[Paper Review] Tight Guarantees for Multi-unit Prophet Inequalities and Online Stochastic Knapsack

Jiashuo Jiang, Will Ma+1 more|arXiv (Cornell University)|Jul 5, 2021

Optimization and Search Problems4 citations

TL;DR

This paper provides tight performance guarantees for multi-unit prophet inequalities and online stochastic knapsack problems using a novel invariant-based analysis. It improves the best-known guarantee for $k$-unit prophet inequalities beyond Alaei's $1 - \frac{1}{\sqrt{k+3}}$, and introduces a new 'best-fit' procedure with a tight 0.319 guarantee for online knapsack, further refined to 0.3557 in the unit-density case, both relative to the LP relaxation.

ABSTRACT

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).

Motivation & Objective

• To characterize the tight performance guarantee of Alaei's $k$-unit prophet inequality procedure, which had been conjectured but not proven.
• To develop a new online algorithm for the general online stochastic knapsack problem with improved performance guarantees.
• To refine analysis techniques beyond traditional large/small item classification, using a utilization invariant across sample paths.
• To establish tight bounds relative to the standard LP relaxation for both multi-unit and knapsack settings.
• To generalize results to multi-resource online assignment problems, including appointment scheduling and e-commerce fulfillment.

Proposed method

• Introduce a new 'best-fit' online algorithm for the knapsack problem that dynamically selects items based on capacity utilization and valuation.
• Use an invariant-based analysis that tracks overall resource utilization across different sample paths, avoiding the need to classify items as large or small.
• Formulate the optimal performance ratio as a solution to a continuous-time optimization problem involving a function $ h_{\gamma_0}(t) $, derived from an ODE system.
• Apply LP duality to prove the optimality of the $\gamma$-Conservative Magician procedure in the $k$-unit setting, deriving the exact optimal $\gamma^*_k$.
• Numerically solve the resulting optimization problem to find the maximum achievable guarantee, yielding 0.3557 in the unit-density case.
• Construct a counterexample to show that no online algorithm can exceed $ \frac{1 - e^{-2}}{2} \approx 0.432 $ relative to the LP upper bound, establishing a theoretical limit.

Figure 1 : Our tight ratios compared to existing lower/upper bounds, plotted up to $k=50$ .

Experimental results

Research questions

• RQ1 What is the true tight performance guarantee of Alaei's $k$-unit prophet inequality procedure, and is it strictly better than $1 - \frac{1}{\sqrt{k+3}}$?
• RQ2 Can a new online algorithm for the stochastic knapsack problem achieve a better performance guarantee than the previously known 0.2?
• RQ3 How does an invariant-based analysis of resource utilization improve upon traditional large/small item classification in online stochastic knapsack?
• RQ4 What is the optimal performance guarantee achievable in the unit-density knapsack case, and can it be improved beyond 0.321?
• RQ5 Is the proposed 'best-fit' procedure tight with respect to the standard LP relaxation in the general knapsack setting?

Key findings

• The tight performance guarantee for Alaei's $k$-unit prophet inequality procedure is strictly greater than $1 - \frac{1}{\sqrt{k+3}}$ for all $k > 1$, resolving a long-standing open question.
• The proposed 'best-fit' algorithm achieves a tight performance guarantee of $\frac{1}{3 + e^{-2}} \approx 0.319$ for the online stochastic knapsack problem, improving over the prior best of 0.2.
• In the unit-density special case, the improved analysis yields a performance guarantee of 0.3557, surpassing the previous best of 0.321.
• The invariant-based analysis technique avoids the need to split items into large and small categories, enabling tighter bounds.
• A counterexample shows that no online algorithm can achieve a better guarantee than $ \frac{1 - e^{-2}}{2} \approx 0.432 $ relative to the LP relaxation, even in the single-knapsack case.
• The results generalize directly to multi-resource online assignment problems, such as multi-knapsack and appointment scheduling, with tight guarantees relative to the LP relaxation.