Testing Intersectingness of Uniform Families

TL;DR

This work studies the property testing complexity of distinguishing whether a uniform family ${\cal F} \subseteq {\binom{[n]}{k}}$ is intersecting or far from intersecting. It develops both two-sided and one-sided non-adaptive testers: a tolerant two-sided tester with query complexity $O(\frac{\ln n}{\varepsilon})$ for suitable $\varepsilon$, and a one-sided canonical tester with $O(\frac{\ln k}{\varepsilon})$ queries under comparable conditions, plus an optimal $O(\frac{1}{\varepsilon})$ tester in the $r=2$ case. The approaches hinge on the Dinur–Friedgut junta structural theorem for large intersecting uniform families and, in the one-sided setting, a careful canonical testing strategy that searches for disjoint witnesses; the Friedgut–Regev results further handle the $n=\Theta(k)$ regime. A matching lower bound of $\Omega(\frac{1}{\varepsilon})$ shows these bounds are tight up to logarithmic factors, highlighting a qualitative difference from non-uniform testing. Overall, the paper advances the understanding of uniform-family property testing, providing nearly tight query complexities and illuminating the impact of structural results on algorithm design.

Abstract

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).

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Theorem 3.1
• Theorem 3.2: DinurF09
• Remark 3.3
• Theorem 4.1
• Lemma 4.2