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 magical procedure of Alaei (2011) with its celebrated performance guarantee of $1-\frac{1}{\sqrt{k+3}}$ has found widespread adoption in mechanism design and other online allocation problems in online advertising, healthcare scheduling, and revenue management. Despite being commonly used for implementing online allocation, the tightness of Alaei's procedure for a given $k$ has remained unknown. In this paper we resolve this question, characterizing the tight bound by identifying the structure of the optimal online implementation, and consequently improving the best-known guarantee for $k$-unit prophet inequalities for all $k>1$. We also consider a more general online stochastic knapsack problem where each individual allocation can consume an arbitrary fraction of the initial capacity. We introduce a new "best-fit" procedure for implementing a fractionally-feasible knapsack solution online, with a performance guarantee of $\frac{1}{3+e^{-2}}\approx0.319$, which we also show is tight. 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). All in all, our results imply \textit{tight} Online Contention Resolution Schemes for $k$-uniform matroids and the knapsack polytope, respectively, which has further implications in mechanism design.
翻译:先知的不平等是设计在线分配程序和将其业绩与最佳离线分配进行比较的有用工具。 在美元单位先知不平等的基本设置中, Alaei (2011年) 的神奇程序发现在机制设计和其他在线分配问题中广泛采用$-frac{1unsurt{k+3 ⁇ {{{{{{{{{{{{{{}在设计在线分配程序并将其业绩与最佳离线分配进行比较。 尽管通常用于实施在线分配, Alaei 的程序对于给定美元单位的精确度仍然不为人所知。 在本文中,我们通过确定最佳在线执行结构,将最著名的Alaei (2011年) (2011年) (2011年) (2011年) (2011年) 3月 (2011年) 美元单位先知不平等保证,所有美元单位 (1美元单位 ) (1美元单位 ) 的神奇的在线分配问题。 ”我们采用新的“最合适的 knampbackack ” 程序在网上实施一个小的精细的精细的精细的Knassack 解决方案, —— —— —— —— 我们用最精确的直径=2xxxxxxxxxxxxxxxxxxx