We study the dynamic fulfillment problem in e-commerce, in which incoming (multi-item) customer orders must be immediately dispatched to (a combination of) fulfillment centers that have the required inventory. A prevailing approach to this problem, pioneered by Jasin and Sinha (2015), is to write a ``deterministic'' linear program that dictates, for each item in an incoming multi-item order from a particular region, how frequently it should be dispatched to each fulfillment center (FC). However, dispatching items in a way that satisfies these frequency constraints, without splitting the order across too many FC's, is challenging. Jasin and Sinha identify this as a correlated rounding problem, and propose an intricate rounding scheme that they prove is suboptimal by a factor of at most $\approx q/4$ on a $q$-item order. This paper provides to our knowledge the first substantially improved scheme for this correlated rounding problem, which is suboptimal by a factor of at most $1+\ln(q)$. We provide another scheme for sparse networks, which is suboptimal by a factor of at most $d$ if each item is stored in at most $d$ FC's. We show both of these guarantees to be tight in terms of the dependence on $q$ or $d$. Our schemes are simple and fast, based on an intuitive idea -- items wait for FC's to ``open'' at random times, but observe them on ``dilated'' time scales. This also implies a new randomized rounding method for the classical Set Cover problem, which could be of general interest. We numerically test our new rounding schemes under the same realistic setups as Jasin and Sinha (2015) and find that they improve runtimes, shorten code, and robustly improve performance. Our code is made publicly available.

翻译：我们研究电商中的动态履约问题，即（多品）客户订单必须立即分配至（多个）具有所需库存的履约中心（FC）。针对该问题，Jasin 与 Sinha（2015）首创主流方法：构建一个“确定性”线性规划，该规划规定了来自特定区域的多品订单中每个商品应向各履约中心（FC）发货的频率。然而，在满足这些频率约束的同时避免将订单拆分至过多 FC 颇具挑战性。Jasin 与 Sinha 将此识别为相关舍入问题，并提出一种精巧的舍入方案，但证明其对 $q$ 品订单的最优性差距至多为 $\approx q/4$ 倍。本文首次提出该相关舍入问题的显著改进方案，其最优性差距至多为 $1+\ln(q)$ 倍。我们还为稀疏网络提出另一种方案：若每种商品存储于至多 $d$ 个 FC 中，则最优性差距至多为 $d$ 倍。我们证明这两个界在 $q$ 或 $d$ 依赖关系上均为紧的。基于直观思想——商品等待 FC 在随机时间“开放”，但以“膨胀”时间尺度观察它们——我们的方案简单高效。这也为经典集合覆盖问题推导出新的随机舍入方法，具有普遍意义。我们采用与 Jasin 和 Sinha（2015）相同的现实场景对新型舍入方案进行数值测试，发现其能缩短运行时间、精简代码并稳健提升性能。相关代码已公开。