On Computing the Frequencies of Induced Subhypergraphs

Abstract

Let $\mathcal{F}$ be an r-uniform hypergraph with f vertices, where $f>r\geq3$. In [Inform. Process. Lett., 99 (2006), pp. 130–134], Yuster posed the problem of whether there exists an algorithm which, for a given r-uniform hypergraph $\mathcal{H}$ with n vertices, computes the number of induced copies of $\mathcal{F}$ in $\mathcal{H}$ in time $o(n^f)$. The analogous question for graphs ($r=2$) was known to hold from an $O(n^{f-\varepsilon})$ time algorithm of Nešetřil and Poljak [Comment. Math. Univ. Carolin., 26 (1985), pp. 415–419] (for a constant $\varepsilon=\varepsilon_f>0$ which is independent of n). Here, we present an algorithm for this problem, when $r\geq3$, with running time $O(n^f/\log_2n)$.