Gabow 的 $O(\sqrt{n}m)$ 最大基数匹配算法再审视
Gabow's $O(\sqrt{n}m)$ Maximum Cardinality Matching Algorithm, Revisited
AI总结：
本文重新审视 Gabow 的 $O(\sqrt{n}m)$ 最大基数匹配算法，提出更直接的实现方法，并提供可运行的实现代码。
摘要：
我们重新审视 Gabow 的 $O(\sqrt{n}m)$ 最大基数匹配算法（《Fundamenta Informaticae》, 2017）。该算法适应了 Gabow 和 Tarjan 的加权匹配算法以用于最大基数匹配。Gabow 的算法是迭代进行的。在每次迭代中，它构造出当前匹配下最长的边不相交的最短增广路径，并进行增广。已知 $O(\sqrt{n})$ 次迭代足够。每次迭代包含三个部分。第一部分计算最短增广路径的长度。第二部分构造一个辅助图 $H$，其性质是 $G$ 中的最短增广路径对应于 $H$ 中的增广路径。第三部分确定 $H$ 中的最大边不相交增广路径集，并将这些路径提升并增广到 $G$。我们为第一部分提出了一种新算法。Gabow 的第一部分算法源于 Edmonds 的加权匹配的对偶算法。我们相信我们的方法更直接，更容易教学。我们已经实现了该算法；实现可在配套网页上获得（https://people.mpi-inf.mpg.de/~mehlhorn/CompanionPageGenMatchingImplementation.html）
英文摘要：
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).
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