How hard is it to satisfy (almost) all roommates?

TL;DR

This paper investigates the computational complexity of finding stable or nearly stable matchings in the Stable Roommates problem, introducing new fixed-parameter tractability results for minimizing dissatisfaction and hardness results for minimizing blocking pairs.

Contribution

It proves that the Egal Stable Roommates problem is fixed-parameter tractable when parameterized by dissatisfaction, while the Min-Block-Pair Stable Roommates problem is W[1]-hard when parameterized by blocking pairs.

Findings

Egal Stable Roommates is fixed-parameter tractable by dissatisfaction.

Min-Block-Pair Stable Roommates is W[1]-hard by blocking pairs.

Complexity results hold even with preference lists of length at most five.

Abstract

The classic Stable Roommates problem (which is the 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 Egal Stable Roommates, which seeks to find a stable matching with a minimum egalitarian cost {\gamma}, i.e. the sum of the dissatisfaction of the 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-Pair Stable Roommates, which seeks to find a matching with a minimum number {\beta} of blocking pairs. Our main result is that Egal Stable Roommates parameterized by {\gamma} is fixed-parameter tractable, while Min-Block-Pair Stable Roommates parameterized by {\beta} is W[1]-hard, even if the length of each preference list is at most five.