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Equality-reducibility characterization of UPP ∩ BPP

Determine whether the intersection of UPP and BPP is exactly the class of communication problems that reduce to Equality.

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Background

UPP consists of problems with constant-cost unbounded-error randomized protocols (equivalently, bounded sign-rank matrices). The paper points to an open question posed in prior work, asking whether UPP ∩ BPP coincides precisely with the set of problems reducible to Equality, which would severely constrain the possibility of finding new BPP examples within UPP.

References

The relationship between UPP and BPP is the subject of several open problems. [HZ24] ask if UPP ∩ BPP is exactly the class of problems which reduce to Equality, which would rule out any new examples in this class.

— Constant-Cost Communication is not Reducible to k-Hamming Distance (2407.20204 - Fang et al., 29 Jul 2024) in Section: Constant-Cost Communication: the Story so Far and Farther (Sign-rank)