0. Matching in Practice: Junior Doctor Allocation in Scotland
David Manlove
Joint work with Rob Irving, Augustine Kwanashie, and Iain McBride

1. Centralised matching schemes
Intending junior doctors must undergo training in hospitals
Applicants rank hospitals in order of preference
Hospitals do likewise with their applicants
Centralised matching schemes (clearinghouses) produce a matching in several countries
US (National Resident Matching Program)
Canada (Canadian Resident Matching Service)
Japan (Japan Residency Matching Program)
Scotland (Scottish Foundation Allocation Scheme)
typically around 750 applicants and 50 hospitals
Stability is the key property of a matching
[Roth, 1984]

2. Hospitals / Residents problem (HR)
Underlying theoretical model: Hospitals / Residents problem (HR)
We have n1 residents r1, r2, …, rn1 and n2 hospitals h1, h2, …, hn2
Each hospital has a capacity
Residents rank hospitals in order of preference, hospitals do likewise
r finds h acceptable if h is on r’s preference list, and unacceptable otherwise (and vice versa)
A matching M is a set of resident-hospital pairs such that:
(r,h)M  r, h find each other acceptable
No resident appears in more than one pair
No hospital appears in more pairs than its capacity

3. HR: example instance r1: h2 h1 r2: h1 h2 Each hospital has capacity 2
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

4. HR: stability 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

5. M = {(r1, h2), (r2, h1), (r3, h1), (r4, h3), (r6, h2)} (size 5)
HR: stable matching
r1: h2 h1
r2: h1 h Each hospital has capacity 2
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, h2), (r2, h1), (r3, h1), (r4, h3), (r6, h2)} (size 5)
r5 is unmatched
h3 is undersubscribed

6. HR: classical results
A stable matching always exists and can be found in linear time [Gale and Shapley, 1962]
There are resident-optimal and hospital-optimal stable matchings
Stable matchings form a distributive lattice [Conway, 1976; Gusfield and Irving, 1989]
“Rural Hospitals Theorem”: for a given instance of HR:
the same residents are assigned in all stable matchings;
each hospital is assigned the same number of residents in all stable matchings;
any hospital that is undersubscribed in one stable matching is assigned exactly the same set of residents in all stable matchings.
[Roth, 1984; Gale and Sotomayor, 1985; Roth, 1986]

7. Hospitals / Residents problem with Ties
In practice, residents’ preference lists are short
Hospitals’ lists are generally long, so ties may be used – Hospitals / Residents problem with Ties (HRT)
A hospital may be indifferent among several residents
E.g., h1: (r1 r3) r2 (r5 r6 r8)
Matching M is stable if there is no pair (r,h) such that:
1. r, h find each other acceptable
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
A matching M is stable in an HRT instance I if and only if M is stable in some instance I of HR obtained from I by breaking the ties [M et al, 1999]

8. HRT: stable matching (1)
r1: h1 h2
r2: h1 h Each hospital has capacity 2
r3: h1 h3
r4: h2 h3 h1: r1 r2 r3 r5 r6
r5: h2 h1 h2: r2 r1 r6(r4 r5)
r6: h1 h2 h3: r4 r3
Resident preferences Hospital preferences

9. HRT: stable matching (1)
r1: h1 h2
r2: h1 h Each hospital has capacity 2
r3: h1 h3
r4: h2 h3 h1: r1 r2 r3 r5 r6
r5: h2 h1 h2: r2 r1 r6(r4 r5)
r6: h1 h2 h3: r4 r3
Resident preferences Hospital preferences
M = {(r1, h1), (r2, h1), (r3, h3), (r4, h2), (r6, h2)} (size 5)

10. HRT: stable matching (2)
r1: h1 h2
r2: h1 h Each hospital has capacity 2
r3: h1 h3
r4: h2 h3 h1: r1 r2 r3 r5 r6
r5: h2 h1 h2: r2 r1 r6(r4 r5)
r6: h1 h2 h3: r4 r3
Resident preferences Hospital preferences
M = {(r1, h1), (r2, h1), (r3, h3), (r4, h3), (r5, h2), (r6, h2)} (size 6)

11. Maximum stable matchings
Stable matchings can have different sizes
A maximum stable matching can be (at most) twice the size of a minimum stable matching
Problem of finding a maximum stable matching (MAX HRT) is NP-hard [Iwama, M et al, 1999], even if (simultaneously):
each hospital has capacity 1 (Stable Marriage problem with Ties and Incomplete Lists)
the ties occur on one side only
each preference list is either strictly ordered or is a single tie
and
either each tie is of length 2 [M et al, 2002]
or each preference list is of length 3 [Irving, M, O’Malley, 2009]
Minimization problem is NP-hard too, for similar restrictions! [M et al, 2002]

