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Stable matchings Indifference Generalized preferences Complexity results Algorithm Pairwise preferences in the stable marriage problem gnes Cseh 1 Attila Juhos 2 1 Hungarian Academy of Sciences 2 Budapest University of Technology and

  1. Stable matchings Indifference Generalized preferences Complexity results Algorithm Pairwise preferences in the stable marriage problem Ágnes Cseh 1 Attila Juhos 2 1 Hungarian Academy of Sciences 2 Budapest University of Technology and Economics Aussois, 10 January 2018 Accepted to STACS’19

  2. Stable matchings Indifference Generalized preferences Complexity results Algorithm Stable matchings

  3. Stable matchings Indifference Generalized preferences Complexity results Algorithm

  4. Stable matchings Indifference Generalized preferences Complexity results Algorithm

  5. Stable matchings Indifference Generalized preferences Complexity results Algorithm 3 2 2 1 3 1 4 1 2 3 2 1 1 2 3 3 2 3 1 2 1 4 2 1

  6. Stable matchings Indifference Generalized preferences Complexity results Algorithm 3 2 2 1 3 1 4 1 2 3 2 1 1 2 3 3 2 3 1 2 1 4 2 1

  7. Stable matchings Indifference Generalized preferences Complexity results Algorithm 3 2 2 1 3 1 4 1 2 3 2 1 1 2 3 3 2 3 1 2 1 4 2 1 Definition Edge mw is blocking if

  8. Stable matchings Indifference Generalized preferences Complexity results Algorithm 3 2 2 1 3 1 4 1 2 3 2 1 1 2 3 3 2 3 1 2 1 4 2 1 Definition Edge mw is blocking if it is not in the matching and

  9. Stable matchings Indifference Generalized preferences Complexity results Algorithm 3 2 2 1 3 1 4 1 2 3 2 1 1 2 3 3 2 3 1 2 1 4 2 1 Definition Edge mw is blocking if it is not in the matching and m prefers w to his wife or he is single and

  10. Stable matchings Indifference Generalized preferences Complexity results Algorithm 3 2 2 1 3 1 4 1 2 3 2 1 1 2 3 3 2 3 1 2 1 4 2 1 Definition Edge mw is blocking if it is not in the matching and m prefers w to his wife or he is single and w prefers m to her husband or she is single.

  11. Stable matchings Indifference Generalized preferences Complexity results Algorithm 3 2 2 1 3 1 4 1 2 3 2 1 1 2 3 3 2 3 1 2 1 4 2 1 Definition Edge mw is blocking if it is not in the matching and m prefers w to his wife or he is single and w prefers m to her husband or she is single. Theorem (Gale, Shapley 1962) A stable matching always exists.

  12. Stable matchings Indifference Generalized preferences Complexity results Algorithm 1 3 3 4 2 1 3 2 2 1 2 1 1 2 3 2 1 1 2 4 1 3 2 2 Definition Edge mw is blocking if it is not in the matching and m prefers w to his wife or he is single and w prefers m to her husband or she is single.

  13. Stable matchings Indifference Generalized preferences Complexity results Algorithm 1 3 3 4 2 1 3 2 2 1 2 1 1 2 3 2 1 1 2 4 1 3 2 2 Definition Edge mw is blocking if it is not in the matching and m prefers w to his wife or he is single and w prefers m to her husband or she is single.

  14. Stable matchings Indifference Generalized preferences Complexity results Algorithm 3 3 4 2 1 3 2 2 2 1 1 2 1 1 2 4 1 3 2 2 Definition Edge mw is blocking if it is not in the matching and m prefers w to his wife or he is single and w prefers m to her husband or she is single.

  15. Stable matchings Indifference Generalized preferences Complexity results Algorithm 3 3 4 2 1 3 2 2 2 1 1 2 1 1 2 4 1 3 2 2 Definition Edge mw is blocking if it is not in the matching and m prefers w to his wife or he is single and w prefers m to her husband or she is single.

  16. Stable matchings Indifference Generalized preferences Complexity results Algorithm 3 3 4 2 1 3 2 2 2 1 2 1 1 2 4 1 2 2 Definition Edge mw is blocking if it is not in the matching and m prefers w to his wife or he is single and w prefers m to her husband or she is single.

  17. Stable matchings Indifference Generalized preferences Complexity results Algorithm 3 3 4 2 1 3 2 2 2 1 2 1 1 2 4 1 2 2 Definition Edge mw is blocking if it is not in the matching and m prefers w to his wife or he is single and w prefers m to her husband or she is single.

  18. Stable matchings Indifference Generalized preferences Complexity results Algorithm 3 3 4 2 1 3 2 2 2 1 2 1 1 2 4 1 2 2 Definition Edge mw is blocking if it is not in the matching and m prefers w to his wife or he is single and w prefers m to her husband or she is single.

  19. Stable matchings Indifference Generalized preferences Complexity results Algorithm 1 3 3 4 2 1 3 2 2 1 2 1 1 2 3 2 1 1 2 4 1 3 2 2 Definition Edge mw is blocking if it is not in the matching and m prefers w to his wife or he is single and w prefers m to her husband or she is single.

