Pareto optimality in house allocation problems

  • Authors:

  • David J. Abraham;Katarína Cechlárová;David F. Manlove;Kurt Mehlhorn

  • Affiliations:

  • Computer Science Department, Carnegie-Mellon University, Pittsburgh, PA;Institute of Mathematics, P.J Šafárik University in Košice, Faculty of Science, Košice, Slovakia;Department of Computing Science, University of Glasgow, Glasgow, UK;Max-Planck-Institut für Informatik, Saarbrücken, Germany

  • Venue:
  • ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
  • Year:
  • 2004

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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.