Dice Question Streamline Icon: https://streamlinehq.com

Large monochromatic rectangles in BPP

Establish whether every N × N communication matrix arising from a problem in the constant-cost randomized communication class BPP contains a monochromatic rectangle of size Ω(N) × Ω(N).

Information Square Streamline Icon: https://streamlinehq.com

Background

Large monochromatic rectangles are a central structural feature linked to efficient communication. The paper notes that all known examples in BPP have such rectangles of linear size and cites prior conjectures asserting this should hold universally, especially for problems solvable via constant queries to RP0.

References

Also, all known N × N matrices in BPP have monochromatic rectangles of size Ω(N) × Ω(N), and [CLV19, HHH22dimfree] conjecture that this is always true (it is true for problems solved by (1) queries to RP0). 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.”