Prophet inequalities are a useful tool for designing online allocation procedures and comparing their performance to the optimal offline allocation. In the basic setting of $k$-unit prophet inequalities, the well-known procedure of Alaei (2011) with its celebrated performance guarantee of $1-\frac{1}{\sqrt{k+3}}$ has found widespread adoption in mechanism design and general online allocation problems in online advertising, healthcare scheduling, and revenue management. Despite being commonly used to derive approximately-optimal algorithms for multi-resource allocation problems, the tightness of Alaei's guarantee has remained unknown. In this paper characterize the tight guarantee in Alaei's setting, which we show is in fact strictly greater than $1-\frac{1}{\sqrt{k+3}}$ for all $k>1$. We also consider the more general online stochastic knapsack problem where each individual allocation can consume an arbitrary fraction of the initial capacity. Here we introduce a new ``best-fit'' procedure with a performance guarantee of $\frac{1}{3+e^{-2}}\approx0.319$, which we also show is tight with respect to the standard LP relaxation. This improves the previously best-known guarantee of 0.2 for online knapsack. Our analysis differs from existing ones by eschewing the need to split items into ``large'' or ``small'' based on capacity consumption, using instead an invariant for the overall utilization on different sample paths. Finally, we refine our technique for the unit-density special case of knapsack, and improve the guarantee from 0.321 to 0.3557 in the multi-resource appointment scheduling application of Stein et al. (2020).

翻译：先知不等式是设计在线分配程序并比较其与最优离线分配性能的有用工具。在$k$单元先知不等式的基本设定中，Alaei (2011) 提出的著名程序以其$1-\frac{1}{\sqrt{k+3}}$的卓越性能保证，已广泛应用于机制设计以及在线广告、医疗排程和收益管理等一般性在线分配问题。尽管该程序常用于推导多资源分配问题的近似最优算法，但其保证的紧性此前仍属未知。本文刻画了Alaei设定下的紧界，证明对所有$k>1$，该紧界实际上严格大于$1-\frac{1}{\sqrt{k+3}}$。我们进一步考虑了更一般的在线随机背包问题，其中每次分配可消耗初始容量的任意比例。在此我们引入一种新的"最佳适配"程序，其性能保证为$\frac{1}{3+e^{-2}}\approx0.319$，并证明该界相对于标准线性规划松弛是紧的。这改进了在线背包问题此前已知的最佳保证0.2。与现有方法不同，我们的分析无需依据容量消耗将物品划分为"大"或"小"，而是针对不同样本路径上的总体利用采用不变式。最后，我们改进技术以处理背包的单位密度特例，并将Stein等人(2020)的多资源预约排程应用中的保证从0.321提升至0.3557。