On Geometric Stable Roommates and Minimum-Weight Matching Robustness (Extended Abstract) ∗ Valentin Polishchuk † Esther M. Arkin ‡ Kobus Barnard § Kevin Coogan § Alon Efrat § Joseph S. B. Mitchell ‡ James B. Orlin ¶ Abstract 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. 1 Introduction and Problems Formulation Both the Stable Matching and the Minimum-Weight Matching problems are classical problems in combinatorial optimization. We attempt to add new twists to the problems. Specifically, as a way of getting “consistent” preference lists, we consider representing people in a stable matching problem instance by points in a metric space—as we show, such instances have appealing properties, which distinguish them from the general ones. For the minimum-weight matching, we give algorithms that help to understand how robust the matching is with respect to changes in the weights of the edges of the graph. 1.1 The Stable Roommates Problem The Stable Marriage problem is a true multidisciplinary one: it is well-studied in economics, com- puter science, and combinatorics. The problem and its numerous extensions continue to receive considerable attention, both from the theoretical point of view and for real-world applications [16]. ∗ Preliminary results presented at KyotoCGGT’2007. Full version is available as technical reports [4, 27]. † Helsinki Institute for Information Technology, University of Helsinki. valentin@compgeom.com. ‡ Applied Math and Statistics, Stony Brook University. {estie,jsbm}@ams.sunysb.edu. § Computer Science, the University of Arizona. {kobus,kpcoogan,alon}@cs.arizona.edu ¶ Operations Research Center, Massachusetts Institute of Technology. jorlin@mit.edu. 1