[论文解读] Stochastic Knapsack: Semi-Adaptivity Gaps and Improved Approximation

Zohar Barak, Inbbal Talgam-Cohen|arXiv (Cornell University)|Feb 27, 2026

Optimization and Search Problems被引用 0

一句话总结

本文研究风险-随机背包问题的半自适应策略，改进自适应性差距界，并提供常数个自适应查询策略，在近似性方面优于以往工作。

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.

研究动机与目标

• 重新考察在溢出时所有累计价值损失的随机背包问题。
• 量化通过0-k与k-n半自适应差距体现的有限自适应能力，特别是对常数k的情况。
• 开发多项式时间算法，在少量自适应查询的条件下实现改进的近似比。
• 在非自适应与完全自适应策略之间提供更平滑的插值。
• 引入一个3步法来分析自适应决策树，该方法可应用于其他随机性问题。

提出的方法

• 为Risky-SK与Non-Risky-SK定义并分析0-k与k-n半自适应差距。
• 设计简单的非自适应策略并通过加入一次自适应选择来提升保证。
• 通过epsilon阈值将物品分成小/大两类，并提出带有k个自适应查询的策略。
• 利用基于LP的上界和简单贪婪法来界定自适应性差距。
• 应用3步法Simplify-Equalize-Optimize框架来界定差距并推导最坏情形实例。
• 通过聚焦高度为k的自适应树的实例族来证明界限。

实验结果

研究问题

• RQ1 限制自适应查询数量对带风险溢出的随机背包问题性能有何影响？
• RQ2 Risky-SK与Non-Risky-SK的0-k与k-n半自适应差距能达到多大？
• RQ3 是否可用常数数量的自适应选择在一个小的加法因子内近似完全自适应策略？
• RQ4 在半自适应下，小物品与大物品如何影响可实现的近似？
• RQ5 单一自适应选择增益（0-1差距）的紧性以及1-n差距的界限有多紧？

主要发现

• Risky-SK的0-n全自适应差距：上界8.47，下界2。
• Risky-SK的0-1半自适应差距：上界与下界均为1.69。
• Risky-SK的1-n半自适应差距：上界8.26，下界1.18。
• Risky-SK的k-n半自适应差距，当k = ~O(1/ε)时：上界6.44 + sqrt(ε)。
• 对于Non-Risky-SK，相关差距得到改进，在ε-噪声伯努利情形下上界为2，相关实例中的下界为1.37。
• 结果表明，固定数量的自适应查询可以显著优于非自适应策略，且0-k与k-n差距的乘积对标准自适应差距提供上界。