1. A Grand Tour of Stable Matching Problems
Progress and Challenges
David Manlove
University of Glasgow
Department of Computing Science
Supported by EPSRC grant GR/R84597/01
and Nuffield Foundation Award NUF-NAL-02

2. The Stable Marriage problem (SM)
Input to the problem consists of:
a set of n men {m1,m2,…,mn}
a set of n women {w1,w2,…,wn}
for each person q, a preference list in which
q ranks all members of the opposite sex in
strict order
A matching M is a set of n disjoint (man,woman)
pairs
Let pM(q) denote person q’s partner in M
A blocking pair of matching M is a
(man,woman) pair (m,w)M such that:
m prefers w to pM(m)
w prefers m to pM(w)
Matching M is stable if M admits no blocking
pair

3. Stable marriage: example instance
m1: w4 w1 w2 w w1: m4 m1 m3 m2
m2: w2 w3 w1 w w2: m1 m3 m2 m4
m3: w2 w4 w3 w w3: m1 m2 m3 m4
m4: w3 w1 w4 w w4: m4 m1 m3 m2
Men's preferences Women's preferences

4. Stable marriage: example instance
m1: w4 w1 w2 w w1: m4 m1 m3 m2
m2: w2 w3 w1 w w2: m1 m3 m2 m4
m3: w2 w4 w3 w w3: m1 m2 m3 m4
m4: w3 w1 w4 w w4: m4 m1 m3 m2
Men's preferences Women's preferences
M={(m1,w1), (m2,w2), (m3,w4) ,(m4,w3)}

5. Stable marriage: example instance
m1: w4 w1 w2 w w1: m4 m1 m3 m2
m2: w2 w3 w1 w w2: m1 m3 m2 m4
m3: w2 w4 w3 w w3: m1 m2 m3 m4
m4: w3 w1 w4 w w4: m4 m1 m3 m2
Men's preferences Women's preferences
M={(m1,w1), (m2,w2), (m3,w4) ,(m4,w3)}
(m3,w2) is a blocking pair, so M is not stable

6. Stable marriage: example instance
m1: w4 w1 w2 w w1: m4 m1 m3 m2
m2: w2 w3 w1 w w2: m1 m3 m2 m4
m3: w2 w4 w3 w w3: m1 m2 m3 m4
m4: w3 w1 w4 w w4: m4 m1 m3 m2
Men's preferences Women's preferences
M={(m1,w1), (m2,w2), (m3,w4) ,(m4,w3)}
(m3,w2) is a blocking pair, so M is not stable
m1: w4 w1 w2 w w1: m4 m1 m3 m2
m2: w2 w3 w1 w w2: m1 m3 m2 m4
m3: w2 w4 w3 w w3: m1 m2 m3 m4
m4: w3 w1 w4 w w4: m4 m1 m3 m2
Men's preferences Women's preferences
M={(m1,w4), (m2,w3), (m3,w2) ,(m4,w1)} is stable

7. The Gale / Shapley algorithm
A stable matching always exists for a given
instance of SM
Such a matching may be found in linear time
using the Gale / Shapley (GS) algorithm
D. Gale and L. Shapley, “College Admissions and
the stability of marriage”, American Mathematical
Monthy, 1962
The set of stable matchings forms a
distributive lattice
Given an instance I of SM, there is a CP
encoding J of I such that the action of an
extended version of the GS algorithm on I is
equivalent (in a precise sense) to establishing
arc consistency in J
I.P. Gent, R.W. Irving, D.F. Manlove, P. Prosser and
B.M. Smith, “A Constraint Programming Approach
to the Stable Marriage Problem”, Proc. CP '01.

8. Stable marriage with unacceptable partners (SMI)
If q appears on p’s list, then p finds q acceptable,
otherwise p finds q unacceptable
Now a matching M is a set of  n disjoint
(man,woman) pairs such that (m,w) M only if:
m finds w acceptable
w finds m acceptable
Now a blocking pair of matching M is a
(man,woman) pair (m,w)M such that:
m and w find each other acceptable
m is unmatched in M or prefers w to pM(m)
w is unmatched in M or prefers m to pM(w)
Matching M is stable if M admits no blocking
pair
Assume that preference lists are consistent, i.e.
p is on q’s list if and only if q is on p’s list