12. MAX HRT: approximability
MAX HRT is not approximable within 33/29 unless P=NP, even if each hospital has capacity 1 [Yanagisawa, 2007]
UGC-hard to approximate MAX HRT within 4/3- [Yanagisawa, 2007]
Trivial upper bound of 2 for MAX HRT
Succession of papers gave improvements, culminating in:
MAX HRT is approximable within 3/2 [McDermid, 2009; Király, 2013; Paluch 2012] and within 19/13 (~1.4615) for ties on one side only [Dean and Jalasutram, 2014]
Experimental comparison of approximation algorithms and heuristics for MAX HRT [Irving and M, 2009]
Integer programming models for MAX HRT [Kwanashie and M, 2014]

13. Couples in HR
Pairs of residents who wish to be matched to geographically close hospitals form couples
Each couple (ri,rj) ranks in order of preference a set of pairs of hospitals (hp,hq) representing the assignment of ri to hp and rj to hq
Stability definition may be extended to this case [Roth, 1984; McDermid and M, 2010; Biró et al, 2011]
Gives the Hospitals / Residents problem with Couples (HRC)
A stable matching need not exist:
(r1,r2): (h1,h1) h1:2: r1 r3 r2
r3: h1

14. Couples in HR
Pairs of residents who wish to be matched to geographically close hospitals form couples
Each couple (ri,rj) ranks in order of preference a set of pairs of hospitals (hp,hq) representing the assignment of ri to hp and rj to hq
Stability definition may be extended to this case [Roth, 1984; McDermid and M, 2010; Biró et al, 2011]
Gives the Hospitals / Residents problem with Couples (HRC)
A stable matching need not exist:
(r1,r2): (h1,h1) h1:2: r1 r3 r2
r3: h1
Stable matchings can have different sizes

15. Couples in HR
The problem of determining whether a stable matching exists in a given HRC instance is NP-complete, even if each hospital has capacity 1 and:
there are no single residents
[Ng and Hirschberg, 1988; Ronn, 1990]
each couple has a preference list of length ≤2
each hospital has a preference list of length ≤2
[Biró, M and McBride, 2014]
Heuristic (Algorithm C) described in [Biró et al, 2011]
Integer programming models for HRC [Biró, M and McBride, 2014]

16. Scottish Foundation Allocation Scheme
Set of applicants and programmes (residents and hospitals)
Each applicant:
ranked 10 programmes in strict order of preference
had a score in the range
from 2009, two applicants could link their applications
Each programme:
had a capacity indicating its number of posts
had a preference list derived from the above scoring function
so ties were possible

17. Empirical results for SFAS instances (1)
SFAS datasets from the matching runs were modelled and solved [Kwanashie and M, 2014]
no couples; ties exist only on hospitals’ preference lists
sizes (|M|) were compared against those obtained from approximation algorithms and heuristics due to [Irving and M, 2009] (|M|)
Year
Residents
Hospitals
|M|
|M|
SFAS
Time
(sec)
2008
748 52
709
705
75.5
2007
781 53
746
744
736
21.8
2006
759
758
754
745
93.0

18. Empirical results for SFAS instances (2)
For SFAS datasets from the matching runs
approx. 5% of applicants involved in a couple – IP model extended to HRC [Biró, M and McBride, 2014]
maximum stable matching sizes (|M|) were compared against those obtained from heuristics due to [Biró et al, 2011] (|M|)
Year
Residents
Couples
Hospitals
Posts
|M|
|M|
SFAS
Time
(sec)
2012
710 17 52
720
683
681
9.6
2011
736 12 ?
688
684
2010
734 20
735
33.9

19. Open problems
Approximation algorithm for MAX HRT with performance guarantee < 3/2?
consider special cases:
ties on one side only (< 19/13?)
master lists
To make an HRC instance solvable, try to find “perturbation” of the hospital capacities. A solvable instance can be found by:
varying the capacity of each hospital by at most 4
increasing the total number of posts by at most 9
[Nguyen and Vohra, 2014]
is it sufficient to vary the capacity of each hospital by a smaller number?

20. Book on Matching Under Preferences

21. References
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22. References
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23. References
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24. References
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27. Nobel prize in Economic Sciences, 2012