Maximum-weight matching

Basic definitions

1. Basic graph definitions
2. Matchings in graphs

Definition

Input:

1. An undirected graph [math]\displaystyle{ G=(V,E) }[/math].
2. A real-valued weight [math]\displaystyle{ c(e) }[/math] for each edge [math]\displaystyle{ e\in E }[/math].

Output: A matching [math]\displaystyle{ M }[/math] in [math]\displaystyle{ G }[/math] such that [math]\displaystyle{ \sum_{e\in M'}c(e)\leq\sum_{e\in M}c(e) }[/math] for any other matching [math]\displaystyle{ M' }[/math] in [math]\displaystyle{ G }[/math].

Known algorithms

1. The Hungarian method for bipartite graphs

Remark

The maximum-weight matching problem restricted to bipartite graphs is usually called the assignment problem.