When Local and Non-Local Meet: Quadratic Improvement for Edge Estimation with Independent Set Queries

2026-01-29

AI Summary

This work addresses the efficiency bottleneck in estimating the number of edges in a graph under query models that support only local or independent set queries. The authors propose a hybrid query model that combines independent set, degree, and neighbor queries, and design a randomized algorithm achieving a $(1\pm\varepsilon)$-approximation of the edge count. This approach yields the first quadratic speedup for edge estimation within a hybrid query framework, surpassing the theoretical lower bounds inherent to single-query models. The algorithm achieves a query complexity of $O\left(\min\left(\sqrt{m}, \sqrt{n/\sqrt{m}}\right) \cdot \frac{\log n}{\varepsilon^{5/2}}\right)$, and the authors establish a nearly matching information-theoretic lower bound, demonstrating that this quadratic improvement is theoretically optimal.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

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 $\widetilde{\Theta}(n/\sqrt{m})$ queries in the same parameter regimes. Our results therefore yield a quadratic improvement in the hybrid model, and no asymptotically better improvement is possible.

Problem

• edge estimation
• independent set queries
• graph algorithms
• query complexity
• hybrid model

Innovation

• edge estimation
• independent set queries
• hybrid query model
• sublinear algorithms
• quadratic improvement