Near-Optimal Fault-Tolerant Strong Connectivity Preservers

2025-10-02

AI Summary

This paper studies the *k-fault-tolerant strongly connected subgraph* problem on directed graphs: given a digraph $G$, construct a spanning subgraph $H$ such that, after deleting any set of at most $k$ edges from $H$, the strong connectivity structure (i.e., the condensation DAG and SCC partition) remains identical to that of $G$. To address the large gap ($Omega(n^{1-1/k})$) between prior upper and lower bounds, we introduce a novel framework combining probabilistic construction, sparse sampling, hierarchical decomposition, and strong connectivity reduction techniques. Our contributions are threefold: (i) a non-constructive upper bound of $O(k 4^k n log n)$ edges; (ii) an explicit construction with only $O(n sqrt{kn})$ edges—breaking the previous exponential dependence on $k$; and (iii) an explicit $O(8^k n log^{5/2} n)$-edge construction computable in $mathrm{poly}(2^k n)$ time. For constant $k$, all results are optimal up to logarithmic factors in $n$.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Large-scale security measurements

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 $ ilde{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 $O(k4^{k}nlog n)$ edges, and we can construct one with $O(8^{k}nlog^{5/2}n)$ edges in $ ext{poly}(2^{k}n)$ time. Our results also improve the state-of-the-art for a closely related object; a extit{$k$-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 $O(k4^{k}nlog n)$ edges, improving the previous best bound of $ ilde{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.

Problem

• Reducing edge count in fault-tolerant strong connectivity preservers for directed graphs
• Closing the gap between upper and lower bounds for directed graph preservers
• Improving construction efficiency for k-connectivity preservers in digraphs

Innovation

• Uses O(k4^kn log n) edges for fault-tolerant connectivity
• Constructs preserver in poly(2^kn) time complexity
• Provides O(n sqrt(kn)) edges alternative construction