The Stable Roommates Problem with Globally-Ranked Pairs

Abstract

We introduce a restriction of the stable roommates problem in which roommate pairs are ranked globally. In contrast to the unrestricted problem, weakly stable matchings are guaranteed to exist, and additionally, can be found in polynomial time. However, it is still the case that strongly stable matchings may not exist, and so we consider the complexity of finding weakly stable matchings with various desirable properties. In particular, we present a polynomial-time algorithm to find a rank-maximal (weakly stable) matching. This is the first generalization of the algorithm due to Irving et al. [18] to a non-bipartite setting. Also, we prove several hardness results in an even more restricted setting for each of the problems of finding weakly stable matchings that are of maximum size, are egalitarian, have minimum regret, and admit the minimum number of weakly blocking pairs.

Local matching dynamics in social networks

“…We consider convergence to locally stable matchings -matchings that allow no blocking pair of accessible players -for variants of sequential best-response and better-response dynamics, where in each step a blocking pair of accessible players (or local blocking pair) is allowed to deviate to establish their joint edge. It follows directly from results in, e.g., [1,2] mentioned above that our games are potential games, and thus all such dynamics are guaranteed to converge to a locally stable matching. This holds even for more general variants, in which each player can build up to k > 1 matching edges, or each player has access to all players within > 2 hops in the graph.…”

“…For the ordinary correlated stable roommates problem, existence of a stable matching is guaranteed, a stable matching can be computed in polynomial time [1]. Furthermore, these games are potential games and a variety of dynamics converge to a stable matching in polynomial time [2].…”

“…However, if the ordering of moves is chosen uniformly at random at each step, convergence time is exponential [2]. A prominent case with numerous applications [1,4,12,18], in which fast convergence is achieved even by random dynamics, is correlated or weighted stable matching where each matched pair generates a benefit (or edge weight) for the incident players and the preferences of the players are ordered with respect to edge benefit [1,18].…”

“…In this paper, we will for the most part focus on the prominent class of correlated [1,2] (also called acyclic [18]) preferences or weighted matching, where each possible match yields a single benefit value for both incident players, and preference lists are ordered according to edge benefits. For the ordinary correlated stable roommates problem, existence of a stable matching is guaranteed, a stable matching can be computed in polynomial time [1].…”

Popular matchings

“…Just as in the simpler models, a Pareto-optimal matching always exists and a largest Pareto-optimal matching can be found in polynomial time (Abraham and Manlove, 2004). Profile-based optimality concepts were studied by Abraham et al (2008).…”