Crude Approximation of Directed Minimum Cut and Arborescence Packing in Almost Linear Time

2025-12-04

AI Summary

This paper studies the dual problems of minimum $s$-rooted cuts and maximum packing of $s$-arborescences in directed graphs. For an $n$-vertex, $m$-edge directed graph with $s$-rooted minimum cut value $k$, we present the first almost-linear-time algorithm—running in $m^{1+o(1)}$ time—that: (1) computes an $s$-rooted cut of size at most $O(k log^5 n)$; and (2) packs $k$ edge-disjoint $s$-arborescences with congestion $n^{o(1)}$, thereby certifying that the minimum $s$-rooted cut is at least $k / n^{o(1)}$. Our approach integrates graph-theoretic duality, weighted divide-and-conquer, and low-congestion tree decomposition techniques. Prior algorithms either incurred superlinear dependence on $m$ or polynomial dependence on $k$, limiting scalability. In contrast, our method achieves simultaneous breakthroughs in both approximation quality and runtime efficiency—establishing, for the first time, an efficient approximate duality between minimum rooted cuts and maximum arborescence packings in directed graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $O(klog^{5} n)$ 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 $ ilde{O}(mk)$ [Gab91] or $ ilde{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 $ ilde{O}(m cdot mathrm{poly}(k))$ time [BHKP08].

Problem

• Approximates rooted minimum cut in directed graphs efficiently
• Packs maximum arborescences with minimal congestion quickly
• Provides near-linear time algorithms for dual directed graph problems

Innovation

• Almost-linear-time algorithm for rooted minimum cut
• Almost-linear-time algorithm for arborescence packing
• Approximation with logarithmic factor and congestion