A Nearly Linear-Time Distributed Algorithm for Exact Maximum Matching

In this paper, we propose a randomized (O) over tilde(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 l with high probability within (O) over tilde (l) 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 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.