AI 中文总结
本文针对带非负单调次模目标的拟阵秘书问题，提出了竞争比为8.699的算法，通过划分学习集、计算元素固定权重并调用线性目标拟阵秘书子程序，完成了精细分析。
AI 中文摘要
我们研究带非负单调次模目标的拟阵秘书问题，元素以均匀随机顺序到达，每次接受决策即时且不可撤销。我们为任意拟阵给出了一个竞争比为8.699的算法：该算法首先拒绝到达序列中随机长度的初始段作为学习集，并对该集合中的所有元素运行次模贪心算法；后续每个元素会获得一个固定权重，等于其插入存储的贪心序列时的边际价值，算法基于这些固定权重运行线性目标拟阵秘书子程序。该算法使用O(nr)次价值查询和O(n²)次独立性查询，其中r为拟阵的秩。
英文摘要
We study the matroid secretary problem with a nonnegative monotone submodular objective. Elements arrive in uniformly random order, and every acceptance decision is immediate and irrevocable. We give an $8.699$-competitive algorithm for arbitrary matroids. The algorithm first rejects a randomly sized initial segment of the arrival sequence as a learning set and runs submodular greedy on all elements in this set. Each later element receives a fixed weight equal to the marginal value it would have when inserted into the stored greedy sequence. On these fixed weights, the algorithm runs a linear-objective matroid secretary subroutine. The algorithm uses $O(nr)$ value queries and $O(n^2)$ independence queries, where $r$ is the matroid rank.
