Stable Roommates with Narcissistic, Single-Peaked, and Single-Crossing Preferences

TL;DR

This paper explores the computational complexity of the Stable Roommates problem under various preference structures, providing more efficient algorithms for certain cases and establishing NP-completeness in others.

Contribution

It introduces more efficient algorithms for Stable Roommates with complete, structured preferences and proves NP-completeness persists with incomplete, tied preferences.

Findings

Efficient algorithms for complete, narcissistic, single-peaked, and single-crossing preferences.

NP-completeness remains for incomplete, tied preferences even with single-peaked and single-crossing structures.

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.