Gabow's $O(\sqrt{n}m)$ Maximum Cardinality Matching Algorithm, Revisited

TL;DR

This paper revisits Gabow's $O(\sqrt{n} m)$ maximum cardinality matching algorithm, providing a new approach for the shortest augmenting path computation, with an implementation available online.

Contribution

It introduces a more direct method for the shortest augmenting path step in Gabow's algorithm, simplifying understanding and implementation.

Findings

The new approach is more direct than Gabow's original method.

Implementation of the algorithm is available online.

The method potentially simplifies teaching of the algorithm.

Abstract

We revisit Gabow's $O(\sqrt{n} m)$ maximum cardinality matching algorithm (The Weighted Matching Approach to Maximum Cardinality Matching, Fundamenta Informaticae, 2017). It adapts the weighted matching algorithm of Gabow and Tarjan~\cite{GT91} to maximum cardinality matching. Gabow's algorithm works iteratively. In each iteration, it constructs a maximal number of edge-disjoint shortest augmenting paths with respect to the current matching and augments them. It is well-known that $O(\sqrt{n})$ iterations suffice. Each iteration consists of three parts. In the first part, the length of the shortest augmenting path is computed. In the second part, an auxiliary graph $H$ is constructed with the property that shortest augmenting paths in $G$ correspond to augmenting paths in $H$. In the third part, a maximal set of edge-disjoint augmenting paths in $H$ is determined, and the paths are lifted to and augmented to $G$. We give a new algorithm for the first part. Gabow's algorithm for the first part is derived from Edmonds' primal-dual algorithm for weighted matching. We believe that our approach is more direct and will be easier to teach. We have implemented the algorithm; the implementation is available at the companion webpage (https://people.mpi-inf.mpg.de/~mehlhorn/CompanionPageGenMatchingImplementation.html).