Testing Intersectingness of Uniform Families

Authors Ishay Haviv, Michal Parnas



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Author Details

Ishay Haviv
  • The Academic College of Tel Aviv-Yaffo, Tel Aviv 61083, Israel
Michal Parnas
  • The Academic College of Tel Aviv-Yaffo, Tel Aviv 61083, Israel

Acknowledgements

We would like to thank the anonymous reviewers for their useful comments.

Cite AsGet BibTex

Ishay Haviv and Michal Parnas. Testing Intersectingness of Uniform Families. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.35

Abstract

A set family F is called intersecting if every two members of F intersect, and it is called uniform if all members of F share a common size. A uniform family F ⊆ binom([n],k) of k-subsets of [n] is ε-far from intersecting if one has to remove more than ε ⋅ binom(n,k) of the sets of F to make it intersecting. We study the property testing problem that given query access to a uniform family F ⊆ binom([n],k), asks to distinguish between the case that F is intersecting and the case that it is ε-far from intersecting. We prove that for every fixed integer r, the problem admits a non-adaptive two-sided error tester with query complexity O((ln n)/ε) for ε ≥ Ω((k/n)^r) and a non-adaptive one-sided error tester with query complexity O((ln k)/ε) for ε ≥ Ω((k²/n)^r). The query complexities are optimal up to the logarithmic terms. For ε ≥ Ω((k²/n)²), we further provide a non-adaptive one-sided error tester with optimal query complexity of O(1/ε). Our findings show that the query complexity of the problem behaves differently from that of testing intersectingness of non-uniform families, studied recently by Chen, De, Li, Nadimpalli, and Servedio (ITCS, 2024).

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Intersecting family
  • Uniform family
  • Property testing

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References

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