Strong Stability in the Hospitals/Residents

Problem

Rob Irving, David Manlove and Sandy

Scott

University of GlasgowDepartment of Computing Science

Supported by EPSRC grant GR/R84597/01,Nuffield Foundation award NUF-NAL-02, and RSE / SEETLLD Personal Research Fellowship

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Hospitals/Residentsproblem (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

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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:

1. (r,h)M implies that r finds h acceptable2. Each resident r is assigned at most one hospital

3. Each hospital h is assigned at most ph residents

Hospitals/Residentsproblem (HR): Definition

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An instance of HR

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

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A matching in HR

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

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Stable matchings in HR

Matching M is stable if M admits no blocking pair

(r,h) is a blocking pair of matching M if:

1. r finds h acceptable and

2. either r is unmatched in M or r prefers h to his assigned hospital in

M and3. either h is undersubscribed in M

or h prefers r to its worst resident assigned in M

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A blocking pair

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

(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

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A stable matching

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

Example shows that, in a stable matching,

• one or more residents may be unmatched• one or more hospitals may be undersubscribed

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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

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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

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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

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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)

An instance of HRT

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Weak stability

Matching M is weakly stable if M admits no blocking pair

(r,h) is a blocking pair of matching M if:

1. r finds h acceptable2. either r is unmatched in M or r

strictly prefers h to his assigned hospital in M3. 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

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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

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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

A second weakly stable matching in HRT

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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

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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

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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

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Super-stability

Matching M is super-stable if M admits no blocking pair

(r,h) is a blocking pair of matching M if:

1. r finds h acceptable2. either r is unmatched in M

or r strictly prefers h to his assigned hospital in M or r is indifferent between them

3. 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

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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)

Super-stable matchings in HRT

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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

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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

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A blocking pair

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

r4 and h2 form a blocking pair

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Another blocking pair

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

r1 and h3 form a blocking pair

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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

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The algorithm in brief

repeatprovisionally assign all free residents to hospitals at head of listform reduced provisional assignment graphfind critical set of residents and make corresponding deletions

until critical set is emptyform a feasible matchingcheck if feasible matching is strongly stable

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An instance of HRT

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

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A provisional assignmentand 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

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A deletion

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

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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

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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

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Provisional assignment graph with one bound resident

r2

r3

r4

r5

r1h1:(3)

h2:(2)

h3:(1)

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Removing a bound resident

r2

r3

r4

r5

r1h1:(3)

h2:(1)

h3:(1)

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Reduced provisional assignment graph

r3

r4

r5

r1

h2:(1)

h3:(1)

35

r3

r4

r5

r1

h2:(1)

h3:(1)

Critical set

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Critical set

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

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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

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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

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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

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Final provisional assignment graph with 4 bound residents

r2

r3

r4

r5

r1h1:(3)

h2:(2)

h3:(1)

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Removing a bound resident

r2

r3

r4

r5

r1h1:(2)

h2:(2)

h3:(1)

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Removing a second bound resident

r2

r3

r4

r5

r1h1:(2)

h2:(1)

h3:(1)

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Removing a third bound resident

r2

r3

r4

r5

r1h1:(2)

h2:(0)

h3:(1)

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Removing a bound resident with an additional provisional assignment

r2

r3

r4

r5

r1h1:(1)

h2:(0)

h3:(1)

45

Final reduced provisional assignment graph

r4h3:(1)

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A feasible matching

This consists of:

1. the bound (resident,hospital) pairs

— (r1,h1), (r2,h2), (r3,h2), (r5,h1)

unioned together with:

2. perfect matching in the final reduced provisional assignment graph

— (r4,h3)

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A strongly stable matching

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

(r1,h1), (r2,h2), (r3,h2), (r4,h3), (r5,h1)

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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 = ;

The algorithm in full (1)

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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;

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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

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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

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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