el problema del matrimonio estable el problema del matrimonio estable roger z. ríos programa de...

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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