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Testing the Independent Set Property in Hypergraphs

Authors Elena Grigorescu , Shreya Nasa , Cameron Seth

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LIPIcs.APPROX-RANDOM.2026.73.pdf
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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Randomness, geometry and discrete structures
  • Theory of computation → Theory and algorithms for application domains
Keywords
  • independent set
  • property testing
  • hypergraph container method
  • hypergraph

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Abstract

The optimal sample complexity of testing if an n-vertex graph has an independent set of size ρ n, or is ε-far from having an independent set of size ρ n, was established to be Õ(ρ³/ε²), in a notable result by Blais and Seth (SICOMP 2025). In contrast, for q-uniform hypergraphs, there is a significant gap between the best known upper and lower bounds, and there has been no progress on the problem for the last two decades. In this work, we prove a new upper bound of Õ(qρ^{2q-3}/{ε²(q-2)!²}) on the sample complexity of testing the ρ-independent set property. The previous best known upper bound was Õ(2^q q! ρ^{2q}/ε³), due to Langberg (RANDOM 2004). This establishes the optimal dependence on ε and gives an exponential improvement in the dependence on q. We prove our result via a new application of the hypergraph container method.

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Elena Grigorescu, Shreya Nasa, and Cameron Seth. Testing the Independent Set Property in Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 73:1-73:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.73

Author Details

Elena Grigorescu
  • Cheriton School of Computer Science, University of Waterloo, Canada
Shreya Nasa
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Cameron Seth
  • Cheriton School of Computer Science, University of Waterloo, Canada

Acknowledgements

We would like to thank Eric Blais for his useful insights and suggestions throughout the project.

References

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