A Nearly Linear-Time Distributed Algorithm for Maximum Cardinality Matching

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 in part to a centralized approach of collecting the entire information of the substructure into a single vertex for constructing a long 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.