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CMPSC360hw4soln

CMPSC360hw4soln - CMPSC 360 Discrete Mathematics for...

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CMPSC 360 Discrete Mathematics for Computer Science, Fall 2008 Homework 4 Solutions Note: These solutions are not necessarily the model answers. Instead, they are tutorial in nature and may contain a little more explanation than an ideal solution. Also, there may be more than one correction solution. 1. (20 pts.) True. In class, we saw the “traditional” stable matching algorithm, which was guaranteed to produce a male optimal and female-pessimal pairing. By switching the men’s and women’s roles, it is clear that we obtain a new algorithm that is guaranteed to produce a female-optimal and male-pessimal pairing. By the very fact that these algorithms exist and have been proven to work in this way, we know that a female-optimal pairing and a female-pessimal pairing must always exist. Since there are three pairings in this particular stable marriage instance, we know that one of them must be female-optimal and one must be female-pessimal. Since W 1 has different partners in each matching, and prefers M 1 above the other stable pairings, only P 1 can be female-optimal by definition of female-optimality. Similarly, since W 1 likes M 3 the least, this must be the female-pessimal pairing. Therefore, again from the definitions of optimal- ity/pessimality, since all women have different partners in the three stable pairings, they must all strictly prefer M 1 to both of the others, and they must all like M 3 strictly less than both of the others. Thus, each womans preference order among the stable pairings must be M 1 > M 2 > M 3 . 2. (22 pts.) To solve the College Admissions Problem, we will extend the Propose & Reject (P&R) algorithm given in the notes. Students will play the role of men in the P&R procedure and universities will play the role of women. Instead of keeping a single person on a string as in the original algorithm, each university will keep a waiting list of size equal to its quota. The extended procedure works as follows: (a) All students apply to their first-choice university. (b) Each university u , with a quota of q u , then places on its waiting list the q u applicants who rank highest (or all the applicants if there are less than q u of them) and rejects all the rest. (c) Rejected applicants then apply to their second-choice university, and again each uni- versity u selects the top q u students from among the new applicants and those on its waiting list; it puts the selected students on its new waiting list, and rejects the rest of its applicants (including those who were previously on its waiting list but now are not).

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CMPSC360hw4soln - CMPSC 360 Discrete Mathematics for...

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