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Apparition11 (talk | contribs) m Reverted 1 edit by NamanZomak123 (talk): Spam |
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Some of the local methods assume that the graph admits a ''perfect matching''; if this is not the case, then some of these methods might run forever.<ref name=":0" />{{Rp|3}} A simple technical way to solve this problem is to extend the input graph to a ''complete bipartite graph,'' by adding artificial edges with very large weights. These weights should exceed the weights of all existing matchings, to prevent appearance of artificial edges in the possible solution.
As shown by Mulmuley, Vazirani and Vazirani,<ref>{{Cite journal|last1=Mulmuley|first1=Ketan|last2=Vazirani|first2=Umesh|author-link2=Umesh Vazirani|last3=Vazirani|first3=Vijay|author-link3=Vijay Vazirani|year=1987|title=Matching is as easy as matrix inversion|journal=Combinatorica|volume=7|issue=1|pages=105–113|doi=10.1007/BF02579206|s2cid=47370049|author-link1=Ketan Mulmuley}}</ref> the problem of minimum weight perfect matching is converted to finding minors in the [[adjacency matrix]] of a graph. Using the [[isolation lemma]], a minimum weight perfect matching in a graph can be found with probability at least
=== Unbalanced assignment ===
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The total weight of the matching is: <math>\sum_{(i,j)\in A\times T} w_{ij}x_{ij}</math>. The goal is to find a maximum-weight perfect matching.
To guarantee that the variables indeed represent a perfect matching, we add constraints saying that each vertex is adjacent to exactly one edge in the matching, i.e.,
<math display="block">\sum_{j\in T}x_{ij}=1\text{ for }i\in A, \,
~~~
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