Connectivity Labeling Schemes for Edge and Vertex Faults via Expander Hierarchies

We consider the problem of assigning short labels to the vertices and edges of a graph $G$ so that given any query $\langle s,t,F\rangle$ with $|F|\leq f$, we can determine whether $s$ and $t$ are still connected in $G-F$, given only the labels of $F\cup\{s,t\}$. This problem has been considered when $F\subset E$ (edge faults), where correctness is guaranteed with high probability (w.h.p.) or deterministically, and when $F\subset V$ (vertex faults), both w.h.p.~and deterministically. Our main results are as follows. [Deterministic Edge Faults.] We give a new deterministic labeling scheme for edge faults that uses $\tilde{O}(\sqrt{f})$-bit labels, which can be constructed in polynomial time. This improves on Dory and Parter's [PODC 2021] existential bound of $O(f\log n)$ (requiring exponential time to compute) and the efficient $\tilde{O}(f^2)$-bit scheme of Izumi, Emek, Wadayama, and Masuzawa [PODC 2023]. Our construction uses an improved edge-expander hierarchy and a distributed coding technique based on Reed-Solomon codes. [Deterministic Vertex Faults.] We improve Parter, Petruschka, and Pettie's [STOC 2024] deterministic $O(f^7\log^{13} n)$-bit labeling scheme for vertex faults to $O(f^4\log^{7.5} n)$ bits, using an improved vertex-expander hierarchy and better sparsification of shortcut graphs. [Randomized Edge/Verex Faults.] We improve the size of Dory and Parter's [PODC 2021] randomized edge fault labeling scheme from $O(\min\{f+\log n, \log^3 n\})$ bits to $O(\min\{f+\log n, \log^2 n\log f\})$ bits, shaving a $\log n/\log f$ factor. We also improve the size of Parter, Petruschka, and Pettie's [STOC 2024] randomized vertex fault labeling scheme from $O(f^3\log^5 n)$ bits to $O(f^2\log^6 n)$ bits, which comes closer to their $\Omega(f)$-bit lower bound.