Stochastic Knapsack -- Semi-Adaptivity Gaps and Improved Approximation

2026-02-27

AI Summary

This study investigates the performance gaps among non-adaptive, partially adaptive, and fully adaptive strategies in the stochastic knapsack problem under a risk-sensitive setting where overflow incurs total loss. Focusing on scenarios limited to a constant number of adaptive queries, the work introduces a novel three-step analytical framework—simplify, balance, and optimize—to systematically quantify partial adaptivity gaps for the first time, including those between 0–1, 1–n, and k–n levels of adaptivity. By integrating techniques from stochastic combinatorial optimization, decision tree modeling, and probabilistic analysis, the paper tightens existing bounds on classical adaptivity gaps, demonstrating that even a single adaptive query yields a significant advantage over non-adaptive approaches. The proposed framework further offers a scalable toolkit for analyzing partial adaptivity across a broad class of stochastic combinatorial optimization problems.

Technology Category

Reasoning under Uncertainty: Stochastic OptimizationSearch and Optimization: Combinatorial OptimizationConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In stochastic combinatorial optimization, algorithms differ in their adaptivity: whether or not they query realized randomness and adapt to it. Dean et al. (FOCS'04) formalize the adaptivity gap, which compares the performance of fully adaptive policies to that of non-adaptive ones. We revisit the fundamental Stochastic Knapsack problem of Dean et al., where items have deterministic values and independent stochastic sizes. A policy packs items sequentially, stopping at the first knapsack overflow or before. We focus on the challenging risky variant, in which an overflow forfeits all accumulated value, and study the problem through the lens of semi-adaptivity: We measure the power of $k$ adaptive queries for constant $k$ through the notions of $0$-$k$ semi-adaptivity gap (the gap between $k$-semi-adaptive and non-adaptive policies), and $k$-$n$ semi-adaptivity gap (between fully adaptive and $k$-semi-adaptive policies). Our first contribution is to improve the classic results of Dean et al. by giving tighter upper and lower bounds on the adaptivity gap. Our second contribution is a smoother interpolation between non-adaptive and fully-adaptive policies, with the rationale that when full adaptivity is unrealistic (due to its complexity or query cost), limited adaptivity may be a desirable middle ground. We quantify the $1$-$n$ and $k$-$n$ semi-adaptivity gaps, showing how well $k$ queries approximate the fully-adaptive policy. We complement these bounds by quantifying the $0$-$1$ semi-adaptivity gap, i.e., the improvement from investing in a single query over no adaptivity. As part of our analysis, we develop a 3-step"Simplify-Equalize-Optimize"approach to analyzing adaptive decision trees, with possible applications to the study of semi-adaptivity in additional stochastic combinatorial optimization problems.

Problem

• Stochastic Knapsack
• Adaptivity Gap
• Semi-Adaptivity
• Combinatorial Optimization
• Approximation

Innovation

• semi-adaptivity
• stochastic knapsack
• adaptivity gap
• approximation algorithm
• decision tree analysis