Abstract
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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References
Abdulkadiroǧlu, A., Sönmez, T.: Random serial dictatorship and the core from random endowments in house allocation problems. Econometrica 66(3), 689–701 (1998)
Abdulkadiroǧlu, A., Sönmez, T.: House allocation with existing tenants. Journal of Economic Theory 88, 233–260 (1999)
Deng, X., Papadimitriou, C., Safra, S.: On the complexity of equilibria. Journal of Computer and System Sciences 67(2), 311–324 (2003)
Fekete, S.P., Skutella, M., Woeginger, G.J.: The complexity of economic equilibria for house allocation markets. Inf. Proc. Lett. 88, 219–223 (2003)
Gabow, H.N., Tarjan, R.E.: Faster scaling algorithms for network problems. SIAM Journal on Computing 18(5), 1013–1036 (1989)
Horton, J.D., Kilakos, K.: Minimum edge dominating sets. SIAM Journal on Discrete Mathematics 6, 375–387 (1993)
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)
Hylland, A., Zeckhauser, R.: The efficient allocation of individuals to positions. Journal of Political Economy 87(2), 293–314 (1979)
Korte, B., Hausmann, D.: An analysis of the greedy heuristic for independence systems. Annals of Discrete Mathematics 2, 65–74 (1978)
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)
Roth, A.E.: Incentive compatibility in a market with indivisible goods. Economics Letters 9, 127–132 (1982)
Roth, A.E., Postlewaite, A.: Weak versus strong domination in a market with indivisible goods. Journal of Mathematical Economics 4, 131–137 (1977)
Roth, A.E., Sotomayor, M.A.O.: Two-sided matching: a study in game-theoretic modeling and analysis. Cambridge University Press, Cambridge (1990)
Shapley, L., Scarf, H.: On cores and indivisibility. Journal of Mathematical Economics 1, 23–37 (1974)
Yuan, Y.: Residence exchange wanted: a stable residence exchange problem. European Journal of Operational Research 90, 536–546 (1996)
Zhou, L.: On a conjecture by Gale about one-sided matching problems. Journal of Economic Theory 52(1), 123–135 (1990)
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Abraham, D.J., Cechlárová, K., Manlove, D.F., Mehlhorn, K. (2004). Pareto Optimality in House Allocation Problems. In: Fleischer, R., Trippen, G. (eds) Algorithms and Computation. ISAAC 2004. Lecture Notes in Computer Science, vol 3341. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30551-4_3
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DOI: https://doi.org/10.1007/978-3-540-30551-4_3
Publisher Name: Springer, Berlin, Heidelberg
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