El Problema del Matrimonio EstableEl Problema del Matrimonio Estable

Roger Z. Ríos

Programa de Posgrado en Ing. de SistemasFacultad de Ing. Mecánica y EléctricaUniversidad Autónoma de Nuevo León

http://yalma.fime.uanl.mx/~pisis

Seminario de Clase “Optimización de Flujo en Redes”PISIS – FIME - UANL

Cd. Universitaria 24 Agosto 2007

Agenda

Problem definition

Solution algorithm

Variations and extensions

Real-world application

Problem Definition

2 disjoint sets of size n (women, men)

1 2 1 4 3

2 4 3 1 2

3 1 4 3 2

4 2 1 4 3

1 2 4 1 3

2 3 1 4 2

3 2 3 1 4

4 4 1 3 2

Men’s preferencesWomen’s preferences

Matching M ={(1,1), (2,3), (3,2), (4,4)}

Blocking pair (4,1)

Problem Definition

2 disjoint sets of size n (women, men)

1 2 1 4 3

2 4 3 1 2

3 1 4 3 2

4 2 1 4 3

1 2 4 1 3

2 3 1 4 2

3 2 3 1 4

4 4 1 3 2

Men’s preferencesWomen’s preferences

Matching M ={(1,1), (2,3), (3,2), (4,4)}

Blocking pair (4,1)

Problem Definition

2 disjoint sets of size n (women, men)

1 2 1 4 3

2 4 3 1 2

3 1 4 3 2

4 2 1 4 3

1 2 4 1 3

2 3 1 4 2

3 2 3 1 4

4 4 1 3 2

Men’s preferencesWomen’s preferences

Matching M ={(1,1), (2,3), (3,2), (4,4)}

Blocking pair (4,1)

Problem Definition (SMP)

Instance of size n (n women, n men) and strictly ordered preference list

Matching: 1-1 correspondence between men and women

If woman w and man m matched in M w and m are partners, w=pM(m), m=pM(w)

(w,m) block a match M if w and m are not partners, but w prefers m to pM(w) and m prefers w to pM(m)

A match for which there is at least one blocking pair is unstable

Solution: Properties

Stability checking (for a given matching M) is easy

Stable matching existence (not obvious) due to Gale and Shapley (1962)

Solution: Stability

for (m:=1 to n) do for (each w such that m prefers w to pM(m))

do if (w prefers m to pM(w)) then

report matching unstable stop endifReport matching stable

for (m:=1 to n) do for (each w such that m prefers w to pM(m))

do if (w prefers m to pM(w)) then

report matching unstable stop endifReport matching stable

Simple stability checking algorithmO(n2)

Solution

assign each person to be freewhile (some man m is free) do { w:=first woman on m’s list to whom m has not yet

proposed if (w is free) then assign m and w to be engaged {to each other} else if (w prefers m to her fiance m’) then assign m and w to be engaged and m’ to be

free else w rejects m {and m remains free}} end while

assign each person to be freewhile (some man m is free) do { w:=first woman on m’s list to whom m has not yet

proposed if (w is free) then assign m and w to be engaged {to each other} else if (w prefers m to her fiance m’) then assign m and w to be engaged and m’ to be

free else w rejects m {and m remains free}} end while

Basic Gale-Shapley man-oriented algorithm

Solution: Proof

Theorem: For any given SMP instance the G-S algorithm terminates with a stable matching

Proof: No man can be rejected by all woman Termination in O(n2) No blocking pairs

If m prefers w to pM(m), w must have rejected m at some point for a man she prefers better

Theorem: For any given SMP instance the G-S algorithm terminates with a stable matching

Proof: No man can be rejected by all woman Termination in O(n2) No blocking pairs

If m prefers w to pM(m), w must have rejected m at some point for a man she prefers better

