Stable Matching with Multilayer Approval Preferences: Approvals can be Harder than Strict Preferences

TL;DR

This paper investigates the computational complexity of stable matching problems with multilayer approval preferences, revealing that approval preferences often do not simplify and can even complicate the problem.

Contribution

It extends prior work by analyzing the complexity of multilayer approval preferences, showing many problems are NP-hard or polynomial, and examining the effects of layers and stability degree.

Findings

Approval preferences do not significantly simplify stability problems.

Some problems become NP-hard under approval preferences.

The number of layers and stability degree influence complexity.

Abstract

We study stable matching problems where agents have multilayer preferences: There are $\ell$ layers each consisting of one preference relation for each agent. Recently, Chen et al. [EC '18] studied such problems with strict preferences, establishing four multilayer adaptions of classical notions of stability. We follow up on their work by analyzing the computational complexity of stable matching problems with multilayer approval preferences. We consider eleven stability notions derived from three well-established stability notions for stable matchings with ties and the four adaptions proposed by Chen et al. For each stability notion, we show that the problem of finding a stable matching is either polynomial-time solvable or NP-hard. Furthermore, we examine the influence of the number of layers and the desired "degree of stability" on the problems' complexity. Somewhat surprisingly, we discover that assuming approval preferences instead of strict preferences does not considerably simplify the situation (and sometimes even makes polynomial-time solvable problems NP-hard).