不相交性的一个强直积定理
A Strong Direct Product Theorem for Disjointness
摘要：
强直积定理表明，如果我们想计算函数的$k$独立实例，使用少于$k$倍的资源来计算一个实例，那么在$k$中，总的成功概率将呈指数级小。我们为不相交性问题的随机通信复杂性建立了这样一个定理，即，利用通信$const\cdot kn$，求解大小为$n$的$k$实例的成功概率在$k$中只能是指数小的。我们表明，这个界限甚至持有有限的模糊性的$AM$通信协议。这也意味着一个新的下界的不相交性在一个有限的3人NOF协议，和最佳的通信空间的权衡布尔矩阵产品。我们的主要结果如下从一个解决方案的对偶的线性规划问题，其可行性来自于一个所谓的交叉采样引理，推广了拉兹博罗夫的结果。
英文摘要：
A strong direct product theorem states that if we want to compute $k$ independent instances of a function, using less than $k$ times the resources needed for one instance, then the overall success probability will be exponentially small in $k$. We establish such a theorem for the randomized communication complexity of the Disjointness problem, i.e., with communication $const\cdot kn$ the success probability of solving $k$ instances of size $n$ can only be exponentially small in $k$. We show that this bound even holds for $AM$ communication protocols with limited ambiguity. This also implies a new lower bound for Disjointness in a restricted 3-player NOF protocol, and optimal communication-space tradeoffs for Boolean matrix product. Our main result follows from a solution to the dual of a linear programming problem, whose feasibility comes from a so-called Intersection Sampling Lemma that generalizes a result by Razborov.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
