🤖 AI Summary
To address the high worst-case round complexity of computing a maximum-cardinality matching in the CONGEST model, this paper presents the first randomized distributed algorithm with round complexity $ ilde{O}(mu(G))$, where $mu(G)$ is the size of an optimal matching. Our method introduces an efficient distributed construction of augmenting paths of length $ell$, completing in $ ilde{O}(ell)$ rounds with high probability—the first such result in the distributed setting. This breakthrough rests on a novel distributed implementation of alternating base trees, resolving a long-standing open problem posed by Ahmadi and Kuhn, and revealing several key independence properties of this structure. The algorithm integrates sparse subgraph identification, randomized path exploration, decentralized information aggregation, and lightweight synchronization protocols. It significantly improves upon prior state-of-the-art bounds, achieving near-linear round complexity in the optimal matching size.
📝 Abstract
In this paper, we propose a randomized $ ilde{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 $ ilde{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.
