Blossom VI: A Practical Minimum Weight Perfect Matching Algorithm

2026-04-22

AI Summary

This work addresses the computational inefficiency of solving large-scale minimum-weight perfect matching problems by proposing an efficient algorithm that first computes a maximum-cardinality unweighted matching. It then introduces cherry trees and cherry blossoms as alternatives to traditional alternating trees and blossoms; by contracting cherry blossoms into supernodes, the algorithm substantially reduces supernode depth. Combined with an improved blossom contraction strategy and a high-performance maximum-cardinality matching solver, the method achieves near-linear time complexity across diverse test instances, significantly outperforming the state-of-the-art Blossom V algorithm.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• minimum weight perfect matching
• graph matching
• combinatorial optimization

Innovation

• minimum weight perfect matching
• cherry trees
• cherry blossoms
• supernodes
• almost-linear runtime