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

TL;DR

This paper explores the relationship between counting and sampling in sublinear graph algorithms, establishing equivalences and lower bounds for uniform edge sampling across different query models, including independent-set and local queries.

Contribution

It extends the known sampling-counting equivalence to hybrid and independent-set query models, matching the complexity of existing edge-count estimation results.

Findings

Sampling-counting equivalence established for hybrid models combining IS and local queries.

Lower bounds for uniform edge sampling match known bounds for approximate edge counting.

Results unify understanding of sampling complexity across different query models.

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 Chen, 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.