We study problems in randomized communication complexity when the protocol is only required to attain some small advantage over purely random guessing, i.e., it produces the correct output with probability at least $$\epsilon$$ ϵ greater than one over the codomain size of the function. Previously, Braverman and Moitra (in: Proceedings of the 45th symposium on theory of computing (STOC), ACM, pp 161–170, 2013) showed that the set-intersection function requires $$\Theta(\epsilon{n})$$ Θ ( ϵ n ) communication to achieve advantage $$\epsilon$$ ϵ . Building on this, we prove the same bound for several variants of set-intersection: (1) the classic “tribes” function obtained by composing with And (provided $$1/\epsilon$$ 1 / ϵ is at most the width of the And ), and (2) the variant where the sets are uniquely intersecting and the goal is to determine partial information about (say, certain bits of the index of) the intersecting coordinate.

中文翻译：

---

通信复杂性小优势

我们研究随机通信复杂性中的问题，当协议只需要获得比纯随机猜测的一些小的优势时，即它产生正确的输出的概率至少为 $$\epsilon$$ϵ 大于 1功能。此前，Braverman 和 Moitra（在：第 45 届计算理论研讨会 (STOC)，ACM，pp 161–170，2013 年）表明集合交集函数需要 $$\Theta(\epsilon{n})$ $ Θ ( ϵ n ) 沟通取得优势 $$\epsilon$$ ϵ 。在此基础上，我们证明了集合交集的几个变体的相同界限：（1）通过与 And 组合获得的经典“部落”函数（假设 $$1/\epsilon$$1 / ϵ 至多是和 ），