背包秘书问题不具有1/e竞争比

中文摘要 AI 辅助

我们证明了背包秘书问题不存在1/e竞争比的算法。背包秘书问题最早由Babaioff等人于2007年提出，此后竞争比虽有诸多改进，但1/e的不可能性障碍仍未改变。许多秘书问题的组合变体，包括背包秘书问题，通过将单选择问题作为特殊情况嵌入，继承了1/e的不可能性。我们为1 - B背包秘书问题（一般背包秘书问题的特殊情况）构造了一族困难实例，以改进现有的不可能性结果。在这种特殊情况下，我们表明竞争比至多为0.36437 < 1/e - 0.0035。我们的构造类似于Abels等人（2022年）所使用的构造，他们证明了对于序数算法（仅知道物品的相对排名），1/(1 + e)是不可能的。我们的工作解决了他们的一个开放问题，表明即使在1 - B背包秘书问题的基数情况下，也无法达到1/e。我们用一个简单的1 - B背包秘书算法补充了我们的不可能性结果，对于每个固定的B≥2，该算法具有(1/5.10 - o(1))的竞争比。这改进了将通用随机顺序背包算法应用于这种特殊情况所获得的保证。

英文摘要

We prove that no algorithm for the knapsack secretary problem can be $1/e$-competitive. The knapsack secretary problem was first introduced by Babaioff, Immorlica, Kempe, and Kleinberg (2007). There have been many improvements to the achievable competitive ratio since then, but the $1/e$ impossibility barrier has remained unchanged. Many combinatorial variants of the secretary problem, including knapsack secretary, inherit the $1/e$ impossibility by embedding the single-choice problem as a special case. We construct a family of hard instances for the $1$-$B$ knapsack secretary problem, which is a special case of the general knapsack secretary problem, to improve the existing impossibility result. We show in this special case that the competitive ratio is at most $0.36437 < \frac{1}{e} - 0.0035$. Our construction is similar to the one used by Abels, Ladewig, Schewior, and Stinzendörfer (2022), for which they show an impossibility of $1/(1+e)$ for ordinal algorithms, where only the relative ranks of the items are known. Our work resolves an open question of theirs by showing that $1/e$ cannot be achieved even in the cardinal case of the $1$-$B$ knapsack secretary problem. We complement our impossibility result with a simple algorithm for $1$-$B$ knapsack secretary that is $(1/5.10-o(1))$-competitive for every fixed $B \geq 2$. This improves the guarantee obtained by applying general-purpose random-order knapsack algorithms to this special case.

补充信息

备注

34 pages, 1 figure