Popularity on the Roommate Diversity Problem

TL;DR

This paper studies the roommate diversity problem, a variant of the stable roommate problem focusing on agents' preferences based on type ratios, and identifies conditions under which popular partitions are computationally feasible or intractable.

Contribution

It proves that for fixed room size two, popular partitions always exist and are computable in polynomial time, while for unrestricted sizes, the problem becomes NP-hard and intractable.

Findings

Popular partitions exist and are polynomial-time computable for room size two.

Existence of popular partitions is NP-hard when coalition size is unrestricted.

Computing a mixed popular partitioning is NP-hard even with dichotomous preferences.

Abstract

A recently introduced restricted variant of the multidimensional stable roommate problem is the roommate diversity problem: each agent belongs to one of two types (e.g., red and blue), and the agents' preferences over the coalitions solely depend on the fraction of agents of their own type among their roommates. There are various notions of stability that defines an optimal partitioning of agents. The notion of popularity has received a lot of attention recently. A partitioning of agents is popular if there does not exist another partitioning in which more agents are better off than worse off. Computing a popular partition in a stable roommate game can be done in polynomial time. When we allow ties the stable roommate problem becomes NP-complete. Determining the existence of a popular solution in the multidimensional stable roommate problem also NP-hard. We show that in the roommate diversity problem with the room size fixed to two, the problem becomes tractable. Particularly, a popular partitioning of agents is guaranteed to exist and can be computed in polynomial time. Additionally a mixed popular partitioning of agents is always guaranteed to exist in any roommate diversity game. By contrast, when there are no restrictions on the coalition size of a roommate diversity game, a popular partitioning may fail to exist and the problem becomes intractable. Our results intractability results are summarized as follows: * Determining the existence of a popular partitioning is co-NP-hard, even if the agents' preferences are trichotomous. * Determining the existence of a strictly popular partitioning is co-NP-hard, even if the agents' preferences are dichotomous. * Computing a mixed popular partitioning of agents in polynomial time is impossible unless P=NP, even if the agents' preferences are dichotomous.