Hyperedge Estimation using Polylogarithmic Subset Queries

A hypergraph ${\cal H}$ is a \emph{set system} $(U({\cal H}),{\cal F}(H))$, where $U({\cal H})$ denotes the set of $n$ vertices and ${\cal F}(H)$, a set of subsets of $U({\cal H})$, denotes the set of hyperedges. A hypergraph ${\cal H}$ is said to be $d$-uniform if every hyperedge in ${\cal H}$ consists of exactly $d$ vertices. The cardinality of the hyperedge set is denoted as $\ |{{\cal F}({\cal H})}\ |=m({\cal H})$. We consider an oracle access to the hypergraph ${\cal H}$ of the following form. Given $d$ (non-empty) pairwise disjoint subsets of vertices $A_1,\ldots,A_d \subseteq U({\cal H})$ of hypergraph ${\cal H}$, the oracle, known as the {\bf Generalized $d$-partite independent set oracle (\gpis)}~(that was introduced in \cite{BishnuGKM018}), answers {\sc Yes} if and only if there exists a hyperedge in ${\cal H}$ having (exactly) one vertex in each $A_i, i \in [d]$. The \gpis{} oracle belongs to the class of oracles for subset queries. The study of subset queries was initiated by Stockmeyer \cite{Stockmeyer85}, and later the model was formalized by Ron and Tsur \cite{RonT16}. Subset queries generalize the set membership queries. In this work we give an algorithm for the \hest problem using the \gpis query oracle to obtain an estimate $\widehat{m}$ for $m({\cal H})$ satisfying $(1 - \epsilon) \cdot m({\cal H}) \leq \widehat{m} \leq (1 + \epsilon) \cdot m({\cal H})$. The number of queries made by our algorithm, assuming $d$ as a constant, is polylogarithmic in the number of vertices of the hypergraph. Our work can be seen as a natural generalization of {\sc Edge Estimation} using {\sc Bipartite Independent Set}({\sc BIS}) oracle \cite{BeameHRRS18} and {\sc Triangle Estimation} using {\sc Tripartite Independent Set}({\sc TIS}) oracle \cite{Bhatta-abs-1808-00691}.