9. Unacceptable partners: example instance
m1: w4 w1 w2 w w1: m4 m1 m3 m2
m2: w2 w3 w1 w w2: m1 m3 m2 m4
m3: w2 w4 w3 w w3: m1 m2 m3 m4
m4: w3 w1 w4 w w4: m4 m1 m3 m2
Men's preferences Women's preferences

10. Unacceptable partners: example instance
m1: w4 w1 w w1: m4 m1 m2
m2: w2 w1 w w2: m3 m2 m4
m3: w2 w4 w w3: m1 m3
m4: w1 w4 w w4: m4 m1 m3 m2
Men's preferences Women's preferences

11. Unacceptable partners: example instance
m1: w4 w1 w w1: m4 m1 m2
m2: w2 w1 w w2: m3 m2 m4
m3: w2 w4 w w3: m1 m3
m4: w1 w4 w w4: m4 m1 m3 m2
Men's preferences Women's preferences
M={(m1,w4), (m2,w2), (m4,w1)}
(m3,w2) and (m3,w3) are blocking pairs, so M is not stable
m1: w4 w1 w w1: m4 m1 m2
m2: w2 w1 w w2: m3 m2 m4
m3: w2 w4 w w3: m1 m3
m4: w1 w4 w w4: m4 m1 m3 m2
Men's preferences Women's preferences
M={(m1,w4), (m3,w2), (m4,w1)} is stable

12. Revised Gale/Shapley algorithm
A stable matching always exists, for a given
instance of SMI
Such a matching may be found in linear time
using the revised Gale/Shapley algorithm
Some people may be unmatched in a stable
matching, but
the same people are unmatched in all
stable matchings
hence all stable matchings have the same size
The distributive lattice property also holds
Constraint programming formulation is also
possible, giving similar structural properties to
those holding for SM

13. Stable Marriage with Ties (SMT)
Assume all preference lists are complete again
(i.e. of length n)
A given person may be indifferent among two
or more other persons, so a preference list
may involve ties
Example instance:
m1: w4 w1 w2 w w1: m4 m1 m3 m2
m2: w2 w3 w1 w w2: m1 (m3 m2) m4
m3: w2 w4 w3 w w3: m1 m2 m3 m4
m4: w3 w1 w4 w w4: m4 m1 m3 m2
Men's preferences Women's preferences
Three possible definitions of stability
A matching M is weakly stable if there is no
blocking pair (m,w)M such that
m strictly prefers w to pM(m)
w strictly prefers m to pM(w)

14. M={(m1,w4), (m2,w3), (m3,w2),(m4,w1)} is weakly stable
Weak stability
m1: w4 w1 w2 w w1: m4 m1 m3 m2
m2: w2 w3 w1 w w2: m1 (m3 m2) m4
m3: w2 w4 w3 w w3: m1 m2 m3 m4
m4: w3 w1 w4 w w4: m4 m1 m3 m2
M={(m1,w4), (m2,w3), (m3,w2),(m4,w1)} is weakly stable

15. M={(m1,w4), (m2,w3), (m3,w2),(m4,w1)} is weakly stable
Weak stability
m1: w4 w1 w2 w w1: m4 m1 m3 m2
m2: w2 w3 w1 w w2: m1 (m3 m2) m4
m3: w2 w4 w3 w w3: m1 m2 m3 m4
m4: w3 w1 w4 w w4: m4 m1 m3 m2
M={(m1,w4), (m2,w3), (m3,w2),(m4,w1)} is weakly stable
A weakly stable matching always exists, given an instance of SMT, and such a matching can be found in linear time:

16. M={(m1,w4), (m2,w3), (m3,w2),(m4,w1)} is weakly stable
Weak stability
m1: w4 w1 w2 w w1: m4 m1 m3 m2
m2: w2 w3 w1 w w2: m1 (m3 m2) m4
m3: w2 w4 w3 w w3: m1 m2 m3 m4
m4: w3 w1 w4 w w4: m4 m1 m3 m2
M={(m1,w4), (m2,w3), (m3,w2),(m4,w1)} is weakly stable
A weakly stable matching always exists, given an instance of SMT, and such a matching can be found in linear time:
Let I be an instance of SMT

