Let G be a directed weighted graph (DiGraph) on n vertices and m edges with source s and sink t. An edge in G is vital if its removal reduces the capacity of (s,t)-mincut. Since the seminal work of Ford and Fulkerson, a long line of work has been done on computing the most vital edge and all vital edges of G. Unfortunately, after 60 years, the existing results are for undirected or unweighted graphs. We present the following result for DiGraph, which solves an open problem stated by Ausiello et al. 1. There is an algorithm that computes all vital edges as well as the most vital edge of G using O(n) maxflow computations. Vital edges play a crucial role in the design of Sensitivity Oracle (SO) for (s,t)-mincut. For directed graphs, the only existing SO is for unweighted graphs by Picard and Queyranne. We present the first and optimal SO for DiGraph. 2. (a) There is an O(n) space SO that can report in O(1) time the capacity of (s,t)-mincut and (b) an O($n^2$) space SO that can report an (s,t)-mincut in O(n) time after failure/insertion of an edge. For unweighted graphs, Picard and Queyranne designed an O(m) space DAG that stores and characterizes all mincuts for all vital edges. Conversely, there is a set containing at most n-1 (s,t)-cuts such that at least one mincut for every vital edge belongs to the set. We generalize these results for DiGraph. 3. (a) There is a set containing at most n-1 (s,t)-cuts such that at least one mincut for every vital edge is present in the set. (b) We design two compact structures for storing and characterizing all mincuts for all vital edges, (i) O(m) space DAG for partial characterization and (ii) O(mn) space structure for complete characterization. To arrive at our results, we develop new techniques, especially a generalization of maxflow-mincut theorem by Ford and Fulkerson, which might be of independent interest.

翻译：设G为具有n个顶点、m条边的有向加权图，源点为s，汇点为t。若移除某条边会降低(s,t)-最小割的容量，则该边称为关键边。自Ford与Fulkerson的开创性工作以来，关于计算图G中最关键边及所有关键边的研究已形成大量工作。然而经过60年，现有结果仅适用于无向图或无权图。本文针对有向图给出如下结果，解决了Ausiello等人提出的开放问题：1. 存在一种算法，通过O(n)次最大流计算即可求出G的所有关键边及最关键边。关键边在(s,t)-最小割的灵敏度预知器设计中起着核心作用。对于有向图，现存唯一的灵敏度预知器是Picard与Queyranne针对无权图提出的。我们提出了有向图首个且最优的灵敏度预知器：2. (a) 存在一种O(n)空间的灵敏度预知器，可在O(1)时间内报告(s,t)-最小割的容量；(b) 另存在一种O(n²)空间的灵敏度预知器，可在单条边失效/插入后于O(n)时间内报告一个(s,t)-最小割。Picard与Queyranne为无权图设计了O(m)空间的有向无环图，用于存储并刻画所有关键边对应的最小割。反之，存在一个至多包含n-1个(s,t)-割的集合，使得每条关键边至少对应一个属于该集合的最小割。我们将这些结果推广至有向图：3. (a) 存在一个至多包含n-1个(s,t)-割的集合，使得每条关键边至少有一个最小割存在于该集合中；(b) 我们设计了两种紧凑结构用于存储并刻画所有关键边对应的最小割：(i) 用于部分刻画的O(m)空间有向无环图，以及(ii) 用于完整刻画的O(mn)空间结构。为得到上述结果，我们发展了新技术，特别是对Ford-Fulkerson最大流-最小割定理的推广，该推广本身可能具有独立的研究价值。