拟阵秘书问题的3.7321竞争比算法
A 3.1462-Competitive Algorithm for Matroid Secretary
- University of Nebraska–Lincoln（内布拉斯加大学林肯分校）
- Rensselaer Polytechnic Institute（伦斯勒理工学院）
- Beijing Normal University–Zhuhai（北京师范大学珠海校区）
- Beijing Normal–Hong Kong Baptist University（北京师范大学-香港浸会大学联合国际学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对拟阵秘书问题，提出一种改进的2+√3竞争比算法，通过修改Singla的可逆参考过程并保留储备，实现至少2-√3的接受概率，使用O(n^2)次独立性查询。
AI中文摘要：
拟阵秘书问题要求在线算法从按均匀随机顺序到达的元素中选择一个高权重的独立集，且决策必须即时且不可撤销。Singla（2026）最近针对任意拟阵给出了一个4竞争比算法，该算法仅使用元素数量和针对已到达元素的独立性查询。遵循他的方法，我们在相同的信息模型下获得了改进的竞争比$2+\sqrt3\approx3.7321$。我们的算法以至少$2-\sqrt3$的概率接受固定规范最优解中的每个元素，并使用$O(n^2)$次独立性查询。该算法修改了Singla的可逆参考过程，保留样本中随机选择的一部分作为储备，其参考贪心解中的成员资格不被冻结。平衡剩余样本和样本后元素保持了可逆性，并允许精确计算交换伙伴阻止目标元素的概率。由此产生的保证具有直接的分析证明。
英文摘要：
The matroid secretary problem asks an online algorithm to select a high-weight independent set from elements arriving in uniformly random order, with immediate and irrevocable decisions. Singla (2026) recently gave a $4$-competitive algorithm for arbitrary matroids using only the number of elements and independence queries on already-arrived elements. Following his approach of maintaining a dynamically updated reference set, we introduce random deletions and a time-dependent acceptance rule, improving the competitive ratio to $C_*\approx3.1462$ in the same information model, where $C_*-\log C_*=2$. Our ordinal algorithm accepts every element of a fixed canonical optimum with probability exactly $1/C_*$ and uses $O(n^2)$ independence queries. The algorithm maintains a greedy reference solution, protects only accepted elements, and randomly deletes unaccepted reference-basis elements. A time-dependent acceptance rule makes the accepted set, conditional on the reference set, an independent thinning of its greedy basis. The resulting guarantee has a direct analytic proof.
