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Stable Matchings a.k.a. the Stable Marriage Problem

Stable Matchings a.k.a. the Stable Marriage Problem. Samia Qader 252a-az CSC 254. The Stable Marriage Problem . Overview Real - life application The stable marriage problem Theorems Termination Correctness of algorithm Open problems.

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Stable Matchings a.k.a. the Stable Marriage Problem

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  1. Stable Matchingsa.k.a. the Stable Marriage Problem Samia Qader 252a-az CSC 254

  2. The Stable Marriage Problem Overview • Real - life application • The stable marriage problem • Theorems • Termination • Correctness of algorithm • Open problems

  3. Real-life application • Residency: after medical school • “The Match” was developed to assign prospective students to residency programs • The students submit a rank-order list (ROL) to the National Residency Match Program (NRMP) • Hospitals rank-order their students • NRMP finds a match.

  4. Simple Problem • We’ve got n boys and n girls. • Each boy submits an ROL of all n girls and the girls do the same • A matching is unstable if a boy A and a girl a, not married to each other, mutually prefer each other to their spouses. • A marriage is a bijection between the boys and girls

  5. Men Preference A c b d a B b a c d C c a d b D c a d b Women Preference a A B D C b C A D B c C B D A d B A C D Example 1: 4 boys (A,B,C,D) married to 4 girls (a,b,c,d) Matching (Aa, Bb, Cc, Dd) Is it stable? Look at Aa and Bb: A(4) + a(1) = 5 B(1) + b(4) = 5 total = 10

  6. Men Preference A c b d a B b a c d C c a d b D c a d b Women Preference a A B D C b C A D B c C B D A d B A C D Ex 1: 4 boys (A,B,C,D) married to 4 girls (a,b,c,d) Check Ab and Ba: A(2) + a(2) = 4 B(2) + b(2) = 4 total = 8

  7. Marriage Algorithm – in words • The boys are all single and free to propose

  8. Marriage Algorithm – in words • The girls are all engaged temporarily to  (the ugly man) Ugly man

  9. Marriage Algorithm – in words • The boys propose to the best girl on their list (who hasn’t rejected them yet). The girls accept - tentatively. • If the girl gets a better offer from a boy that she prefers to her fiancé, she will break off the engagement. • STOP when all the boys are engaged

  10. ROL

  11. Results 3 happy couples:

  12. ROL Brown man wants blonde girl – But she doesn’t want him

  13. ROL Scratch off blonde from brown’s list

  14. New Results 3 different couples:

  15. More detail…. n= no.of men & women k = no. of couples formed X = man suitor x = woman  = ugly man while (k<n) do begin X = (k+1)st man; while (X ! = ) do { x= best remaining choice on X’s list if (x prefers X to her fiancé) { engage X and x X = former fiancé of } // end if }// end while if (X != ) withdraw x from X’s list end k = k+1; } // end while

  16. Some Theorems • The order in which the boys propose does not matter • The final matching is stable • The boys get to marry the best girl the could possibly get in a stable marriage

  17. Termination: the algorithm has to end • In what situation would the algorithm run forever - if a boy B keeps getting rejected by all girls • This can never happen: a boy in only rejected by all girls if all the girls are engaged to someone better Impossible - since there are n girls and n-1 boys that are better than B

  18. Correctness of the Algorithm • Is the final marriage stable? • Proof by contradiction: • There exists some pair (b,g) that mutually prefer each other to whomever they are matched to. • Since b proposed in the order of his ROL, he must have proposed to g already

  19. Correctness continued... • But if g is not engaged to b, she must have rejected him • g would only reject b if she is already engaged to someone she prefers • Therefore, g’s fiancé must be ranked more highly than b • CONTRADICTION

  20. Boy Optimality • The marriage returned by this algorithm assigns every boy to his favorite stable partner. • Consider matching each girl to her least favorite choice. The matching produced is stable and is the same as the matching returned by the algorithm described.

  21. Residency Problem • Who gets to play the boy? • Answer : the hospitals

  22. Bipartite matching • In a complete bipartite graph, each node would have an ROL of the nodes that it prefers. 1 1 a 2 2 b 3 3 c 4 4 d

  23. Open Problems… • Maximum number of stable matches grows exponentially with n. What is the instance of size n with the largest possible number of stable matchings? • Sex-equal matching: where the sum of the male scores is equal to the sum of the female scores. • Is it possible to test the stability of every matching in a set of matchings faster (significantly) than by checking each of the matchings from scratch?

  24. References • Asratian A.S, Denley T.M.J, Bipartite Graphs and their Applications, Cambridge University Press, N.Y. (1998). • Gusfield, D., Irving R.W., The Stable Marriage Problem :Structure and Algorithms, The MIT Press, Cambridge (1989). • Knuth D.E., Stable Marriage and its Relation to Other Combinatorial Problems, American Mathematical Society, Providence (1997). • Littman M.L., http://www.cs.duke.edu/education/courses/cps130/fall98 lectures/lect13/lect13.html, (1998). • Lovasz L., Plummer M.D., Matching Theory, North-Holland, North- Holland (1986).

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