  Schelling's model considers $k$ types of agents each of whom needs to select
a vertex on an undirected graph, where every agent prefers to neighbor agents
of the same type. We are motivated by a recent line of work that studies
solutions that are optimal with respect to notions related to the welfare of
the agents. We explore the parameterized complexity of computing such
solutions. We focus on the well-studied notions of social welfare (WO) and
Pareto optimality (PO), alongside the recently proposed notions of
group-welfare optimality (GWO) and utility-vector optimality (UVO), both of
which lie between WO and PO. Firstly, we focus on the fundamental case where
$k=2$ and there are $r$ red agents and $b$ blue agents. We show that all
solution-notions we consider are $\textsf{NP}$-hard to compute even when $b=1$
and that they are $\textsf{W}[1]$-hard when parameterized by $r$ and $b$. In
addition, we show that WO and GWO are $\textsf{NP}$-hard even on cubic graphs.
We complement these negative results by an $\textsf{FPT}$ algorithm
parameterized by $r, b$ and the maximum degree of the graph. For the general
case with $k$ types of agents, we prove that for any of the notions we consider
the problem is $\textsf{W}[1]$-hard when parameterized by $k$ for a large
family of graphs that includes trees. We accompany these negative results with
an $\textsf{XP}$ algorithm parameterized by $k$ and the treewidth of the graph.
