In this paper, we propose a randomized $\tilde{O}(\Mmax)$-round algorithm for the maximum cardinality matching problem in the CONGEST model, where $\Mmax$ 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：本文提出一种随机化的 $\tilde{O}(\Mmax)$ 轮算法，用于解决 CONGEST 模型中的最大基数匹配问题，其中 $\Mmax$ 表示输入图 $G$ 的最大匹配规模。该算法显著改进了当前最坏情况下的最优运行时间。关键的技术创新在于一种新的随机化算法，可在 $\tilde{O}(\ell)$ 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] 先前工作中遗留的开放问题。我们的增广路径算法思想基于 Kitamura 与 Izumi [IEICE Trans.'22] 的最新成果，该成果通过引入名为"交替基树"的新概念，高效识别输入图中包含增广路径的稀疏子结构。然而，他们的算法采用集中式方法，需将子结构的全部信息收集到单个顶点以构建增广路径。本文的技术亮点在于提供这种集中式方法的完全分布式对应实现。为开发该算法，我们证明了交替基树的若干新结构性质，这些性质本身也具有独立研究价值。