Hyperedge Estimation using Polylogarithmic Subset Queries

08/12/2019
by   Anup Bhattacharya, et al.
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A hypergraph H is a set system (U( H), F(H)), where U( H) denotes the set of n vertices and F(H), a set of subsets of U( H), denotes the set of hyperedges. A hypergraph H is said to be d-uniform if every hyperedge in H consists of exactly d vertices. The cardinality of the hyperedge set is denoted as | F( H) |=m( H). We consider an oracle access to the hypergraph H of the following form. Given d (non-empty) pairwise disjoint subsets of vertices A_1,...,A_d ⊆ U( H) of hypergraph H, the oracle, known as the Generalized d-partite independent set oracle () (that was introduced in BishnuGKM018), answers Yes if and only if there exists a hyperedge in H having (exactly) one vertex in each A_i, i ∈ [d]. The oracle belongs to the class of oracles for subset queries. The study of subset queries was initiated by Stockmeyer Stockmeyer85, and later the model was formalized by Ron and Tsur RonT16. Subset queries generalize the set membership queries. In this work we give an algorithm for the problem using the query oracle to obtain an estimate m for m( H) satisfying (1 - ϵ) · m( H) ≤m≤ (1 + ϵ) · m( 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 Edge Estimation using Bipartite Independent Set( BIS) oracle BeameHRRS18 and Triangle Estimation using Tripartite Independent Set( TIS) oracle Bhatta-abs-1808-00691.

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