On Geometric Stable Roommates and Minimum-Weight Matching

This paper consists of two parts, both of which address stability of perfect matchings. In the first part we consider instances of the Stable Roommates problem that arise from geometric representation of participants preferences: a participant is a point in Euclidean 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. In the second part we study the “robustness” of the minimum-weight perfect matching. The robustness measures how much the edge weights of a graph are allowed to be distorted before the minimum-weight matching changes. We consider two cases: when the edge weights are changed adversarially and when they are changed at random. We provide algorithms for computing the robustness in both cases.