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

TL;DR

The paper introduces a novel structure theorem for shortest alternating paths in general graphs that avoids the intricate blossom machinery by leveraging alternating base trees (ABTs) and an alternating-base DAG. Building on this framework, it develops a modular MCM algorithm that computes a maximal set of disjoint shortest augmenting paths via a double-path transformation, achieving a sequential runtime of $O(m\sqrt{n}\log n)$ with a simpler correctness proof than MV. It further shows how the ABT-based approach yields $(1-\varepsilon)$-approximation algorithms in the CONGEST and semi-streaming models, with improved round and pass bounds through a unified amplifier framework. The work hints at broad applicability to distributed, streaming, and dynamic graph settings, and opens avenues for further refinements and extensions beyond unweighted MCM in general graphs.

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 $(1 - ε)$-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.

Figures (6)

• Figure 1: An example of ABT and edge volume. The tree edges are colored with blue. The pair of values assigned to each vertex means the values of $\mathrm{dist}^{\mathrm{odd}}(\cdot)$ and $\mathrm{dist}^{\mathrm{even}}(\cdot)$. The value assigned to each non-tree edge is its volume. The path indiceted by a red arrow is a shortest augmenting path.
• Figure 2: Proof of Lemma \ref{['lma:volumeBound']}.
• Figure 3: Proof of Lemma \ref{['lma:oddOrthodoxProp']}
• Figure 4: Proof of Theorem \ref{['thm:IKY24']} (S1).
• Figure 5: Proof of Theorem \ref{['thm:IKY24']} (S2)

Theorems & Definitions (72)

• Theorem 1.1: informal
• Theorem 1.2
• Definition 2.1: Alternating Base Tree
• Definition 2.2: Edge volume, height, and level
• Theorem 2.3: Izumi, Kitamura, and YamaguchiIKY24, rephrased
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3