lab7 - ENGRI 115 Engineering Applications of OR Fall 2007...

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ENGRI 115 Engineering Applications of OR Fall 2007 The Assignment Problem Lab 7 Name: Objectives: Show how to find maximum weight matching in bipartite graphs using the Hungarian Algorithm. Key Ideas: weighted matching, maximum weighted matching maximum weighted matching in bipartite graphs = assignment problem weighted node cover, node weights exposed node with respect to a matching, alternating tree rooted at an exposed node outer nodes, inner nodes in an alternating tree Prelab Exercise: Please write your answer on the back of this sheet (or an attached paper). Cardinality matching is a special case of weighted matching where the edges in the graph have weights 1 and the edges not in the graph have weights 0. (Convince yourself that this is true.) We have already seen how to find a minimum node cover for a bipartite graph. The Hungarian Algorithm for finding a maximum weighted matching relies on something called the weighted node cover which is a generalization of the node cover for the cardinality case. The goal of this prelab exercise is to draw a parallel between the cardinality and the weighted case. 1
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A-3, B-1, C-4 and E-2 is a maximum (cardinality) matching for the graph below (matching edges are bold). a. Draw an alternating tree to show that the indicated matching is maximum. b. Find a minimum node cover (circle the nodes in the minimum node cover in the original graph). What is the size of the minimum node cover? c. Is there an exposed node (wrt the matching) which is in the minimum node cover? Why? Is there a matching edge with both endpoints in the minimum node cover? Why? A B C D E 1 2 3 4 5 2
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#1 - First example for the assignment problem In the assignment problem we are given n individuals and n jobs, and so called weights c ij indicating how experienced (well qualified, productive, etc.) individual i is in job j . We can tabulate the data as you can see below. Our goal is to assign people to jobs (each person is assigned to at most one job and at most one person can work on each job, as usual) so that the total experience level for the assignment (that is the sum of the c ij ’s for all person-job pairs in the assignment) is maximized. In other words, we are looking for a maximum weighted matching in the bipartite graph where the two sets of nodes V 1 and V 2 correspond to the individuals and jobs, respectively; and the c ij ’s are weights on the edges. Note that we might not assign people to all the jobs, even if a perfect (cardinality) matching is possible. 1 2 3 4 5 a 4 5 6 1 5 b 3 5 7 1 6 c 1 1 5 0 5 d 5 4 8 2 7 e 4 5 9 3 9 What is the weight of assignment a-1, b-2, c-4, d-3, e-5? What is the weight of assignment a-4, b-1, c-3, d-5, e-2? What does it mean that c c 4 is 0? At every stage of the computation we will have a
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This note was uploaded on 06/01/2008 for the course ENGRI 1101 taught by Professor Trotter during the Spring '05 term at Cornell.

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lab7 - ENGRI 115 Engineering Applications of OR Fall 2007...

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