The disjointness problem - where Alice and Bob are given two subsets of $\{1, \dots, n\}$ and they have to check if their sets intersect - is a central problem in the world of communication complexity. While both deterministic and randomized communication complexities for this problem are known to be $\Theta(n)$, it is also known that if the sets are assumed to be drawn from some restricted set systems then the communication complexity can be much lower. In this work, we explore how communication complexity measures change with respect to the complexity of the underlying set system. The complexity measure for the set system that we use in this work is the Vapnik-Chervonenkis (VC) dimension. More precisely, on any set system with VC dimension bounded by $d$, we analyze how large can the deterministic and randomized communication complexities be, as a function of $d$ and $n$. In this paper, we construct two natural set systems of VC dimension $d$, motivated from geometry. Using these set systems we show that the deterministic and randomized communication complexity can be $\widetilde{\Theta}\left(d\log \left( n/d \right)\right)$ for set systems of VC dimension $d$ and this matches the deterministic upper bound for all set systems of VC dimension $d$. We also study the deterministic and randomized communication complexities of the set intersection problem when sets belong to a set system of bounded VC dimension. We show that there exists set systems of VC dimension $d$ such that both deterministic and randomized (one-way and multi-round) complexity for the set intersection problem can be as high as $\Theta\left( d\log \left( n/d \right) \right)$, and this is tight among all set systems of VC dimension $d$.

中文翻译：

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Vapnik-Chervonenkis 维度视角下的不相交：稀疏性和超越

不相交问题——爱丽丝和鲍勃被赋予 $\{1, \dots, n\}$ 的两个子集，他们必须检查他们的集合是否相交——是通信复杂性领域的核心问题。虽然已知这个问题的确定性和随机通信复杂度都是 $\Theta(n)$，但也知道如果假设集合是从一些受限集合系统中提取的，那么通信复杂度会低得多。在这项工作中，我们探索了通信复杂性度量如何随着底层集合系统的复杂性而变化。我们在这项工作中使用的集合系统的复杂性度量是 Vapnik-Chervonenkis (VC) 维度。更准确地说，在任何 VC 维以 $d$ 为界的集合系统上，我们分析了确定性和随机化的通信复杂度有多大，作为 $d$ 和 $n$ 的函数。在本文中，我们构建了两个 VC 维 $d$ 的自然集合系统，其动机来自几何。使用这些集合系统，我们表明对于 VC 维 $d$ 的集合系统，确定性和随机通信复杂度可以是 $\widetilde{\Theta}\left(d\log \left( n/d \right)\right)$这与 VC 维 $d$ 的所有集合系统的确定性上限相匹配。当集合属于有界 VC 维的集合系统时，我们还研究了集合交集问题的确定性和随机通信复杂性。我们表明存在 VC 维 $d$ 的集合系统，使得集合相交问题的确定性和随机（单向和多轮）复杂度可以高达 $\Theta\left( d\log \left ( n/d \right) \right)$,