No Constant-Cost Protocol for Point--Line Incidence

TL;DR

This paper proves that the randomized communication complexity of the Point--Line Incidence problem is logarithmic in n, confirming a conjecture and providing a unique example of a problem with constant support-rank but super-constant complexity.

Contribution

It establishes the first example of a communication problem with constant support-rank yet super-constant randomized complexity.

Findings

Randomized complexity of Point--Line Incidence is Θ(log n).

Confirms a conjecture that the problem's complexity is super-constant.

Provides the first such example linking support-rank and complexity.

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 $\Theta(\log n)$. 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.