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(2 4r m.poly(r,logn)) queries with high probability. The queries can be made in O(min(2 r r 2 log 2 n, r 3 log 3 n)) rounds. We also give an algorithm that learns a non-uniform hypergraph whose minimum edge size is r 1 and maximum edge size is r 2 using O(f 1 (r 1 ,r 2 ).m (r 2 -r 1 +2)/2 . poly(log n)) queries with high probability, and give a lower bound of Ω(f 2 (r 1 ,r 2 ).m (r 2 -r 1 +2)/2 for this class of hypergraphs, where fi and f 2 are functions depending only on r 1 and r 2 . The queries can also be made in O(min(2 r 2r 2 2 log 2 n, r 3 2 log 3 n)) rounds.