Non-adaptive Learning of Random Hypergraphs with Queries

TL;DR

This paper investigates non-adaptive algorithms for learning random hypergraphs through specific queries, extending previous graph results to hypergraphs and linking the problem to group testing.

Contribution

It generalizes the non-adaptive hypergraph learning results from graphs to k-uniform hypergraphs using a novel equivalence with group testing.

Findings

Established a connection between hyperedge learning and group testing.

Extended previous graph-based results to hypergraphs.

Provided theoretical bounds for non-adaptive hypergraph learning.

Abstract

We study the problem of learning a hidden hypergraph $G=(V,E)$ by making a single batch of queries (non-adaptively). We consider the hyperedge detection model, in which every query must be of the form: ``Does this set $S\subseteq V$ contain at least one full hyperedge?'' In this model, it is known that there is no algorithm that allows to non-adaptively learn arbitrary hypergraphs by making fewer than $\Omega(\min\{m^2\log n, n^2\})$ even when the hypergraph is constrained to be $2$-uniform (i.e. the hypergraph is simply a graph). Recently, Li et al. overcame this lower bound in the setting in which $G$ is a graph by assuming that the graph learned is sampled from an Erd\H{o}s-R\'enyi model. We generalize the result of Li et al. to the setting of random $k$-uniform hypergraphs. To achieve this result, we leverage a novel equivalence between the problem of learning a single hyperedge and the standard group testing problem. This latter result may also be of independent interest.