17. M={(m1,w4), (m2,w3), (m3,w2),(m4,w1)} is weakly stable
Weak stability
m1: w4 w1 w2 w w1: m4 m1 m3 m2
m2: w2 w3 w1 w w2: m1 (m3 m2) m4
m3: w2 w4 w3 w w3: m1 m2 m3 m4
m4: w3 w1 w4 w w4: m4 m1 m3 m2
M={(m1,w4), (m2,w3), (m3,w2),(m4,w1)} is weakly stable
A weakly stable matching always exists, given an instance of SMT, and such a matching can be found in linear time:
Let I be an instance of SMT
Break the ties arbitrarily to form an SM
instance J
m2: w2 w3 w1 w w2: m1 m3 m2 m4

18. M={(m1,w4), (m2,w3), (m3,w2),(m4,w1)} is weakly stable
Weak stability
m1: w4 w1 w2 w w1: m4 m1 m3 m2
m2: w2 w3 w1 w w2: m1 (m3 m2) m4
m3: w2 w4 w3 w w3: m1 m2 m3 m4
m4: w3 w1 w4 w w4: m4 m1 m3 m2
M={(m1,w4), (m2,w3), (m3,w2),(m4,w1)} is weakly stable
A weakly stable matching always exists, given an instance of SMT, and such a matching can be found in linear time:
Let I be an instance of SMT
Break the ties arbitrarily to form an SM
instance J
Use the Gale / Shapley algorithm to find a
stable matching M in J
m2: w2 w3 w1 w w2: m1 m3 m2 m4

19. M={(m1,w4), (m2,w3), (m3,w2),(m4,w1)} is weakly stable
Weak stability
m1: w4 w1 w2 w w1: m4 m1 m3 m2
m2: w2 w3 w1 w w2: m1 (m3 m2) m4
m3: w2 w4 w3 w w3: m1 m2 m3 m4
m4: w3 w1 w4 w w4: m4 m1 m3 m2
M={(m1,w4), (m2,w3), (m3,w2),(m4,w1)} is weakly stable
A weakly stable matching always exists, given an instance of SMT, and such a matching can be found in linear time:
Let I be an instance of SMT
Break the ties arbitrarily to form an SM
instance J
Use the Gale / Shapley algorithm to find a
stable matching M in J
M is then weakly stable in I
M={(m1,w4), (m2,w3), (m3,w2),(m4,w1)} is same weakly stable matching as above

20. Strong stability
Assume we are given an instance of SMT
A matching M is strongly stable if there is no
blocking pair (x,y)M such that
x strictly prefers y to pM(x)
y strictly prefers x to pM(y) or is indifferent
between them
A strongly stable matching is weakly stable
An instance of SMT may not admit a strongly
stable matching:
m1: w1 w w1:(m1 m2)
m2: w1 w w2:(m1 m2)
Men's preferences Women's preferences

21. Strong stability
Assume we are given an instance of SMT
A matching M is strongly stable if there is no
blocking pair (x,y)M such that
x strictly prefers y to pM(x)
y strictly prefers x to pM(y) or is indifferent
between them
A strongly stable matching is weakly stable
An instance of SMT may not admit a strongly
stable matching:
m1: w1 w w1:(m1 m2)
m2: w1 w w2:(m1 m2)
Men's preferences Women's preferences

22. Strong stability
Assume we are given an instance of SMT
A matching M is strongly stable if there is no
blocking pair (x,y)M such that
x strictly prefers y to pM(x)
y strictly prefers x to pM(y) or is indifferent
between them
A strongly stable matching is weakly stable
An instance of SMT may not admit a strongly
stable matching:
m1: w1 w w1:(m1 m2)
m2: w1 w w2:(m1 m2)
Men's preferences Women's preferences
(m2,w1) is a blocking pair

23. Strong stability
Assume we are given an instance of SMT
A matching M is strongly stable if there is no
blocking pair (x,y)M such that
x strictly prefers y to pM(x)
y strictly prefers x to pM(y) or is indifferent
between them
A strongly stable matching is weakly stable
An instance of SMT may not admit a strongly
stable matching:
m1: w1 w w1:(m1 m2)
m2: w1 w w2:(m1 m2)
Men's preferences Women's preferences

