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prelab6 - them all off but the trouble was that whatever...

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ENGRI 115 Engineering Applications of OR Fall 2007 Maximum Matchings in Bipartite Graphs Prelab 6 Name: Objectives: Show how to find maximum cardinality matching in bipartite graphs using maximum flows. Demonstrate the Augmenting Path Algorithm for finding a maximum cardinality matching. Key Ideas: bipartite graphs matching, maximum matching, perfect matching alternating path, augmenting path Prelab Exercise: Please write your answer on the back of this sheet (or an attached paper). A matching in a bipartite graph is called perfect if the matching edges cover all the nodes. Observe that we can have a perfect matching only if the number of nodes in the two parts of the node set is the same. The following classic story was adapted from Matching Theory, by Lov´ asz and Plummer . In the court of King Arthur there dwelt 150 knights and 150 ladies-in-waiting. Arthur decided to marry
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Unformatted text preview: them all off, but the trouble was that whatever pairing he tried there were always some couples that didn’t like each other. So he summoned Merlin, the Wizard and ordered him to find a pairing in which every pair was willing to marry. Merlin had supernatural powers (or maybe just knew a little about optimization. ..who knows?) and he saw at once that no such pairing was possible. How could he convince the king (who would surely order Merlin to be beheaded if the wizard started babbling about bipartite graphs, flows and cuts)? Below is a simple example with 5 ladies and 5 knights. Convince yourself that there is no perfect matching in this case. How could you convince King Arthur about this? (Lines join nodes corresponding to ladies and knights who are willing to be paired with one another.) Ladies Knights...
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