Packing Dijoins in Weighted Chordal Digraphs

TL;DR

The paper addresses a min-max relation between the minimum weight of a dicut and the maximum packing of dijoins in weighted digraphs, focusing on conditions under which the Edmonds-Giles conjecture holds. It proves that the conjecture is true for digraphs whose underlying undirected graph is chordal and provides a strongly polynomial-time algorithm to construct a packing achieving the bound. The approach hinges on a perfect elimination scheme: remove a simplicial vertex, transfer incident weights to its neighbor clique to keep the minimum dicut weight, and recursively solve on the smaller digraph, then map the solution back to the original graph. This result broadens the classes where the conjecture holds, connects to Woodall’s unweighted conjecture and poset comparability digraphs, and yields an explicit $O(m^2+n)$-time algorithm.

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.