A 3.1462-Competitive Algorithm for Matroid Secretary
TL;DR
The ordinal algorithm accepts every element of a fixed canonical optimum with probability exactly $1/C_*$ and uses O(n^2)$ independence queries and the resulting guarantee has a direct analytic proof.
Abstract
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.