A Min-Max Relation on Dicuts and Dijoins in Weighted Chordal Digraphs
It resolves a special case of a known conjecture for a specific graph class, providing algorithmic implications for chordal digraphs.
The paper proves that the Edmonds-Giles conjecture on dicuts and dijoins holds for weighted chordal digraphs, and provides a strongly polynomial-time algorithm for constructing a packing of dijoins achieving the maximum size.
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.