Learning a Hidden Hypergraph

We consider the problem of learning a hypergraph using edge-detecting queries. In this model, the learner may query whether a set of vertices induces an edge of the hidden hypergraph or not. We show that an r-uniform hypergraph with m edges and n vertices is learnable with O(24rm · poly(r, logn)) queries with high probability. The queries can be made in O(min(2r(logm + r)2, (logm + r)3)) rounds. We also give an algorithm that learns an almost uniform hypergraph of dimension r using O(2O((1+ ∆ 2 )r) ·m1+ ∆ 2 · poly(logn)) queries with high probability, where ∆ is the difference between the maximum and the minimum edge sizes. This upper bound matches our lower bound of Ω(( m 1+ 2 )1+ ∆ 2 ) for this class of hypergraphs in terms of dependence on m. The queries can also be made in O((1+∆) ·min(2r(logm+ r)2,(logm+ r)3)) rounds.