, such as the directed pathwidth and the cycle-rank can be computed efficiently using directed modular width as a parameter.