Pareto Optimality in House Allocation Problems

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.

[1]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[2]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[3]  L. Shapley,et al.  On cores and indivisibility , 1974 .

[4]  Andrew Postlewaite,et al.  Weak Versus Strong Domination in a Market with Indivisible Goods , 1977 .

[5]  B. Korte,et al.  An Analysis of the Greedy Heuristic for Independence Systems , 1978 .

[6]  R. Zeckhauser,et al.  The Efficient Allocation of Individuals to Positions , 1979, Journal of Political Economy.

[7]  A. Roth.  Incentive compatibility in a market with indivisible goods , 1982 .

[8]  Robert E. Tarjan,et al.  Faster Scaling Algorithms for Network Problems , 1989, SIAM J. Comput..

[9]  Alvin E. Roth,et al.  Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis , 1990 .

[10]  Lin Zhou.  On a conjecture by gale about one-sided matching problems , 1990 .

[11]  Joseph Douglas Horton,et al.  Minimum Edge Dominating Sets , 1993, SIAM J. Discret. Math..

[12]  Yufei Yuan,et al.  Residence exchange wanted: A stable residence exchange problem , 1996 .

[13]  Atila Abdulkadiroglu,et al.  RANDOM SERIAL DICTATORSHIP AND THE CORE FROM RANDOM ENDOWMENTS IN HOUSE ALLOCATION PROBLEMS , 1998 .

[14]  Atila Abdulkadiroglu,et al.  HOUSE ALLOCATION WITH EXISTING TENANTS , 1999 .

[15]  Xiaotie Deng,et al.  On the complexity of equilibria , 2002, STOC '02.

[16]  Gerhard J. Woeginger,et al.  The complexity of economic equilibria for house allocation markets , 2003, Inf. Process. Lett..

[17]  Xiaotie Deng,et al.  On the complexity of price equilibria , 2003, J. Comput. Syst. Sci..

[18]  Kurt Mehlhorn,et al.  Rank-maximal matchings , 2004, TALG.