Two-dijoin packing for large minimum dicut weight

Determine whether every weighted digraph whose minimum dicut weight is sufficiently large admits a packing of two dijoins.

Background

The minimum weight of a dicut, denoted by τ\tau, is an upper bound on the number of dijoins that can be packed under the arc weights. The paper notes that, despite progress toward the Edmonds–Giles conjecture, the existence of even two packed dijoins remains unresolved when τ\tau is sufficiently large. This question concerns general weighted digraphs and is not settled by the paper’s result for digraphs with chordal underlying graphs.

References

Despite many efforts, it is still not known whether there exists a packing of $2$ dijoins in a weighted digraph when $\tau$ is large enough , in contrast to the existence of a packing of ${\frac{\tau}{6}$ dijoins in unweighted digraphs .

— Packing Dijoins in Weighted Chordal Digraphs  (2501.10918 - Cornuéjols et al., 19 Jan 2025) in Section 1, Introduction