An Algorithm for a Super-Stable Roommates Problem

In this paper we describe an efficient algorithm that decides if a stable matching exists for a generalized stable roommates problem, where, instead of linear preferences, agents have partial preference orders on potential partners. Furthermore, we may forbid certain partnerships, that is, we are looking for a matching such that none of the matched pairs is forbidden, and yet, no blocking pair (forbidden or not) exists. To solve the above problem, we generalize the first algorithm for the ordi- nary stable roommates problem

The stable roommates problem with globally-ranked pairs

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, they 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 an algorithm due to [Irving et al. 06] to a nonbipartite setting. Also, we describe 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

An algorithm for a super-stable roommates problem

In this paper, we describe an efficient algorithm that decides if a stable matching exists for a generalized stable roommates problem, where, instead of linear preferences, agents have partial preference orders on potential partners. Furthermore, we may forbid certain partnerships, that is, we are looking for a matching such that none of the matched pairs is forbidden, and yet, no blocking pair (forbidden or not) exists. To solve the above problem, we generalize the first algorithm for the ordinary stable roommates problem

Representing Roomates' Preferences with Symmetric Utilities

In the context of the stable roommates problem, it is shown that acyclicity of preferences is equivalent to the existence of symmetric utility functions, i.e. the utility of agent i when matched with j is the same as j 's utility when matched with i .

"Almost stable" matchings in the Roommates problem

An instance of the classical Stable Roommates problem (SR) need not admit a stable matching. This motivates the problem of finding a matching that is “as stable as possible”, i.e. admits the fewest number of blocking pairs. In this paper we prove that, given an SR instance with n agents, in which all preference lists are complete, the problem of finding a matching with the fewest number of blocking pairs is NP-hard and not approximable within n^{\frac{1}{2}-\varepsilon}, for any \varepsilon>0, unless P=NP. If the preference lists contain ties, we improve this result to n^{1-\varepsilon}. Also, we show that, given an integer K and an SR instance I in which all preference lists are complete, the problem of deciding whether I admits a matching with exactly K blocking pairs is NP-complete. By contrast, if K is constant, we give a polynomial-time algorithm that finds a matching with at most (or exactly) K blocking pairs, or reports that no such matching exists. Finally, we give upper and lower bounds for the minimum number of blocking pairs over all matchings in terms of some properties of a stable partition, given an SR instance I