随机背包问题：半适应性间隙与改进的近似
Stochastic Knapsack: Semi-Adaptivity Gaps and Improved Approximation
AI总结：
研究随机背包问题中的半适应性间隙，改进适应性间隙的上下界，并通过三步方法分析适应决策树。
摘要：
在随机组合优化中，算法的适应性不同：是否查询已实现的随机性并适应它。Dean等人（FOCS '04）正式化了适应性间隙，比较完全适应策略与非适应策略的性能。我们重新审视Dean等人提出的根本随机背包问题，其中物品有确定性价值和独立的随机尺寸。一种策略按顺序打包物品，在第一个背包溢出或之前停止。我们关注具有挑战性的风险变种，其中溢出将放弃所有累积价值，并通过半适应性视角研究该问题：我们通过0-k半适应性间隙（k-半适应策略与非适应策略之间的差距）和k-n半适应性间隙（完全适应与k-半适应策略之间的差距）的概念来衡量k次适应查询的权力。我们的第一个贡献是通过给出更紧的上界和下界来改进Dean等人的经典结果。我们的第二个贡献是开发一种更平滑的非适应与完全适应策略之间的插值，其理由是当完全适应性不现实（由于其复杂性或查询成本）时，有限的适应性可能是一个有吸引力的中间地带。我们量化了1-n和k-n半适应性间隙，显示k次查询如何近似完全适应策略。我们通过量化0-1半适应性间隙来补充这些界限，即投资一次查询相对于无适应性的改进。作为分析的一部分，我们开发了“简化-均等-优化”三步方法来分析适应决策树，可能应用于其他随机组合优化问题的半适应性研究。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