24. Strong stability
Assume we are given an instance of SMT
A matching M is strongly stable if there is no
blocking pair (x,y)M such that
x strictly prefers y to pM(x)
y strictly prefers x to pM(y) or is indifferent
between them
A strongly stable matching is weakly stable
An instance of SMT may not admit a strongly
stable matching:
m1: w1 w w1:(m1 m2)
m2: w1 w w2:(m1 m2)
Men's preferences Women's preferences
(m1,w1) is a blocking pair

25. Assume we are given an instance of SMT
Strong stability
Assume we are given an instance of SMT
A matching M is strongly stable if there is no
blocking pair (x,y)M such that
x strictly prefers y to pM(x)
y strictly prefers x to pM(y) or is indifferent
between them
A strongly stable matching is weakly stable
An instance of SMT may not admit a strongly
stable matching:
m1: w1 w w1:(m1 m2)
m2: w1 w w2:(m1 m2)
Men's preferences Women's preferences
There is an O(n4) algorithm to determine
whether an instance of SMT has a strongly
stable matching, and to find one if one does
R.W. Irving, “Stable marriage and indifference”,
Discrete Applied Maths, 1994

26. The set of strongly stable matchings forms a distributive lattice
D.F. Manlove, “The structure of stable marriage
with indifference”, Discrete Applied Maths, 2002
Open questions
Is there a faster (e.g. O(n3)) algorithm for
finding a strongly stable matching if one exists,
given an instance of SMT?
Is there an efficient algorithm for finding a
weakly stable matching with the minimum
number of blocking pairs of the strong
stability type?
matching M of maximum cardinality such that
M is strongly stable with respect to the
matched men and women?
If a strongly stable matching does not exist, is
there a succinct certificate of this fact?

27. Super-stability
Assume we are given an instance of SMT
A matching M is super-stable if there is no
blocking pair (m,w)M such that
m strictly prefers w to pM(m) or is
indifferent between them
w strictly prefers m to pM(w) or is
A super-stable matching is strongly stable, and
a strongly stable matching is weakly stable

28. Super-stability
Assume we are given an instance of SMT
A matching M is super-stable if there is no
blocking pair (m,w)M such that
m strictly prefers w to pM(m) or is
indifferent between them
w strictly prefers m to pM(w) or is
A super-stable matching is strongly stable, and
a strongly stable matching is weakly stable
A stable marriage instance with ties may not
admit a super-stable matching:
m1: (w1 w2) w1:(m1 m2)
m2: (w1 w2) w2:(m1 m2)
Men's preferences Women's preferences

29. Super-stability
Assume we are given an instance of SMT
A matching M is super-stable if there is no
blocking pair (m,w)M such that
m strictly prefers w to pM(m) or is
indifferent between them
w strictly prefers m to pM(w) or is
A super-stable matching is strongly stable, and
a strongly stable matching is weakly stable
A stable marriage instance with ties may not
admit a super-stable matching:
m1: (w1 w2) w1:(m1 m2)
m2: (w1 w2) w2:(m1 m2)
Men's preferences Women's preferences
(m2,w1) is a blocking pair

30. Super-stability
Assume we are given an instance of SMT
A matching M is super-stable if there is no
blocking pair (m,w)M such that
m strictly prefers w to pM(m) or is
indifferent between them
w strictly prefers m to pM(w) or is
A super-stable matching is strongly stable, and
a strongly stable matching is weakly stable
A stable marriage instance with ties may not
admit a super-stable matching:
m1: (w1 w2) w1:(m1 m2)
m2: (w1 w2) w2:(m1 m2)
Men's preferences Women's preferences
(m1,w1) is a blocking pair

31. There is an O(n2) algorithm to determine
whether an instance has a super-stable
matching, and to find one if one does
R.W. Irving, “Stable marriage and indifference”,
Discrete Applied Maths, 1994
The set of super-stable matchings forms a
distributive lattice
Open questions
Is there an efficient algorithm for finding a
weakly / strongly stable matching with the
minimum number of blocking pairs of the
super-stability type?
matching M of maximum cardinality such that
M is super-stable with respect to the matched
men and women?
If a super-stable matching does not exist, is
there a succinct certificate of this fact?

