AI 中文总结
研究在仅能访问最优解比较信息时恢复装箱线性规划目标的问题，聚焦分数背包问题，开发多项式时间算法，用$O(n \log(1/\delta)+B^2)$次比较查询恢复物品价值，给出下界，还扩展算法到利润最大化。
AI 中文摘要
我们研究了在算法仅能访问关于不同约束矩阵下最优解的比较信息时，恢复装箱线性规划目标的问题。受比较预言机优化和偏好反馈的启发，通过用序数查询替代最优解的直接观测来强化逆优化框架。聚焦分数背包问题，其装箱线性规划有由物品价格指定的单个预算约束，目标由物品价值决定。算法用两个价格向量查询预言机，返回哪个最优解有更大的总装箱或目标值。对于分数背包，我们开发了一个多项式时间算法，使用$O(n \log(1/\delta)+B^2)$次比较查询来恢复物品价值到比例，其中$n$是物品数量，$B$是背包容量，$\delta$是价值网格分辨率。我们用一个$\Omega(n \log(1/\delta))$的下界进行补充。关键见解是在比较预言机模型中，分数背包和装箱线性规划一样通用。我们的算法通过将约束矩阵行视为价格向量并将其余部分归零来解决装箱设置。对于装箱线性规划，$\Omega(n \log(1/\delta))$的下界仍然成立，使我们的上界在线性因子差距内基本是最优的。最后，我们将算法扩展到利润最大化，得到了Amin等人（AAAI 2015）的显示偏好结果的比较预言机类似物。
英文摘要
We study the problem of recovering the objective of a packing linear program when the algorithm accesses only comparison information about optimal solutions under varying constraint matrices. Motivated by optimization with comparison oracles (Cohen-Addad et al., STOC 2026) and preference feedback (Kaufmann et al., TMLR 2025), this strengthens inverse-optimization frameworks by replacing direct observations of optimal solutions with ordinal queries. We focus on the fractional knapsack problem, where the packing linear program (LP) has a single budget constraint specified by item prices, and the objective is determined by item values. This captures monopoly-pricing where a seller infers a buyer's unknown valuations for divisible items from comparison information. The algorithm queries an oracle with two price vectors, returning which optimal solution has the larger total packing or objective value. Such oracles abstract discrete-choice surveys of buyers choosing between differently priced alternatives. For fractional knapsack, we develop a polynomial-time algorithm recovering item values up to scale using $O(n \log(1/δ)+B^2)$ comparison queries, where $n$ is the number of items, $B$ is the knapsack capacity, and $δ$ is the value grid resolution. We complement this with an $Ω(n \log(1/δ))$ lower bound. A key insight is that in the comparison-oracle model, fractional knapsack is as general as packing LPs. Our algorithm solves the packing setting by treating a constraint matrix row as the price vector and zeroing the rest. The $Ω(n \log(1/δ))$ lower bound continues to hold for packing LPs, making our upper bound essentially best possible, up to a linear-factor gap. Finally, we extend our algorithm to profit-maximization, yielding a comparison-oracle analogue of the revealed-preference result of Amin et al. (AAAI 2015).
