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 femalepessimal 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 femaleoptimal and malepessimal pairing.
By the very fact that these algorithms exist
and have been proven to work in this way, we know that a femaleoptimal pairing and a
femalepessimal pairing must always exist.
Since there are three pairings in this particular stable marriage instance, we know that one
of them must be femaleoptimal and one must be femalepessimal.
Since
W
1
has different
partners in each matching, and prefers
M
1
above the other stable pairings, only
P
1
can be
femaleoptimal by definition of femaleoptimality.
Similarly, since
W
1
likes
M
3
the least,
this must be the femalepessimal 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 firstchoice 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 secondchoice 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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 Fall '08
 HAULLGREN
 mod mod mod, waiting list, improvement lemma

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