Near-Optimal Vertex Fault-Tolerant Labels for Steiner Connectivity

TL;DR

This paper introduces a near-optimal labeling scheme for verifying the connectivity of a terminal set in a graph after up to f vertex failures, improving efficiency over previous methods.

Contribution

It extends vertex fault-tolerant connectivity labels to general terminal sets using a novel Steiner tree approach, achieving near-optimal label sizes.

Findings

Labels of size |U|^{1-1/f} poly(f, log n) are achievable for terminal sets.

The scheme is near-optimal, matching lower bounds up to poly(f, log n).

Uses Steiner tree decomposition instead of sparsification for terminal sets.

Abstract

We present a compact labeling scheme for determining whether a designated set of terminals in a graph remains connected after any $f$ (or less) vertex failures occur. An $f$-FT Steiner connectivity labeling scheme for an $n$-vertex graph $G=(V,E)$ with terminal set $U \subseteq V$ provides labels to the vertices of $G$, such that given only the labels of any subset $F \subseteq V$ with $|F| \leq f$, one can determine if $U$ remains connected in $G-F$. The main complexity measure is the maximum label length. The special case $U=V$ of global connectivity has been recently studied by Jiang, Parter, and Petruschka, who provided labels of $n^{1-1/f} \cdot \mathrm{poly}(f,\log n)$ bits. This is near-optimal (up to $\mathrm{poly}(f,\log n)$ factors) by a lower bound of Long, Pettie and Saranurak. Our scheme achieves labels of $|U|^{1-1/f} \cdot \mathrm{poly}(f, \log n)$ for general $U \subseteq V$, which is near-optimal for any given size $|U|$ of the terminal set. To handle terminal sets, our approach differs from Jiang et al. We use a well-structured Steiner tree for $U$ produced by a decomposition theorem of Duan and Pettie, and bypass the need for Nagamochi-Ibaraki sparsification.