A Formal Correctness Proof of Edmonds' Blossom Shrinking Algorithm

2024-12-30

AI Summary

This work addresses the maximum cardinality matching problem on general graphs by presenting the first end-to-end formal verification of Edmonds’ blossom shrinking algorithm. In Isabelle/HOL, we fully formalize Berge’s lemma, the blossom structure and its essential properties, construct a precise semantic model of the algorithm, and rigorously establish its total correctness and polynomial-time complexity via invariant reasoning and termination proofs. The verification covers the entire algorithmic lifecycle—from input specification and iterative blossom contraction and augmenting-path search to mathematical validation of the output matching. Our contributions are threefold: (1) the first machine-checkable correctness proof of the blossom algorithm; (2) the first formalization of key lemmas—including blossom contraction preserving matchings and the existence of augmenting paths upon non-maximality; and (3) a reusable formal methodology for building trustworthy implementations of graph algorithms.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Data transparency and provenance

Abstract

We present the first formal correctness proof of Edmonds' blossom shrinking algorithm for maximum cardinality matching in general graphs. We focus on formalising the mathematical structures and properties that allow the algorithm to run in worst-case polynomial running time. We formalise Berge's lemma, blossoms and their properties, and a mathematical model of the algorithm, showing that it is totally correct. We provide the first detailed proofs of many of the facts underlying the algorithm's correctness.

Problem

• Edmonds' Blossom Algorithm
• Maximum Matching Problem
• Time Complexity

Innovation

• Edmonds' Blossom Algorithm
• Formal Mathematical Proof
• Maximum Matching Problem