No Constant-Cost Protocol for Point--Line Incidence

2026-04-04

AI Summary

This study addresses the randomized communication complexity of the point-line incidence problem: given two parties holding integer pairs $(x, y)$ and $(a, b)$ respectively, they must determine whether $y = ax + b$. By integrating techniques from communication complexity theory and information theory, the authors establish for the first time a tight bound of $\Theta(\log n)$ on the randomized communication complexity of this problem, thereby proving a super-constant lower bound. This result rules out the existence of any constant-communication protocol for the task and, notably, provides the first explicit example of a communication problem with constant support rank yet super-constant randomized communication complexity, confirming a conjecture by Cheung et al.

Abstract

Alice and Bob are given $n$-bit integer pairs $(x,y)$ and $(a,b)$, respectively, and they must decide if $y=ax+b$. We prove that the randomised communication complexity of this Point--Line Incidence problem is $Θ(\log n)$. This confirms a conjecture of Cheung, Hatami, Hosseini, and Shirley (CCC 2023) that the complexity is super-constant, and gives the first example of a communication problem with constant support-rank but super-constant randomised complexity.

Problem

• Point--Line Incidence
• communication complexity
• randomised complexity
• support-rank

Innovation

• communication complexity
• Point--Line Incidence
• support-rank
• randomised protocols
• super-constant complexity