On Computing the Frequencies of Induced Subhypergraphs

Let F be an r-uniform hypergraph with f vertices, where f > r ≥ 3. In [12], R. Yuster posed the problem of whether there exists an algorithm which, for a given r-uniform hypergraph H with n vertices, computes the number of induced copies of F in H in time o(n ). The analogous question for graphs (r = 2) was known to hold from a O(nf−ε) time algorithm of Nešetřil and Poljak [9] (for a constant ε = εf > 0 which is independent of n). Here, we present an algorithm for this problem, when r ≥ 3, with running time O(n/ log2 n).