  The class of assignment problems is a fundamental and well-studied class in
the intersection of Social Choice, Computational Economics and Discrete
Allocation. In a general assignment problem, a group of agents expresses
preferences over a set of items, and the task is to allocate items to agents in
an "optimal" way. A verification variant of this problem includes an allocation
as part of the input, and the question becomes whether this allocation is
"optimal". In this paper, we generalize the verification variant to the setting
where each agent is equipped with multiple incomplete preference lists: Each
list (called a layer) is a ranking of items in a possibly different way
according to a different criterion.
  In particular, we introduce three multi-layer verification problems, each
corresponds to an optimality notion that weakens the notion of global
optimality (that is, pareto optimality in multiple layers) in a different way.
Informally, the first notion requires that, for each group of agents whose size
is exactly some input parameter k, the agents in the group will not be able to
trade their assigned items among themselves and benefit in at least alpha
layers; the second notion is similar, but it concerns all groups of size at
most k rather than exactly k; the third notion strengthens these notions by
requiring that groups of k agents will not be part of possibly larger groups
that benefit in at least alpha layers. We study the three problems from the
perspective of parameterized complexity under several natural parameterizations
such as the number of layers, the number of agents, the number of items, the
number of allocated items, the maximum length of a preference list, and more.
We present an almost comprehensive picture of the parameterized complexity of
the problems with respect to these parameters.
