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

TL;DR

This paper investigates how the Vapnik-Chervonenkis (VC) dimension influences the communication complexity of the disjointness problem, providing tight bounds and constructions for set systems with bounded VC dimension.

Contribution

It establishes tight bounds on deterministic and randomized communication complexities for set systems with bounded VC dimension, and constructs natural set systems demonstrating these bounds.

Findings

Communication complexity can be as high as rac{d}{ ext{log}(n/d)} for VC dimension d.

Constructed set systems of VC dimension d with complexities matching upper bounds.

Tight bounds for set intersection problem complexities in set systems with bounded VC dimension.

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 $n$. 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 $d$.