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原文摘要

A $k$-edge-failure strong connectivity preserver of a directed graph on $N$ vertices is a spanning subgraph that has the same strongly connected components as the input after each set of at most $k$ edge deletions. The lower-bound construction of Baswana, Choudhary, and Roditty [BCR18], adapted to strong connectivity in [CC20], gives the previous best bound of $Ω(2^kN)$ necessary edges. Hoppenworth, Saranurak, and Wang [HSW25] prove an $O(k4^kN\log N)$ upper bound, leaving a logarithmic gap for fixed $k$. We close this gap by giving an explicit strongly connected graph on $N=2^d$ vertices with $N-1+(N/2)\log_2N$ edges, every one of which is necessary in a two-failure preserver. Together with the upper bound of [HSW25], this proves that the optimal size in the worst case is $Θ_k(N\log N)$ for every fixed $k\geq2$, whereas it is $Θ(N)$ for $k=1$ [GIP20]. The same graph has a one-failure preserver with at most $3(N-1)$ edges. We also give a randomized algorithm that either returns a spanning subgraph or reports failure. For every fixed $c,\varepsilon>0$, with probability at least $1-N^{-c}$ it returns a correct preserver for $k=2$ with $O(N\log N)$ edges, in $\widetilde O(N^{(ω+3)/2+\varepsilon})$ time and $\widetilde O(N^2)$ space. Here $ω$ is the matrix multiplication exponent over a fixed finite field.