Geometric stable roommates

We consider instances of the Stable Roommates problem that arise from geometric representation of participants' preferences: a participant is a point ill 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 alpha-stable matching: the participants are willing to switch partners only for a (multiplicative) improvement of at least alpha. We prove that, in general, finding alpha-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. (C) 2008 Elsevier B.V. All rights reserved.