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The joint replenishment problem (JRP) is a classical inventory managementproblem. We consider a natural generalization with outliers, where we areallowed to reject (that is, not service) a subset of demand points. In thispaper, we are motivated by issues of fairness - if we do not serve all of thedemands, we wish to ``spread out the pain'' in a balanced way among customers,communities, or any specified market segmentation. One approach is to constrainthe rejections allowed, and to have separate bounds for each given customer. Inour most general setting, we consider a set of C features, where each demandpoint has an associated rejection cost for each feature, and we have a givenbound on the allowed rejection cost incurred in total for each feature. Thisgeneralizes a model of fairness introduced in earlier work on the Colorfulk-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 thefairness-constrained JRP with a constant number of features; specifically, wegive a 2.86-approximation algorithm in this case. Even for the special case inwhich we bound the total (weighted) number of outliers, this performanceguarantee improves upon bounds previously known for this case. Our approach isan LP-based algorithm that splits the instance into two subinstances. One issolved by a novel iterative rounding approach and the other by pipage-basedrounding. The standard LP relaxation has an unbounded integrality gap, andhence another key element of our algorithm is to strengthen the relaxation bycorrectly guessing key attributes of the optimal solution, which aresufficiently concise, so that we can enumerate over all possible guesses inpolynomial time - albeit exponential in C, the number of features.

Improved Approximation Algorithms for the Joint Replenishment Problem with Outliers, and with Fairness Constraints