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：令G为具有源点s和汇点t的n个顶点、m条边的有向加权图。若删除某条边会降低(s,t)-最小割的容量，则称该边为关键边。自Ford与Fulkerson开创性工作以来，计算G中最关键边及所有关键边的问题已获长期研究。然而历经60年，现有结果仅适用于无向图或无权图。本文针对有向图提出以下成果，解决了Ausiello等人提出的公开问题：(1) 存在算法可通过O(n)次最大流计算，求得所有关键边及最关键边。关键边在(s,t)-最小割的敏感性预言设计中起核心作用。针对有向图，现有唯一敏感性预言仅适用于Picard与Queyranne提出的无权图。本文首次提出有向图最优敏感性预言：(2)(a) 存在O(n)空间的敏感性预言，可在O(1)时间内报告(s,t)-最小割容量；(b) 存在O(n²)空间的敏感性预言，可在边失效/插入后O(n)时间内报告(s,t)-最小割。对于无权图，Picard与Queyranne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，存在含至多n-1个(s,t)-割的集合，使得每个关键边至少有一个最小割属于该集合。本文将上述结论推广至有向图：(3)(a) 存在含至多n-1个(s,t)-割的集合，使得每个关键边至少有一个最小割出现在该集合中；(b) 我们设计了两种紧凑结构用于存储和刻画所有关键边对应的最小割：(i) 部分刻画的O(m)空间有向无环图，(ii) 完整刻画的O(mn)空间结构。为获得上述成果，我们发展了新技术，特别是对Ford与Fulkerson最大流最小割定理的推广，该技术可能具有独立研究价值。