Almost-Uniform Edge Sampling: Leveraging Independent-Set and Local Graph Queries

2026-03-15

AI Summary

This work investigates how to achieve approximately uniform edge sampling in sublinear time under a hybrid query model that combines independent set queries and local graph queries. It establishes, for the first time, a tight two-way reduction between edge sampling and approximate edge counting in this model, demonstrating that the two problems share matching upper and lower bounds in query complexity. The proposed sampling algorithm matches the query complexity of the current best-known edge counting algorithms, while the analysis yields tight lower bounds for each constituent query model. These results highlight the pivotal role of independent set queries in enabling efficient graph sampling tasks, revealing a fundamental equivalence between sampling and counting in terms of computational hardness under the hybrid query framework.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

A central theme in sublinear graph algorithms is the relationship between counting and sampling: can the ability to approximately count a combinatorial structure be leveraged to sample it nearly uniformly at essentially the same cost? We study (i) independent-set (IS) queries, which return whether a vertex set $S$ is edge-free, and (ii) two standard local queries: degree and neighbor queries. Eden and Rosenbaum (SOSA `18) proved that in the local-query model, uniform edge sampling is no harder than approximate edge counting. We extend this phenomenon to new settings. We establish sampling-counting equivalence for the hybrid model that combines IS and local queries, matching the complexity of edge-count estimation achieved by Adar, Hotam and Levi (2026), and an analogous equivalence for IS queries, matching the complexity of edge-count estimation achieved by Xi, Levi and Waingarten (SODA `20). For each query model, we show lower bounds for uniform edge sampling that essentially coincide with the known bounds for approximate edge counting.

Problem

• edge sampling
• independent-set queries
• local graph queries
• sublinear algorithms
• sampling-counting equivalence

Innovation

• edge sampling
• independent-set queries
• local graph queries
• sublinear algorithms
• sampling-counting equivalence