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

Abstract

The disjointness problem—where Alice and Bob are given twosubsets of {1, dots, n} and they have to check if their setsintersect—is a central problem in the world of communicationcomplexity. While both deterministic and randomized communicationcomplexities for this problem are known to be Theta(n), it isalso known that if the sets are assumed to be drawn from somerestricted set systems then the communication complexity can be muchlower. In this work, we explore how communication complexitymeasures change with respect to the complexity of the underlying setsystem. The complexity measure for the set system that we use inthis work is the Vapnik—Chervonenkis (VC) dimension. Moreprecisely, on any set system with VC dimension bounded by d, weanalyze how large can the deterministic and randomized communicationcomplexities be, as a function of d and n. The d-sparse setdisjointness problem, where the sets have size at most d, is onesuch set system with VC dimension d. The deterministic and therandomized communication complexities of the d-sparse setdisjointness problem have been well studied and are known to beTheta left( d log left({n}/{d}right)right) and Theta(d),respectively, in the multi-round communication setting. In thispaper, we address the question of whether the randomizedcommunication complexity of the disjointness problem is always upperbounded by a function of the VC dimension of the set system, anddoes there always exist a gap between the deterministic andrandomized communication complexities of the disjointness problemfor 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(dlog 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 Thetaleft( dlog left( n/d right) right).