Preference profiles determining the proposals in the Gale–Shapley algorithm for stable matching problems

  • Noriyoshi Sukegawa
  • Yoshitsugu Yamamoto
Original Paper Area 1


Concerning the strategic manipulability of the stable matching produced by the Gale–Shapley algorithm, Kobayashi and Matsui recently considered the existence problem of a preference profile of women, that is, given a preference profile of men, find a preference profile of women that makes the Gale–Shapley algorithm produce the prescribed complete matching of men and women. Reformulating this problem by introducing the set of proposals to be made through the execution of the algorithm, and switching the roles of men and women, we consider the existence problem of a preference profile of men and show that the problem is reduced to a problem of checking if a directed graph is a rooted tree and it is solvable in polynomial time. We also show that the existence problem of preference profiles of both sexes when a set of proposals is given is solvable in polynomial time.


Stable matching Gale–Shapley algorithm Preference profile Strategic manipulability Rooted spanning tree Matroid intersection 

Mathematics Subject Classification

90C27 91A40 91B08 91B68 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berge C.: Graphs and Hypergraphs. North-Holland, Amsterdam (1973)zbMATHGoogle Scholar
  2. 2.
    Fujishige S.: A primal approach to the independent assignment problem. J. Oper. Res. Soc. Jpn. 20, 1–15 (1977)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Gabow H.N., Tarjan R.E.: Efficient algorithms for a family of matroid intersection problems. J. Algorithms 5, 80–131 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Gale D., Shapley L.S.: College admissions and stability of marriage. Am. Math. Monthly 69, 9–15 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Gusfield D., Irving R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Cambridge (1989)zbMATHGoogle Scholar
  6. 6.
    Knuth, D.E.: Mariages stables et leurs relations avec d’autres problèmes combinatoires, Les Presses de l’Universitè de Montrèal (1976). English translation Goldstein, M.: Stable marriage and its relation to other combinatorial problems. In: CRM Proceedings and Lecture Notes, vol. 10. American Mathematical Society, Providence (1997)Google Scholar
  7. 7.
    Kobayashi H., Matsui T.: Successful manipulation in stable marriage model with complete preference lists. IEICE Trans. Inf. Syst. E E92-D, 116–119 (2009)CrossRefGoogle Scholar
  8. 8.
    Kobayashi H., Matsui T.: Cheating strategies for the Gale–Shapley algorithm with complete preference lists. Algorithmica 58, 151–169 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Lawler E.L.: Combinatorial Optimization: Network and Matroids, Holt. Rinehart and Winston, New York (1976)Google Scholar
  10. 10.
    Teo C.-T., Sethuraman J., Tan W.-P.: Gale–Shapley stable marriage problem revisited: strategic issues and applications. Manag. Sci. 47, 1252–1267 (2001)zbMATHCrossRefGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer 2012

Authors and Affiliations

  1. 1.Graduate School of Systems and Information EngineeringUniversity of TsukubaTsukubaJapan
  2. 2.Faculty of Engineering, Information and SystemsUniversity of TsukubaTsukubaJapan

Personalised recommendations