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, 509–518 (1995)] showed that there exists an (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 (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 (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 (n2) size data structure that, given a pair of vertices {u,v} that are not separated by an (s,t)-mincut, can determine in (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 (|B|) time. (3)Sensitivity oracle: There exists an (n2) size data structure that solves the dual-edge sensitivity problem for (s,t)-mincuts. It takes (1) time to report the capacity of a resulting (s,t)-mincut (A,B) and (|B|) time to report the cut. (4)Lower bounds: For the data structure problems addressed in results (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, 827–847 (2011)] and Goldstein et al. [WADS, 421–436 (2017)], this leads to \(\tilde{\Omega }(n^2)\) lower bounds on the space for the latter two p