Pareto Optimality in House Allocation Problems

  • David J. Abraham
  • Katarína Cechlárová
  • David F. Manlove
  • Kurt Mehlhorn
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3341)


We study Pareto optimal matchings in the context of house allocation problems. We present an \(O(\sqrt{n}m)\) algorithm, based on Gale’s Top Trading Cycles Method, for finding a maximum cardinality Pareto optimal matching, where n is the number of agents and m is the total length of the preference lists. By contrast, we show that the problem of finding a minimum cardinality Pareto optimal matching is NP-hard, though approximable within a factor of 2. We then show that there exist Pareto optimal matchings of all sizes between a minimum and maximum cardinality Pareto optimal matching. Finally, we introduce the concept of a signature, which allows us to give a characterization, checkable in linear time, of instances that admit a unique Pareto optimal matching.


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  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)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abdulkadiroǧlu, A., Sönmez, T.: House allocation with existing tenants. Journal of Economic Theory 88, 233–260 (1999)CrossRefGoogle Scholar
  3. 3.
    Deng, X., Papadimitriou, C., Safra, S.: On the complexity of equilibria. Journal of Computer and System Sciences 67(2), 311–324 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Fekete, S.P., Skutella, M., Woeginger, G.J.: The complexity of economic equilibria for house allocation markets. Inf. Proc. Lett. 88, 219–223 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Gabow, H.N., Tarjan, R.E.: Faster scaling algorithms for network problems. SIAM Journal on Computing 18(5), 1013–1036 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Horton, J.D., Kilakos, K.: Minimum edge dominating sets. SIAM Journal on Discrete Mathematics 6, 375–387 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Hopcroft, J.E., Karp, R.M.: A n 5/2 Algorithm for Maximum Matchings in Bipartite Graphs. SIAM Journal on Computing 2, 225–231 (1973)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Hylland, A., Zeckhauser, R.: The efficient allocation of individuals to positions. Journal of Political Economy 87(2), 293–314 (1979)CrossRefGoogle Scholar
  9. 9.
    Korte, B., Hausmann, D.: An analysis of the greedy heuristic for independence systems. Annals of Discrete Mathematics 2, 65–74 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Irving, R.W., Kavitha, T., Mehlhorn, K., Michail, D., Paluch, K.: Rank-maximal matchings. In: Proceedings of SODA 2004, pp. 68–75. ACM-SIAM, New York (2004)Google Scholar
  11. 11.
    Roth, A.E.: Incentive compatibility in a market with indivisible goods. Economics Letters 9, 127–132 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Roth, A.E., Postlewaite, A.: Weak versus strong domination in a market with indivisible goods. Journal of Mathematical Economics 4, 131–137 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Roth, A.E., Sotomayor, M.A.O.: Two-sided matching: a study in game-theoretic modeling and analysis. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  14. 14.
    Shapley, L., Scarf, H.: On cores and indivisibility. Journal of Mathematical Economics 1, 23–37 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Yuan, Y.: Residence exchange wanted: a stable residence exchange problem. European Journal of Operational Research 90, 536–546 (1996)zbMATHCrossRefGoogle Scholar
  16. 16.
    Zhou, L.: On a conjecture by Gale about one-sided matching problems. Journal of Economic Theory 52(1), 123–135 (1990)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • David J. Abraham
    • 1
  • Katarína Cechlárová
    • 2
  • David F. Manlove
    • 3
  • Kurt Mehlhorn
    • 4
  1. 1.Computer Science DepartmentCarnegie-Mellon UniversityPittsburghUSA
  2. 2.Institute of MathematicsP.J. Šafárik University in Košice, Faculty of ScienceKošiceSlovakia
  3. 3.Department of Computing ScienceUniversity of GlasgowGlasgowUK
  4. 4.Max-Planck-Institut für InformatikSaarbrückenGermany

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