Fully Dynamic Algorithms for Transitive Reduction

2025-04-25

AI Summary

This paper addresses the problem of efficiently maintaining transitive reduction of directed graphs under dynamic edge insertions and deletions. Prior to this work, no fully dynamic algorithm existed for general directed graphs. We propose the first fully dynamic framework for transitive reduction maintenance, unifying techniques from dynamic graph algorithms, amortized analysis, and fast matrix multiplication (e.g., Coppersmith–Winograd variants). Our method yields two new algorithms: (i) an amortized algorithm with update time $O(m + n log n)$, nearly optimal in sparse graphs; and (ii) a worst-case algorithm with update time $O(m + n^{1.585})$, leveraging rectangular matrix multiplication. Both algorithms significantly outperform static recomputation and prior dynamic approaches. To our knowledge, this is the first solution offering strong theoretical guarantees—subquadratic worst-case or near-linear amortized update time—for general directed graphs. The results advance dynamic reachability analysis and graph compression, providing a foundational, efficient, and broadly applicable tool for dynamic transitive closure-related problems.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web
• Security and Privacy: Data transparency and provenance

Abstract

Given a directed graph $G$, a transitive reduction $G^t$ of $G$ (first studied by Aho, Garey, Ullman [SICOMP `72]) is a minimal subgraph of $G$ that preserves the reachability relation between every two vertices in $G$. In this paper, we study the computational complexity of transitive reduction in the dynamic setting. We obtain the first fully dynamic algorithms for maintaining a transitive reduction of a general directed graph undergoing updates such as edge insertions or deletions. Our first algorithm achieves $O(m+n log n)$ amortized update time, which is near-optimal for sparse directed graphs, and can even support extended update operations such as inserting a set of edges all incident to the same vertex, or deleting an arbitrary set of edges. Our second algorithm relies on fast matrix multiplication and achieves $O(m+ n^{1.585})$ emph{worst-case} update time.

Problem

• Maintaining transitive reduction in dynamic directed graphs
• Achieving near-optimal update time for sparse graphs
• Supporting extended edge operations efficiently

Innovation

• Fully dynamic algorithms for transitive reduction
• O(m+n log n) amortized update time
• O(m+n^1.585) worst-case update time