KMW界的一个简洁证明
A Breezing Proof of the KMW Bound
摘要：
在他们2004年的开创性论文中，Kuhn，Moscibroda和Wattenhofer（KMW）证明了一个困难的结果，适用于EQUIPMENT模型中的几个基本图形问题：任何（随机化）算法，存在具有$n$节点和最大度$Δ$的输入图，其中$Ω(\min\{\sqrt{\log n/\log \log n},\log Δ/\log \log Δ\})$需要（预期的）通信回合来获得对最小顶点覆盖，最小支配集，或最大匹配。通过约简，这种困难扩展到对称破缺任务，如寻找最大独立集或最大匹配。$15$多年后的今天，仍然没有证明这个结果对读者来说很容易。为了改变这一点，在这项工作中，我们提供了一个完全独立的和$\mathit{simple}$证明的KMW下限。关键参数是算法，它依赖于一个不变量，可以很容易地从下界图的生成规则进行验证。
英文摘要：
In their seminal paper from 2004, Kuhn, Moscibroda, and Wattenhofer (KMW) proved a hardness result for several fundamental graph problems in the LOCAL model: For any (randomized) algorithm, there are input graphs with $n$ nodes and maximum degree $Δ$ on which $Ω(\min\{\sqrt{\log n/\log \log n},\log Δ/\log \log Δ\})$ (expected) communication rounds are required to obtain polylogarithmic approximations to a minimum vertex cover, minimum dominating set, or maximum matching. Via reduction, this hardness extends to symmetry breaking tasks like finding maximal independent sets or maximal matchings. Today, more than $15$ years later, there is still no proof of this result that is easy on the reader. Setting out to change this, in this work, we provide a fully self-contained and $\mathit{simple}$ proof of the KMW lower bound. The key argument is algorithmic, and it relies on an invariant that can be readily verified from the generation rules of the lower bound graphs.
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