1. Strong Stability in the Hospitals/Residents Problem
Rob Irving, David Manlove and Sandy Scott
University of Glasgow
Department of Computing Science
Supported by EPSRC grant GR/R84597/01,
Nuffield Foundation award NUF-NAL-02, and
RSE / SEETLLD Personal Research Fellowship

2. Hospitals/Residents problem (HR): Motivation
Graduating medical students or residents seek hospital appointments
Free-for-all markets are chaotic
Centralised matching schemes are in operation
Schemes produce stable matchings of residents to hospitals
National Resident Matching Program (US)
other large-scale matching schemes, both educational and vocational

3. Hospitals/Residents problem (HR): Definition
A set H of hospitals, a set R of residents
Each resident r ranks a subset of H in strict order of preference
r finds h acceptable if h appears on r’s preference list
Each hospital h has ph posts, and ranks in strict order those residents who find h acceptable
A matching M is a subset of R×H such that:
(r,h)M implies that r finds h acceptable
Each resident r is assigned at most one hospital
Each hospital h is assigned at most ph residents

4. An instance of HR r1: h2 h3 h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3
h1:3: r2 r1 r3 r5
h2:2: r3 r2 r1 r4 r5
h3:1: r4 r5 r1 r3 r6

5. A matching in HR r1: h2 h3 h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3
h1:3: r2 r1 r3 r5
h2:2: r3 r2 r1 r4 r5
h3:1: r4 r5 r1 r3 r6

6. Stable matchings in HR
Matching M is stable if M admits no blocking pair
(r,h) is a blocking pair of matching M if:
r finds h acceptable and
either r is unmatched in M or r prefers h to his assigned hospital in M and
either h is undersubscribed in M or h prefers r to its worst resident assigned in M

7. A blocking pair r1: h2 h3 h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3
h1:3: r2 r1 r3 r5
h2:2: r3 r2 r1 r4 r5
h3:1: r4 r5 r1 r3 r6
(r4, h2) is a blocking pair of matching M since:
r4 finds h2 acceptable
r4 is unmatched in M
h2 prefers r4 to its worst resident assigned in M

8. A stable matching r1: h2 h3 h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3
h1:3: r2 r1 r3 r5
h2:2: r3 r2 r1 r4 r5
h3:1: r4 r5 r1 r3 r6
Example shows that, in a stable matching,
one or more residents may be unmatched
one or more hospitals may be undersubscribed

9. Algorithmic results for HR
Every instance of HR admits at least one stable matching
Such a matching can be found in linear time using the Gale / Shapley algorithm
Resident-oriented version finds the resident-optimal stable matching
Each resident obtains the best hospital that he could obtain in any stable matching
Hospital-oriented version finds the hospital-optimal stable matching
Each hospital obtains the best set of residents that it could obtain in any stable matching

10. The Rural Hospitals Theorem
For a given instance of HR, there may be more than one stable matching, but:
All stable matchings have the same size
The same set of residents are matched in all stable matchings
Any hospital that is undersubscribed in one stable matching has exactly same residents in all stable matchings
This is the Rural Hospitals Theorem

11. Ties in the preference lists
Participants may wish to express ties in their preferences lists:
h1 : r7 (r1 r3) r5
version of HR with ties is HRT
more general form of indifference involves partial orders
version of HR with partial orders is HRP
Three stability definitions are possible:
Weak stability
Strong stability
Super-stability

12. An instance of HRT r1:(h2 h3) h1 h1:3: r2 (r1 r3) r5 r2: h2 h1

13. Weak stability
Matching M is weakly stable if M admits no blocking pair
(r,h) is a blocking pair of matching M if:
r finds h acceptable
either r is unmatched in M or r strictly prefers h to his assigned hospital in M
either h is undersubscribed in M or h strictly prefers r to its worst resident assigned in M
That is, if (r,h) joined together, both would be better off

14. A weakly stable matching in HRT
r1:(h2 h3) h1
r2: h2 h1
r3: h3 (h1 h2)
r4: h2 h3
r5: h2 (h1 h3)
r6: h3
h1:3: r2 (r1 r3) r5
h2:2: r3 r2 (r1 r4 r5)
h3:1:(r4 r5) (r1 r3 r6)
Weakly stable matching has size 5

15. A second weakly stable matching in HRT
r1:(h2 h3) h1
r2: h2 h1
r3: h3 (h1 h2)
r4: h2 h3
r5: h2 (h1 h3)
r6: h3
h1:3: r2 (r1 r3) r5
h2:2: r3 r2 (r1 r4 r5)
h3:1:(r4 r5) (r1 r3 r6)
Weakly stable matching has size 6

16. Weak stability in HRT: algorithmic results
Weakly stable matching always exists and can be found in linear time
Gusfield and Irving, 1989
Weakly stable matchings can have different sizes
Problem of finding a maximum cardinality weakly stable matching is NP-hard, even if each hospital has 1 post (i.e. instance of stable marriage problem with ties and incomplete lists)
Iwama, Manlove, Miyazaki and Morita, ICALP ’99
Result holds even if the ties occur on one side only, at most one tie per list, and each tie is of length 2
Manlove, Irving, Iwama, Miyazaki, Morita, TCS 2002
Result also holds for minimum weakly stable matchings

