Matchings in graphs: Difference between revisions
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## Each edge of <math>p</math> that was in <math>M</math> immediately before this augmentation step, is removed from <math>M</math>. | ## Each edge of <math>p</math> that was in <math>M</math> immediately before this augmentation step, is removed from <math>M</math>. | ||
## Each edge of <math>p</math> that was ''not'' in <math>M</math> immediately before this augmentation step, is inserted in <math>M</math>. | ## Each edge of <math>p</math> that was ''not'' in <math>M</math> immediately before this augmentation step, is inserted in <math>M</math>. | ||
:Clearly, the size of <math>M</math> is increased by one. | |||
== Cardinality-maximal matching == | == Cardinality-maximal matching == |
Revision as of 14:21, 17 November 2014
Definitions
- Let [math]\displaystyle{ G=(V,E) }[/math] be an undirected graph. A matching in [math]\displaystyle{ G }[/math] is a set [math]\displaystyle{ M }[/math] of edges such that no two edges in [math]\displaystyle{ M }[/math] are incident.
- A node [math]\displaystyle{ v\in V }[/math] is matched with respect to a matching [math]\displaystyle{ M }[/math] if it is incident to a member of [math]\displaystyle{ M }[/math]; otherwise, [math]\displaystyle{ v }[/math] is called free or exposed.
- A path [math]\displaystyle{ p }[/math] in an undirected graph [math]\displaystyle{ G=(V,E) }[/math] is called alternating with respect to some matching [math]\displaystyle{ M }[/math] if, for any two subsequent edges on [math]\displaystyle{ p }[/math], exactly one of them belongs to [math]\displaystyle{ M }[/math]. Consequently, the edges in [math]\displaystyle{ M }[/math] and the edges not in [math]\displaystyle{ M }[/math] appear strictly alternatingly on [math]\displaystyle{ p }[/math].
- A path [math]\displaystyle{ p }[/math] in an undirected graph [math]\displaystyle{ G=(V,E) }[/math] is called augmenting with respect to some matching [math]\displaystyle{ M }[/math] if [math]\displaystyle{ p }[/math] is alternating and both of its end nodes are exposed.
- Augmenting a matching [math]\displaystyle{ M }[/math] along an augmenting path [math]\displaystyle{ p }[/math] means:
- Each edge of [math]\displaystyle{ p }[/math] that was in [math]\displaystyle{ M }[/math] immediately before this augmentation step, is removed from [math]\displaystyle{ M }[/math].
- Each edge of [math]\displaystyle{ p }[/math] that was not in [math]\displaystyle{ M }[/math] immediately before this augmentation step, is inserted in [math]\displaystyle{ M }[/math].
- Clearly, the size of [math]\displaystyle{ M }[/math] is increased by one.
Cardinality-maximal matching
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:
Maximum-weight matching
Input:
- An undirected graph [math]\displaystyle{ G=(V,E) }[/math].
- A weight [math]\displaystyle{ w(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'}w(e)\leq\sum_{e\in M}w(e) }[/math] for any other matching [math]\displaystyle{ M' }[/math] in [math]\displaystyle{ G }[/math].
Variations
- Bipartite matching: The graph [math]\displaystyle{ G=(V,E) }[/math] is bipartite.
- Perfect matching: A variant on the cardinality-maximal matching problem, in which [math]\displaystyle{ |V| }[/math] is even and the output is void if the maximal cardinality of any matching is strictly smaller than the upper bound [math]\displaystyle{ |V|/2 }[/math]. So the output is a matching if, and only if, it is of size [math]\displaystyle{ |V|/2 }[/math]. Such a matching is usually called a perfect matching.
Remark: The maximum-weight matching problem restricted to bipartite graphs is usually called the assignment problem.