A Nearly Linear-Time Distributed Algorithm for Maximum Cardinality Matching

TL;DR

The paper tackles computing a maximum matching in general graphs within the CONGEST model. It introduces a randomized algorithm that runs in $\tilde{O}(\mu(G))$ rounds with high probability by achieving $O(\ell)$-round augmentation for a path of length $\ell$ and integrating a fully-decentralized augmenting-path construction based on alternating base trees. The approach blends Edmonds' blossom ideas, Hopcroft–Karp analysis, and a refined sparse-certificate technique to enable scalable, distributed augmentation without gathering global input. This yields a substantial improvement over prior bounds and provides new structural tools (alternating base trees and edge levels) that may inform future sublinear-time distributed matching and related problems.

Abstract

In this paper, we propose a randomized $\tilde{O}(μ(G))$-round algorithm for the maximum cardinality matching problem in the CONGEST model, where $μ(G)$ 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.

Figures (4)

• Figure 1: Examples of alternating base trees (from KI22). Solid lines are matching edges, and broken lines are non-matching edges.
• Figure 2: An example of edge levels. Two values paired by parentheses respectively mean the values of $\mathrm{dist}^{\mathrm{odd}}(\cdot)$ and $\mathrm{dist}^{\mathrm{even}}(\cdot)$. The values paired by brackets represent edge levels. All tree edges have level $[\infty, \infty]$, which is not explicitly described in the figure.
• Figure 3: A sketch of Lemma \ref{['lma:twopaths']}.
• Figure 4: Illustration of Case 2 in the proof of Lemma \ref{['lma:levelbound']}.

Theorems & Definitions (38)

• Theorem 1
• Theorem 2: Hopcroft and Karp HK73
• Lemma 1: Ahmadi, Kuhn, and OshmanAKO18
• Lemma 2: Ahmadi and KuhnAK20, Kitamura and IzumiKI22
• Definition 1: Alternating base tree
• Definition 2: Edge level
• Definition 3: Minimum outgoing edge (MOE) of $T_v$
• Lemma 3
• Definition 4
• Lemma 4