A $k$-fault-tolerant connectivity preserver of a directed $n$-vertex graph $G$ is a subgraph $H$ such that, for any edge set $F \subseteq E(G)$ of size $|F| \le k$, the strongly connected components of $G - F$ and $H - F$ are the same. While some graphs require a preserver with $\Omega(2^{k}n)$ edges [BCR18], the best-known upper bound is $\tilde{O}(k2^{k}n^{2-1/k})$ edges [CC20], leaving a significant gap of $\Omega(n^{1-1/k})$. In contrast, there is no gap in undirected graphs; the optimal bound of $\Theta(kn)$ has been well-established since the 90s [NI92]. We nearly close the gap for directed graphs; we prove that there exists a $k$-fault-tolerant connectivity preserver with $O(k4^{k}n\log n)$ edges, and we can construct one with $O(8^{k}n\log^{5/2}n)$ edges in $\text{poly}(2^{k}n)$ time. Our results also improve the state-of-the-art for a closely related object; a \textit{$k$-connectivity preserver} of $G$ is a subgraph $H$ where, for all $i \le k$, the strongly $i$-connected components of $G$ and $H$ agree. By a known reduction, we obtain a $k$-connectivity preserver with $O(k4^{k}n\log n)$ edges, improving the previous best bound of $\tilde{O}(k2^{k}n^{2-1/(k-1)})$ [CC20]. Therefore, for any constant $k$, our results are optimal to a $\log n$ factor for both problems. Lastly, we show that the exponential dependency on $k$ is not inherent for $k$-connectivity preservers by presenting another construction with $O(n \sqrt{kn})$ edges.

翻译：对于有向 $n$ 顶点图 $G$，其 $k$ 容错连通性保持子图 $H$ 满足：对于任意规模 $|F| \le k$ 的边集 $F \subseteq E(G)$，$G - F$ 与 $H - F$ 的强连通分量完全相同。虽然某些图需要 $\Omega(2^{k}n)$ 条边的保持子图 [BCR18]，但已知最佳上界为 $\tilde{O}(k2^{k}n^{2-1/k})$ 条边 [CC20]，存在 $\Omega(n^{1-1/k})$ 的显著间隙。相比之下，无向图中不存在此间隙；自 90 年代起 $\Theta(kn)$ 的最优界已得到确立 [NI92]。我们几乎完全弥合了有向图中的间隙；证明存在具有 $O(k4^{k}n\log n)$ 条边的 $k$ 容错连通性保持子图，并能在 $\text{poly}(2^{k}n)$ 时间内构造出具有 $O(8^{k}n\log^{5/2}n)$ 条边的保持子图。我们的结果还改进了密切相关对象的最新进展；$G$ 的 \textit{$k$ 连通性保持子图} $H$ 满足：对于所有 $i \le k$，$G$ 与 $H$ 的强 $i$ 连通分量保持一致。通过已知规约，我们获得具有 $O(k4^{k}n\log n)$ 条边的 $k$ 连通性保持子图，改进了先前最佳界 $\tilde{O}(k2^{k}n^{2-1/(k-1)})$ [CC20]。因此，对于任意常数 $k$，我们的结果对两个问题均达到 $\log n$ 因子内的最优性。最后，我们通过提出另一种具有 $O(n \sqrt{kn})$ 条边的构造，证明 $k$ 连通性保持子图中对 $k$ 的指数依赖并非固有性质。