Stochastic Knapsack with Costs: On Adaptivity and Return-on-Investment
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arXiv
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| Format: | Preprint |
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2025
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| author | Barak, Zohar Berlin, Asnat Cohen, Ilan Reuven Eden, Alon Porat, Omri Talgam-Cohen, Inbal |
| author_facet | Barak, Zohar Berlin, Asnat Cohen, Ilan Reuven Eden, Alon Porat, Omri Talgam-Cohen, Inbal |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05956 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stochastic Knapsack with Costs: On Adaptivity and Return-on-Investment Barak, Zohar Berlin, Asnat Cohen, Ilan Reuven Eden, Alon Porat, Omri Talgam-Cohen, Inbal Computer Science and Game Theory 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. |
| title | Stochastic Knapsack with Costs: On Adaptivity and Return-on-Investment |
| topic | Computer Science and Game Theory |
| url | https://arxiv.org/abs/2509.05956 |