  Hedonic diversity games are a variant of the classical Hedonic games designed
to better model a variety of questions concerning diversity and fairness.
Previous works mainly targeted the case with two diversity classes (represented
as colors in the model) and provided some initial complexity-theoretic and
existential results concerning Nash and individually stable outcomes. Here, we
design new algorithms accompanied with lower bounds which provide a complete
parameterized-complexity picture for computing Nash and individually stable
outcomes with respect to the most natural parameterizations of the problem.
Crucially, our results hold for general Hedonic diversity games where the
number of colors is not necessarily restricted to two, and show that -- apart
from two trivial cases -- a necessary condition for tractability in this
setting is that the number of colors is bounded by the parameter. Moreover, for
the special case of two colors we resolve an open question asked in previous
work (Boehmer and Elkind, AAAI 2020).
