# sol1 - CSE 202 Solution Set 1 Spring 2010 7 April 2010...

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Unformatted text preview: CSE 202 Solution Set 1 Spring 2010 7 April 2010 Chapter 1, Problem 1: The statement is false, which is evident due to the following counterexample with 2 men ( m 1 and m 2 ) and 2 women ( w 1 and w 2 ). The preferences are as follows: • m 1 prefers w 1 to w 2 , • m 2 prefers w 2 to w 1 , • w 1 prefers m 2 to m 1 , and • w 2 prefers m 1 to m 2 . A stable matching in this instance is { ( m 1 ,w 1 ) , ( m 2 ,w 2 ) } . Note that this matching does not include a pair ( m,w ) where m and w are ranked first on each other’s preference list. Chapter 1, Problem 2: The statement is true, which can be proved by contradiction. Suppose that the Gale-Shapley algorithm returns a stable matching where m and w are not matched, but they are first on each other’s preference lists. This means that m is matched with some other woman w , and w is matched with some other woman m . But the preference lists imply that m prefers w to w , and w prefers m to m . This is an instability, which contradicts the fact that the Gale-Shapley algorithm always returns a stable matching. Therefore, ( m,w ) must be included in the matching. Chapter 1, Problem 4: We give the following algorithm that matches students to hospitals. Then algo- rithm and its analysis are every similar to the original Gale-Shapley algorithm for the stable matching problem: Initially all students s ∈ S and hospital positions h ∈ H are free While there is a hospital h that has a free position and hasn’t extended an offer to every student: Choose such a hospital h Let s be the highest-ranked student in h ’s preference list that hasn’t received an offer from...
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## This note was uploaded on 04/20/2010 for the course CSE 202 taught by Professor Hu during the Spring '06 term at UCSD.

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sol1 - CSE 202 Solution Set 1 Spring 2010 7 April 2010...

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