Tutorial questionsCOMP9007 Algorithms
2019 Semester 1 Tutorial 1
The University of Sydney
School of Computer Science
Problem 1
Consider the problem of assigning four players {A, B, C, D}to four teams {W, X, Y, Z }. Each player has
a preference for which team they want to be assigned to, and each team has a preference for which players
they take. These are listed in the preference tables below:
Player 1st 2nd 3rd 4th Team 1st 2nd 3rd 4th
A X W Z Y W B D C A
B Z Y X W X B D A C
C Y X Z W Y A B C D
D X Z W Y Z B C D A
a) Compute the stable matching determined by the Gale-Shapley algorithm for these preferences when the
Players propose to the Teams.
b) Compute the stable matching determined by the G-S algorithm when the Teams propose to the Players.
c) What, if anything, can these two answers tell us about the sets of valid partners of the players and teams?
Solution:
a) A:W,B:Z,C:Y,D:X
b) W:C,X:D,Y:A,Z:B
c) Recall that a pairing is valid if there is a stable matching where the pair are matched.
An important observation about the G-S algorithm is that proposers always end up with
their best (under their preference) valid partner and proposees always end up with their
worst valid partner. So the answer given in a) matches each team with their worst valid
player (and each player with their best valid team). Since Bis the best (valid or invalid)
partner for Z, this immediately tells us that no other partner for Zcan be valid – so Z’s
set of valid partners is exactly {B}. Combining the rest of the results with the answer
for b) – where each team is matched with their best valid partner – tells us that:
•W’s set of valid partners must lie between Aand C(inclusive), so is exactly {A, C}.
•X’s set of valid partners must lie between Dand D, so it is exactly {D}.
•Y’s set of valid partners must lie between Cand A, so it is a subset of {A, B, C }.
We can refine this with a bit of extra reasoning: since Bmust be matched with Z
in any stable matching, and Aand Cdo occur as partners for Y, we conclude that
Y’s set of valid partners is exactly {A, C}.
A similar argument for the players (or by observing the symmetry of the situation) yields:
•A’s set of valid partners is {W,Y }.
•B’s set of valid partners is {Z}.
•C’s set of valid partners is {W,Y }.
•D’s set of valid partners is {X}.
1
