Blossom VI: A Practical Minimum Weight Perfect Matching Algorithm

TL;DR

This paper presents a practical implementation of a minimum weight perfect matching algorithm that significantly outperforms existing algorithms like Blossom V, achieving near-linear runtime on tested instances.

Contribution

The authors introduce a new algorithm leveraging cherry blossom structures and supernode shrinking, improving efficiency over prior methods.

Findings

Algorithm outperforms Blossom V on instances with superlinear complexity

Achieves almost-linear runtime on tested families of instances

Uses cherry trees and supernode shrinking for efficiency

Abstract

We implement an algorithm for solving the minimum weight perfect matching problem. Our code significantly outperforms the current state-of-the-art Blossom V algorithm on those families of instances where Blossom V takes superlinear time. In practice, our implementation shows almost-linear runtime on every family of instances on which we have tested it. Our algorithm relies on solving the maximum-cardinality unweighted matching problems during its primal phase. Following the state-of-the-art cherry blossom algorithm, we use cherry trees instead of traditional alternating trees and cherry blossoms instead of traditional blossoms. We shrink cherry blossoms rather than traditional blossoms into supernodes. This strategy allows us to deal with much shallower supernodes.