Abstract: 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) and $F\subset V$ (vertex faults), where correctness is guaranteed with high probability (w.h.p.) or deterministically.
In this talk, I will introduce 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.
Based on joint work (https://arxiv.org/abs/2410.18885) with Seth Pettie and Thatchaphol Saranurak.