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Non-adaptive Learning of Random Hypergraphs with Queries

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.

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