  We study the computational complexity of finding fair allocations of
indivisible goods in the setting where a social network on the agents is given.
Notions of fairness in this context are "localized", that is, agents are only
concerned about the bundles allocated to their neighbors, rather than every
other agent in the system. We comprehensively address the computational
complexity of finding locally envy-free and Pareto efficient allocations in the
setting where the agents have binary valuations for the goods and the
underlying social network is modeled by an undirected graph. We study the
problem in the framework of parameterized complexity.
  We show that the problem is computationally intractable even in fairly
restricted scenarios, for instance, even when the underlying graph is a path.
We show NP-hardness for settings where the graph has only two distinct
valuations among the agents. We demonstrate W-hardness with respect to the
number of goods or the size of the vertex cover of the underlying graph. We
also consider notions of proportionality that respect the structure of the
underlying graph and show that two natural versions of this notion have
different complexities: allocating according to the notion that accounts for
locality to the greatest degree turns out to be computationally intractable,
while for other notions, the allocation problem can be modeled as a structured
ILP which can be solved efficiently.
