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).
@InProceedings{haviv_et_al:LIPIcs.APPROX/RANDOM.2024.35, author = {Haviv, Ishay and Parnas, Michal}, title = {{Testing Intersectingness of Uniform Families}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {35:1--35:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.35}, URN = {urn:nbn:de:0030-drops-210288}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.35}, annote = {Keywords: Intersecting family, Uniform family, Property testing} }
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