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One-sided-error completeness for BPP

Determine whether every constant-cost randomized two-party communication problem (the class BPP in the paper’s sense) can be computed by a constant-cost deterministic oracle protocol that makes a constant number of queries to constant-cost one-sided-error communication problems (the class RP^0).

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Background

The paper studies constant-cost randomized communication problems (denoted BPP). All known examples can be solved using constant many queries to problems with one-sided error (RP0). Whether this is universally true for BPP has significant implications for structural properties of constant-cost communication and for the search of complete problems or hierarchies.

References

All known problems in BPP can be solved with (1) queries to constant-cost problems with one-sided error (ie the class RP0). Is this always true? (See also [HH24]). We mentioned above that finding a complete problem (or hierarchy) could have answered many questions about BPP (depending on the properties of the complete problem itself); these two open problems are examples of that.

— Constant-Cost Communication is not Reducible to k-Hamming Distance (2407.20204 - Fang et al., 29 Jul 2024) in Section 1.5 (Constant-Cost Communication: the Story so Far and Farther), paragraph “One-sided error, and large rectangles.”