Multilevel Dynamic Thinning for Matroid Secretary

In the matroid secretary problem, weighted elements arrive in random order, and an online algorithm must irrevocably accept elements forming a high-weight independent set. Dynamic Thinning is a recent, conceptually simple $3.1462$-competitive algorithm for the matroid secretary problem that maintains a random reference set. Given this reference set, the elements in its maximum-weight independent subset have been accepted independently with a time-dependent probability. We extend this approach by replacing the single reference set with a finite hierarchy of nested reference sets. For every fixed $\text{eps}>0$, the resulting algorithm is $(e+\text{eps})$-probability-competitive for arbitrary matroids, with $O_{\text{eps}}(n^2)$ queries in the worst case.