We revisit the Stochastic Knapsack problem, where a policy-maker chooses an execution order for jobs with fixed values and stochastic running-times, aiming to maximize the value completed by a deadline. Dean et al. (FOCS'04) show that simple non-adaptive policies can approximate the (highly adaptive) optimum, initiating the study of adaptivity gaps. We introduce an economically motivated generalization in which each job also carries an execution cost. This uncovers new applications, most notably a new and natural variant of contract design: jobs are processed by agents who choose among effort levels that induce different processing-time distributions, and the principal must decide which jobs to run and what payments induce the intended effort. With costs, the objective becomes mixed-sign: value from completed jobs must be balanced against costs of execution, and running a job that misses the deadline can create negative utility. This changes the algorithmic picture, and the adaptivity gap is no longer constant. We give an economic explanation: the performance of non-adaptive policies is governed by jobs' return on investment (ROI) -- utility over cost -- which can be arbitrarily small. We introduce a hierarchy of increasingly adaptive policies, trading off simplicity and adaptivity. We prove near-tight guarantees across the hierarchy, showing that with costs the adaptivity gap is $Θ(α)$, where $1/α$ is the ROI. Higher in the hierarchy, we identify an efficiently computable policy with limited adaptivity that is approximately-optimal. Analogous to the centrality of ROI in economics, we believe our ROI-based, simple-vs-optimal approach to adaptivity may be useful for additional stochastic optimization and online problems with mixed-sign objectives.

翻译：我们重新审视随机背包问题，其中决策者需为具有固定价值和随机运行时间的任务选择执行顺序，目标是在截止期限前最大化已完成任务的总价值。Dean等人（FOCS'04）的研究表明，简单的非适应性策略可以逼近（高度适应性的）最优解，由此开启了适应性差距的研究。本文提出一种受经济学启发的推广形式：每个任务还附带执行成本。这一推广揭示了新的应用场景，其中最突出的是合约设计的一种新颖且自然的变体：任务由代理人执行，代理人可在引致不同处理时间分布的努力水平中进行选择，委托方必须决定运行哪些任务以及支付何种报酬以诱导预期努力水平。引入成本后，目标函数变为混合符号：已完成任务的价值需与执行成本相权衡，而运行超期任务可能产生负效用。这改变了算法格局，适应性差距不再保持常数。我们给出经济学解释：非适应性策略的性能由任务的投资回报率（ROI）——效用与成本之比——决定，该比值可能任意小。我们构建了适应性逐级增强的策略层次结构，在简洁性与适应性之间进行权衡。我们证明了该层次结构中近乎紧确的保证界，表明在存在成本时适应性差距为$Θ(α)$，其中$1/α$即投资回报率。在更高层次中，我们提出一种具有有限适应性且可高效计算的近似最优策略。类比于投资回报率在经济学中的核心地位，我们相信这种基于投资回报率的、在简单性与最优性间权衡的适应性研究方法，可能对处理其他具有混合符号目标的随机优化和在线问题具有借鉴价值。