The communication complexity of the inevitable intersection problem

TL;DR

This paper investigates the communication complexity of a new relational problem called inevitable intersection, which challenges existing lower bound methods in communication complexity theory.

Contribution

It introduces the inevitable intersection problem, a natural variation of set disjointness, designed to evade current lower bound techniques and motivate new analytical approaches.

Findings

Identifies limitations of existing lower bound methods for certain problems.

Proposes inevitable intersection as a challenging test case for communication complexity.

Highlights the need for novel techniques to analyze this problem.

Abstract

Set disjointness is a central problem in communication complexity. Here Alice and Bob each receive a subset of an n-element universe, and they need to decide whether their inputs intersect or not. The communication complexity of this problem is relatively well understood, and in most models, including $-$ most famously $-$ interactive randomised communication with bounded error, the problem requires much communication. In this work we were looking for a variation of the set disjointness problem, as natural and simple as possible, for which the known lower bound methods would fail, and thus a new approach would be required in order to understand its complexity. The problem that we have found is a relational one: each player receives a subset as input, and the goal is to find an element that belongs to both players. We call it inevitable intersection.