The Strong Secretary Conjecture is True for Linear Matroids

2026-09-17

AI Summary

该研究解决了线性拟阵上的秘书问题，证明了强秘书猜想，采用的方法保证了每个最优基元素被选中的概率至少为1/e。

Abstract

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.

Problem

• matroid secretary problem
• linear matroids
• strong secretary conjecture

Innovation

• Strong Secretary Conjecture
• Linear Matroids
• Matroid Secretary Problem
• Finitary Modular Extension