Cardinality-maximal matching
Basic definitions
Definition
Input: An undirected graph [math]\displaystyle{ G=(V,E) }[/math].
Output: A matching [math]\displaystyle{ M }[/math] in [math]\displaystyle{ G }[/math] such that [math]\displaystyle{ |M'|\leq|M| }[/math] for any other matching [math]\displaystyle{ M' }[/math] in [math]\displaystyle{ G }[/math].
Known algorithms:
- Classical bipartite cardinality matching (bipartite graphs [math]\displaystyle{ G }[/math] only)
- Maximum matching by Edmonds
Berge's theorem
Statement: A matching [math]\displaystyle{ M }[/math] in an undirected graph [math]\displaystyle{ G=(V,E) }[/math] is cardinality-maximal if, and only if, [math]\displaystyle{ M }[/math] admits no augmenting path.
Proof: Clearly, if [math]\displaystyle{ M }[/math] admits an augmenting path, [math]\displaystyle{ M }[/math] is not cardinality-maximal. So consider the case that [math]\displaystyle{ M }[/math] is not cardinality-maximal. We have to show that [math]\displaystyle{ M }[/math] admits an augmenting path.
By assumption, there is a matching [math]\displaystyle{ M' }[/math] in [math]\displaystyle{ G }[/math] such that [math]\displaystyle{ |M'|\gt |M| }[/math]. Let [math]\displaystyle{ \Delta:=(M\setminus M')\cup(M'\setminus M) }[/math] denote the symmetric difference of [math]\displaystyle{ M }[/math] and [math]\displaystyle{ M' }[/math]. Obviously, any node [math]\displaystyle{ v\in V }[/math] is incident to at most two edges in [math]\displaystyle{ \Delta }[/math]. Consequently, [math]\displaystyle{ \Delta }[/math] decomposes into node-disjoint paths (some of them may be cycles). These paths are alternating. Since [math]\displaystyle{ |M'|\gt |M| }[/math], at least one of these paths must be augmenting.
Remark: As a by-product, the following result is proved as well: Let [math]\displaystyle{ k }[/math] be the difference between [math]\displaystyle{ |M| }[/math] and the maximal cardinality of a matching. Then there are at least [math]\displaystyle{ k }[/math] node-disjoint augmenting paths.