New Extremal bounds for Reachability and Strong-Connectivity Preservers under failures
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In this paper, we consider the question of computing sparse subgraphs for any input directed graph G=(V,E) on n vertices and m edges, that preserves reachability and/or strong connectivity structures. We show O(n+min{| P|√(n),n√(| P|)}) bound on a subgraph that is an 1-fault-tolerant reachability preserver for a given vertex-pair set P⊆ V× V, i.e., it preserves reachability between any pair of vertices in P under single edge (or vertex) failure. Our result is a significant improvement over the previous best O(n | P|) bound obtained as a corollary of single-source reachability preserver construction. We prove our upper bound by exploiting the special structure of single fault-tolerant reachability preserver for any pair, and then considering the interaction among such structures for different pairs. In the lower bound side, we show that a 2-fault-tolerant reachability preserver for a vertex-pair set P⊆ V× V of size Ω(n^ϵ), for even any arbitrarily small ϵ, requires at least Ω(n^1+ϵ/8) edges. This refutes the existence of linear-sized dual fault-tolerant preservers for reachability for any polynomial sized vertex-pair set. We also present the first sub-quadratic bound of at most Õ(k 2^k n^2-1/k) size, for strong-connectivity preservers of directed graphs under k failures. To the best of our knowledge no non-trivial bound for this problem was known before, for a general k. We get our result by adopting the color-coding technique of Alon, Yuster, and Zwick [JACM'95].
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