17. Weak stability: lower bounds
Problem of finding a maximum cardinality weakly stable matching in HRT is APX-complete, even if each hospital has 1 post
Halldórsson, Iwama, Miyazaki, Morita, LATIN 2002
Ties on both sides, ties of length 2
Ties on one side only, ties of length ≤3
Halldórsson, Iwama, Miyazaki, Yanagisawa, ESA 2003
Ties on one side only, ties of length 2
NP-hard to approximate maximum within 21/19
Halldórsson, Irving, Iwama, Manlove, Miyazaki, Morita, Scott, TCS 2003
Ties on one side only, preference lists of constant length
Result also holds for minimum weakly stable matchings

18. Weak stability: upper bounds
Problem of finding a maximum or minimum cardinality weakly stable matching is approximable within 2
Manlove, Irving, Iwama, Miyazaki, Morita, TCS 2002
If each hospital has 1 post, problem of finding a maximum cardinality weakly stable matching is approximable within:
2/(1+L-2) – ties on one side only, ties of length ≤L
13/7 – ties on both sides, ties of length 2
Halldórsson, Iwama, Miyazaki, Yanagisawa, ESA 2003
10/7 (expected) – ties on one side only, ≤1 tie per list, ties of length 2
7/4 (expected) – ties on both sides, ≤n ties, ties of length 2
Halldórsson, Iwama, Miyazaki, Yanagisawa, COCOON 2003
Let s+ and s- denote the sizes of a maximum and minimum cardinality weakly stable matching, and let t be the number of lists with ties. If each hospital has 1 post, any weakly stable matching M satisfies:
s+ - t ≤ |M| ≤ s- + t
Halldórsson, Irving, Iwama, Manlove, Miyazaki, Morita, Scott, TCS 2003

19. Super-stability
Matching M is super-stable if M admits no blocking pair
(r,h) is a blocking pair of matching M if:
r finds h acceptable
either r is unmatched in M or r strictly prefers h to his assigned hospital in M or r is indifferent between them
either h is undersubscribed in M or h strictly prefers r to its worst resident assigned in M or h is indifferent between them
That is, if (r,h) joined together, neither would be worse off

20. Super-stable matchings in HRT
A matching is super-stable in an instance of HRT if it is stable in every instance of HR obtained by breaking the ties
A super-stable matching is weakly stable
“Rural Hospitals Theorem” also holds for HRT under super-stability
Irving, Manlove, Scott, SWAT 2000
A super-stable matching may not exist:
r1:(h1 h2) h1:1:(r1 r2)
r2:(h1 h2) h2:1:(r1 r2)

21. Finding a super-stable matching
Stable marriage problem with ties (i.e. HRT with |R|=|H|=n and each hospital has 1 post):
O(n2) algorithm
Irving, Discrete Applied Mathematics, 1994
General HRT case
O(L) algorithm, where L is total length of preference lists
Algorithm can be extended to HRP case
Irving, Manlove, Scott, SWAT 2000

22. Strong stability in HRT
A matching M is strongly stable unless there is an acceptable pair (r,h)M such that, if they joined together, one would be better off and the other no worse off
Such a pair constitutes a blocking pair
A super-stable matching is strongly stable, and a strongly stable matching is weakly stable
Ties on one side only  super-stability = strong stability
An instance of HRT may admit no strongly stable matching:
r1: h1 h2 h1:1:(r1 r2)
r2: h1 h2 h2:1: r1 r2

23. A blocking pair r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3
h1:3: r2 (r1 r3) r5
h2:2: r3 r2 (r1 r4 r5)
h3:1: (r4 r5) (r1 r3) r6
r4 and h2 form a blocking pair

24. Another blocking pair r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3
h1:3: r2 (r1 r3) r5
h2:2: r3 r2 (r1 r4 r5)
h3:1: (r4 r5) (r1 r3) r6
r1 and h3 form a blocking pair

25. Finding a strongly stable matching
Stable marriage problem with ties (i.e. HRT with |R|=|H|=n and each hospital has 1 post):
O(n4) algorithm
Irving, Discrete Applied Mathematics, 1994
General HRT case
O(L2) algorithm, where L is total length of preference lists
HRP case
Deciding whether a strongly stable matching exists is NP-complete
Irving, Manlove, Scott, STACS 2003

26. The algorithm in brief repeat
provisionally assign all free residents to hospitals at head of list
form reduced provisional assignment graph
find critical set of residents and make corresponding deletions
until critical set is empty
form a feasible matching
check if feasible matching is strongly stable

27. An instance of HRT r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3
h1:3: r2 (r1 r3) r5
h2:2: r3 r2 (r1 r4 r5)
h3:1: (r4 r5) (r1 r3) r6

28. A provisional assignment and a dominated resident
r1: (h2 h3) h1
r2: h2 h1
r3: h3 h2 h1
r4: h2 h3
r5: h2 (h1 h3)
r6: h3
h1:3: r2 (r1 r3) r5
h2:2: r3 r2 (r1 r4 r5)
h3:1: (r4 r5) (r1 r3) r6

