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的有用工具，因为它允许我们从一般图设置转移到低直径图设置，其中可以有效地进行暴力信息收集。最近，Chang和Su [PODC 2022]表明，任何排除固定矿物的高导电网络都包含一个高度节点，因此可以使用扩展器路由在CONGEST模型中将整个图拓扑有效地聚集到一个节点。因此，在排除固定矿物的网络中，许多可以通过低直径分解在LOCAL中有效地解决问题也可以通过扩展器分解在CONGEST中有效地解决。在这项工作中，我们展示了排除固定矿物的网络中改进的分解和路由算法。我们的算法确定地需要多项式的轮数，复杂度为$\text{poly}(\log n,1/\epsilon)$。对于有限度图，我们的算法在$O(\epsilon^{-1}\log n)+\epsilon^{-O(1)}$轮内完成。我们的算法具有广泛的应用，包括CONGEST中的以下结果。1.可以在排除固定矿物的网络中确定地计算（1-ε）-逼近最大独立集，复杂度为$O(\epsilon^{-1}\log^\ast n)+\epsilon^{-O(1)}$轮，接近Lenzen和Wattenhofer [DISC 2008]的$\Omega(\epsilon^{-1}\log^\ast n)$下限。2.如果ε是一个常数或最大度数$\Delta$是一个常数，可以在$O(\log n)$轮内或者在$O(\epsilon^{-1}\log n) + \epsilon^{-O(1)}$轮内确定地完成任何添加矿物排斥属性的性质测试，接近于Levi, Medina和Ron [PODC 2018]的$\Omega(\epsilon^{-1}\log n)$下限。