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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.11504 |
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| _version_ | 1866910532931420160 |
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| author | Haviv, Ishay Parnas, Michal |
| author_facet | Haviv, Ishay Parnas, Michal |
| contents | A set family ${\cal F}$ is called intersecting if every two members of ${\cal F}$ intersect, and it is called uniform if all members of ${\cal F}$ share a common size. A uniform family ${\cal F} \subseteq \binom{[n]}{k}$ of $k$-subsets of $[n]$ is $\varepsilon$-far from intersecting if one has to remove more than $\varepsilon \cdot \binom{n}{k}$ of the sets of ${\cal F}$ to make it intersecting. We study the property testing problem that given query access to a uniform family ${\cal F} \subseteq \binom{[n]}{k}$, asks to distinguish between the case that ${\cal F}$ is intersecting and the case that it is $\varepsilon$-far from intersecting. We prove that for every fixed integer $r$, the problem admits a non-adaptive two-sided error tester with query complexity $O(\frac{\ln n}{\varepsilon})$ for $\varepsilon \geq Ω( (\frac{k}{n})^r)$ and a non-adaptive one-sided error tester with query complexity $O(\frac{\ln k}{\varepsilon})$ for $\varepsilon \geq Ω( (\frac{k^2}{n})^r)$. The query complexities are optimal up to the logarithmic terms. For $\varepsilon \geq Ω( (\frac{k^2}{n})^2)$, we further provide a non-adaptive one-sided error tester with optimal query complexity of $O(\frac{1}{\varepsilon})$. 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11504 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Testing Intersectingness of Uniform Families Haviv, Ishay Parnas, Michal Data Structures and Algorithms A set family ${\cal F}$ is called intersecting if every two members of ${\cal F}$ intersect, and it is called uniform if all members of ${\cal F}$ share a common size. A uniform family ${\cal F} \subseteq \binom{[n]}{k}$ of $k$-subsets of $[n]$ is $\varepsilon$-far from intersecting if one has to remove more than $\varepsilon \cdot \binom{n}{k}$ of the sets of ${\cal F}$ to make it intersecting. We study the property testing problem that given query access to a uniform family ${\cal F} \subseteq \binom{[n]}{k}$, asks to distinguish between the case that ${\cal F}$ is intersecting and the case that it is $\varepsilon$-far from intersecting. We prove that for every fixed integer $r$, the problem admits a non-adaptive two-sided error tester with query complexity $O(\frac{\ln n}{\varepsilon})$ for $\varepsilon \geq Ω( (\frac{k}{n})^r)$ and a non-adaptive one-sided error tester with query complexity $O(\frac{\ln k}{\varepsilon})$ for $\varepsilon \geq Ω( (\frac{k^2}{n})^r)$. The query complexities are optimal up to the logarithmic terms. For $\varepsilon \geq Ω( (\frac{k^2}{n})^2)$, we further provide a non-adaptive one-sided error tester with optimal query complexity of $O(\frac{1}{\varepsilon})$. 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). |
| title | Testing Intersectingness of Uniform Families |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2404.11504 |