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 $\text{poly}(\log n, 1/\epsilon)$ rounds deterministically. For bounded-degree graphs, our algorithms finish in $O(\epsilon^{-1}\log n) + \epsilon^{-O(1)}$ rounds. Our algorithms have a wide range of applications, including the following results in CONGEST. 1. A $(1-\epsilon)$-approximate maximum independent set in a network excluding a fixed minor can be computed deterministically in $O(\epsilon^{-1}\log^\ast n) + \epsilon^{-O(1)}$ rounds, nearly matching the $\Omega(\epsilon^{-1}\log^\ast n)$ 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 $\epsilon$ is a constant or $O(\epsilon^{-1}\log n) + \epsilon^{-O(1)}$ rounds if the maximum degree $\Delta$ is a constant, nearly matching the $\Omega(\epsilon^{-1}\log n)$ lower bound of Levi, Medina, and Ron [PODC 2018].

翻译：在LOCAL模型中，低直径分解是设计算法时的重要工具，因为它使我们能够从一般图场景转换到低直径图场景，在此场景下可以高效进行暴力信息收集。近期，Chang与Su [PODC 2022] 证明，不含固定子图的任意高电导网络必存在高度数顶点，因此通过扩展器路由可在CONGEST模型中将整个图拓扑高效收集至单个顶点。由此，在不含固定子图的网络中，许多原需通过低直径分解在LOCAL模型中高效解决的问题，现也可通过扩展器分解在CONGEST模型中得到高效解决。本研究针对CONGEST模型中不含固定子图的网络，提出了改进的分解与路由算法。算法代价确定性为 $\text{poly}(\log n, 1/\epsilon)$ 轮。对于有界度图，算法可在 $O(\epsilon^{-1}\log n) + \epsilon^{-O(1)}$ 轮内完成。本算法具有广泛的应用，包括以下CONGEST模型中的成果：1. 在不含固定子图的网络中，可通过确定性算法在 $O(\epsilon^{-1}\log^\ast n) + \epsilon^{-O(1)}$ 轮内计算 $(1-\epsilon)$ 近似最大独立集，接近Lenzen与Wattenhofer [DISC 2008] 提出的 $\Omega(\epsilon^{-1}\log^\ast n)$ 下界；2. 对于任意加法型子图闭合性质的属性测试，在 $\epsilon$ 为常数时可通过确定性算法在 $O(\log n)$ 轮内完成；若最大度 $\Delta$ 为常数，则需 $O(\epsilon^{-1}\log n) + \epsilon^{-O(1)}$ 轮，接近Levi、Medina与Ron [PODC 2018] 提出的 $\Omega(\epsilon^{-1}\log n)$ 下界。