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

Tomer Adar, Amit Levi

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.

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

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 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.
Paper Structure (23 sections, 4 theorems, 4 equations, 4 figures)

This paper contains 23 sections, 4 theorems, 4 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our results
  3. Related work
  4. Conceptual overview
  5. Factors
  6. Canceling the factors
  7. Combining the category-specific samplers
  8. The IS algorithm
  9. Additional details
  10. Estimating an indicator inverse
  11. Finding an edge in a non-independent set
  12. Preliminaries
  13. Notation scheme
  14. Graph notations
  15. Set of neighbors
  16. ...and 8 more sections

Key Result

theorem 1

There exists an algorithm $procname:sample-edge-hyb\xspace(G,\varepsilon)$ where $G$ is a graph over $n$ vertices (known to the algorithm) and $m$ edges (unknown to the algorithm) and $\varepsilon > 0$, that makes $O(R \log n \log (n/\varepsilon))$ degree, neighbor and independent-set queries, where

Theorems & Definitions (4)

  • theorem 1: note=Rephrase of Lemma \ref{['lemma:sample-edge-hyb']}, label=th:ubnd-hyb
  • theorem 2: note=Rephrase of Lemma \ref{['lemma:lbnd-sample-edge-hyb']}
  • theorem 3: note=Rephrase of Lemma \ref{['lemma:sample-edge-IS']}, label=th:ubnd-IS
  • theorem 4: note=Rephrase of Lemma \ref{['lemma:lbnd-sample-edge-IS']}