Stochastic Knapsack with Costs: On Adaptivity and Return-on-Investment

Abstract

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.

Figures (4)

• Figure 1: Hierarchy of policy adaptivity levels showing the approximation gaps between different classes.
• Figure 2: Visualization of the removal of double-dominated points. The hollow circles ($j', j"$) represent two choices for an item. The red point ($j$) is "double-dominated" because it lies above the line segment representing the convex combination of $j'$ and $j"$. In the LP relaxation, this dominated point can be replaced by its vertical projection (the green point) onto the segment, which yields a solution that is no worse in objective value and potentially better in terms of feasibility ($\mu$).
• Figure 3: Visualization of the set of undominated points, $\Gamma_{i}$. The blue line represents the efficient (Pareto) frontier formed by undominated choices and their convex combinations. An arbitrary combination of choices that lies above this frontier (the red point) is inefficient. It can be strictly improved by projecting it horizontally onto the frontier (the green point), which achieves the same $\mu$ value for a greater $w$ value.
• Figure 4: For an optimal solution $x$, items $i_1$ and $i_2$ each have two choices with nonzero values. The solution for each item is a convex combination of the two choices, corresponding to the red dots in the graphs $\Gamma_{i_1}$ and $\Gamma_{i_2}$, which lie on the segments connecting the two choices. As the slope of the first item is larger, the modified solution indicated by the dashed arrows illustrates one possible update that maintains feasibility and increases the objective value.

Theorems & Definitions (67)

• Proposition 1.0
• Definition 2.1: ROI and $\alpha$ Parameters
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 3.1
• proof : Proof sketch
• Theorem 3.2
• Lemma 3.3