No Complete Problem for Constant-Cost Randomized Communication

Yuting Fang; Lianna Hambardzumyan; Nathaniel Harms; Pooya Hatami

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arXiv · 2404.00812

No Complete Problem for Constant-Cost Randomized Communication

Yuting Fang·Lianna Hambardzumyan·Nathaniel Harms·Pooya Hatami

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Abstract

We prove that the class of communication problems with public-coin randomized constant-cost protocols, called $BPP^0$, does not contain a complete problem. In other words, there is no randomized constant-cost problem $Q \in BPP^0$, such that all other problems $P \in BPP^0$ can be computed by a constant-cost deterministic protocol with access to an oracle for $Q$. We also show that the $k$-Hamming Distance problems form an infinite hierarchy within $BPP^0$. Previously, it was known only that Equality is not complete for $BPP^0$. We introduce a new technique, using Ramsey theory, that can prove lower bounds against arbitrary oracles in $BPP^0$, and more generally, we show that $k$-Hamming Distance matrices cannot be expressed as a Boolean combination of any constant number of matrices which forbid large Greater-Than subproblems.

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Yuting Fang, Lianna Hambardzumyan, Nathaniel Harms, Pooya Hatami. 2024-03-31. No Complete Problem for Constant-Cost Randomized Communication. https://arxiv.org/abs/2404.00812

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