29. A deletion   r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3
h1:3: r2 (r1 r3) r5
h2:2: r3 r2 (r1 r4 r5)
h3:1: (r4 r5) (r1 r3) r6  

30. Another provisional assignment
r1: (h2 h3) h1
r2: h2 h1
r3: h3 h2 h1
r4: h2 h3
r5: h2 (h1 h3)
r6: h3
h1:3: r2 (r1 r3) r5
h2:2: r3 r2 (r1 r4 r5)
h3:1: (r4 r5) (r1 r3) r6  

31. Several provisional assignments
r1: (h2 h3) h1
r2: h2 h1
r3: h3 h2 h1
r4: h2 h3
r5: h2 (h1 h3)
r6: h3
h1:3: r2 (r1 r3) r5
h2:2: r3 r2 (r1 r4 r5)
h3:1: (r4 r5) (r1 r3) r6  

32. Provisional assignment graph with one bound resident

33. Removing a bound resident
h1:(3) r2
h2:(1) r3 r4
h3:(1) r5

34. Reduced provisional assignment graph

35. Critical set r1
h2:(1) r3 r4
h3:(1) r5

36. Critical set   r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3
h1:3: r2 (r1 r3) r5
h2:2: r3 r2 (r1 r4 r5)
h3:1: (r4 r5) (r1 r3) r6  

37. Deletions from the critical set, end of loop iteration
r1: (h2 h3) h1
r2: h2 h1
r3: h3 h2 h1
r4: h2 h3
r5: h2 (h1 h3)
r6: h3  
h1:3: r2 (r1 r3) r5
h2:2: r3 r2 (r1 r4 r5)
h3:1: (r4 r5) (r1 r3) r6          

38. Second loop iteration, starting with a provisional assignment
r1: (h2 h3) h1
r2: h2 h1
r3: h3 h2 h1
r4: h2 h3
r5: h2 (h1 h3)
r6: h3  
h1:3: r2 (r1 r3) r5
h2:2: r3 r2 (r1 r4 r5)
h3:1: (r4 r5) (r1 r3) r6          

39. Several provisional assignments
r1: (h2 h3) h1
r2: h2 h1
r3: h3 h2 h1
r4: h2 h3
r5: h2 (h1 h3)
r6: h3  
h1:3: r2 (r1 r3) r5
h2:2: r3 r2 (r1 r4 r5)
h3:1: (r4 r5) (r1 r3) r6          

40. Final provisional assignment graph with 4 bound residents

41. Removing a bound resident
h1:(2) r2
h2:(2) r3 r4
h3:(1) r5

42. Removing a second bound resident
h1:(2) r2
h2:(1) r3 r4
h3:(1) r5

43. Removing a third bound resident

44. Removing a bound resident with an additional provisional assignment

45. Final reduced provisional assignment graph

46. A feasible matching This consists of:
the bound (resident,hospital) pairs
(r1,h1), (r2,h2), (r3,h2), (r5,h1)
unioned together with:
perfect matching in the final reduced provisional assignment graph
(r4,h3)

47. A strongly stable matching
(r1,h1), (r2,h2), (r3,h2), (r4,h3), (r5,h1)
r1: (h2 h3) h1
r2: h2 h1
r3: h3 h2 h1
r4: h2 h3
r5: h2 (h1 h3)
r6: h3
h1:3: r2 (r1 r3) r5
h2:2: r3 r2 (r1 r4 r5)
h3:1: (r4 r5) (r1 r3) r6

48. The algorithm in full (1)
repeat {
while some resident r is free and has a non-empty list
for each hospital h at the head of r’s list {
provisionally assign r to h;
if h is fully-subscribed or over-subscribed {
for each resident r' dominated on h’s list
delete the pair (r',h); } }
form the reduced provisional assignment graph;
find the critical set Z of residents;
for each hospital h  N(Z)
for each resident r in the tail of h’s list
delete the pair (r,h);
} until Z = ;

49. The algorithm in full (2)
let G be the final provisional assignment graph;
let M be a feasible matching in G;
if M is strongly stable
output M;
else
no strongly stable matching exists;

50. Theoretical results
Algorithm has complexity O(L2), where L is total length of preference lists
Bounded below by complexity of finding a perfect matching in a bipartite graph
Matching produced by the algorithm is resident-optimal
“Rural Hospitals Theorem” holds for HRT under strong stability

51. Strong stability in HRP
HRP under strong stability is NP-complete
even if each hospital has 1 post and all pairs are acceptable
Reduction from RESTRICTED 3-SAT:
Boolean formula B in CNF where each variable v occurs in exactly two clauses as literal v, and exactly two clauses as literal ~v

52. Conclusions
O(nL) algorithm, where n is number of residents and L is total length of preference lists
Kavitha, Mehlhorn, Michail, Paluch, STACS 2004
Find a weakly stable matching with minimum number of strongly stable blocking pairs
Size of strongly stable matchings relative to possible sizes of weakly stable matchings
Hospital-oriented algorithm