Assortment Optimization over Dense Universe is Easy

Assortment optimization is an important problem in revenue management arising in industries such as online advertising, retailing and airline ticketing. We study assortment optimization under an arbitrary mixture of multi-nomial logit (MNL) models, when the universe of products is dense. In general, it is NP-hard to approximate assortment optimization under mixture of MNLs model within any reasonable factor. However, if the universe of products is dense, we show that assortment optimization has an optimal solution that is well approximated by a nested-by-revenue policy. We argue this by considering an alternate, continuous-space model where product features follow a density on a continuous space, and offer set can be any open subset on the feature space. In this model, we show that there exists a nested-by-revenue policy that is optimal. This characterization holds regardless of the mixture distribution (including discrete and continuous) and the dimension of feature space. Through this characterization and linking the continuous-space to the standard discrete-item model via a "fractional" relaxation, we also show that, in the finite and discrete item case, nested-by-revenue performs nearly optimally. In particular, we derive an optimality gap, explicitly dependent on the model parameters, that shrinks to zero under suitable scaling as the number of items grows.