Formal correctness of the Micali–Vazirani maximum matching algorithm

Establish a formal correctness proof for the Micali–Vazirani algorithm for maximum cardinality matching in general graphs, which achieves O(√(|V|)|E|) running time by using shortest augmenting paths in phases and avoiding blossom shrinking.

Background

The Micali–Vazirani algorithm is the fastest known algorithm for maximum cardinality matching in general graphs, using phases based on shortest augmenting paths to obtain an O(√(|V|)|E|) running time. Despite its importance, multiple attempts over decades to fully prove its correctness have encountered difficulties and gaps, and a complete, fully verified proof remains elusive. The authors emphasize this as the most interesting future direction following their formal verification of Edmonds' blossom-shrinking algorithm.

The algorithm differs fundamentally from Edmonds' approach by avoiding blossom shrinking, relying instead on structural properties of minimal-length alternating paths. A formal correctness proof would require pinning down complex case analyses and invariants analogous to those developed in this paper, but adapted to the specific mechanisms of Micali–Vazirani.