Approximately Packing Dijoins via Nowhere-Zero Flows

TL;DR

The paper addresses Woodall's conjecture on packing disjoint dijoins by establishing approximate packing bounds via nowhere-zero flows. It develops a reduction that augments a digraph to a strongly connected framework and uses nowhere-zero (circular) $k$-flows to partition the augmented graph into two $\left\lfloor\frac{\tau}{k}\right\rfloor$-strongly-connected digraphs, enabling the extraction of $\left\lfloor\frac{\tau}{k}\right\rfloor$ disjoint dijoins through Edmonds' disjoint arborescences. For $k=6$, Seymour's result guarantees a nowhere-zero $6$-flow in $2$-edge-connected graphs, yielding $\left\lfloor\frac{\tau}{6}\right\rfloor$ disjoint dijoins, with a polynomial-time algorithmic construction; more generally, $6p$-edge-connected graphs admit nowhere-zero circular $(2+\frac{1}{p})$-flows, giving $\left\lfloor\frac{\tau p}{2p+1}\right\rfloor$ disjoint dijoins. The article also reformulates Woodall's conjecture in terms of packing strongly connected orientations, linking to strengthenings and SCOs, and discusses extensions to decompositions of nowhere-zero $ au$-SCDs. Overall, the work advances approximate packing of dijoins via flow-based decompositions, clarifies the role of connectivity, and provides polynomial-time methods for key cases, moving toward resolving Woodall's conjecture in broad classes of 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 each dicut. Woodall conjectured in 1976 that in every digraph, the minimum size of a dicut equals to the maximum number of disjoint dijoins. However, prior to our work, it was not even known whether at least $3$ disjoint dijoins exist in an arbitrary digraph whose minimum dicut size is sufficiently large. By building connections with nowhere-zero (circular) $k$-flows, we prove that every digraph with minimum dicut size $τ$ contains $\left\lfloor\fracτ{k}\right\rfloor$ disjoint dijoins if the underlying undirected graph admits a nowhere-zero (circular) $k$-flow. The existence of nowhere-zero $6$-flows in $2$-edge-connected graphs (Seymour 1981) directly leads to the existence of $\left\lfloor\fracτ{6}\right\rfloor$ disjoint dijoins in a digraph with minimum dicut size $τ$, which can be found in polynomial time as well. The existence of nowhere-zero circular $\frac{2p+1}{p}$-flows in $6p$-edge-connected graphs (Lovász et al. 2013) directly leads to the existence of $\left\lfloor\frac{τp}{2p+1}\right\rfloor$ disjoint dijoins in a digraph with minimum dicut size $τ$ whose underlying undirected graph is $6p$-edge-connected. We also discuss reformulations of Woodall's conjecture into packing strongly connected orientations.