AI 中文总结
本文解决随机通信复杂度中直积猜想的核心形式，证明任意函数的摊销期望随机通信复杂度等于其零错误信息复杂度，还将该结果应用于集合不交性问题并反驳了DFHL18的相关猜想。
AI 中文摘要
计算复杂度中的直积问题研究的是求解n个独立的计算任务实例是否必然需要求解单个实例所需资源的n倍。本文中，我们解决了随机通信复杂度中直积猜想的核心形式。具体而言，我们证明了任意函数的“摊销期望随机通信复杂度”恰好等于其“零错误信息复杂度”——该度量用于衡量通信双方为无错误计算函数必须泄露的关于其输入的精确信息量。这一结果还为摊销的“最坏情况”随机通信复杂度提供了至多一个常数因子的紧刻画。为实现这一精确刻画，我们引入了一种新的单实例协议嵌入方法，该方法配备前缀验证机制以准确定位全局错误。此外，我们将新的结构定理应用于基础的集合不交性（Set-Disjointness）问题，得到的集合不交性问题的精确渐近界成功反驳了DFHL18中关于其缩放行为的一项猜想。
英文摘要
The direct sum problem in computational complexity asks whether solving $n$ independent instances of a computational task inherently requires $n$ times the resources needed to solve a single instance. In this paper, we resolve a central form of the direct sum conjecture in randomized communication complexity. Specifically, we prove that the "amortized expected randomized communication complexity" of any function is exactly equal to its "zero-error information complexity"---a measure of the precise amount of information the communicating parties must reveal about their inputs to compute the function without error. This result also provides a tight characterization of the amortized "worst-case" randomized communication complexity up to a constant factor. To achieve our exact characterization, we introduce a new single-instance protocol embedding equipped with a prefix-verification mechanism to accurately localize global errors. Furthermore, we apply our new structural theorems to the fundamental Set-Disjointness problem. Our resulting exact asymptotic bounds for Set-Disjointness successfully refute a conjecture in DFHL18 regarding its scaling behavior.
