No Complete Problem for Constant-Cost Randomized Communication
<|reference_start|>No Complete Problem for Constant-Cost Randomized Communication: 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.<|reference_end|>
@article{fang2024no,
    title={No Complete Problem for Constant-Cost Randomized Communication},
    author={Yuting Fang, Lianna Hambardzumyan, Nathaniel Harms, Pooya Hatami},
    journal={arXiv preprint arXiv:2404.00812},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.00812},
    primaryClass={cs.CC}
}