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.

翻译：给定一个有向图 $G$，其传递归约 $G^t$（由 Aho、Garey、Ullman [SICOMP `72] 首次研究）是 $G$ 的一个极小子图，它保持了 $G$ 中每对顶点之间的可达性关系。本文研究了动态设置下传递归约的计算复杂性。我们首次提出了完全动态算法，用于维护一个经历边插入或删除等更新的通用有向图的传递归约。我们的第一个算法实现了 $O(m+n \log n)$ 的摊还更新时间，这对于稀疏有向图是接近最优的，并且甚至可以支持扩展更新操作，例如插入一组全部关联于同一顶点的边，或删除任意一组边。我们的第二个算法依赖于快速矩阵乘法，实现了 $O(m+ n^{1.585})$ 的 \emph{最坏情况} 更新时间。