Stable Roommates Problem with Random Preferences

Matching under preferences is a topic of great practical importance, deep mathematical structure, and elegant algorithmics [Manlove 2013; Gusfield and Irving 1989]. The most famous example is the stable marriage problem, where n men and n women compete with each other in the “marriage market.” Each man ranks all the women according to his individual preferences, and each woman does the same with all men. Everybody wants to get married to someone at the top of his or her list, but mutual attraction is not symmetric and frustration and compromises are unavoidable. A minimum requirement is a matching of men and women such that no man and woman would agree to leave their assigned partners in order to marry each other. Such a matching is called stable since no individual has an icentive to break it. The problem then is to find such a stable matching. The stable marriage problem was introduced David Gale and Lloyd Shapley in 1962 [Gale and Shapley 1962]. In their seminal paper they proved that each instance of the marriage problem has at least one stable solution, and they presented an efficient algorithm to find it. Since then, the Gale-Shapley algorithm has been applied to many real-world problems, not by dating agencies but by central bodies that organize twosided markets like the assignment of students to colleges or residents to hospitals [Roth and Sotomayor 1990]. The salient feature of the stable marriage problem is its bipartite structure: the agents form two groups (men and women), and matchings are only allowed between these groups but not within a group. This is adequate for two-sided markets. But what about one-sided markets, like the formation of cockpit crews from a pool of pilots or the