All-Pairs $2$-reachability in $\widetilde{O}(n^{\omega})$ Time

In the 2-reachability problem we are given a directed graph G and we wish to determine if there are two (edge or vertex) disjoint paths from u to v, for a given pair of vertices u and v. In this paper, we present an algorithm that computes 2-reachability information for all pairs of vertices in O(nω log n) time, where n is the number of vertices and ω is the matrix multiplication exponent. Hence, we show that the running time of all-pairs 2-reachability is only within a log factor of transitive closure. Moreover, our algorithm produces a witness (i.e., a separating edge or a separating vertex) for all pair of vertices where 2-reachability does not hold. By processing these witnesses, we can compute all the edgeand vertex-dominator trees of G in O(n2) additional time, which in turn enables us to answer various connectivity queries in O(1) time. For instance, we can test in constant time if there is a path from u to v avoiding an edge e, for any pair of query vertices u and v, and any query edge e, or if there is a path from u to v avoiding a vertex w, for any query vertices u, v, and w.