Disjointness through the Lens of Vapnik–Chervonenkis Dimension: Sparsity and Beyond

Abstract

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. The d-sparse set disjointness problem, where the sets have size at most d, is one such set system with VC dimension d. The deterministic and the randomized communication complexities of the d-sparse set disjointness problem have been well studied and are known to be $\Theta \left( d \log \left({n}/{d}\right)\right)$ and $\Theta(d)$, respectively, in the multi-round communication setting. In this paper, we address the question of whether the randomized communication complexity of the disjointness problem is always upper bounded by a function of the VC dimension of the set system, and does there always exist a gap between the deterministic and randomized communication complexities of the disjointness problem for set systems with small VC dimension.

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 exist set systems of VC dimension d such that both deterministic and randomized (one-way and multi-round) complexities for the set intersection problem can be as high as $\Theta\left( d\log \left( n/d \right) \right)$.