We study the problem of estimating the number of edges in an unknown graph. We consider a hybrid model in which an algorithm may issue independent set, degree, and neighbor queries. We show that this model admits strictly more efficient edge estimation than either access type alone. Specifically, we give a randomized algorithm that outputs a $(1\pm\varepsilon)$-approximation of the number of edges using $O\left(\min\left(\sqrt{m}, \sqrt{\frac{n}{\sqrt{m}}}\right)\cdot\frac{\log n}{\varepsilon^{5/2}}\right)$ queries, and prove a nearly matching lower bound. In contrast, prior work shows that in the local query model (Goldreich and Ron, \textit{Random Structures \& Algorithms} 2008) and in the independent set query model (Beame \emph{et al.} ITCS 2018, Chen \emph{et al.} SODA 2020), edge estimation requires $\widetildeΘ(n/\sqrt{m})$ queries in the same parameter regimes. Our results therefore yield a quadratic improvement in the hybrid model, and no asymptotically better improvement is possible.

翻译：我们研究未知图中边数估计的问题。考虑一种混合模型，其中算法可以执行独立集查询、度查询和邻居查询。我们证明该模型相比单一查询类型能实现严格更高效的边数估计。具体而言，我们提出一种随机算法，使用$O\left(\min\left(\sqrt{m}, \sqrt{\frac{n}{\sqrt{m}}}\right)\cdot\frac{\log n}{\varepsilon^{5/2}}\right)$次查询即可输出边数的$(1\pm\varepsilon)$近似值，并给出了近乎匹配的下界。相比之下，先前研究表明在局部查询模型（Goldreich和Ron，\textit{Random Structures \& Algorithms} 2008）和独立集查询模型（Beame等，ITCS 2018；Chen等，SODA 2020）中，相同参数范围内边数估计需要$\widetildeΘ(n/\sqrt{m})$次查询。因此我们的结果在混合模型中实现了二次改进，且该改进在渐近意义上已达到最优。