Minimum+1 (s,t)-cuts and Dual Edge Sensitivity Oracle

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 an (s, t)-mincut upon failure or insertion 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. For undirected multi-graphs, Dinitz and Nutov [STOC, pages 509-518, 1995] showed that there exists an \({\mathcal {O}}(n) \) size 2-level cactus model that stores all global cuts of capacity minimum+1. However, for minimum+1 (s, t)-cuts, no such compact structure exists till date. We present the following structural and algorithmic results on minimum+1 (s, t)-cuts. (1) Structure: There is an \({\mathcal {O}}(m) \) size 2-level DAG structure that stores all minimum+1 (s, t)-cuts of G such that each minimum+1 (s, t)-cut appears as 3-transversal cut – it intersects any path in this structure at most thrice. We also show that there is an \({\mathcal {O}}(mn) \) size structure for storing and characterizing all minimum+1 (s, t)-cuts in terms of 1-transversal cuts. (2) Data structure: There exists an \({\mathcal {O}}(n^2) \) size data structure that, given a pair of vertices {u, v} which are not separated by an (s, t)-mincut, can determine in \({\mathcal {O}}(1) \) 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 \({\mathcal {O}}(|B|) \) time.(3) Sensitivity oracle: There exists an \({\mathcal {O}}(n^2) \) size data structure that solves the dual edge sensitivity problem for (s, t)-mincuts. It takes \({\mathcal {O}}(1) \) time to report the capacity of a resulting (s, t)-mincut (A, B) and \({\mathcal {O}}(|B|) \) time to report the cut. (4) Lower bounds: 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 the problem of reporting the capacity of ({x, y}, {u, v})-mincut for any four vertices x, y, u, v in G. Assuming the Directed Reachability Hypothesis by Patrascu [SIAM J. Computing, pages 827–847, 2011] and Goldstein et al. [WADS, pages 421-436, 2017], this leads to \(\tilde{\Omega }(n^2) \) lower bounds on the space for the latter two problems.