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

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 Erd\H{o}s--R\'enyi (ER) model, in which each potential hyperedge appears independently with probability $q = \Theta(n^{-3(1-\theta)})$ for $\theta \in (0;1)$. 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 $\Omega(n^3)$, 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 $\theta > \dfrac{2}{3}$ and $O(\bar{m}^{5/3}\log^2{\bar{m}}\log n)$ for the case $\theta \leq \dfrac{2}{3}$. In particular, the decoding runtime is subcubic in $n$ whenever $\theta<\frac{3}{5}$, providing a new test-decoding tradeoff compared with existing schemes.