A Fast Hierarchical Splitting Approach for Non-Adaptive Learning of Random Hypergraphs

2026-05-11

AI Summary

This work addresses the problem of efficiently recovering the hyperedge set of an unknown 3-uniform Erdős–Rényi random hypergraph using a minimal number of non-adaptive edge-detection queries. To this end, the authors introduce a novel hierarchical binary splitting framework, which extends this strategy to the setting of random hypergraph learning for the first time. The proposed method achieves optimal query complexity $O(\bar{m} \log n)$ while significantly reducing decoding time: with high probability, it recovers the hyperedge set in $O(\bar{m}^{5/3} \log n)$ time when $\theta > 2/3$, and in $O(\bar{m}^{5/3} \log^2 \bar{m} \log n)$ time when $\theta \leq 2/3$. This breakthrough overcomes the previous $\Omega(n^3)$ decoding bottleneck.

Abstract

This work focuses on the problem of learning an unknown $3$-uniform hypergraph using edge-detecting queries. Our goal is to design a querying strategy that recovers the hyperedge set using as few queries as possible. We restrict our attention to random hypergraphs under the Erdős--Rényi (ER) model, in which each potential hyperedge appears independently with probability $q = Θ(n^{-3(1-θ)})$ for $θ\in (0;1)$. Prior work [Austhof-Reyzin-Tani, ISIT 2025] presents a testing-decoding scheme that uses $O(\bar{m}\log n)$ tests but requires a decoding time of $Ω(n^3)$, where $\bar{m} = q\binom{n}{3}$ denotes the expected number of hyperedges. In this work, we extend the binary splitting framework and adapt it to the $3$-uniform hypergraph setting. We obtain a testing-decoding scheme that recovers the hyperedge set with high probability using $O(\bar{m} \log n)$ tests and achieves decoding time $O(\bar{m}^{5/3}\log n)$ for the case $θ> \dfrac{2}{3}$ and $O(\bar{m}^{5/3}\log^2{\bar{m}}\log n)$ for the case $θ\leq \dfrac{2}{3}$. Thus, compared with prior work, our result significantly improves the decoding complexity while maintaining optimal query complexity.

Problem

• hypergraph learning
• edge-detecting queries
• random hypergraphs
• Erdős–Rényi model
• 3-uniform hypergraph

Innovation

• hierarchical splitting
• non-adaptive learning
• random hypergraphs
• edge-detecting queries
• decoding complexity