  We define a new class of set functions that in addition to being monotone and
subadditive, also admit a very limited form of submodularity defined over a
permutation of the ground set. We refer to this permutation as a submodular
order. This class of functions includes monotone submodular functions as a
sub-family. We give fast algorithms with strong approximation guarantees for
maximizing submodular order functions under a variety of constraints and show a
nearly tight upper bound on the highest approximation guarantee achievable by
algorithms with polynomial query complexity. Applying this new notion to the
problem of constrained assortment optimization in fundamental choice models, we
obtain new algorithms that are both faster and have stronger approximation
guarantees (in some cases, first algorithm with constant factor guarantee). We
also show an intriguing connection to the maximization of monotone submodular
functions in the streaming model, where we recover best known approximation
guarantees as a corollary of our results.
