Constrained Assortment Optimization Under the Markov Chain–based Choice Model

Assortment optimization is an important problem that arises in many practical applications such as retailing and online advertising. The fundamental goal is to select a subset of items to offer from a universe of substitutable items to maximize expected revenue when customers exhibit a random substitution behavior captured by a choice model. We study assortment optimization under the Markov chain choice model in the presence of capacity constraints that arise naturally in many applications. The Markov chain choice model considers item substitutions as transitions in a Markov chain and provides a good approximation for a large class of random utility models, thereby addressing the challenging problem of model selection in choice modeling. In this paper, we present constant factor approximation algorithms for the cardinality- and capacity-constrained assortment-optimization problem under the Markov chain model. We show that this problem is APX-hard even when all item prices are uniform, meaning that, unless P = NP, it is not possible to obtain an approximation better than a particular constant. Our algorithmic approach is based on a new externality adjustment paradigm that exactly captures the externality of adding an item to a given assortment on the remaining set of items, thereby allowing us to linearize a nonlinear, nonsubmodular, and nonmonotone revenue function and to design an iterative algorithm that iteratively builds up a provably good assortment. This paper was accepted by Yinyu Ye, optimization.