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s2008.prelim3

s2008.prelim3 - PRELIM 3 Closed book exam Justify all work...

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PRELIM 3 ORIE 321/521 April 15, 2008 Closed book exam. Justify all work. 1. (a) (20 points) Suppose the members of n committees C 1 , . . . , C n are drawn from a common pool of individuals i = 1 , . . . , m ; i.e., C j ⊆ { 1 , . . ., m } , for j = 1 , . . . , n , where C j denotes the set of individuals who are on the j th committee. From the membership of each committee we would like to select a person to serve as committee chair. Moreover, in order to distribute the workload evenly, we wish to know whether it is possible to select distinct chairs for the committees, i.e., n different individuals i 1 , . . . , i n with i j C j for j = 1 , . . . , n . Indicate a bipartite cardinality matching model which can be used to determine whether such a selection of individuals to chair the n committees is possible. You must interpret the vertices and edges of your bipartite graph and explain clearly why the solution for your model actually determines the individuals to chair the various committees or proves that a complete selection of n distinct chairs is impossible.
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