  Assortment optimization refers to the problem of designing a slate of
products to offer potential customers, such as stocking the shelves in a
convenience store. The price of each product is fixed in advance, and a
probabilistic choice function describes which product a customer will choose
from any given subset. We introduce the combinatorial assortment problem, where
each customer may select a bundle of products. We consider a model of consumer
choice where the relative value of different bundles is described by a
valuation function, while individual customers may differ in their absolute
willingness to pay, and study the complexity of the resulting optimization
problem. We show that any sub-polynomial approximation to the problem requires
exponentially many demand queries when the valuation function is XOS, and that
no FPTAS exists even for succinctly-representable submodular valuations. On the
positive side, we show how to obtain constant approximations under a
"well-priced" condition, where each product's price is sufficiently high. We
also provide an exact algorithm for $k$-additive valuations, and show how to
extend our results to a learning setting where the seller must infer the
customers' preferences from their purchasing behavior.