32. Stable marriage: unacceptable partners and ties (SMTI)
Assume that persons may express both
unacceptable partners and ties in their lists
Example instance:
m1: w4 w1 w w1: m4 m1 m2
m2: w2 w1 w w2:(m3 m2) m4
m3: w2 w4 w w3: m1 m3
m4: w1 w4 w w4: m4 m1 m3 m2
Men's preferences Women's preferences
Again, a matching can only contain mutually
acceptable (man,woman) pairs
A matching M is weakly stable if there is no
blocking pair (m,w)M such that
m and w find each other acceptable
m is unmatched or strictly prefers w to pM(m)
w is unmatched or strictly prefers m to pM(w)

33. Weakly stable matchings
For a given SMTI instance, the weakly stable matchings could be of different sizes

34. Weakly stable matchings
For a given SMTI instance, the weakly stable matchings could be of different sizes
Example instance:
m1: w4 w1 w w1: m4 m1 m2 m2: w2 w1 w w2:(m3 m2) m4
m3: w2 w4 w w3: m1 m3
m4: w1 w4 w w4: m4 m1 m3 m2
Men's preferences Women's preferences
M1={(m1,w4), (m2,w2), (m3,w3),(m4,w1)} is a weakly stable matching of size 4

35. Weakly stable matchings
For a given SMTI instance, the weakly stable matchings could be of different sizes
Example instance:
m1: w4 w1 w w1: m4 m1 m2
m2: w2 w1 w w2:(m3 m2) m4
m3: w2 w4 w w3: m1 m3
m4: w1 w4 w w4: m4 m1 m3 m2
Men's preferences Women's preferences
M1={(m1,w4), (m2,w2), (m3,w3),(m4,w1)} is a weakly stable matching of size 4
M2={(m1,w4), (m3,w2),(m4,w1)} is a weakly stable matching of size 3

36. Maximum / minimum weakly stable matchings
Do efficient algorithms exist for finding maximum or minimum cardinality weakly stable matchings?
Unfortunately, this is unlikely
Each of the problems of finding a maximum or minimum weakly stable matching is NP-hard,
and the result is true even if:
1. the ties occur in the preference lists of one sex
only, and
2. any tie occurs at the tail of some person's
preference list, and
3. any tie is of length 2
K. Iwama, D. Manlove, S. Miyazaki & Y. Morita:
“Stable marriage with incomplete lists and ties”,
Proc. ICALP ’99
D.F. Manlove, R.W. Irving, K. Iwama, S. Miyazaki &
Y. Morita: “Hard Variants of Stable Marriage”,
Theoretical Computer Science, 2002

37. Approximation algorithms
An approximation algorithm with performance
guarantee c is a polynomial-time algorithm
that finds a solution within a factor of c from
optimal
Given an SMTI instance, any weakly stable
matching has size:
at least half the size of a maximum
cardinality weakly stable matching
at most twice the size of a minimum
So both problems admit an approximation
algorithm with performance guarantee 2
D.F. Manlove, R.W. Irving, K. Iwama, S. Miyazaki &
Y. Morita: “Hard Variants of Stable Marriage”,
Theoretical Computer Science, 2002

38. Approximation algorithms (continued)
Given an SMTI instance, any weakly stable
matching has size
at least the size of a maximum cardinality
weakly stable matching minus the number
of preference lists with ties
at most the size of a minimum cardinality
weakly stable matching plus the number
The problem of finding a maximum
cardinality weakly stable matching is not
approximable within δ, for some δ>1
M. Halldórsson, K. Iwama, S. Miyazaki &
Y. Morita: “Inapproximability results on stable
marriage problems”, Proc. LATIN 2002
M. Halldórsson, R.W. Irving, K. Iwama, D.F. Manlove,
S. Miyazaki, Y. Morita and S. Scott: “Approximability
results for stable marriage problems with ties”,
Submitted for publication, 2002

39. SMTI under strong stability and super-stability
Each of the definitions of a strongly stable
matching and a super-stable matching can be
extended to SMTI
All strongly stable (respectively super-stable)
matchings are of the same size, for a given
instance of SMTI
There are polynomial-time algorithms for
finding a strongly stable matching and / or a
super-stable matching, if they exist, given an
SMTI instance
D.F. Manlove, “Stable marriage with ties and
unacceptable partners”, DCS Tech Report, 1999
If an instance I of SMTI admits a strongly
stable matching M, then |M |  ⅔ |M' |, where
M' is a weakly stable matching of maximum
cardinality in I
S. Scott, “On the size of strongly stable matchings”,
manuscript, 2002

