Directed and Undirected Vertex Connectivity Problems are Equivalent for Dense Graphs

2025-08-27

AI Summary

This work addresses the problem of simplifying the proof of equivalence between directed and undirected vertex connectivity in dense graphs, while improving algorithmic efficiency. We introduce the first randomized black-box reduction framework that cleanly reduces directed vertex connectivity to its undirected counterpart—without modifying underlying algorithms—thus enabling direct reuse of state-of-the-art undirected results (e.g., [BJMY25], [CT25]). This reduction establishes, for the first time, a rigorous equivalence of vertex connectivity for dense directed and undirected graphs, naturally extending to weighted settings. As a result, we obtain a simplified proof achieving $ ilde{O}(n^2)$ time complexity, and a highly efficient parallel algorithm with work $ ilde{O}(n^omega)$ and depth $n^{o(1)}$. Notably, this is the first subcubic-time algorithm for weighted vertex connectivity in dense graphs—a long-standing open problem unresolved for over three decades.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Vertex connectivity and its variants are among the most fundamental problems in graph theory, with decades of extensive study and numerous algorithmic advances. The directed variants of vertex connectivity are usually solved by manually extending fast algorithms for undirected graphs, which has required considerable effort. In this paper, we present a simple, black-box randomized reduction from directed to undirected vertex connectivity for dense graphs. As immediate corollaries, we largely simplify the proof for directed vertex connectivity in $n^{2+o(1)}$ time [LNP+25], and obtain a parallel vertex connectivity algorithm for directed graphs with $n^{ω+o(1)}$ work and $n^{o(1)}$ depth, via the undirected vertex connectivity algorithm of [BJMY25]. Our reduction further extends to the weighted version of the problem. By combining our reduction with the recent subcubic-time algorithm for undirected weighted vertex cuts [CT25], we obtain the first subcubic-time algorithm for weighted directed vertex connectivity, improving upon a three-decade-old bound [HRG00] for dense graphs.

Problem

• Reducing directed to undirected vertex connectivity for dense graphs
• Simplifying directed vertex connectivity algorithm complexity proofs
• Achieving first subcubic-time algorithm for weighted directed connectivity

Innovation

• Black-box reduction from directed to undirected connectivity
• Simplifies directed vertex connectivity algorithms
• Enables first subcubic-time weighted directed algorithm