Packing Dijoins in Weighted Chordal Digraphs

TL;DR

This paper proves the Edmonds-Giles conjecture for weighted digraphs with chordal underlying graphs and provides a polynomial-time algorithm to find maximum dijoins, advancing understanding in digraph connectivity and packing problems.

Contribution

It establishes the conjecture's validity for chordal graphs and introduces an efficient algorithm for packing dijoins in this class.

Findings

Confirmed the conjecture for chordal graphs

Developed a strongly polynomial algorithm for packing dijoins

Extended understanding of dicut and dijoin relationships in weighted digraphs

Abstract

In a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects every dicut. Edmonds and Giles conjectured that in a weighted digraph, the minimum weight of a dicut is equal to the maximum size of a packing of dijoins. This has been disproved. However, the unweighted version conjectured by Woodall remains open. We prove that the Edmonds-Giles conjecture is true if the underlying undirected graph is chordal. We also give a strongly polynomial time algorithm to construct such a packing.