背包秘书问题比秘书问题严格更难
The Knapsack Secretary Problem is Strictly Harder Than the Secretary Problem
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中文总结 AI 辅助
研究背包秘书问题是否能达到经典秘书问题的最优1/e竞争比，通过构造困难实例族分析得出不存在(1/e - 0.0001)竞争比的算法，并给出一种将最佳已知竞争比从0.153提高到0.178的算法。
中文摘要 AI 辅助
背包秘书问题是经典秘书问题的推广，其中被接受的物品必须满足背包约束。一系列工作为此问题开发了常数竞争算法，当前最佳竞争比为0.153。一个自然的开放问题是背包秘书问题是否也能达到经典秘书问题的最优1/e竞争比。我们通过证明不存在(1/e - 0.0001)竞争比的算法来否定回答此问题。对所构造的困难实例族的分析分三步进行。首先，将这些实例上的基数问题简化为几乎序数问题。其次，制定一个线性规划来捕捉几乎序数算法在此实例族上的性能。最后，展示一个目标值严格低于1/e的可行对偶解。我们还给出了一种将最佳已知竞争比从0.153提高到0.178的算法。
英文摘要
The knapsack secretary problem is a generalization of the classical secretary problem where the accepted items must satisfy a knapsack constraint. A line of work has developed constant-competitive algorithms for this problem, with successive improvements culminating in the current best-known competitive ratio of $0.153$. A natural open question was whether the optimal $1/e$ competitive ratio for the classical secretary problem is also achievable for the knapsack secretary problem. We answer this question negatively by showing that no $(1/e - 0.0001)$-competitive algorithm exists for the knapsack secretary problem. The analysis of the family of hard instances we construct proceeds in three steps. First, we reduce the cardinal problem on these instances to an almost-ordinal problem. Second, we formulate a linear program that captures the performance of almost-ordinal algorithms on this instance family. Finally, we exhibit a feasible dual solution whose objective value is strictly below $1/e$. We also give an algorithm that improves the best-known competitive ratio from $0.153$ to $0.178$.
