Computing the (k+2)-Edge-Connected Components in k-Edge-Connected Digraphs in Subquadratic Time

2026-04-30

AI Summary

This work addresses the long-standing challenge of efficiently computing $(k+2)$-edge-connected components in directed graphs, breaking the classical $O(mn)$ time barrier. By introducing novel structural insights into directed edge cuts, a randomized strategy, and an improved graph decomposition technique, the authors present the first randomized algorithm that outperforms the classic Nagamochi–Watanabe algorithm for any constant $k > 3$, achieving a running time of $O(k^2 m \sqrt{n} \log n)$ when $k = o(n^{1/4}/\sqrt{\log n})$. The approach not only simplifies and generalizes the existing framework for 3-edge-connected components but also enables, for the first time, the computation of 4-edge-connected components in general directed graphs within $O(m \sqrt{n} \log n)$ time, yielding significant efficiency gains on sparse and moderately dense graphs.

Abstract

Computing edge-connected components in directed and undirected graphs is a fundamental and well-studied problem in graph algorithms. In a very recent breakthrough, Korhonen [STOC 2025] showed that for any fixed $k$, the $k$-edge connected components of an undirected graph can be computed in linear time. In contrast, the directed case remains significantly more challenging: linear-time algorithms are only known for $k \le 3$, and for any fixed $k > 3$, the best known bound for sparse or moderately dense graphs is still the $O(mn)$-time algorithm of Nagamochi and Watanabe (1993). In this paper, we break the $O(mn)$ barrier for all $k = o(n^{1/4}/\sqrt{\log{n}})$. We present a randomized algorithm that computes the $(k+2)$-edge-connected components of a $k$-edge-connected directed graph in $O(k^2 m \sqrt{n} \log n)$ time, for any~$k$. This constitutes the first improvement over the classic Nagamochi--Watanabe bound for any constant $k > 3$. Our approach introduces new structural insights into directed edge-cuts and combines these with both new and existing techniques. A central contribution of our work is a substantial simplification and generalization of the framework introduced in~\cite{GKPP:3ECC}, which achieved an $\widetilde{O}(m\sqrt{m})$ bound for computing the $3$-edge-connected components of a digraph. In addition, we develop a variant of our algorithm that achieves the same $O(m \sqrt{n} \log n)$ running time for computing the $4$-edge-connected components of a \emph{general} directed graph.

Problem

• edge-connected components
• directed graphs
• graph algorithms
• k-edge-connected
• subquadratic time

Innovation

• edge-connected components
• directed graphs
• subquadratic algorithm
• randomized algorithm
• graph connectivity