Slides12B_col - Lecture 12 Matchings in Bipartite Graphs...

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1 Lecture 12 Matchings in Bipartite Graphs Application Suppose in your company are three programmers Alice , Bob and Charlie . Together they should write a commercial software product to solve Maximum Flow problems. There are three different parts that have to be written: (1) The graphical interface (2) Routines for graph manipulations (3) The Ford-Fulkerson algorithm Not every person is qualified to do each of the three jobs. Every person can only be assigned to one job, and each job can be carried out by only one person.
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2 Application How can we solve this task? Question: How can we “match” persons and jobs (according to their qualification) such that as many jobs are carried out as possible? Model For each person we know which of the jobs they could carry out. Alice, for example, can only write the graph routines. We model these qualifications via an undirected graph: • Edges (i,j) indicate that person i is qualified for job j. •Th isisa bipartite graph, because the nodes are partitioned into 2 parts {A,B,C} and {1,2,3} such that there is no edge between two nodes of the same part. C A B 3 1 2
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3 Model The following set of edges (bold) is not a valid solution to our problem. Why? C A B 3 1 2 Model A solution ( matching ), for example, could be the following set of edges: (This matching has a larger cardinality – contains more edges) C A B 3 1 2 or C A B 3 1 2 Matching covers {B,1} Matching covers {A,B,2,3}
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4 Formal Definitions • A graph G=(V,E) is called bipartite if we have: V=V 1 V 2 with V 1 V 2 = , and (i,j) E implies i V 1 , j V 2 or i V 2 , j V 1 .
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Slides12B_col - Lecture 12 Matchings in Bipartite Graphs...

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