Fault-tolerant subgraph for single-source reachability: General and optimal

Let G be a directed graph with n vertices, m edges, and a designated source vertex s. We address the problem of single-source reachability (SSR) from s in the presence of failures of vertices/edges. We show that for every k ≥ 1, there is a subgraph H of G with at most 2^kn 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 v ∈ V (G) is reachable from s in G \ F if and only if v is reachable from s in H \F. We call H a k-fault tolerant reachability subgraph (k- FTRS). We also prove a matching lower bound of Ω(2^kn) edges for such subgraphs that holds for all n, k with 2^k ≤ n. Our results extend to vertex failures without any extra overhead. The 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 and undirected graphs, Demetrescu et al. showed that there is a lower bound of Ω(m) edges even for a single edge failure [SIAM J. Comput., 37 (2008), pp. 1299–1318]. In the case of unweighted graphs Parter and Peleg gave a lower bound of Ω(n³/²) edges, again, even for a single edge failure [Proc. Algorithms—21st Annual European Symposium, 2013, pp. 779–790]. From the Algorithms perspective it implies fault-tolerant algorithms for other interesting problems, namely, (i) verifying if the strong connectivity of a graph is preserved after k edge or vertex failures, and (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 [Flows in Networks, Princeton University Press, 1962; reprinted 2011]. 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.