Many online platforms, ranging from online retail stores to social media platforms, employ algorithms to optimize their offered assortment of items (e.g., products and contents). These algorithms tend to prioritize the platforms' short-term goals by solely featuring items with the highest popularity or revenue. However, this practice can then lead to undesirable outcomes for the rest of the items, making them leave the platform, and in turn hurting the platform's long-term goals. Motivated by that, we introduce and study a fair assortment planning problem, which requires any two items with similar quality/merits to be offered similar outcomes. We show that the problem can be formulated as a linear program (LP), called (FAIR), that optimizes over the distribution of all feasible assortments. To find a near-optimal solution to (FAIR), we propose a framework based on the Ellipsoid method, which requires a polynomial-time separation oracle to the dual of the LP. We show that finding an optimal separation oracle to the dual problem is an NP-complete problem, and hence we propose a series of approximate separation oracles, which then result in a $1/2$-approx. algorithm and a PTAS for the original Problem (FAIR). The approximate separation oracles are designed by (i) showing the separation oracle to the dual of the LP is equivalent to solving an infinite series of parameterized knapsack problems, and (ii) taking advantage of the structure of the parameterized knapsack problems. Finally, we conduct a case study using the MovieLens dataset, which demonstrates the efficacy of our algorithms and further sheds light on the price of fairness.

翻译：许多在线平台（从零售网站到社交媒体）都采用算法来优化其提供的商品组合（例如产品与内容）。这些算法往往仅展示最受欢迎或收益最高的商品，从而优先实现平台的短期目标。然而，这种做法可能导致其余商品获得不佳结果，促使它们离开平台，进而损害平台的长期利益。受此启发，我们提出并研究了公平商品组合规划问题，该问题要求任何两个质量/价值相似的商品应获得相似的结果。我们证明该问题可建模为一个线性规划（LP），称为（FAIR），该规划在全部可行商品组合的分布上进行优化。为求得（FAIR）的近似最优解，我们提出一个基于椭球法的框架，该框架需要为对偶线性规划设计多项式时间分离判定算法。我们发现求解对偶问题的最优分离判定算法是NP难问题，因此提出一系列近似分离判定算法，从而得到原始问题（FAIR）的一个1/2-近似算法和一个多项式时间近似方案（PTAS）。这些近似分离判定算法的设计通过以下两步实现：（i）证明对偶线性规划的分离判定等价于求解无穷级参数化背包问题；（ii）利用参数化背包问题的结构特性。最后，我们使用MovieLens数据集进行案例研究，验证了所提算法的有效性，并进一步揭示了公平性代价。