Learning a Hidden Hypergraph

13 years 10 months ago

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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(24r m · poly(r, log n)) queries with high probability. The queries can be made in O(min(2r (log m+r)2 , (log m+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(log n)) 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 dependency on m. The queries can also be made in O((1 + ∆) · min(2r (log m + r)2 , (log m + r)3 )) rounds.

Dana Angluin, Jiang Chen

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