No Constant-Cost Protocol for Point--Line Incidence

Mika Göös; Nathaniel Harms; Florian K. Richter; Anastasia Sofronova

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arXiv · 2604.03805

No Constant-Cost Protocol for Point--Line Incidence

Mika Göös·Nathaniel Harms·Florian K. Richter·Anastasia Sofronova

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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.

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Mika Göös, Nathaniel Harms, Florian K. Richter, Anastasia Sofronova. 2026-04-04. No Constant-Cost Protocol for Point--Line Incidence. https://arxiv.org/abs/2604.03805

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