Incremental Directed Minimum Cut by Dynamizing Gabow's Algorithm

2026-08-17

AI Summary

This study addresses the absence of efficient incremental algorithms for global minimum cuts in directed graphs, where existing methods are restricted to k≤2 or undirected settings. We propose the first incremental algorithm supporting arbitrary k values by dynamizing Gabow’s static algorithm and designing a deterministic update strategy that explicitly maintains the minimum cut or verifies its value is at least k during edge insertions. Achieving O(km log n) total update time without asymptotic loss, this approach overcomes prior limitations and fills a critical theoretical gap in strictly incremental extensions for directed graphs. Consequently, our method demonstrates significant performance improvements over previous state-of-the-art techniques while maintaining optimal complexity bounds for general k-connectivity verification in dynamic directed networks.

Abstract

We give the first incremental algorithm for directed global minimum cut. Given a directed graph with $n$ vertices undergoing $m$ edge insertions, our deterministic algorithm explicitly maintains a global minimum cut or certifies that its value is at least $k$ in $O(km\log n)$ total update time. Prior work required either that $k\le2$ or that the graph is undirected. Our algorithm is a strict incremental extension of Gabow's state-of-the-art static algorithm (JCSS 1995), with no asymptotic loss in running time over the entire insertion sequence.

Problem

• Directed Global Minimum Cut
• Incremental Algorithm
• Dynamic Graph
• Edge Insertions

Innovation

• Incremental Algorithm
• Directed Global Minimum Cut
• Gabow's Algorithm
• Dynamic Graph
• Edge Insertions