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We study the dynamic fulfillment problem in e-commerce, in which incoming(multi-item) customer orders must be immediately dispatched to (a combinationof) 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 inan incoming multi-item order from a particular region, how frequently it shouldbe dispatched to each fulfillment center (FC). However, dispatching items in away that satisfies these frequency constraints, without splitting the orderacross too many FC's, is challenging. Jasin and Sinha identify this as acorrelated rounding problem, and propose an intricate rounding scheme that theyprove 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 schemefor this correlated rounding problem, which is suboptimal by a factor of atmost $1+\ln(q)$. We provide another scheme for sparse networks, which issuboptimal 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 dependenceon $q$ 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 forthe classical Set Cover problem, which could be of general interest.  We numerically test our new rounding schemes under the same realistic setupsas Jasin and Sinha (2015) and find that they improve runtimes, shorten code,and robustly improve performance. Our code is made publicly available.

Order-optimal Correlated Rounding for Fulfilling Multi-item E-commerce Orders