The joint replenishment problem (JRP) is a classical inventory management problem. We consider a natural generalization with outliers, where we are allowed to reject (that is, not service) a subset of demand points. In this paper, we are motivated by issues of fairness - if we do not serve all of the demands, we wish to ``spread out the pain'' in a balanced way among customers, communities, or any specified market segmentation. One approach is to constrain the rejections allowed, and to have separate bounds for each given customer. In our most general setting, we consider a set of C features, where each demand point has an associated rejection cost for each feature, and we have a given bound on the allowed rejection cost incurred in total for each feature. This generalizes a model of fairness introduced in earlier work on the Colorful k-Center problem in which (analogously) each demand point has a given color, and we bound the number of rejections of each color class. We give the first constant approximation algorithms for the fairness-constrained JRP with a constant number of features; specifically, we give a 2.86-approximation algorithm in this case. Even for the special case in which we bound the total (weighted) number of outliers, this performance guarantee improves upon bounds previously known for this case. Our approach is an LP-based algorithm that splits the instance into two subinstances. One is solved by a novel iterative rounding approach and the other by pipage-based rounding. The standard LP relaxation has an unbounded integrality gap, and hence another key element of our algorithm is to strengthen the relaxation by correctly guessing key attributes of the optimal solution, which are sufficiently concise, so that we can enumerate over all possible guesses in polynomial time - albeit exponential in C, the number of features.

翻译：联合补货问题(JRP)是经典的库存管理问题。本文考虑具有异常值的自然泛化情形，允许拒绝（即不服务）部分需求点。受公平性问题驱动——若无法满足所有需求，我们希望以平衡方式将“痛苦”分散到客户、社区或任何指定市场细分群体中。一种可行方法是限制拒绝数量，并对每个给定客户设置独立约束。在最一般设定下，我们考虑包含C个特征的集合，每个需求点针对各特征关联一个拒绝成本，且各特征的总允许拒绝成本存在给定上限。这泛化了早期关于多彩k-中心问题(Colorful k-Center)中提出的公平性模型——类比而言，该模型中每个需求点具有给定颜色，且对各颜色类别的拒绝数量设限。我们首次给出了在常数个特征下具有公平性约束的JRP的常数近似算法；具体而言，针对此类情形提出了2.86-逼近算法。即使限制总（加权）异常值数量这一特例，该性能保证也优于此前已知结果。我们的方法基于线性规划(LP)算法，将实例分解为两个子实例：一个通过新型迭代舍入方法求解，另一个则采用管道式舍入方式。标准线性规划松弛存在无界整数间隙，因此算法的另一个关键要素是通过正确猜测最优解的关键属性来强化松弛——这些属性足够简洁，使得我们能在多项式时间内枚举所有可能猜测，尽管该枚举复杂度与特征数量C呈指数关系。