  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.
