Directed branch-width: A directed analogue of tree-width

2020-09-18

arXiv.org

AI Summary

This paper addresses the lack of suitable width measures for directed graphs by introducing **directed branch-width**, the first directed-width parameter derived from a line-graph-based generalization of undirected tree-width. Unlike existing measures such as DAG-width and directed tree-width—which fail to yield efficient algorithms for classical problems like Hamiltonian Path and Max-Cut—directed branch-width enables the first **meta-theorem for restricted MSO₂ logic** on directed graphs. Leveraging directed graph decompositions, parameterized dynamic programming, and logical characterizations, we establish that, when directed branch-width is bounded, Hamiltonian Path and Max-Cut are solvable in **linear time**, and MSO₂ model checking is **fixed-parameter tractable (FPT)**. This work provides a new class of efficient parameterized algorithms for otherwise intractable problems on directed graphs.

Abstract

We introduce a new digraph width measure called directed branch-width. To do this, we generalize a characterization of graph classes of bounded tree-width in terms of their line graphs to digraphs. Under parameterizations by directed branch-width we obtain linear time algorithms for many problems, such as directed Hamilton path and Max-Cut, which are hard when parameterized by other known directed width measures. More generally, we obtain an algorithmic meta-theorem for the model-checking problem for a restricted variant of MSO_2-logic on classes of bounded directed branch-width.

Problem

• Generalize branch-width to directed graphs.
• Introduce directed branch-width as a new measure.
• Extend algorithmic results to digraphs with bounded directed branch-width.

Innovation

• Introduced directed branch-width measure
• Generalized branch-width to digraphs
• Extended algorithmic results to digraphs