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

by   Jiashuo Jiang, et al.

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 1-1/√(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 1/3+e^-2≈0.319, 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 tight Online Contention Resolution Schemes for k-uniform matroids and the knapsack polytope, respectively, which has further implications in mechanism design.


page 1

page 2

page 3

page 4


Tightness without Counterexamples: A New Approach and New Results for Prophet Inequalities

Prophet inequalities consist of many beautiful statements that establish...

Optimal Budget-Feasible Mechanisms for Additive Valuations

In this paper, we obtain tight approximation guarantee for budget-feasib...

Optimal Online Contention Resolution Schemes via Ex-Ante Prophet Inequalities

Online contention resolution schemes (OCRSs) were proposed by Feldman, S...

Tight Analysis of the Smartstart Algorithm for Online Dial-a-Ride on the Line

The online Dial-a-Ride problem is a fundamental online problem in a metr...

Improved Guarantees for Offline Stochastic Matching via new Ordered Contention Resolution Schemes

Matching is one of the most fundamental and broadly applicable problems ...

Randomized and Deterministic Maximin-share Approximations for Fractionally Subadditive Valuations

We consider the problem of guaranteeing a fraction of the maximin-share ...

Online Unrelated-Machine Load Balancing and Generalized Flow with Recourse

We consider the online unrelated-machine load balancing problem with rec...

Please sign up or login with your details

Forgot password? Click here to reset