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 strictly positive 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 complete bipartite graphs.

Remarks

1. The maximum-weight matching problem restricted to bipartite graphs is usually called the assignment problem.
2. If the graph is bipartite, [math]\displaystyle{ G=(V_1\dot\cup V_2,E) }[/math] and [math]\displaystyle{ |V_1|=|V_2| }[/math], the perfect matching of minimum weight can be found as follows:
   1. Let [math]\displaystyle{ C:=\max\{c(e)|e\in E\} }[/math].
   2. For each edge [math]\displaystyle{ e\in E }[/math], set [math]\displaystyle{ c'(e):=C-c(e) }[/math].
   3. Find a maximum-weight matching with respect to [math]\displaystyle{ c' }[/math].
3. If [math]\displaystyle{ |V_1|\neq|V_2| }[/math], say [math]\displaystyle{ |V_1|\lt |V_2| }[/math], we may simply add as many as [math]\displaystyle{ |V_2|-|V_1| }[/math] nodes to [math]\displaystyle{ V_1 }[/math] and for each new node [math]\displaystyle{ v }[/math] one edge [math]\displaystyle{ \{v,w\} }[/math] to every node [math]\displaystyle{ w\in V_2 }[/math] with:
   1. [math]\displaystyle{ c(\{v,w\}):=0 }[/math] if a maximum-weight matching is to be computed.
   2. [math]\displaystyle{ c(\{v,w\}) }[/math] sufficiently large if a minimum-weight perfect matching is to be computed.