Dynamic Matching Markets and Voting Paths

  • David J. Abraham
  • Telikepalli Kavitha
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4059)


We consider a matching market, in which the aim is to maintain a popular matching between a set of applicants and a set of posts, where each applicant has a preference list that ranks some subset of acceptable posts. The setting is dynamic: applicants and posts can enter and leave the market, and applicants can also change their preferences arbitrarily. After any change, the current matching may no longer be popular, in which case, we are required to update it. However, our model demands that we can switch from one matching to another only if there is consensus among the applicants to agree to the switch. Hence, we need to update via a voting path, which is a sequence of matchings, each more popular than its predecessor, that ends in a popular matching. In this paper, we show that, as long as some popular matching exists, there is a 2-step voting path from any given matching to some popular matching. Furthermore, given any popular matching, we show how to find a shortest-length such voting path in linear time.


Linear Time Maximum Match Stable Match Preference List Blocking Pair 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abdulkadiroǧlu, A., Sönmez, T.: Random serial dictatorship and the core from random endowments in house allocation problems. Econometrica 66(3), 689–701 (1998)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Abraham, D.J., Cechlárová, K., Manlove, D.F., Mehlhorn, K.: Pareto Optimality in House Allocation Problems. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 3–15. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Abraham, D.J., Irving, R.W., Kavitha, T., Mehlhorn, K.: Popular matchings. In: Proc. of 16th SODA, pp. 424–432 (2005)Google Scholar
  4. 4.
    Diamantoudi, E., Miyagawa, E., Xue, L.: Random paths to stability in the roommate problem. Games and Economic Behavior 48(1), 18–28 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Fekete, S.P., Skutella, M., Woeginger, G.J.: The complexity of economic equilibria for house allocation markets. Information Processing Letters 88, 219–223 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. American Mathematical Monthly 69, 9–15 (1962)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gardenfors, P.: Match Making: assignments based on bilateral preferences. Behavioural Sciences 20, 166–173 (1975)CrossRefGoogle Scholar
  8. 8.
    Gusfield, D., Irving, R.W.: The Stable Marriage Problem: Structure and Algorithms. MIT Press, Cambridge (1989)zbMATHGoogle Scholar
  9. 9.
    Hylland, A., Zeckhauser, R.: The efficient allocation of individuals to positions. Journal of Political Economy 87(2), 293–314 (1979)CrossRefGoogle Scholar
  10. 10.
    Irving, R.W.: An efficient algorithm for the ”stable roommates” problem. Journal of Algorithms 6, 577–596 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Irving, R.W., Kavitha, T., Mehlhorn, K., Michail, D., Paluch, K.: Rank-maximal matchings. In: Proc. of 15th SODA, pp. 68–75 (2004)Google Scholar
  12. 12.
    Knuth, D.E.: Stable marriage and its relation to other combinatorial problems. In: CRM Proceedings and Lecture Notes, vol. 10 (1976)Google Scholar
  13. 13.
    Landau, H.: On dominance relations and the structure of animal societies, III: the condition for secure structure. Bulletin of Math. Biophysics 15(2), 143–148 (1953)CrossRefGoogle Scholar
  14. 14.
    Mahdian, M.: Random popular matchings. In: ACM-EC (to appear, 2006)Google Scholar
  15. 15.
    Mestre, J.: Weighted Popular Matchings. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 715–726. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Netflix DVD Rental: see:
  17. 17.
    Roth, A.E., Postlewaite, A.: Weak versus strong domination in a market with indivisible goods. Journal of Mathematical Economics 4, 131–137 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Roth, A.E., Vande Vate, J.H.: Random paths to stability in two-sided matching. Econometrica 58, 1475–1480 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Tamura, A.: Transformation from arbitrary matchings to stable matchings. Journal of Combinatorial Theory, Series A 62(2), 310–323 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Yuan, Y.: Residence exchange wanted: a stable residence exchange problem. European Journal of Operational Research 90, 536–546 (1996)zbMATHCrossRefGoogle Scholar
  21. 21.
    Zhou, L.: On a conjecture by Gale about one-sided matching problems. Journal of Economic Theory 52(1), 123–135 (1990)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • David J. Abraham
    • 1
  • Telikepalli Kavitha
    • 2
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityUSA
  2. 2.Indian Institute of ScienceBangaloreIndia

Personalised recommendations