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- W3038069570 abstract "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 $O(n+min{|{cal P}|sqrt{n},nsqrt{|{cal P}|}})$ bound on a subgraph that is an $1$-fault-tolerant reachability preserver for a given vertex-pair set ${cal P}subseteq Vtimes V$, i.e., it preserves reachability between any pair of vertices in ${cal P}$ under single edge (or vertex) failure. Our result is a significant improvement over the previous best $O(n |{cal P}|)$ 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 ${cal P}subseteq Vtimes V$ of size $Omega(n^epsilon)$, for even any arbitrarily small $epsilon$, requires at least $Omega(n^{1+epsilon/8})$ 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 $tilde{O}(k 2^k n^{2-1/k})$ 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]." @default.
- W3038069570 created "2020-07-02" @default.
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- W3038069570 date "2020-04-01" @default.
- W3038069570 modified "2023-09-24" @default.
- W3038069570 title "New Extremal bounds for Reachability and Strong-Connectivity Preservers under failures" @default.
- W3038069570 hasPublicationYear "2020" @default.
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