In the LOCAL model, low-diameter decomposition is a useful tool in designing algorithms, as it allows us to shift from the general graph setting to the low-diameter graph setting, where brute-force information gathering can be done efficiently. Recently, Chang and Su [PODC 2022] showed that any high-conductance network excluding a fixed minor contains a high-degree vertex, so the entire graph topology can be gathered to one vertex efficiently in the CONGEST model using expander routing. Therefore, in networks excluding a fixed minor, many problems that can be solved efficiently in LOCAL via low-diameter decomposition can also be solved efficiently in CONGEST via expander decomposition. In this work, we show improved decomposition and routing algorithms for networks excluding a fixed minor in the CONGEST model. Our algorithms cost $\text{poly}(\log n, 1/ε)$ rounds deterministically. For bounded-degree graphs, our algorithms finish in $O(ε^{-1}\log n) + ε^{-O(1)}$ rounds. Our algorithms have a wide range of applications, including the following results in CONGEST. 1. A $(1-ε)$-approximate maximum independent set in a network excluding a fixed minor can be computed deterministically in $O(ε^{-1}\log^\ast n) + ε^{-O(1)}$ rounds, nearly matching the $Ω(ε^{-1}\log^\ast n)$ lower bound of Lenzen and Wattenhofer [DISC 2008]. 2. Property testing of any additive minor-closed property can be done deterministically in $O(\log n)$ rounds if $ε$ is a constant or $O(ε^{-1}\log n) + ε^{-O(1)}$ rounds if the maximum degree $Δ$ is a constant, nearly matching the $Ω(ε^{-1}\log n)$ lower bound of Levi, Medina, and Ron [PODC 2018].

翻译：在LOCAL模型中，低直径分解是算法设计中的有用工具，它允许我们从一般图场景转移到低直径图场景，从而高效执行暴力信息收集。近期，Chang 与 Su [PODC 2022] 表明，任何排除固定子图的高导通网络均包含高度数顶点，因此可通过扩展器路由在CONGEST模型中高效地将整个图拓扑汇聚至单一顶点。由此，在排除固定子图的网络中，许多在LOCAL模型中凭借低直径分解可高效解决的问题，亦可通过扩展器分解在CONGEST模型中高效求解。本研究针对CONGEST模型中排除固定子图的网络，提出了改进的分解与路由算法。我们的算法以确定性方式需$\text{poly}(\log n, 1/ε)$轮。对于有界度图，算法在$O(ε^{-1}\log n) + ε^{-O(1)}$轮内完成。这些算法具有广泛的应用，包括以下CONGEST模型中的成果： 1. 在排除固定子图的网络中，可确定性计算$(1-ε)$-近似最大独立集，所需轮数为$O(ε^{-1}\log^\ast n) + ε^{-O(1)}$，几乎匹配Lenzen与Wattenhofer [DISC 2008]提出的$Ω(ε^{-1}\log^\ast n)$下界。 2. 针对任意加性子图封闭性质的属性测试，若$ε$为常数则可确定性在$O(\log n)$轮内完成；若最大度$Δ$为常数则需$O(ε^{-1}\log n) + ε^{-O(1)}$轮，几乎匹配Levi、Medina与Ron [PODC 2018]提出的$Ω(ε^{-1}\log n)$下界。