Geometric stable roommates

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 sorted distances to the other participants. We observe that, unlike in the general case, if there are no ties in the preference lists, there always exists a unique stable matching; a simple greedy algorithm finds the matching efficiently. We show that, also contrary to the general case, the problem admits polynomial-time solution even in the case when ties are present in the preference lists. We define the notion of α-stable matching: the participants are willing to switch partners only for the improvement of at least α. We prove that in general, finding α-stable matchings is not easier than finding matchings, 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.