  Motivated by applications in e-retail and online advertising, we study the
problem of assortment optimization under visibility constraints, that we refer
to as APV. We are given a universe of substitutable products and a stream of T
customers. The objective is to determine the optimal assortment of products to
offer to each customer in order to maximize the total expected revenue, subject
to the constraint that each product is required to be shown to a minimum number
of customers. The minimum display requirement for each product is given
exogenously and we refer to these constraints as visibility constraints. We
assume that customer choices follow a Multinomial Logit model (MNL). We provide
a characterization of the structure of the optimal assortments and present an
efficient polynomial time algorithm for solving APV. To accomplish this, we
introduce a novel function called the ``expanded revenue" of an assortment and
establish its supermodularity. Our algorithm takes advantage of this structural
property. Additionally, we demonstrate that APV can be formulated as a compact
linear program. Next, we consider APV with cardinality constraints, which we
prove to be strongly NP-hard and not admitting a Fully Polynomial Time
Approximation Scheme (FPTAS), even in the special case where all the products
have identical prices. Subsequently, we devise a Polynomial Time Approximation
Scheme (PTAS) for APV under cardinality constraints with identical prices. Our
algorithm starts by linearizing the objective function through a carefully
crafted guessing procedure, then solves the linearized program, and finally
randomly rounds the obtained solution to derive a near optimal solution. We
also examine the revenue loss resulting from the enforcement of visibility
constraints, comparing it to the unconstrained version of the problem and
propose a novel strategy to offset the loss.
