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Let G be a directed weighted graph on n vertices and m edges with designated source and sink vertices s and 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 [CJM 1956], a long line of work has been done on computing the most vital edge and all vital edges of G. However, even after 60 years, the existing results are for either undirected or unweighted graphs. We present the following result for directed weighted graphs that also solves an open problem by Ausiello, Franciosa, Lari, and Ribichini [NETWORKS 2019]. 1. Algorithmic Results: There is an algorithm that computes all vital edges as well as the most vital edge of G using {O}(n) maximum (s,t)-flow computations. Vital edges play a crucial role in the design of sensitivity oracle for (s,t)-mincut - a compact data structure for reporting (s,t)-mincut after insertion/failure of any edge. For directed graphs, the only existing sensitivity oracle is for unweighted graphs by Picard and Queyranne [MPS 1982]. We present the first and optimal sensitivity oracle for directed weighted graphs as follows. 2. Sensitivity Oracles: a) There is an optimal O(n²) space data structure that can report an (s,t)-mincut C in O(|C|) time after the failure/insertion of any edge. b) There is an O(n) space data structure that can report the capacity of (s,t)-mincut after failure or insertion of any edge e in O(1) time if the capacity of edge e is known. A mincut for a vital edge e is an (s,t)-cut of the least capacity in which edge e is outgoing. For unweighted graphs, in a classical work, Picard and Queyranne [MPS 1982] designed an O(m) space directed acyclic graph (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 directed weighted graphs as follows. 3. Structural & Combinatorial Results: a) There is a set M containing at most n-1 (s,t)-cuts such that at least one mincut for every vital edge belongs to the set. This bound is tight as well. We also show that set M can be computed using O(n) maximum (s,t)-flow computations. b) We design two compact structures for storing and characterizing all mincuts for all vital edges - (i) an O(m) space DAG for partial and (ii) an 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 [CJM 1956], which might be of independent interest.

Let G be a directed multi-graph on n vertices and m edges with a designated source vertex s and a designated sink vertex t. We study the (s,t)-cuts of capacity minimum+1 and as an important application of them, we give a solution to the dual edge sensitivity for (s,t)-mincuts - reporting the (s,t)-mincut upon failure or addition of any pair of edges. Picard and Queyranne [Mathematical Programming Studies, 13(1):8-16, 1980] showed that there exists a directed acyclic graph (DAG) that compactly stores all minimum (s,t)-cuts of G. This structure also acts as an oracle for the single edge sensitivity of minimum (s,t)-cut. Dinitz and Nutov [STOC, pages 509-518, 1995] showed that there exists an 𝒪(n) size 2-level cactus model that stores all global cuts of capacity minimum+1. However, for minimum+1 (s,t)-cuts, no such compact structures exist till date. We present the following structural and algorithmic results on minimum+1 (s,t)-cuts. 1) There exists a pair of DAGs of size O(m) that compactly store all minimum+1 (s,t)-cuts of G. Each minimum+1 (s,t)-cut appears as a (s,t)-cut in one of the 2 DAGs and is 3-transversal - it intersects any path in the DAG at most thrice. 2) There exists an O(n²) size data structure that, given a pair of vertices {u,v} which are not separated by an (s,t)-mincut, can determine in 𝒪(1) time if there exists a minimum+1 (s,t)-cut, say (A,B), such that {s,u} ∈ A and {v,t} ∈ B; the corresponding cut can be reported in 𝒪(|B|) time. 3) There exists an O(n²) size data structure that solves the dual edge sensitivity problem for (s,t)-mincuts. It takes 𝒪(1) time to report the value of a resulting (s,t)-mincut (A,B) and 𝒪(|B|) time to report the cut. 4) For the data structure problems addressed in (2) and (3) above, we also provide a matching conditional lower bound. We establish a close relationship among three seemingly unrelated problems – all-pairs directed reachability problem, the dual edge sensitivity problem for (s,t)-mincuts, and 2× 2 maximum flow. Assuming the directed reachability hypothesis, this leads to Ω(n²) lower bounds on the space for the latter two problems.