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.

翻译：战略性的产品陈列在实体店铺及线上平台均能显著影响顾客的购买行为。受此启发，我们研究如何优化可替代产品在指定陈列位置的布局，以最大化卖方的预期收益。我们将顾客行为建模为两个阶段：首先，顾客根据浏览分布访问部分陈列位置；其次，顾客根据选择模型，从这些位置陈列的产品中至多选择一件。我们的目标是设计一种通用算法，能够针对任意浏览分布和选择模型，最优地选择并陈列产品，该问题被称为"布局问题"。我们提出一种随机化算法，该算法利用基数约束品类优化的α-近似算法，对于具有m个陈列位置的布局问题，输出一个（期望意义上）为最优值至少$\frac{\Omega(\alpha)}{\log m}$倍的解，且该界在最坏情况下是紧的。针对某些特例，我们还给出具有更强保证的算法：具体而言，针对马尔可夫选择模型，我们给出一个确定性的$\frac{\Omega(1)}{\log m}$-近似算法；当产品价格相同时，我们给出一个紧的$(1-1/e)$-近似算法。