40. SMTI – further results and open problems
Constraint programming formulations of
SMTI under weak / strong / super-stability:
I.P. Gent and P. Prosser, “SAT encodings of
the stable marriage problem with ties and
incomplete lists”, Proc. SAT 2002
I.P. Gent and P. Prosser, “An empirical study of
incomplete lists”, Proc. ECAI 2002
Open problems
Is there an approximation algorithm with
performance guarantee <2 for the problem of
finding a maximum / minimum cardinality
weakly stable matching, given an instance of
SMTI?
Possibly easier to consider restricted instances,
e.g. involving
bounded length ties
ties on one side only
at most one tie per list

41. Hospitals/Residents problem (HR)
n residents r1, r2, …, rn
m hospitals h1, h2, …, hm
Hospital hi has capacity ci
Each resident ranks a subset of the hospitals in
strict order of preference
Each hospital ranks its applicants in strict
order of preference
r finds h acceptable if h is on r’s preference list; r finds h unacceptable otherwise (and vice
versa)
A matching M in an instance of HR is an
allocation of residents to hospitals such that:
1) (r,h)M  r,h find each other acceptable
2) No resident receives more than one post
3) No hospital exceeds its capacity

42. Hospitals/Residents problem: example
r1: h2 h1
r2: h1 h Each hospital has 2 posts
r3: h1 h3
r4: h2 h3 h1: r1 r3 r2 r5 r6
r5: h2 h1 h2: r2 r6 r1 r4 r5
r6: h1 h2 h3: r4 r3
Resident preferences Hospital preferences

43. Hospitals/Residents problem: matching
r1: h2 h1
r2: h1 h Each hospital has 2 posts
r3: h1 h3
r4: h2 h3 h1: r1 r3 r2 r5 r6
r5: h2 h1 h2: r2 r6 r1 r4 r5
r6: h1 h2 h3: r4 r3
Resident preferences Hospital preferences
M = {(r1, h1), (r2, h2), (r3, h3), (r5, h2), (r6, h1)}
(size 5)

44. Hospitals/Residents problem: matching
r1: h2 h1
r2: h1 h Each hospital has 2 posts
r3: h1 h3
r4: h2 h3 h1: r1 r3 r2 r5 r6
r5: h2 h1 h2: r2 r6 r1 r4 r5
r6: h1 h2 h3: r4 r3
Resident preferences Hospital preferences
Matching M is stable if M admits no blocking pair
(r,h) is a blocking pair of matching M if:
1) r, h find each other acceptable
and
2) either r is unmatched in M
or r prefers h to his/her assigned hospital in M
3) either h is undersubscribed in M
or h prefers r to its worst resident assigned in M

45. HR: unstable matching r1: h2 h1 r2: h1 h2 Each hospital has 2 posts
r4: h2 h3 h1: r1 r3 r2 r5 r6
r5: h2 h1 h2: r2 r6 r1 r4 r5
r6: h1 h2 h3: r4 r3
Resident preferences Hospital preferences
Matching M is stable if M admits no blocking pair
(r,h) is a blocking pair of matching M if:
1) r, h find each other acceptable
and
2) either r is unmatched in M
or r prefers h to his/her assigned hospital in M
3) either h is undersubscribed in M
or h prefers r to its worst resident assigned in M
Matching above is unstable as e.g. (r2,h1), (r4,h2) and (r4,h3) are blocking pairs

46. HR: unstable matching r1: h2 h1 r2: h1 h2 Each hospital has 2 posts
r4: h2 h3 h1: r1 r3 r2 r5 r6
r5: h2 h1 h2: r2 r6 r1 r4 r5
r6: h1 h2 h3: r4 r3
Resident preferences Hospital preferences
Matching M is stable if M admits no blocking pair
(r,h) is a blocking pair of matching M if:
1) r, h find each other acceptable
and
2) either r is unmatched in M
or r prefers h to his/her assigned hospital in M
3) either h is undersubscribed in M
or h prefers r to its worst resident assigned in M
Matching above is unstable as e.g. (r2,h1), (r4,h2) and (r4,h3) are blocking pairs

