The classic lower bound of Kuhn, Moscibroda and Wattenhofer [JACM 2016] states that approximate maximum matching and approximate vertex cover (among other problems) in the LOCAL model require $Ω(\min\{\sqrt{\frac{\log n}{\log\log n}}, \frac{\log Δ}{\log\log Δ}\})$ rounds, for any polylogarithmic or smaller approximation ratio. As a function of $Δ$, this complexity was subsequently matched for constant-approximate weighted vertex cover [Bar-Yehuda, Censor-Hillel and Schwartzman, JACM 2017] and constant-approximate maximum matching [Bar-Yehuda, Censor-Hillel, Ghaffari and Schwartzman, PODC 2017]. One might expect, therefore, that the true complexity should be $Θ(\frac{\log Δ}{\log\log Δ})$, and the $n$-dependent term in the lower bound is just an artefact of the proof method. We show that this is not the case, and a term dependent on $n$ is in fact required. Specifically, we show randomized algorithms for $2+\varepsilon$-approximate maximum matching and approximate (weighted) minimum vertex cover taking $O(\frac{\log n}{\log^2 \log n})$ rounds. Our algorithms are based on a novel graph decomposition result generalizing the method of Miller, Peng and Xu [SPAA 2013], which we use to reduce the `effective' degree of high-degree graphs. We expect that this decomposition may be of further use for other problems.

翻译：Kuhn、Moscibroda和Wattenhofer [JACM 2016] 的经典下界指出，在LOCAL模型中，近似最大匹配与近似顶点覆盖（以及其他问题）对于任意多对数或更小的近似比，都需要$Ω(\min\{\sqrt{\frac{\log n}{\log\log n}}, \frac{\log Δ}{\log\log Δ}\})$轮复杂度。作为$Δ$的函数，该复杂度随后被常数近似加权顶点覆盖 [Bar-Yehuda、Censor-Hillel和Schwartzman，JACM 2017] 与常数近似最大匹配 [Bar-Yehuda、Censor-Hillel、Ghaffari和Schwartzman，PODC 2017] 所匹配。因此，人们可能预期其真实复杂度应为$Θ(\frac{\log Δ}{\log\log Δ})$，而下界中与$n$相关的项仅是证明方法的人为产物。本文证明情况并非如此，实际上需要与$n$相关的项。具体而言，我们提出了$2+\varepsilon$近似最大匹配与近似（加权）最小顶点覆盖的随机算法，其轮复杂度为$O(\frac{\log n}{\log^2 \log n})$。我们的算法基于一个推广Miller、Peng和Xu [SPAA 2013] 方法的新型图分解结果，用以降低高度数图的"有效"度数。我们预期该分解可进一步应用于其他问题。