Chapter

Algorithms and Computation

Volume 3341 of the series Lecture Notes in Computer Science pp 3-15

Pareto Optimality in House Allocation Problems

  • David J. AbrahamAffiliated withLancaster UniversityComputer Science Department, Carnegie-Mellon University
  • , Katarína CechlárováAffiliated withLancaster UniversityInstitute of Mathematics, P.J. Šafárik University in Košice, Faculty of Science
  • , David F. ManloveAffiliated withCarnegie Mellon UniversityDepartment of Computing Science, University of Glasgow
  • , Kurt MehlhornAffiliated withCarnegie Mellon UniversityMax-Planck-Institut für Informatik

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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.