Stable matching with multilayer approval preferences: Approvals can be harder than strict preferences

Abstract

We study stable matching problems where agents have multilayer preferences: There are ℓ layers each consisting of one preference order for each agent. Recently, Chen et al. [EC '18] studied such problems with strict preferences, establishing four multilayer adaptations of classical notions of stability. We follow up on their work by analyzing the computational complexity of stable matching problems with multilayer approval preferences, which leads to problems that are incomparable to the previously studied ones. We consider eleven stability notions derived from three well-established stability notions for stable matchings with ties and the four adaptations 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.