Set-merging for the Matching Algorithm of Micali and Vazirani

TL;DR

This paper details the application of an incremental-tree set-merging algorithm to improve the matching algorithm of Micali and Vazirani, achieving maximum cardinality matching efficiently.

Contribution

It provides detailed implementation insights for integrating set-merging algorithms into the Micali-Vazirani matching algorithm, which were previously omitted.

Findings

Achieves $O(\sqrt n m)$ time complexity for maximum matching

Provides detailed methodology for set-merging in matching algorithms

Enhances understanding of algorithm integration

Abstract

The algorithm of Micali and Vazirani \cite{MV} finds a maximum cardinality matching in time $O(\sqrt n m)$ if an efficient set-merging algorithm is used. The latter is provided by the incremental-tree set-merging algorithm of \cite{GabTar}. Details of this application to matching were omitted from \cite{GabTar} and are presented in this note.