# Stable Roommate Problem with Diversity Preferences

IJCAI 2020, pp. 1780-1782, 2020.

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Abstract:

In the multidimensional stable roommate problem, agents have to be allocated to rooms and have preferences over sets of potential roommates. We study the complexity of finding good allocations of agents to rooms under the assumption that agents have diversity preferences (Bredereck, Elkind, Igarashi, AAMAS'19): each agent belongs to one o...More

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Introduction

- They have agreed on the gift registry and the music to be played, but they still need to decide on the seating plan for the wedding reception
- They expect 120 guests, and the reception venue has 20 tables, with each table seating 6 guests.
- Bob’s friend Charlie wonders if the hapless couple may benefit from consulting the literature on the stable roommate problem
- In this problem, the goal is to find a stable assignment of 2n agents into n rooms of size 2, where every agent has a preference relation over her possible roommates.
- For the s-dimensional stable roommate problem, where each room has size s and agents have preferences over all s − 1-subsets of agents as their potential roommates, even the core non-emptiness problem for strict preferences is NP-complete for s ≥ 3 [Huang, 2007, Ng and Hirschberg, 1991]

Highlights

- Alice and Bob are planning their wedding
- While Irving [1985] proved that it is possible to decide in time linear in the size of the input if an instance of the roommate problem with strict preferences admits a core stable outcome, many other algorithmic problems for core and exchange stability are computationally hard [Cechlarovaand Manlove, 2005, Ronn, 1990]
- We investigate the multidimensional stable roommate problem with diversity preferences; we refer to the resulting problem as the roommate diversity problem
- We have proposed the roommate diversity problem, which is an interesting special case of the multidimensional stable roommate problem
- While we have answered many questions that arise in this context, the complexity of deciding whether a roommate diversity problem admits an exchange stable outcome remains open
- By parameterizing our computational problems by the size of the rooms, we showed that diversity preferences are a powerful restriction, as all studied existence problems lie in FPT with respect to this parameter

Conclusion

**CONCLUSIONS AND FUTURE DIRECTIONS**

In this paper, the authors have proposed the roommate diversity problem, which is an interesting special case of the multidimensional stable roommate problem.- While the authors have answered many questions that arise in this context, the complexity of deciding whether a roommate diversity problem admits an exchange stable outcome remains open.
- It is unclear if every instance with dichotomous preferences admits an exchange stable outcome.
- It would be desirable to obtain parameterized algorithms that are combinatorial rather than ILP-based, since ILP-based algorithms tend to be slow in practice

Summary

## Introduction:

They have agreed on the gift registry and the music to be played, but they still need to decide on the seating plan for the wedding reception- They expect 120 guests, and the reception venue has 20 tables, with each table seating 6 guests.
- Bob’s friend Charlie wonders if the hapless couple may benefit from consulting the literature on the stable roommate problem
- In this problem, the goal is to find a stable assignment of 2n agents into n rooms of size 2, where every agent has a preference relation over her possible roommates.
- For the s-dimensional stable roommate problem, where each room has size s and agents have preferences over all s − 1-subsets of agents as their potential roommates, even the core non-emptiness problem for strict preferences is NP-complete for s ≥ 3 [Huang, 2007, Ng and Hirschberg, 1991]
## Conclusion:

**CONCLUSIONS AND FUTURE DIRECTIONS**

In this paper, the authors have proposed the roommate diversity problem, which is an interesting special case of the multidimensional stable roommate problem.- While the authors have answered many questions that arise in this context, the complexity of deciding whether a roommate diversity problem admits an exchange stable outcome remains open.
- It is unclear if every instance with dichotomous preferences admits an exchange stable outcome.
- It would be desirable to obtain parameterized algorithms that are combinatorial rather than ILP-based, since ILP-based algorithms tend to be slow in practice

- Table1: Overview of complexity results for different solution concepts and restrictions on the preference relations. For each solution concept and restriction, we indicate whether every instance satisfying this restriction is guaranteed to admit an outcome with the respective property (Gu.), the complexity of deciding if an instance admits such an outcome (Ex.) and of finding one if it exists (Co.). The number in brackets is the number of the respective theorem. For all solution concepts, the problem of verifying whether a given outcome has the desired property is in P except for Pareto optimality, for which this problem is coNP-complete. We prove that all existence problems in this table are in FPT with respect to the room size

Related work

- The stable roommate problem was proposed by Gale and Shapley [1962] and has been studied extensively since then [Cechlarova, 2002, Huang, 2007, Irving, 1985, Irving and Manlove, 2002, Ronn, 1990]. It can be seen as a special case of hedonic coalition formation [Bogomolnaia and Jackson, 2002], where agents have to split into groups (with no prior constraints on the group sizes) and have preferences over groups that they can be part of; precisely, a stable roommate problem with room size s is a hedonic coalition formation problem where each agent considers all coalitions of size other than s unacceptable.

In contrast to the closely related stable marriage problem [Gale and Shapley, 1962], for the stable roommate problem, it is not guaranteed that the core is not empty. While Irving [1985] proved that it is possible to check in time linear in the input whether a roommate problem admits a core stable outcome if the preferences are restricted to be strict, Ronn [1990] showed that this problem becomes NP-complete if ties in the preference relations are allowed. As in practice a group deviation usually requires some regrouping, Alcalde [1994] initiated the study of local stability notions that do not require reallocating non-deviating agents, by introducing the notion of exchange stability. Subsequently, Cechlarovaand Manlove [2005] proved that it is NP-complete to decide whether an instance of the roommate problem with strict preferences admits an exchange stable outcome. Another concept that is relevant for the roommate problem is Pareto optimality [Abraham and Manlove, 2004, Cseh et al, 2019, Sotomayor, 2011]. Morrill [2010] proved that for room size two, it is possible to check if a given outcome is Pareto optimal, and to find a Pareto improvement if it exists, in time cubic in the number of agents. This implies that a Pareto optimal outcome can be found in polynomial time. Researchers have also considered various notions of fairness in the context of the roommate problem [Abdulkadiroglu and Sonmez, 2003, Aziz and Klaus, 2019]. One such notion is envy-freeness: an outcome is said to be envy-free if no agent wants to take the place of another agent.

Funding

- This work was supported by a DFG project “MaMu”, NI 369/19 (Boehmer) and by an ERC Starting Grant ACCORD under Grant Agreement 639945 (Elkind)

Reference

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