New Oracles and Labeling Schemes for Vertex Cut Queries

TL;DR

This paper advances the understanding of vertex cut queries by developing new labeling schemes and oracles that are nearly optimal, addressing open problems for global cut queries in graphs.

Contribution

It introduces the first significant labeling schemes and oracles for global vertex cut queries, nearly matching known lower bounds and establishing their optimality under certain conjectures.

Findings

Vertex cut labeling schemes with labels of size O(n^{1-1/f}) bits.

Vertex cut oracles with O(n) space and O(2^f) query time for f=O(log n).

Optimality of the proposed oracles up to n^{o(1)} factors under plausible complexity assumptions.

Abstract

We study the succinct representations of vertex cuts by centralized oracles and labeling schemes. For an undirected $n$-vertex graph $G = (V,E)$ and integer parameter $f \geq 1$, the goal is supporting vertex cut queries: Given $F \subseteq V$ with $|F| \leq f$, determine if $F$ is a vertex cut in $G$. In the centralized data structure setting, it is required to preprocess $G$ into an $f$-vertex cut oracle that can answer such queries quickly, while occupying only small space. In the labeling setting, one should assign a short label to each vertex in $G$, so that a cut query $F$ can be answered by merely inspecting the labels assigned to the vertices in $F$. While the ``$st$ cut variants'' of the above problems have been extensively studied and are known to admit very efficient solutions, the basic (global) ``cut query'' setting is essentially open (particularly for $f > 3$). This work achieves the first significant progress on these problems: [$f$-Vertex Cut Labels:] Every $n$-vertex graph admits an $f$-vertex cut labeling scheme, where the labels have length of $\tilde{O}(n^{1-1/f})$ bits (when $f$ is polylogarithmic in $n$). This nearly matches the recent lower bound given by Long, Pettie and Saranurak (SODA 2025). [$f$-Vertex Cut Oracles:] For $f=O(\log n)$, every $n$-vertex graph $G$ admits $f$-vertex cut oracle with $\tilde{O}(n)$ space and $\tilde{O}(2^f)$ query time. We also show that our $f$-vertex cut oracles for every $1 \leq f \leq n$ are optimal up to $n^{o(1)}$ factors (conditioned on plausible fine-grained complexity conjectures). If $G$ is $f$-connected, i.e., when one is interested in \emph{minimum} vertex cut queries, the query time improves to $\tilde{O}(f^2)$, for any $1 \leq f \leq n$.