关于通信复杂度中LogRank猜想的一则注记
A Note on the LogRank Conjecture in Communication Complexity
AI总结：
针对通信复杂度领域著名的LogRank猜想这一35年未解决的开放问题，本文提出一种通信开销符合猜想量级的协议，其计算量与原布尔函数通过合数模低次多线性多项式关联，为解决该猜想提供了潜在思路。
摘要：
Lovász与Saks于1988年提出的LogRank猜想是通信复杂度理论中最著名的开放问题。该猜想表述如下：假设两名参与者希望计算布尔函数$f(x,y)$，其中第一名参与者已知$x$，第二名已知$y$，双方可以收发以比特编码的消息，那么他们能够通过交换$(\text{log rank}(M_f))^c$比特完成$f(x,y)$的计算，其中$M_f$是由函数$f$确定的布尔矩阵。该问题至今完全未被解决且广受关注，在过去35年间经受了大量研究尝试的冲击，目前已知的最优上界相对于猜想的界仍呈指数关系。遗憾的是，我们无法证明该猜想，但我们提出了一种通信开销为$(\text{log rank}(M_f))^c$比特的通信协议，该协议计算的量与$f(x,y)$存在一定关联，这种关联通过合数模下的低次多线性多项式表示来刻画。我们的这一结果或许有助于解决这一长期悬而未决的猜想。
英文摘要：
The LogRank conjecture of Lovász and Saks from 1988 is the most famous open problem in the communication complexity theory. The statement is as follows: Suppose that two players intend to compute a Boolean function $f(x,y)$ when $x$ is known for the first and $y$ for the second player, and they may send and receive messages encoded with bits, then they can compute $f(x,y)$ with exchanging $(\log \rank (M_f))^c $ bits, where $M_f$ is a Boolean matrix, determined by function $f$. The problem is widely open and very popular, and it has resisted numerous attacks in the last 35 years. The best upper bound is still exponential in the bound of the conjecture. Unfortunately, we cannot prove the conjecture, but we present a communication protocol with $(\log \rank (M_f))^c $ bits, which computes a -- somewhat -- related quantity to $f(x,y)$. The relation is characterized by a representation of low-degree, multi-linear polynomials modulo composite numbers. This result of ours may help to settle this long-time open conjecture.
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