arXiv:2609.20797 · v2

The Strong Secretary Conjecture is True for Linear Matroids

首次提交 2026-09-18 01:54

版本更新 2026-11-06 07:56

雷达观测 2026-11-09 10:16

摘要

We prove a $1/e$ guarantee for Matroid Secretary on linear matroids: each element of a fixed optimal basis is selected with probability at least $1/e$. This settles the strong secretary conjecture for this class of matroids. Our result holds both when the matroid is known in advance and when a linear representation is given online. In the known-matroid model, the result extends to fully modular extendable matroids. We also obtain $1/e$ guarantees for a more general model on linear matroids, both when the matroid is known and when its representation is revealed online. In XOS Matroid Secretary, agents with XOS valuations over the elements arrive in uniformly random order, and the algorithm must assign each arriving agent a bundle of elements. The bundles must be pairwise disjoint, and their union must be an independent set of the matroid. We obtain an expected total value of at least $1/e$ times the offline optimum. This model contains Group Matroid Secretary, in which the elements are adversarially partitioned into groups that arrive in uniformly random order, and the algorithm selects a subset of each arriving group. For this special case, we select each element of a fixed optimal basis with probability at least $1/e$