Maximum-weight matching: Difference between revisions
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'''Input:''' | '''Input:''' | ||
# An undirected graph <math>G=(V,E)</math>. | # An undirected graph <math>G=(V,E)</math>. | ||
# A real-valued '''weight''' <math> | # A real-valued '''weight''' <math>c(e)</math> for each edge <math>e\in E</math>. | ||
'''Output:''' | '''Output:''' | ||
A matching <math>M</math> in <math>G</math> such that <math>\sum_{e\in M'} | A matching <math>M</math> in <math>G</math> such that <math>\sum_{e\in M'}c(e)\leq\sum_{e\in M}c(e)</math> for any other matching <math>M'</math> in <math>G</math>. | ||
== Known algorithms == | == Known algorithms == | ||
Revision as of 18:12, 22 November 2014
Basic definitions
Definition
Input:
- An undirected graph [math]\displaystyle{ G=(V,E) }[/math].
- 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
- The Hungarian method for bipartite graphs
Remark
The maximum-weight matching problem restricted to bipartite graphs is usually called the assignment problem.