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

Authors Debarshi Chanda , Buddha Dev Das , Arijit Ghosh , Gopinath Mishra

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Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Property Testing
  • Edge Estimation

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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 ± ε)-approximation to the number of edges with constant success probability using Õ(log² 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 Θ̃(min {√m, n/√m}) queries, while direct random edge-sample access requires Θ̃(√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.

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Debarshi Chanda, Buddha Dev Das, Arijit Ghosh, and Gopinath Mishra. From Decision to Random Certificates: Exponential Separation for Edge Estimation with Independent Set Queries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 41:1-41:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.41

Author Details

Debarshi Chanda
  • Indian Statistical Institute, Kolkata, India
Buddha Dev Das
  • Indian Statistical Institute, Kolkata, India
Arijit Ghosh
  • Indian Statistical Institute, Kolkata, India
Gopinath Mishra
  • Institute of Mathematical Sciences, Chennai, India

Funding

  • Ghosh, Arijit: Arijit Ghosh acknowledges partial support from the Science and Engineering Research Board (SERB), Government of India, through the MATRICS grant MTR/2023/001527, and from the Department of Science and Technology (DST), Government of India, through grant TPN-104427.

Acknowledgements

We thank an anonymous reviewer for their suggestions in improving the query complexity from log³ m (in an earlier version) to log² m.

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