Approximating Directed Minimum Cut and Arborescence Packing via Directed Expander Hierarchies

TL;DR

This paper introduces almost-linear-time algorithms for approximating rooted minimum cuts and maximum arborescence packings in directed graphs, significantly improving computational efficiency over previous methods.

Contribution

It presents novel algorithms that approximate rooted minimum cuts and pack arborescences efficiently in directed graphs, with near-linear runtime and improved approximation guarantees.

Findings

Algorithms run in $m^{1+o(1)}$ time for large graphs.

Achieves $O(k ext{log}^5 n)$ approximation for rooted minimum cut.

Provides $k$ arborescences with $n^{o(1)}$ congestion, certifying cut bounds.

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].