Geometric stable roommates

  • Authors:

  • Esther M. Arkin;Sang Won Bae;Alon Efrat;Kazuya Okamoto;Joseph S. B. Mitchell;Valentin Polishchuk

  • Affiliations:

  • Applied Mathematics and Statistics, Stony Brook University, USA;Computer Science, Korea Advanced Institute of Science and Technology, South Korea;Computer Science, The University of Arizona, USA;Graduate School of Informatics, Kyoto University, Japan;Applied Mathematics and Statistics, Stony Brook University, USA;Helsinki Institute for Information Technology, University of Helsinki and Helsinki University of Technology, Finland

  • Venue:
  • Information Processing Letters
  • Year:
  • 2009

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Abstract

We consider instances of the Stable Roommates problem that arise from geometric representation of participants' preferences: a participant is a point in a metric space, and his preference list is given by the sorted list of distances to the other participants. We show that contrary to the general case, the problem admits a polynomial-time solution even in the case when ties are present in the preference lists. We define the notion of an @a-stable matching: the participants are willing to switch partners only for a (multiplicative) improvement of at least @a. We prove that, in general, finding @a-stable matchings is not easier than finding matchings that are stable in the usual sense. We show that, unlike in the general case, in a three-dimensional geometric stable roommates problem, a 2-stable matching can be found in polynomial time.