Non-adaptive Learning of Random Hypergraphs with Queries
Bethany Austhof, Lev Reyzin, Erasmo Tani
TL;DR
This work addresses non-adaptive learning of Erdős-Rényi random $k$-uniform hypergraphs $G^{(k)}(n,q)$ under hyperedge-detection queries, aiming to recover $G=(V,E)$ with few non-adaptive tests. It introduces Hypergraph-GROTESQUE, a three-step decoder (bundles, multiplicity tests, and a location test reduced to group testing) that achieves $O\left(k\overline{m}\log^{2}\overline{m} + k^{2}\overline{m}\log \overline{m}\log^{2} n\right)$ queries and decoding time $O\left(k\overline{m}\log^{2}\overline{m} + k^{3}\overline{m}\log \overline{m}\log^{2} n\right)$ with probability $\Omega(1)$. It also shows that non-adaptive Bernoulli-query schemes based on COMP, DD, and SSS can recover $G$ with $O(\overline{m}\log n)$ queries at the cost of higher decoding time, leveraging a documented equivalence between single hyperedge learning and group testing. A key theoretical contribution is establishing a link between learning a single hyperedge and group testing, enabling efficient location tests; the results hold for typical Erdős-Rényi instances with $q=o(1)$ and provide practical decoding strategies for batch-query settings. These results advance understanding of non-adaptive hypergraph learning and offer near-linear-query methods matching the sparse regime.
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 $Ω(\min\{m^2\log n, n^2\})$ 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 Erdős-Rényi 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.