Fault tolerant subgraph for single source reachability: generic and optimal

Let G=(V,E) be an n-vertices m-edges directed graph. Let s∈ V be any designated source vertex. We address the problem of single source reachability (SSR) from s in presence of failures of vertices/edges. We show that for every k≥ 1, there is a subgraph H of G with at most 2^k n edges that preserves the reachability from s even after the failure of any k edges. Formally, given a set F of k edges, a vertex u∈ V is reachable from s in G∖ F if and only if u is reachable from s in H∖ F. We call H a k-Fault Tolerant Reachability Subgraph (k-FTRS). We prove also a matching lower bound of Ω(2^kn) for such subgraphs. Our results extend to vertex failures without any extra overhead. The general construction of k-FTRS is interesting from several different perspectives. From the Graph theory perspective it reveals a separation between SSR and single source shortest paths (SSSP) in directed graphs. More specifically, in the case of SSSP in weighted directed graphs, there is a lower bound of Ω(m) even for a single edge failure. In the case of unweighted graphs there is a lower bound of Ω(n^3/2) edges, again, even for a single edge failure. There is also a matching upper bound but nothing is known for two or more failures in the directed graphs. From the Algorithms perspective it implies fault tolerant solutions to other interesting problems, namely, (i) verifying if the strong connectivity of a graph is preserved after k edge or vertex failures, (ii) computing a dominator tree of a graph after k-failures. From the perspective of Techniques it makes an interesting usage of the concept of farthest min-cut which was already introduced by Ford and Fulkerson in their pioneering work on flows and cuts. We show that there is a close relationship between the farthest min-cut and the k-FTRS. We believe that our new technique is of independent interest.