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DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.35

Testing Intersectingness of Uniform Families

Authors: Ishay Haviv and Michal Parnas

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

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).

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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)

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@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}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2022.56

On the Binary and Boolean Rank of Regular Matrices

Authors: Ishay Haviv and Michal Parnas

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract

A 0,1 matrix is said to be regular if all of its rows and columns have the same number of ones. We prove that for infinitely many integers k, there exists a square regular 0,1 matrix with binary rank k, such that the Boolean rank of its complement is k^Ω̃(log k). Equivalently, the ones in the matrix can be partitioned into k combinatorial rectangles, whereas the number of rectangles needed for any cover of its zeros is k^Ω̃(log k). This settles, in a strong form, a question of Pullman (Linear Algebra Appl., 1988) and a conjecture of Hefner, Henson, Lundgren, and Maybee (Congr. Numer., 1990). The result can be viewed as a regular analogue of a recent result of Balodis, Ben-David, Göös, Jain, and Kothari (FOCS, 2021), motivated by the clique vs. independent set problem in communication complexity and by the (disproved) Alon-Saks-Seymour conjecture in graph theory. As an application of the produced regular matrices, we obtain regular counterexamples to the Alon-Saks-Seymour conjecture and prove that for infinitely many integers k, there exists a regular graph with biclique partition number k and chromatic number k^Ω̃(log k).

Cite as

Ishay Haviv and Michal Parnas. On the Binary and Boolean Rank of Regular Matrices. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{haviv_et_al:LIPIcs.MFCS.2022.56,
  author =	{Haviv, Ishay and Parnas, Michal},
  title =	{{On the Binary and Boolean Rank of Regular Matrices}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{56:1--56:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.56},
  URN =		{urn:nbn:de:0030-drops-168545},
  doi =		{10.4230/LIPIcs.MFCS.2022.56},
  annote =	{Keywords: Binary rank, Boolean rank, Regular matrices, Non-deterministic communication complexity, Biclique partition number, Chromatic number}
}

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Documents authored by Parnas, Michal

Testing Intersectingness of Uniform Families

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On the Binary and Boolean Rank of Regular Matrices

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