Presentation is loading. Please wait.

Presentation is loading. Please wait.

El Problema del Matrimonio Estable El Problema del Matrimonio Estable Roger Z. Ríos Programa de Posgrado en Ing. de Sistemas Facultad de Ing. Mecánica.

Similar presentations


Presentation on theme: "El Problema del Matrimonio Estable El Problema del Matrimonio Estable Roger Z. Ríos Programa de Posgrado en Ing. de Sistemas Facultad de Ing. Mecánica."— Presentation transcript:

1 El Problema del Matrimonio Estable El Problema del Matrimonio Estable Roger Z. Ríos Programa de Posgrado en Ing. de Sistemas Facultad de Ing. Mecánica y Eléctrica Universidad Autónoma de Nuevo León Seminario de Clase “Optimización de Flujo en Redes” PISIS – FIME - UANL Cd. Universitaria24 Agosto 2007

2 Agenda  Problem definition  Solution algorithm  Variations and extensions  Real-world application

3 Problem Definition  2 disjoint sets of size n (women, men) Men’s preferencesWomen’s preferences Matching M ={(1,1), (2,3), (3,2), (4,4)} Blocking pair (4,1)

4 Problem Definition  2 disjoint sets of size n (women, men) Men’s preferencesWomen’s preferences Matching M ={(1,1), (2,3), (3,2), (4,4)} Blocking pair (4,1)

5 Problem Definition  2 disjoint sets of size n (women, men) Men’s preferencesWomen’s preferences Matching M ={(1,1), (2,3), (3,2), (4,4)} Blocking pair (4,1)

6 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=p M (m), m=p M (w)  (w,m) block a match M if w and m are not partners, but w prefers m to p M (w) and m prefers w to p M (m)  A match for which there is at least one blocking pair is unstable

7 Solution: Properties  Stability checking (for a given matching M) is easy  Stable matching existence (not obvious) due to Gale and Shapley (1962)

8 Solution: Stability for (m:=1 to n) do for (each w such that m prefers w to p M (m)) do if (w prefers m to p M (w)) then report matching unstable stop endif Report matching stable for (m:=1 to n) do for (each w such that m prefers w to p M (m)) do if (w prefers m to p M (w)) then report matching unstable stop endif Report matching stable Simple stability checking algorithm O(n 2 )

9 Solution assign each person to be free while (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 free while (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

10 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(n 2 )  No blocking pairs  If m prefers w to p M (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(n 2 )  No blocking pairs  If m prefers w to p M (m), w must have rejected m at some point for a man she prefers better

11 Solution: Example Men’s preferencesWomen’s preferences Man 1 proposes to woman 4 (accepted) Partial matching M ={(4,1)}

12 Solution: Example 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)}

13 Solution: Example 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)}

14 Solution: Example 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)}

15 Solution: Example 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)}

16 Solution: Example 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)}

17 Solution: Example 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)}

18 Solution: Example 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)

19 Solution: Example 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)

20 Extensions/Variations  Stable Marriage Problem  Sets of unequal size  Unnacceptable partners  Indifference  Stable Roommate Problem  Stable Resident/Hospital Problem

21 Real-World Application Stable Resident/Hospital Problem (SRHP) Largest and best-known application of SMP Used by the National Resident Matching Program (NRMP)

22 SRHP Application  R (residents), H (hospitals), q i := # 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

23 SRHP Application History 1 st dilemma: Early proposals 2 nd dilemma: Tight acceptance start -2

24 SRHP Application  Solution:  Transformation into a SMP  Specialized algorithm SRHP Instance Residents r 1  (h 1, h 2, h 3, …) … Hospitals h 1  (r 1, r 2, r 3, …), q 1 =3 … SMP Instance Residents r 1  (h 11, h 12, h 13, h 2, h 3, …) … Hospitals h 11  (r 1, r 2, r 3, …) h 12  (r 1, r 2, r 3, …) h 13  (r 1, r 2, r 3, …) …

25 Related Hard Problems  Finding all different stable matchings  Maximum number of stable matchings  Parallelization

26 Conclusions Stable matchings Gale-Shapley algorithm Math/Computer Science/Economics Scientific support to decision-making

27 Questions? Acknowledgements: Racing - Barça El Sardinero This Sunday 12:00 CDT


Download ppt "El Problema del Matrimonio Estable El Problema del Matrimonio Estable Roger Z. Ríos Programa de Posgrado en Ing. de Sistemas Facultad de Ing. Mecánica."

Similar presentations


Ads by Google