Abstract
The classical Stable Roommates problem is to decide whether there exists a matching of an even number of agents such that no two agents which are not matched to each other would prefer to be with each other rather than with their respectively assigned partners. We investigate Stable Roommates with complete (i.e., every agent can be matched with any other agent) or incomplete preferences, with ties (i.e., two agents are considered of equal value to some agent) or without ties. It is known that in general allowing ties makes the problem NP-complete. We provide algorithms for Stable Roommates that are, compared to those in the literature, more efficient when the input preferences are complete and have some structural property, such as being narcissistic, single-peaked, and single-crossing. However, when the preferences are incomplete and have ties, we show that being single-peaked and single-crossing does not reduce the computational complexity—Stable Roommates remains NP-complete.
Highlights
Given 2n agents, each having preferences with regard to how suitable the other agents are as potential partners, the Stable Roommates problem is to decide whether there exists a matching, i.e., a set of disjoint pairs of the agents, without inducing a blocking pair
While it is quite straightforward to see that stable matchings may not always exist, it is not trivial to see whether an existing stable matching can be found in polynomial time, even when the input preference orders are complete and do not contain ties
We showed that for complete preferences with ties, assuming narcissism and single-peakedness guarantees the existence of stable matchings, which can be found in linear time; by comparison, Stable Roommates with complete preferences and ties is NP-complete [48]
Summary
Given 2n agents, each having preferences with regard to how suitable the other agents are as potential partners, the Stable Roommates problem is to decide whether there exists a matching, i.e., a set of disjoint pairs of the agents, without inducing a blocking pair. Bartholdi III and Trick [4] studied Stable Roommates with narcissistic and single-peaked preferences They showed that for the case with linear orders 3, we show that for complete preference orders, structurally restricted preferences such as being narcissistic and single-crossing or being narcissistic and single-peaked guarantee the existence of stable matchings. We demonstrate that the known algorithm of Bartholdi III and Trick [4] can be extended to always find a stable matching in two new cases: The algorithm works (1) when the preferences are complete, narcissistic, single-crossing, and may contain ties as well as (2) when the preferences are complete, narcissistic, single-peaked, and may contain ties. 4 we study the case where the preferences are incomplete and may contain ties, and prove that Stable Roommates becomes NP-complete, even when the preferences are narcissistic, single-peaked, and (tie-sensitive) single-crossing.
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