通信复杂性中的矩形大小界和阈值覆盖
Rectangle Size Bounds and Threshold Covers in Communication Complexity
摘要：
我们调查的权力最重要的下界技术随机通信复杂性，这是基于一个评估的最大尺寸的近似单色矩形，最小化的所有分布的输入。虽然已知该界限的0错误版本对于确定性通信是多项式紧的，但是对于恒定错误和随机通信复杂性，在该方向上没有任何已知的。我们首先研究了这个界的一个片面的版本，并得到它的价值在于之间的MA-和AM-所考虑的功能的复杂性。因此，下限实际上适用于MA帽co-MA和AM帽co-AM之间的（通信复杂性）类。我们还证明了不相交性问题的MA-复杂性是Omega（sqrt（n））。在此之后，我们考虑的猜想，下界方法是多项式紧随机通信复杂性。首先，我们反驳了这个猜想的分布版本。然后，我们给出了一个组合特征的下限方法的价值，在所有分布的优化是缺席。这个特征是通过我们所说的均匀阈值覆盖来完成的。我们还研究了这个概念的松弛，即近似多数覆盖和多数覆盖，并比较这三个概念的功率，表现出指数分离。这些覆盖中的每一个都捕获了先前用于随机化通信复杂性的下限方法。
英文摘要：
We investigate the power of the most important lower bound technique in randomized communication complexity, which is based on an evaluation of the maximal size of approximately monochromatic rectangles, minimized over all distributions on the inputs. While it is known that the 0-error version of this bound is polynomially tight for deterministic communication, nothing in this direction is known for constant error and randomized communication complexity. We first study a one-sided version of this bound and obtain that its value lies between the MA- and AM-complexities of the considered function. Hence the lower bound actually works for a (communication complexity) class between MA cap co-MA and AM cap co-AM. We also show that the MA-complexity of the disjointness problem is Omega(sqrt(n)). Following this we consider the conjecture that the lower bound method is polynomially tight for randomized communication complexity. First we disprove a distributional version of this conjecture. Then we give a combinatorial characterization of the value of the lower bound method, in which the optimization over all distributions is absent. This characterization is done by what we call a uniform threshold cover. We also study relaxations of this notion, namely approximate majority covers and majority covers, and compare these three notions in power, exhibiting exponential separations. Each of these covers captures a lower bound method previously used for randomized communication complexity.
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