Efficient Randomized Communication Without Large Monochromatic Rectangles

2026-09-17

AI Summary

本文通过结合作弊纸框架与完全线性PCP方法，构建了一个具有高效随机通信协议的布尔函数，解决了BPP不包含于P^NP的问题。

Abstract

In this paper, we construct a total Boolean function with $\widetilde{O}(\log n)$ randomized communication protocol, while any monochromatic rectangle has density at most $O(2^{-\mathrm{poly}(n)})$. As a corollary, it gives the first total function separation for $\mathrm{BPP}\not\subseteq\mathrm{P}^{\mathrm{NP}}$ in the communication world. Inspired by Gavinsky's recent work (arXiv:2608.18784), our construction combines the cheat-sheet framework with fully linear PCPs.

Problem

• randomized communication
• Boolean function
• monochromatic rectangles
• communication complexity

Innovation

• randomized communication protocol
• monochromatic rectangles
• cheat-sheet framework
• fully linear PCPs