Strong Matroid Secretary Conjecture for General Matroids

Determine whether every matroid admits a matroid secretary algorithm that selects each element of a maximum-weight basis with probability at least the optimal rank-one guarantee of 1/e.

Background

The paper studies the strong version of the Matroid Secretary Problem, in which the guarantee is required separately for every element of the maximum-weight basis rather than only in expectation over the total weight. The benchmark guarantee is 1/e, which is optimal for the classical rank-one secretary problem.

The paper proves this 1/e probability-competitive guarantee for linear matroids, including both the known-matroid and online-representation models, and extends it in the known-matroid model to matroids admitting a finitary modular extension. However, the corresponding statement for arbitrary matroids is explicitly identified as unresolved. Singla's result provides a constant 1/4 guarantee for arbitrary matroids but does not establish the conjectured 1/e bound.

References

This answers the long-standing question of whether arbitrary matroids admit a constant-factor competitive guarantee, while the strong Matroid Secretary Conjecture remains open.

— The Strong Secretary Conjecture is True for Linear Matroids  (2609.20797 - Bérczi et al., 17 Sep 2026) in Section 1, Related Work, paragraph “Singla's algorithm”

The strong matroid secretary conjecture asserts that every matroid admits a $1/e$-competitive secretary algorithm, matching the classical single-choice guarantee.

— On the Strong Matroid Secretary Conjecture and Beyond  (2609.19118 - Abdi et al., 16 Sep 2026) in Abstract; Introduction, Conjecture 1 (Strong matroid secretary conjecture)