A Formal Correctness Proof of Edmonds' Blossom Shrinking Algorithm

TL;DR

This paper provides the first formal correctness proof of Edmonds' blossom shrinking algorithm, rigorously formalising its mathematical foundations and demonstrating its correctness and polynomial runtime.

Contribution

It offers the first detailed formal proof of Edmonds' blossom shrinking algorithm's correctness and runtime, formalising key mathematical structures involved.

Findings

Formal proof of correctness for Edmonds' blossom algorithm

Formalisation of Berge's lemma and blossom properties

Verification of polynomial worst-case runtime

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.