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DOI: 10.4230/LIPIcs.APPROX/RANDOM.2026.41

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

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

Published in: LIPIcs, Volume 392, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)

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)

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@InProceedings{chanda_et_al:LIPIcs.APPROX/RANDOM.2026.41,
  author =	{Chanda, Debarshi and Das, Buddha Dev and Ghosh, Arijit and Mishra, Gopinath},
  title =	{{From Decision to Random Certificates: Exponential Separation for Edge Estimation with Independent Set Queries}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
  pages =	{41:1--41:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-449-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{392},
  editor =	{Singh, Mohit and Gur, Tom},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.41},
  URN =		{urn:nbn:de:0030-drops-277584},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2026.41},
  annote =	{Keywords: Property Testing, Edge Estimation}
}

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

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