Edge search in hypergraphs

Consider an undirected hypergraph H = (X, E) with a probability distribution P on the set E of its hyperedges. We investigate the average case complexity L(H, P) of finding an unknown hyperedge e∗ ϵ E, chosen according to P, if only tests are allowed that check, whether e∗ is contained in the induced subhypergraphs H[Y] for Y ⊂ X, or not. In this note we prove that for hypergraphs with average size of the hyperedges r H(P) ⪯ L(H, P) < H(P) + 3r. Here H(·) denotes the usual entropy function. The lower bound is the information theoretic bound, whereas the upper bound stems from a construction presented in this note. More precisely, our construction gives search lengths which are bounded from above by −log2 P(e∗) + 3 · ¦e∗¦ for all possible choices of e∗.