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

TL;DR

This paper introduces a new framework for maximum matching algorithms that simplifies the structure of shortest alternating paths, leading to more implementable algorithms and improved approximation methods in distributed and streaming contexts.

Contribution

It proposes a novel structure theorem that simplifies the analysis of alternating paths, and develops new algorithms with better efficiency and correctness verification.

Findings

New structure theorem for shortest alternating paths without blossoms

A more implementable and easier-to-verify algorithm for maximum matching

Improved deterministic approximation algorithms in distributed and streaming models

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 $O(m\sqrt{n})$ 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.