Multidimensional Stable Roommates with Master List

TL;DR

This paper investigates the computational complexity of Multidimensional Stable Roommates when preferences are derived from a central master list, a scenario common in applications with objective scoring, extending prior two-dimensional studies to the multidimensional case.

Contribution

It provides the first systematic analysis of Multidimensional Stable Roommates with master lists, exploring tractability and complexity in this specific setting.

Findings

Master lists create potential for tractability in multidimensional stable matchings.

The study identifies conditions under which the problem remains NP-hard or becomes tractable.

It applies parameterized algorithm techniques to analyze the problem's complexity.

Abstract

Since the early days of research in algorithms and complexity, the computation of stable matchings is a core topic. While in the classic setting the goal is to match up two agents (either from different "gender" (this is Stable Marriage) or "unrestricted" (this is Stable Roommates)), Knuth [1976] triggered the study of three- or multidimensional cases. Here, we focus on the study of Multidimensional Stable Roommates, known to be NP-hard since the early 1990's. Many NP-hardness results, however, rely on very general input instances that do not occur in at least some of the specific application scenarios. With the quest for identifying islands of tractability for Multidimensional Stable Roommates, we study the case of master lists. Here, as natural in applications where agents express their preferences based on "objective" scores, one roughly speaking assumes that all agent preferences are "derived from" a central master list, implying that the individual agent preferences shall be similar. Master lists have been frequently studied in the two-dimensional (classic) stable matching case, but seemingly almost never for the multidimensional case. This work, also relying on methods from parameterized algorithm design and complexity analysis, performs a first systematic study of Multidimensional Stable Roommates under the assumption of master lists.