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arXiv · 1907.04381

Nearly optimal edge estimation with independent set queries

Xi Chen·Amit Levi·Erik Waingarten
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Abstract

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 $S\subseteq [n]$ 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 $(1+\epsilon)$-approximation of the number of edges $m$ of the graph using $\min(\sqrt{m},n / \sqrt{m})\cdot\textrm{poly}(\log n,1/\epsilon)$ independent set queries. This improves the upper bound of $\min(\sqrt{m},n^2/m)\cdot\textrm{poly}(\log n,1/\epsilon)$ by Beame et al. \cite{BHRRS18}. Our second main result shows that ${\min(\sqrt{m},n/\sqrt{m}))/\textrm{polylog}(n)}$ independent set queries are necessary, thus establishing that our algorithm is optimal up to a factor of $\textrm{poly}(\log n, 1/\epsilon)$.

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Xi Chen, Amit Levi, Erik Waingarten. 2019-07-09. Nearly optimal edge estimation with independent set queries. https://arxiv.org/abs/1907.04381

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