Tight Guarantees for Multi-unit Prophet Inequalities and Online Stochastic Knapsack

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.