Adaptive Learning a Hidden Hypergraph

TL;DR

This paper introduces an adaptive testing algorithm for identifying localized hypergraphs with minimal tests, matching theoretical bounds, and generalizing classical group testing to hypergraph structures.

Contribution

The paper presents a new adaptive algorithm for hypergraph learning that achieves optimal test complexity, extending group testing to hypergraph detection.

Findings

Algorithm matches the information theory bound in worst case

Number of tests is proportional to sℓ log t

Applicable to localized hypergraphs with bounded edges

Abstract

Learning a hidden hypergraph is a natural generalization of the classical group testing problem that consists in detecting unknown hypergraph $H_{un}=H(V,E)$ by carrying out edge-detecting tests. In the given paper we focus our attention only on a specific family $\mathcal{F}(t,s,\ell)$ of localized hypergraphs for which the total number of vertices $|V| = t$, the number of edges $|E|\le s$, $s\ll t$, and the cardinality of any edge $|e|\le\ell$, $\ell\ll t$. Our goal is to identify all edges of $H_{un}\in \mathcal{F}(t,s,\ell)$ by using the minimal number of tests. We provide an adaptive algorithm that matches the information theory bound, i.e., the total number of tests of the algorithm in the worst case is at most $s\ell\log_2 t(1+o(1))$.