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 (Lovász 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.