Directed Global Minimum Cut in Almost-Linear Time

We give a randomized reduction from global minimum cut in a directed graph with $n$ vertices and $m$ weighted edges to maximum-flow computations on graphs of total size $\tilde{O}(m)$. For polynomially bounded integral weights, this yields the first almost-linear $m^{1+o(1)}$-time algorithm [vdBCK+23], improving the previous $m^{1+o(1)}\min \{\sqrt{n},n\mathrm{/}{m}^{1\mathrm{/}3}\}$ bound [CLN+22]. Previous work reduced finding the minimum cut to constructing a 1-respecting arborescence, a directed spanning tree with exactly one edge crossing a minimum cut [CLN+22]. They then construct such a tree by sampling from a $(1+\epsilon )$-approximate arborescence packing, which currently requires superlinear time. Our algorithm sidesteps constructing the packing entirely and relies on a novel iterative arborescence sampling procedure. We show that, after $O(\log n)$ rounds of sampling, our final arborescence 1-respects the minimum cut with constant probability.