Jointly Stable Matchings

Authors Shuichi Miyazaki, Kazuya Okamoto

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Shuichi Miyazaki
Kazuya Okamoto

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Shuichi Miyazaki and Kazuya Okamoto. Jointly Stable Matchings. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


In the stable marriage problem, we are given a set of men, a set of women, and each person's preference list. Our task is to find a stable matching, that is, a matching admitting no unmatched (man, woman)-pair each of which improves the situation by being matched together. It is known that any instance admits at least one stable matching. In this paper, we consider a natural extension where k (>= 2) sets of preference lists L_i (1 <= i <= k) over the same set of people are given, and the aim is to find a jointly stable matching, a matching that is stable with respect to all L_i. We show that the decision problem is NP-complete already for k=2, even if each person's preference list is of length at most four, while it is solvable in linear time for any k if each man's preference list is of length at most two (women's lists can be of unbounded length). We also show that if each woman's preference lists are same in all L_i, then the problem can be solved in linear time.
  • stable marriage problem
  • stable matching
  • NP-completeness
  • linear time algorithm


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  1. Bengt Aspvall, Michael F. Plass, and Robert Endre Tarjan. A linear-time algorithm for testing the truth of certain quantified boolean formulas. Inf. Process. Lett., 8(3):121-123, 1979. URL:
  2. Shimon Even, Alon Itai, and Adi Shamir. On the complexity of timetable and multicommodity flow problems. SIAM J. Comput., 5(4):691-703, 1976. URL:
  3. David Gale and Lloyd S. Shapley. College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9-15, 1962. URL:
  4. David Gale and Marilda Sotomayor. Some remarks on the stable matching problem. Discrete Applied Mathematics, 11(3):223-232, 1985. URL:
  5. Dan Gusfield and Robert W. Irving. The Stable marriage problem - structure and algorithms. Foundations of computing series. MIT Press, 1989. Google Scholar
  6. Robert W. Irving. Stable marriage and indifference. Discrete Applied Mathematics, 48(3):261-272, 1994. URL:
  7. Robert W. Irving and Paul Leather. The complexity of counting stable marriages. SIAM J. Comput., 15(3):655-667, 1986. URL:
  8. David F. Manlove. Stable marriage with ties and unacceptable partners. Technical Report TR-1999-29, University of Glasgow, Department of Computing Science, 1999. Google Scholar
  9. David F. Manlove. The structure of stable marriage with indifference. Discrete Applied Mathematics, 122(1-3):167-181, 2002. URL:
  10. David F. Manlove. Algorithmics of Matching Under Preferences, volume 2 of Series on Theoretical Computer Science. WorldScientific, 2013. URL:
  11. Jay Sethuraman and Chung-Piaw Teo. A polynomial-time algorithm for the bistable roommates problem. J. Comput. Syst. Sci., 63(3):486-497, 2001. URL:
  12. Boris Spieker. The set of super-stable marriages forms a distributive lattice. Discrete Applied Mathematics, 58(1):79-84, 1995. URL:
  13. Edward G. Thurber. Concerning the maximum number of stable matchings in the stable marriage problem. Discrete Mathematics, 248(1-3):195-219, 2002. URL:
  14. Bob P. Weems. Bistable versions of the marriages and roommates problems. J. Comput. Syst. Sci., 59(3):504-520, 1999. URL: