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 often focus exclusively on achieving the platforms' objectives, highlighting items with the highest popularity or revenue. This approach, however, can compromise the equality of opportunities for the rest of the items, in turn leading to less content diversity and increased regulatory scrutiny for the platform. 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 an FPTAS 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) leveraging 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（从电商平台到社交媒体）都采用算法来优化其提供的商品组合（如产品与内容）。这些算法往往仅聚焦于实现平台目标，突出展示最具人气或收益最高的商品。然而，这种方式可能损害其余商品获得平等机会的权利，进而降低内容多样性并增加平台的监管风险。受此启发，我们提出并研究了一个公平的商品组合规划问题，该问题要求任何两个质量/价值相似的商品应获得相近的展示结果。研究表明，该问题可表述为一个线性规划（LP），称为(FAIR)，其优化对象为所有可行商品组合的分布。为求解(FAIR)的近似最优解，我们提出基于椭球法的框架，该框架需要为LP的对偶问题设计多项式时间分离神谕。我们发现，为对偶问题寻找最优分离神谕是NP完全问题，因此提出一系列近似分离神谕，进而为原问题(FAIR)得到1/2近似算法和FPTAS。近似分离神谕的设计思路为：（i）证明LP对偶问题的分离神谕等价于求解无穷序列的参数化背包问题，（ii）利用参数化背包问题的结构特性。最后，我们基于MovieLens数据集进行案例研究，验证了算法的有效性，并进一步揭示了公平性的代价。