47. HR: unstable matching r1: h2 h1 r2: h1 h2 Each hospital has 2 posts
r4: h2 h3 h1: r1 r3 r2 r5 r6
r5: h2 h1 h2: r2 r6 r1 r4 r5
r6: h1 h2 h3: r4 r3
Resident preferences Hospital preferences
Matching M is stable if M admits no blocking pair
(r,h) is a blocking pair of matching M if:
1) r, h find each other acceptable
and
2) either r is unmatched in M
or r prefers h to his/her assigned hospital in M
3) either h is undersubscribed in M
or h prefers r to its worst resident assigned in M
Matching above is unstable as e.g. (r2,h1), (r4,h2) and (r4,h3) are blocking pairs

48. HR: stable matching r1: h2 h1 r2: h1 h2 Each hospital has 2 posts
r4: h2 h3 h1: r1 r3 r2 r5 r6
r5: h2 h1 h2: r2 r6 r1 r4 r5
r6: h1 h2 h3: r4 r3
Resident preferences Hospital preferences
Matching M is stable if M admits no blocking pair
(r,h) is a blocking pair of matching M if
1) r, h find each other acceptable
and
2) either r is unmatched in M
or r prefers h to his/her assigned hospital in M
3) either h is undersubscribed in M
or h prefers r to its worst resident assigned in M
Example shows that, in a given stable matching,
one or more residents may be unmatched
one or more hospitals may be undersubscribed

49. HR: structure and algorithms
SM is a special case of HR
A stable matching always exists for a given
instance of HR
Such a matching may be found in linear time
D. Gusfield and R.W. Irving, “The Stable Marriage
Problem: Structure and Algorithms”, MIT Press, 1989
This algorithm is at the heart of centralised matching schemes such as the NRMP and SPA
There may be more than one stable matching,
but:
All stable matchings have the same size
The same residents are assigned in all stable
matchings
Any hospital that is undersubscribed in
one stable matching is assigned exactly the
same residents in all stable matchings
(Rural Hospitals Theorem)

50. Hospitals/Residents problem with Ties (HRT)
Participants may wish to express ties in their
preference lists
SMTI is a special case of HRT
NP-hardness and inapproximability results for
SMTI under weak stability also apply to HRT
Each of the problems of finding a maximum
and minimum cardinality weakly stable
matching in HRT is approximable within 2
There are polynomial-time algorithms for
finding a strongly stable matching and / or a
super-stable matching, if they exist, given an
HRT instance
The Rural Hospitals Theorem holds for HRT
under strong stability and super-stability
R.W. Irving, D.F. Manlove and S. Scott, “The Hospitals /
Residents Problem with Ties”, Proc. SWAT 2000
R.W. Irving, D.F. Manlove and S. Scott, “Strong
stability in the Hospitals/Residents Problem”,
Submitted for publication, 2002

51. Stable Roommates (SR) Input: 2n persons; each person ranks all 2n-1
other persons in strict order
Output: a stable matching
Definitions
A matching is a set of n disjoint pairs of persons
A blocking pair of a matching M is a pair of persons {p,q}M such that:
p prefers q to his partner in M, and
q prefers p to his partner in M
A matching is stable if it admits no blocking pair

52. Stable Roommates (SR) Input: 2n persons; each person ranks all 2n-1
other persons in strict order
Output: a stable matching
Definitions
A matching is a set of n disjoint pairs of persons
A blocking pair of a matching M is a pair of persons {p,q}M such that:
p prefers q to his partner in M, and
q prefers p to his partner in M
A matching is stable if it admits no blocking pair
Example SR instance: 1: 2: 3: 4:

53. Stable Roommates (SR) Input: 2n persons; each person ranks all 2n-1
other persons in strict order
Output: a stable matching
Definitions
A matching is a set of n disjoint pairs of persons
A blocking pair of a matching M is a pair of persons {p,q}M such that:
p prefers q to his partner in M, and
q prefers p to his partner in M
A matching is stable if it admits no blocking pair
Example SR instance: 1: 2: 3: 4:

54. Stable Roommates (SR) Input: 2n persons; each person ranks all 2n-1
other persons in strict order
Output: a stable matching
Definitions
A matching is a set of n disjoint pairs of persons
A blocking pair of a matching M is a pair of persons {p,q}M such that:
p prefers q to his partner in M, and
q prefers p to his partner in M
A matching is stable if it admits no blocking pair
Example SR instance: 1: 2: 3: 4:
The matching is not stable as {1,3} blocks.

