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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.

Sparse process flexibility and long chain has became an important concept in design flexible manufacturing systems since the seminal paper of Jordan and Graves (1995). In this paper, we study the performance of long chain in comparison to all designs with at most 2n edges over n supply and n demand nodes. We show that, surprisingly, long chain is not optimal in this class of networks even for i.i.d. demand distributions. In particular, we present a family of instances where a disconnected network with 2n edges has a strictly better performance than long chain even for i.i.d. demand distributions. This is quite surprising and contrary to the intuition that a connected design performs better than a disconnected one for symmetric distributions. Moreover, our family of instances show that the optimal design depends on the particular demand distribution. We also study the performance of long chain in comparison to connected designs with at most 2n arcs. We show that long chain is optimal in this class of designs for exchangeable demand distributions. Our proof is based on a coupling argument and a combinatorial analysis of the structure of maximum flow in directed networks. The analysis provides useful insights towards not just understanding the optimality of long chain but also towards designing more general sparse flexibility networks.

Assortment optimization is an important problem arising in various applications. In many practical settings, the assortment is subject to a capacity constraint. In “Capacitated Assortment Optimization: Hardness and Approximation,” Désir, Goyal, and Zhang study the capacitated assortment optimization problem. The authors first show that adding a general capacity constraint makes the problem NP-hard even for the simple multinomial logit model. They also show that under the mixture of multinomial logit model, even the unconstrained problem is hard to approximate within any reasonable factor when the number of mixtures is not constant. In view of these hardness results, the authors present near-optimal algorithms for a large class of parametric choice models including the mixture of multinomial logit, Markov chain, nested logit, and d-level nested logit choice models. In fact, their approach extends to a large class of objective functions that depend only on a small number of linear functions.