Approximately Packing Dijoins via Nowhere-Zero Flows

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 $\tau$ contains $\left\lfloor\frac{\tau}{k}\right\rfloor$ 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 $\left\lfloor\frac{\tau}{6}\right\rfloor$ disjoint dijoins in a digraph with minimum dicut size $\tau$, 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 (Lov\'asz et al. 2013) directly leads to the existence of $\left\lfloor\frac{\tau p}{2p+1}\right\rfloor$ disjoint dijoins in a digraph with minimum dicut size $\tau$ whose underlying undirected graph is $6p$-edge-connected. We also discuss reformulations of Woodall's conjecture into packing strongly connected orientations.