In this paper, we propose a randomized $\tilde{O}(\mu(G))$-round algorithm for the maximum cardinality matching problem in the CONGEST model, where $\mu(G)$ means the maximum size of a matching of the input graph $G$. The proposed algorithm substantially improves the current best worst-case running time. The key technical ingredient is a new randomized algorithm of finding an augmenting path of length $\ell$ with high probability within $\tilde{O}(\ell)$ rounds, which positively settles an open problem left in the prior work by Ahmadi and Kuhn [DISC'20]. The idea of our augmenting path algorithm is based on a recent result by Kitamura and Izumi [IEICE Trans.'22], which efficiently identifies a sparse substructure of the input graph containing an augmenting path, following a new concept called \emph{alternating base trees}. Their algorithm, however, resorts to a centralized approach of collecting the entire information of the substructure into a single vertex for constructing an augmenting path. The technical highlight of this paper is to provide a fully-decentralized counterpart of such a centralized method. To develop the algorithm, we prove several new structural properties of alternating base trees, which are of independent interest.

翻译：本文针对CONGEST模型中的最大基数匹配问题，提出一种随机化算法，其运行时间为$\tilde{O}(\mu(G))$轮，其中$\mu(G)$表示输入图$G$的最大匹配规模。所提算法显著改进了当前最坏情况下的最优运行时间。关键技术突破在于一种新型随机化算法——能够在$\tilde{O}(\ell)$轮内以高概率找到长度为$\ell$的增广路径，这正面解决了Ahmadi与Kuhn[DISC'20]前期工作中遗留的开放问题。该增广路径算法的设计思路源于Kitamura与Izumi[IEICE Trans.'22]的最新成果：通过名为"交替基树"的新概念，高效识别包含增广路径的输入图稀疏子结构。然而其算法采用集中式方法，需将子结构全部信息收集至单个顶点来构造增广路径。本文的技术亮点在于提供这种集中式方法的完全去中心化等价方案。为构建该算法，我们证明了交替基树若干新的结构性质，这些性质本身具有独立研究价值。