55. Stable Roommates (SR) Input: 2n persons; each person ranks all 2n-1
other persons in strict order
Output: a stable matching
Definitions
A matching is a set of n disjoint pairs of persons
A blocking pair of a matching M is a pair of persons {p,q}M such that:
p prefers q to his partner in M, and
q prefers p to his partner in M
A matching is stable if it admits no blocking pair
Example SR instance: 1: 2: 3: 4:
The matching is not stable as {1,3} blocks.
This instance does not admit a stable matching!

56. Stable Roommates: algorithmic results
SM is a special case of SR
There is a linear-time algorithm for finding a
stable matching if one exists, given an SR
instance
R.W. Irving “An efficient algorithm for the ‘Stable
Roommates’ Problem”, Journal of Algorithms, 1985
If preference lists have ties, we obtain an
instance of SRT (Stable Roommates with Ties)
The problem of deciding whether a given
instance of SRT admits a weakly stable
matching is NP-complete, even if:
each tie has length 2
there is at most one tie per list
E. Ronn, “NP-complete stable matching problems”,
Journal of Algorithms, 1990
R.W. Irving and D.F. Manlove, “The Stable
Roommates Problem with Ties”, Journal of
Algorithms, 2002

57. SR & SRT: further results and open problems
There is a linear-time algorithm for finding a
super-stable matching if one exists, given an
SRT instance
R.W. Irving and D.F. Manlove, “The Stable
Roommates Problem with Ties”, Journal of
Algorithms, 2002
Open problems
Complexity of SRT under strong stability?
Is there a polynomial-time algorithm to find
a matching with the minimum number of
blocking pairs, given an instance of SR?
Is there a polynomial reduction from SR to
SM, or from SR to SMT?

58. Stable Fixtures Problem (SF)
Choose a schedule of fixtures based on teams
having preferences over each other
Two given teams can play each other at most
once
Each team i has a capacity ci , indicating the
maximum number of fixtures it can play
A fixture allocation A is a set of unordered
pairs of teams {i, j} such that, for each i,
team i is scheduled to play at most ci fixtures
Example SF instance:
team capacity preferences 1: 2: 3: 4:

59. Stable Fixtures Problem (SF)
Choose a schedule of fixtures based on teams
having preferences over each other
Two given teams can play each other at most
once
Each team i has a capacity ci , indicating the
maximum number of fixtures it can play
A fixture allocation A is a set of unordered
pairs of teams {i, j} such that, for each i,
team i is scheduled to play at most ci fixtures
Example SF instance:
team capacity preferences 1: 2: 3: 4:
The above fixture allocation is:
{1, 2}, {1, 4}, {2, 3}, {3, 4}

60. A blocking pair of a fixture allocation A is a
pair of teams {i , j}A such that:
i is assigned fewer than ci teams in A
or i prefers j to at least one of its assignees
and
j is assigned fewer than cj teams in A
or j prefers i to at least one of its assignees
A fixture allocation is stable if it admits no
blocking pair
Example SF instance:
team capacity preferences 1: 2: 3: 4:
The above fixture allocation is not stable as
{1,3} is a blocking pair

61. team capacity preferences 1: 2 2 3 4 2: 2 1 3 4 3: 2 1 2 4 4: 2 1 2 3
Example SF instance:
team capacity preferences 1: 2: 3: 4:
The above fixture allocation is stable
Each of SM, HR and SR is a special case of SF
There is an efficient algorithm to find a stable
fixtures allocation, if one exists, given
an instance of SF
R.W. Irving and S. Scott, “The Stable Fixtures
Problem”, submitted for publication
This algorithm has been extended to cover
Stable Fixtures with Ties (SFT) under super-
stability
Open problem
Complexity of SFT under strong stability?

62. Ties – super-stability
Summary of complexity results for stable matching problems with ties
Problem
No ties
Ties –
weak stability
strong
stability
Ties – super-stability
SMTI P
NPC (max)
HRT
SRT
NPC ?
SFT
SMTI – Stable Marriage with Ties and Incomplete Lists
HRT – Hospitals / Residents with Ties
SRT – Stable Roommates with Ties
SFT – Stable Fixtures with Ties
P – problem of finding a relevant matching if one exists is polynomial-time solvable
NPC – problem of deciding whether relevant matching exists is NP-complete