1. Stable Matchings a.k.a. the Stable Marriage Problem
Samia Qader
252a-az
CSC 254

2. The Stable Marriage Problem
Overview
Real - life application
The stable marriage problem
Theorems
Termination
Correctness of algorithm
Open problems

3. Real-life application
Residency: after medical school
“The Match” was developed to assign prospective students to residency programs
The students submit a rank-order list (ROL) to the National Residency Match Program (NRMP)
Hospitals rank-order their students
NRMP finds a match.

4. Simple Problem We’ve got n boys and n girls.
Each boy submits an ROL of all n girls and the girls do the same
A matching is unstable if a boy A and a girl a, not married to each other, mutually prefer each other to their spouses.
A marriage is a bijection between the boys and girls

5. Example 1: 4 boys (A,B,C,D) married to 4 girls (a,b,c,d)
Men Preference
A c b d a
B b a c d
C c a d b
D c a d b
Women Preference
a A B D C
b C A D B
c C B D A
d B A C D
Matching (Aa, Bb, Cc, Dd) Is it stable?
Look at Aa and Bb: A(4) + a(1) = 5
B(1) + b(4) = 5
total = 10

6. Ex 1: 4 boys (A,B,C,D) married to 4 girls (a,b,c,d)
Women Preference
a A B D C
b C A D B
c C B D A
d B A C D
Men Preference
A c b d a
B b a c d
C c a d b
D c a d b
Check Ab and Ba: A(2) + a(2) = 4
B(2) + b(2) = 4
total = 8

7. Marriage Algorithm – in words
The boys are all single and free to propose

8. Marriage Algorithm – in words
The girls are all engaged temporarily to  (the ugly man)
Ugly man

9. Marriage Algorithm – in words
The boys propose to the best girl on their list (who hasn’t rejected them yet). The girls accept - tentatively.
If the girl gets a better offer from a boy that she prefers to her fiancé, she will break off the engagement.
STOP when all the boys are engaged

10. ROL

11. Results
3 happy couples:

12. ROL
Brown man wants blonde girl – But she doesn’t want him

13. ROL
Scratch off blonde from brown’s list

14. New Results
3 different couples:

15. More detail…. n= no.of men & women k = no. of couples formed
X = man suitor
x = woman
 = ugly man
while (k<n) do
begin X = (k+1)st man;
while (X ! = ) do {
x= best remaining choice on X’s list
if (x prefers X to her fiancé) {
engage X and x
X = former fiancé of
} // end if
}// end while
if (X != )
withdraw x from X’s list
end
k = k+1;
} // end while

16. Some Theorems The order in which the boys propose does not matter
The final matching is stable
The boys get to marry the best girl the could possibly get in a stable marriage

17. Termination: the algorithm has to end
In what situation would the algorithm run forever - if a boy B keeps getting rejected by all girls
This can never happen:
a boy in only rejected by all girls if all the girls are engaged to someone better
Impossible - since there are n girls and n-1 boys that are better than B

18. Correctness of the Algorithm
Is the final marriage stable?
Proof by contradiction:
There exists some pair (b,g) that mutually prefer each other to whomever they are matched to.
Since b proposed in the order of his ROL, he must have proposed to g already

19. Correctness continued...
But if g is not engaged to b, she must have rejected him
g would only reject b if she is already engaged to someone she prefers
Therefore, g’s fiancé must be ranked more highly than b
CONTRADICTION

20. Boy Optimality
The marriage returned by this algorithm assigns every boy to his favorite stable partner.
Consider matching each girl to her least favorite choice. The matching produced is stable and is the same as the matching returned by the algorithm described.

21. Residency Problem
Who gets to play the boy?
Answer : the hospitals

22. Bipartite matching
In a complete bipartite graph, each node would have an ROL of the nodes that it prefers. 1 1 a 2 2 b 3 3 c 4 4 d

23. Open Problems…
Maximum number of stable matches grows exponentially with n. What is the instance of size n with the largest possible number of stable matchings?
Sex-equal matching: where the sum of the male scores is equal to the sum of the female scores.
Is it possible to test the stability of every matching in a set of matchings faster (significantly) than by checking each of the matchings from scratch?

24. References
Asratian A.S, Denley T.M.J, Bipartite Graphs and their Applications,
Cambridge University Press, N.Y. (1998).
Gusfield, D., Irving R.W., The Stable Marriage Problem :Structure and
Algorithms, The MIT Press, Cambridge (1989).
Knuth D.E., Stable Marriage and its Relation to Other Combinatorial
Problems, American Mathematical Society, Providence (1997).
Littman M.L.,
lectures/lect13/lect13.html, (1998).
Lovasz L., Plummer M.D., Matching Theory, North-Holland, North-
Holland (1986).