How hard is it to satisfy (almost) all roommates?
The classical Stable Roommates problem (which is a non-bipartite generalization of the well-known Stable Marriage problem) asks whether there is a stable matching for a given set of agents (i.e. a partitioning of the agents into disjoint pairs such that no two agents induce a blocking pair). Herein, each agent has a preference list denoting who it prefers to have as a partner, and two agents are blocking if they prefer to be with each other rather than with their assigned partners. Since stable matchings may not be unique, we study an NP-hard optimization variant of Stable Roommates, called Egalitarian Stable Roommates, which seeks to find a stable matching with a minimum egalitarian cost γ, (i.e. the sum of the dissatisfaction of the matched agents is minimum). The dissatisfaction of an agent is the number of agents that this agent prefers over its partner if it is matched; otherwise it is the length of its preference list. We also study almost stable matchings, called Min-Block-Pairs Stable Roommates, which seeks to find a matching with a minimum number β of blocking pairs. Our main result is that Egalitarian Stable Roommates parameterized by γ is fixed-parameter tractable while Min-Block-Pairs Stable Roommates parameterized by β is W-hard.READ FULL TEXT