A Note on Matchings Constructed during Edmonds' Weighted Perfect Matching Algorithm

TL;DR

This paper proves that matchings constructed during Edmonds' weighted perfect matching algorithm are optimal among matchings of the same size, simplifying the approach to solving weighted matching problems without auxiliary graphs.

Contribution

It provides a new proof that the matchings generated are optimal within their size class, eliminating the need for auxiliary graphs in weighted matching solutions.

Findings

Matchings during Edmonds' algorithm are optimal among same-sized matchings.

No need for auxiliary graphs of doubled size to solve weighted matching problems.

The result was known but not widely documented in modern literature.

Abstract

We reprove that all the matchings constructed during Edmonds' weighted perfect matching algorithm are optimal among those of the same cardinality (provided that certain mild restrictions are obeyed on the choices the algorithm makes). We conclude that in order to solve a weighted matching problem it is not needed to solve a weighted perfect matching problem in an auxiliary graph of doubled size. This result was known before, e.g., posed as an exercise in see Lawler's book from 1976, but is not present in several modern books on combinatorial optimization.