Nearly optimal edge estimation with independent set queries

07/09/2019

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by   Xi Chen, et al.

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We study the problem of estimating the number of edges of an unknown, undirected graph G=([n],E) with access to an independent set oracle. When queried about a subset S⊆ [n] of vertices the independent set oracle answers whether S is an independent set in G or not. Our first main result is an algorithm that computes a (1+ϵ)-approximation of the number of edges m of the graph using (√(m),n / √(m))·poly( n,1/ϵ) independent set queries. This improves the upper bound of (√(m),n^2/m)·poly( n,1/ϵ) by Beame et al. BHRRS18. Our second main result shows that (√(m),n/√(m)))/polylog(n) independent set queries are necessary, thus establishing that our algorithm is optimal up to a factor of poly( n, 1/ϵ).