  Strategic product placement can have a strong influence on customer purchase
behavior in physical stores as well as online platforms. Motivated by this, we
consider the problem of optimizing the placement of substitutable products in
designated display locations to maximize the expected revenue of the seller. We
model the customer behavior as a two-stage process: first, the customer visits
a subset of display locations according to a browsing distribution; second, the
customer chooses at most one product from the displayed products at those
locations according to a choice model. Our goal is to design a general
algorithm that can select and place the products optimally for any browsing
distribution and choice model, and we call this the Placement problem. We give
a randomized algorithm that utilizes an $\alpha$-approximate algorithm for
cardinality constrained assortment optimization and outputs a
$\frac{\Theta(\alpha)}{\log m}$-approximate solution (in expectation) for
Placement with $m$ display locations, i.e., our algorithm outputs a solution
with value at least $\frac{\Omega(\alpha)}{\log m}$ factor of the optimal and
this is tight in the worst case. We also give algorithms with stronger
guarantees in some special cases. In particular, we give a deterministic
$\frac{\Omega(1)}{\log m}$-approximation algorithm for the Markov choice model,
and a tight $(1-1/e)$-approximation algorithm for the problem when products
have identical prices.