  20. Stable matchings Indifference Generalized preferences Complexity results Algorithm Motivation for allowing ties requiring strict lists is often unnatural

  21. Stable matchings Indifference Generalized preferences Complexity results Algorithm Motivation for allowing ties requiring strict lists is often unnatural college admission systems (Hungary, Chile)

  22. Stable matchings Indifference Generalized preferences Complexity results Algorithm Motivation for allowing ties requiring strict lists is often unnatural college admission systems (Hungary, Chile) hospital-resident matching programs (NRMP, SFAS)

  23. Stable matchings Indifference Generalized preferences Complexity results Algorithm Motivation for allowing ties requiring strict lists is often unnatural college admission systems (Hungary, Chile) hospital-resident matching programs (NRMP, SFAS) Preference lists with ties [Irving, ’94] → indifference or incomparability between acceptable partners

  24. Stable matchings Indifference Generalized preferences Complexity results Algorithm Motivation for allowing ties requiring strict lists is often unnatural college admission systems (Hungary, Chile) hospital-resident matching programs (NRMP, SFAS) Preference lists with ties [Irving, ’94] → indifference or incomparability between acceptable partners m m 1 1 2 2 w 1 w 2 w 3 w 4 w 1 w 2 w 3 w 4

  25. Stable matchings Indifference Generalized preferences Complexity results Algorithm Indifference

  26. Stable matchings Indifference Generalized preferences Complexity results Algorithm Definition Question How to define stability if we have ties?

  27. Stable matchings Indifference Generalized preferences Complexity results Algorithm Definition Question How to define stability if we have ties? m w blocking edge wrt WEAK stability

  28. Stable matchings Indifference Generalized preferences Complexity results Algorithm Definition Question How to define stability if we have ties? m m w w blocking edge wrt blocking edge wrt WEAK stability STRONG stability

  29. Stable matchings Indifference Generalized preferences Complexity results Algorithm Definition Question How to define stability if we have ties? m m m w w w blocking edge wrt blocking edge wrt blocking edge wrt WEAK stability STRONG stability SUPER-stability

  30. Stable matchings Indifference Generalized preferences Complexity results Algorithm m 1 m 2 m 3 m 4 1 1 2 1 1 1 2 2 1 3 2 1 1 2 2 1 2 2 2 1 2 1 3 2 w 1 w 2 w 3 w 4 Initial instance.

  31. Stable matchings Indifference Generalized preferences Complexity results Algorithm m 1 m 2 m 3 m 4 1 1 2 1 1 1 2 2 1 3 2 1 1 2 2 1 2 2 2 1 2 1 3 2 w 1 w 2 w 3 w 4 Random matching.

  32. Stable matchings Indifference Generalized preferences Complexity results Algorithm m 1 m 2 m 3 m 4 1 1 2 1 1 1 2 2 1 3 2 1 1 2 2 1 2 2 2 1 2 1 3 2 w 1 w 2 w 3 w 4 Not weakly stable!

  33. Stable matchings Indifference Generalized preferences Complexity results Algorithm m 1 m 2 m 3 m 4 1 1 2 1 1 1 2 2 1 3 2 1 1 2 2 1 2 2 2 1 2 1 3 2 w 1 w 2 w 3 w 4 Weakly stable matching?

  34. Stable matchings Indifference Generalized preferences Complexity results Algorithm m 1 m 2 m 3 m 4 1 1 2 1 1 1 2 2 1 3 2 1 1 2 2 1 2 2 2 1 2 1 3 2 w 1 w 2 w 3 w 4 Not strongly stable!

  35. Stable matchings Indifference Generalized preferences Complexity results Algorithm m 1 m 2 m 3 m 4 1 1 2 1 1 1 2 2 1 3 2 1 1 2 2 1 2 2 2 1 2 1 3 2 w 1 w 2 w 3 w 4 Strongly stable matching.

  36. Stable matchings Indifference Generalized preferences Complexity results Algorithm m 1 m 2 m 3 m 4 1 1 2 1 1 1 2 2 1 3 2 1 1 2 2 1 2 2 2 1 2 1 3 2 w 1 w 2 w 3 w 4 Not super stable!

  37. Stable matchings Indifference Generalized preferences Complexity results Algorithm m 1 m 2 m 3 m 4 1 1 2 1 1 1 2 2 1 3 2 1 1 2 2 1 2 2 2 1 2 1 3 2 w 1 w 2 w 3 w 4 Super stable matching.

  38. Stable matchings Indifference Generalized preferences Complexity results Algorithm Existence Weakly stable matchings always exist. [Irving’94] Strongly and super stable matchings do not. [Irving’94]

  39. Stable matchings Indifference Generalized preferences Complexity results Algorithm Existence Weakly stable matchings always exist. [Irving’94] Strongly and super stable matchings do not. [Irving’94]

  40. Stable matchings Indifference Generalized preferences Complexity results Algorithm Generalized preferences

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