On Multistage Learning a Hidden Hypergraph

TL;DR

This paper introduces an adaptive testing algorithm for identifying all edges in a localized hypergraph with minimal tests, matching the theoretical lower bound, and discusses a probabilistic extension.

Contribution

The paper presents a new adaptive algorithm for hypergraph learning that achieves the information-theoretic lower bound on the number of tests.

Findings

Algorithm matches the theoretical lower bound of sℓ log t tests.

Provides an efficient method for hypergraph edge detection.

Discusses a probabilistic generalization of the hypergraph learning problem.

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 $F(t,s,\ell)$ 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 $H_{un}\in F(t,s,\ell)$ by using the minimal number of tests. We develop 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 $s\ell\log_2 t(1+o(1))$. We also discuss a probabilistic generalization of the problem.