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Assignment Problems

---Bipartite Matching--- define bipartite graph.

Timetabling and Matching example (1): Let x1, x2, and x3 represent 3 teachers teaching 4 classes , , , and for these time periods:

 

A teaching schedule with the smallest possible number of periods can be set up as follows:

 

So in general the above teaching schedule can be represented by a bipartite graph G. A teaching schedule for one period corresponds to a ``matching'' in G.

Timetabling and Matching example (2): Suppose we have n classes and m teachers. Let the sign X to indicate which class to be taught by which teacher(s) (one lesson of one hour a day).

 

Next, construct a bipartite graph G. Fixing the time periods, the problem becomes a matching problem. See Figure 14.

  
Figure 14: Teachers-classes bipartite graph.



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