### 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 α-stable matching: the participants are willing to switch partners only for a (multiplicative) improvement of at least α. We prove that, in general, finding α-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.

Original language | English (US) |
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Pages (from-to) | 219-224 |

Number of pages | 6 |

Journal | Information Processing Letters |

Volume | 109 |

Issue number | 4 |

DOIs | |

State | Published - Jan 31 2009 |

### Keywords

- Algorithms
- Computational geometry
- Consistent preferences
- Graph algorithms
- Stable roommates with ties
- α-Stable matching

### ASJC Scopus subject areas

- Theoretical Computer Science
- Signal Processing
- Information Systems
- Computer Science Applications

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## Cite this

*Information Processing Letters*,

*109*(4), 219-224. https://doi.org/10.1016/j.ipl.2008.10.003