Near-Optimal Fault-Tolerant Strong Connectivity Preservers

TL;DR

The paper advances the theory of connectivity preservers in directed graphs by showing near-optimal constructions for $k$-FT connectivity preservers with size $O(k4^{k}n\log n)$ and providing a $O(8^{k}n\log^{5/2}n)$-time construction. It also yields improved bounds for $k$-connectivity preservers, including an all-pairs result $O(n\sqrt{nk})$ edges, and establishes a strong separation between FT and non-FT variants. The core techniques deploy a directed expander hierarchy and new insights on important separators and unbreakable sets, enabling reductions to structured subgraphs and controlled decomposition across levels. The results substantially shrink the gap between upper and lower bounds for directed preservers (constant $k$) and pave the way for further refinements, including potential near-linear-time constructions. The work also characterizes the limitations under stronger failure models, showing fundamental differences between directed and undirected graphs in this context.

Abstract

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 $Ω(2^{k}n)$ edges [BCR18], the best-known upper bound is $\tilde{O}(k2^{k}n^{2-1/k})$ edges [CC20], leaving a significant gap of $Ω(n^{1-1/k})$. In contrast, there is no gap in undirected graphs; the optimal bound of $Θ(kn)$ 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.

Figures (5)

• Figure 1: Illustration of the giant component analysis in Case 1.
• Figure 2: Illustration of the anti-isolation argument in Case 2.
• Figure 3: An $\Omega(n2^{k/2})$ Lower Bound for $s$-$t$$k$-FT Preserver
• Figure 4: An $\Omega(n^2)$ lower bound for $1$-faulty-degree preservers.
• Figure 5: An $\Tilde{\Omega}(n^2)$ lower bound for $1$-color fault preservers. When the edges in color class $3$ fail, every $(s, y)$-path must pass through vertex $x_3$.

Theorems & Definitions (124)

• theorem 1.1
• corollary 1.2
• theorem 1.3
• definition 2.0: unbreakable
• claim 2.1: Giant component
• lemma 2.1: Improved anti-isolation
• definition 3.1: Reachable Cuts
• claim 3.2
• proof
• lemma 3.3: Farthest Minimum Cuts ford2015flows