Maximum Matching in General Graphs Without Explicit Consideration of Blossoms Revisited

TL;DR

This paper presents a novel approach to maximum matching in general graphs by reducing blossom analysis to reachability problems, simplifying algorithms for maximum and weighted matchings.

Contribution

It introduces a new perspective that avoids explicit blossom analysis, enabling more straightforward algorithms for maximum and weighted matchings in general graphs.

Findings

Efficient realization of Hopcroft-Karp for maximum matching

A variant of Edmonds' primal-dual algorithm for weighted matching

Simplified algorithms without explicit blossom analysis

Abstract

We reduce the problem of finding an augmenting path in a general graph to a reachability problem in a directed bipartite graph. A slight modification of depth-first search leads to an algorithm for finding such paths. Although this setting is equivalent to the traditional terminology of blossoms due to Edmonds, there are some advantages. Mainly, this point of view enables the description of algorithms for the solution of matching problems without explicit analysis of blossoms, nested blossoms, and so on. Exemplary, we describe an efficient realization of the Hopcroft-Karp approach for the computation of a maximum cardinality matching in general graphs and a variant of Edmonds' primal-dual algorithm for the maximum weighted matching problem.