Approximating Directed Minimum Cut and Arborescence Packing via Directed Expander Hierarchies
Yonggang Jiang, Yaowei Long, Thatchaphol Saranurak, Benyu Wang
TL;DR
This work delivers the first almost-linear-time algorithms for two fundamental directed graph problems: the $s$-rooted minimum cut and the packing of $s$-rooted arborescences. It introduces directed expander hierarchies as a core tool, achieving an $O(k\log^5 n)$-approximate $s$-rooted min-cut in $m^{1+o(1)}$ time (extendable to weighted graphs) and, alternatively, a near-optimal arborescence packing with $n^{o(1)}$ congestion in the same running time, or a certificate of a lower bound on connectivity. The methods leverage a hierarchical decomposition, precise sampling, max-flow subroutines, and expander routing to transform global problems into tractable local subproblems. This yields significant speedups for directed connectivity problems and provides a framework that could adapt to related routing and packing challenges in directed graphs.
Abstract
We give almost-linear-time algorithms for approximating rooted minimum cut and maximum arborescence packing in directed graphs, two problems that are dual to each other [Edm73]. More specifically, for an $n$-vertex, $m$-edge directed graph $G$ whose $s$-rooted minimum cut value is $k$, our first algorithm computes an $s$-rooted cut of size at most $O(k\log^{5} n)$ in $m^{1+o(1)}$ time, and our second algorithm packs $k$ $s$-rooted arborescences with $n^{o(1)}$ congestion in $m^{1+o(1)}$ time, certifying that the $s$-rooted minimum cut is at least $k / n^{o(1)}$. Our first algorithm also works for weighted graphs. Prior to our work, the fastest algorithms for computing the $s$-rooted minimum cut were exact but had super-linear running time: either $\tilde{O}(mk)$ [Gab91] or $\tilde{O}(m^{1+o(1)}\min\{\sqrt{n},n/m^{1/3}\})$ [CLN+22]. The fastest known algorithms for packing $s$-rooted arborescences had no congestion, but required $\tilde{O}(m \cdot \mathrm{poly}(k))$ time [BHKP08].