Forgetting Alternation and Blossoms: A New Framework for Fast Matching Augmentation and Its Applications to Sequential/Distributed/Streaming Computation

2025-11-11

AI Summary

Maximum-cardinality matching in general graphs is a classical problem. Although the Micali–Vazirani (MV) algorithm achieves optimal time complexity, its correctness proof is notoriously intricate due to the tight coupling between nested blossoms and shortest alternating paths. This paper introduces a novel structural theorem that eliminates explicit reliance on edge alternation (matched/unmatched) and avoids explicit blossom construction. It generalizes and formalizes the notion of Alternating Base Trees (ABTs), establishing a unified analytical framework. Leveraging this framework, we design an exact matching algorithm with strong verifiability and practical implementability. Furthermore, by integrating deterministic approximation amplification techniques, we obtain deterministic $(1-varepsilon)$-approximate algorithms for both the distributed and semi-streaming models, achieving strictly improved running times over prior state-of-the-art results.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Finding a maximum cardinality matching in a graph is one of the most fundamental problems. An algorithm proposed by Micali and Vazirani (1980) is well-known to solve the problem in $O(msqrt{n})$ time, which is still one of the fastest algorithms in general. While the MV algorithm itself is not so complicated and is indeed convincing, its correctness proof is extremely challenging, which can be seen from the history: after the first algorithm paper had appeared in 1980, Vazirani has made several attempts to give a complete proof for more than 40 years. It seems, roughly speaking, caused by the nice but highly complex structure of the shortest alternating paths in general graphs that are deeply intertwined with the so-called (nested) blossoms. In this paper, we propose a new structure theorem on the shortest alternating paths in general graphs without taking into the details of blossoms. The high-level idea is to forget the alternation (of matching and non-matching edges) as early as possible. A key ingredient is a notion of alternating base trees (ABTs) introduced by Izumi, Kitamura, and Yamaguchi (2024) to develop a nearly linear-time distributed algorithm. Our structure theorem refines the properties of ABTs exploited in their algorithm, and we also give simpler alternative proofs for them. Based on our structure theorem, we propose a new algorithm, which is slightly slower but more implementable and much easier to confirm its correctness than the MV algorithm. As applications of our framework, we also present new $(1 - epsilon)$-approximation algorithms in the distributed and semi-streaming settings. Both algorithms are deterministic, and substantially improve the best known upper bounds on the running time. The algorithms are built on the top of a novel framework of amplifying approximation factors of given matchings, which is of independent interest.

Problem

• Simplifying the complex structure of shortest alternating paths in graphs
• Developing a more implementable matching algorithm with easier correctness proof
• Creating efficient approximation algorithms for distributed and streaming settings

Innovation

• Forgets alternation to simplify matching augmentation
• Uses alternating base trees for structure theorem
• Amplifies approximation factors for distributed streaming