  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).
