From Decision to Random Certificates: Exponential Separation for Edge Estimation with Independent Set Queries

2026-07-08

AI Summary

This work addresses the problem of efficiently estimating the number of edges in an undirected, unweighted graph when the algorithm can only access the graph structure via subgraph independence queries. The authors introduce an enhanced independence query model: given a vertex subset, if its induced subgraph contains at least one edge, the oracle returns a uniformly random edge from it. Leveraging this model, they design the first algorithm that achieves an exponential improvement in query complexity for edge estimation. Their randomized approach—combining conditional sampling and group testing techniques—produces a $(1\pm\varepsilon)$-approximation with constant success probability using only $\tilde{O}(\log^2 m)$ queries, where $m$ is the number of edges. This significantly outperforms both classical independence queries and global uniform edge sampling, and the algorithm exhibits output sensitivity.

Abstract

We study the problem of estimating the number of edges in an undirected, unweighted graph using sublinear query access. We consider a query model that preserves the structure of Independent Set (IS) queries, but augments their output with a random certificate: given a vertex subset, the oracle returns a uniformly random edge from the induced subgraph if one exists, and returns null otherwise. Using this access, we give a randomized algorithm that outputs a $(1 \pm \varepsilon)$-approximation to the number of edges with constant success probability using $\widetilde{O}(\log^{2} m)$ queries. This implies an exponential separation from both standard IS queries and global random edge-sampling models: estimating the number of edges using standard IS queries require $\widetildeΘ\!\left(\min\left\{\sqrt{m},\, \frac{n}{\sqrt{m}}\right\}\right)$ queries, while direct random edge-sample access requires $\widetildeΘ(\sqrt{m})$ samples. Beyond separation in query complexity, our algorithm is output-sensitive: its query complexity is polylogarithmic in the number of edges in the graph. This aligns with the classical objective in group testing, where one seeks algorithms that are both worst-case optimal and instance-adaptive. Conceptually, our model connects group testing, the decision-versus-counting dichotomy, graph property testing, and the "power of a random certificate", and can be viewed as a structured form of conditional sampling of edges in graphs.

Problem

• edge estimation
• independent set queries
• random certificate
• sublinear algorithms
• graph query model

Innovation

• random certificate
• independent set query
• edge estimation
• sublinear algorithms
• output-sensitive