Solution: Example

1 4 1 3 2

2 1 3 2 4

3 1 2 3 4

4 4 1 3 2

1 4 1 2 3

2 2 3 1 4

3 2 4 3 1

4 3 1 4 2

Men’s preferencesWomen’s preferences

Man 1 proposes to woman 4 (accepted)Partial matching M ={(4,1)}

Solution: Example

1 4 1 3 2

2 1 3 2 4

3 1 2 3 4

4 4 1 3 2

1 4 1 2 3

2 2 3 1 4

3 2 4 3 1

4 3 1 4 2

Men’s preferencesWomen’s preferences

Man 1 proposes to woman 4 (accepted)Partial matching M ={(4,1)}

Man 2 proposes to woman 2 (accepted)Partial matching M ={(2,2), (4,1)}

Solution: Example

1 4 1 3 2

2 1 3 2 4

3 1 2 3 4

4 4 1 3 2

1 4 1 2 3

2 2 3 1 4

3 2 4 3 1

4 3 1 4 2

Men’s preferencesWomen’s preferences

Man 1 proposes to woman 4 (accepted)Partial matching M ={(4,1)}

Man 2 proposes to woman 2 (accepted)Partial matching M ={(2,2), (4,1)}

Solution: Example

1 4 1 3 2

2 1 3 2 4

3 1 2 3 4

4 4 1 3 2

1 4 1 2 3

2 2 3 1 4

3 2 4 3 1

4 3 1 4 2

Men’s preferencesWomen’s preferences

Man 3 proposes to woman 2(accepted and women 2 rejects man 2)Partial matching M ={(2,3), (4,1)}

Solution: Example

1 4 1 3 2

2 1 3 2 4

3 1 2 3 4

4 4 1 3 2

1 4 1 2 3

2 2 3 1 4

3 2 4 3 1

4 3 1 4 2

Men’s preferencesWomen’s preferences

Man 3 proposes to woman 2(accepted and women 2 rejects man 2)Partial matching M ={(2,3), (4,1)}

Man 2 proposes to woman 3 (accepted)Partial matching M ={(2,3), (3,2), (4,1)}

Solution: Example

1 4 1 3 2

2 1 3 2 4

3 1 2 3 4

4 4 1 3 2

1 4 1 2 3

2 2 3 1 4

3 2 4 3 1

4 3 1 4 2

Men’s preferencesWomen’s preferences

Man 3 proposes to woman 2(accepted and women 2 rejects man 2)Partial matching M ={(2,3), (4,1)}

Man 2 proposes to woman 3 (accepted)Partial matching M ={(2,3), (3,2), (4,1)}

Solution: Example

1 4 1 3 2

2 1 3 2 4

3 1 2 3 4

4 4 1 3 2

1 4 1 2 3

2 2 3 1 4

3 2 4 3 1

4 3 1 4 2

Men’s preferencesWomen’s preferences

Man 4 proposes to woman 3(rejected, woman 3 prefers man 2)Partial matching M ={(2,3), (3,2), (4,1)}

Solution: Example

1 4 1 3 2

2 1 3 2 4

3 1 2 3 4

4 4 1 3 2

1 4 1 2 3

2 2 3 1 4

3 2 4 3 1

4 3 1 4 2

Men’s preferencesWomen’s preferences

Man 4 proposes to woman 3(rejected, woman 3 prefers man 2)Partial matching M ={(2,3), (3,2), (4,1)}

Man 4 proposes to woman 1 (accepted)Final Matching M ={(1,4), (2,3), (3,2), (4,1)}(Stable matching)

Solution: Example

1 4 1 3 2

2 1 3 2 4

3 1 2 3 4

4 4 1 3 2

1 4 1 2 3

2 2 3 1 4

3 2 4 3 1

4 3 1 4 2

Men’s preferencesWomen’s preferences

Man 4 proposes to woman 3(rejected, woman 3 prefers man 2)Partial matching M ={(2,3), (3,2), (4,1)}

Man 4 proposes to woman 1 (accepted)Final Matching M ={(1,4), (2,3), (3,2), (4,1)}(Stable matching)

Extensions/Variations

Stable Marriage Problem Sets of unequal size Unnacceptable partners Indifference

Stable Roommate Problem

Stable Resident/Hospital Problem

Real-World Application

Stable Resident/Hospital Problem (SRHP)

Largest and best-known application of SMP

Used by the National Resident Matching Program

(NRMP)

SRHP Application

R (residents), H (hospitals), qi:= # of available spots in hospital i

(r,h) is a blocking pair if r prefers h to his/her current hospital and h prefers r to at least one of its assigned residents

Solution: Transformation into a SMP Specialized algorithm

SRHP Application

History 1st dilemma: Early proposals

2nd dilemma: Tight acceptance

start

start

-1

-2

SRHP Application

Solution: Transformation into a SMP Specialized algorithm

SRHP InstanceResidents r1 (h1, h2, h3, …)

…

Hospitals h1 (r1, r2, r3, …), q1=3

…

SMP InstanceResidents r1 (h11, h12, h13, h2, h3, …)

…

Hospitals h11 (r1, r2, r3, …)

h12 (r1, r2, r3, …)

h13 (r1, r2, r3, …)

…

Related Hard Problems

Finding all different stable matchings

Maximum number of stable matchings

Parallelization

Conclusions

Stable matchings

Gale-Shapley algorithm

Math/Computer Science/Economics

Scientific support to decision-making

Questions?

http://yalma.fime.uanl.mx/~roger

roger@yalma.fime.uanl.mx

AcknowledgementAcknowledgements:s:

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