强秘书猜想对线性拟阵成立

中文摘要 AI 辅助

我们证明了线性拟阵上的拟阵秘书问题具有 $1/e$ 的保证，从而解决了这类拟阵中的强秘书猜想。该结果在拟阵预先已知以及在线给出有限域上的线性表示这两种情况下均成立。在已知拟阵模型中，该结果更广泛地适用于允许有限模扩张的拟阵。固定最优基中的每个元素被选中的概率至少为 $1/e$。$\mathbf{\text{并发发现披露：}}$ 本手稿中主要结果的证明是在 2026 年 9 月 15 日星期二太平洋夏令时凌晨 1:02 与 ChatGPT-6 Astra 的对话中获得的。随后我们准备了这份手稿以供公开发布，并打算在 2026 年 9 月 17 日星期四上午上传。在 9 月 17 日凌晨完成提交时，我们发现了 Abdi、Banihashem、Hajiaghayi 和 Mittal 于 2026 年 9 月 16 日上传的手稿 https://arxiv.org/abs/2609.19118，该手稿通过本质上相同的方法包含了相同的结果。尽管如此，我们仍分享我们的手稿，以防我们的阐述对社区具有独立效用，并希望这一经历能引发关于人工智能时代并发发现的更广泛讨论。

英文摘要

We prove a $1/e$ guarantee for the matroid secretary problem on linear matroids, therefore settling the strong secretary conjecture in this class of matroids. The result holds both when the matroid is known in advance and when a linear representation over a finite field is given online. In the known-matroid model, the result extends more generally to matroids admitting a finitary modular extension. Each element of a fixed optimal basis is selected with probability at least $1/e$. $\mathbf{\text{Concurrent Discovery Disclosure:}}$ The proof of the main result in this manuscript was obtained in a conversation with ChatGPT-6 Astra on Tuesday, September 15, 2026 at 1:02 AM PDT. We then prepared this manuscript for public release, with the intent of uploading it on the morning of Thursday, September 17, 2026. In the early morning hours of September 17, while finalizing the submission, we discovered the manuscript https://arxiv.org/abs/2609.19118 of Abdi, Banihashem, Hajiaghayi, and Mittal, uploaded on September 16, 2026, which contains the same result via an essentially identical approach. We are sharing our manuscript nonetheless in case our exposition is of independent utility to the community, and we hope this experience stimulates broader discussion about concurrent discovery in the AI era.

发表机构

• HUN-REN–ELTE Egerváry Research Group, Department of Operations Research, Eötvös Loránd University（匈牙利科学院-罗兰大学埃格瓦里研究组，运筹学系，罗兰大学）
• HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）
• Department of Computer Science, University of Southern California（南加州大学计算机科学系）
• Department of Mathematical Engineering and Center for Mathematical Modeling, Universidad de Chile（智利大学数学工程系与数学建模中心）
• Institute for Mathematical and Computational Engineering, and Department of Industrial and Systems Engineering, Pontificia Universidad Católica de Chile（智利天主教大学数学与计算工程研究所，工业与系统工程系）

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