Edge Estimation with Independent Set Oracles

TL;DR

This paper introduces algorithms for estimating the number of edges in a graph using only independent set oracle queries, with query complexities that are polylogarithmic and sublinear in the number of vertices.

Contribution

The paper presents two novel algorithms for edge estimation in graphs utilizing independent set oracles, advancing understanding of query complexity in graph property testing.

Findings

Polylogarithmic query complexity algorithm for edge estimation.

Sublinear query complexity algorithm with n^{2/3} polylog(n) queries.

Applications to decision versus counting problem complexity.

Abstract

We study the task of estimating the number of edges in a graph with access to only an independent set oracle. Independent set queries draw motivation from group testing and have applications to the complexity of decision versus counting problems. We give two algorithms to estimate the number of edges in an $n$-vertex graph, using (i) $\mathrm{polylog}(n)$ bipartite independent set queries, or (ii) ${n}^{2/3} \cdot\mathrm{polylog}(n)$ independent set queries.