The $k$-Fold Matroid Secretary Problem
Rishi Gujjar, Kevin Hua, Robert Kleinberg, Frederick V. Qiu
TL;DR
The paper studies the matroid secretary problem under the constraint of a $k$-fold matroid union, where elements arrive in random order and must be selected to form a set in $\mathcal M^k$. It develops a multi-phase thresholding algorithm that uses sampling and a covering-number framework to manage feasibility, leveraging a Nash-Williams–style bound for analysis. The main result shows a $1 - O(\sqrt{\log n / k})$ competitive ratio for any matroid $\mathcal M$, with improvements possible when exploiting parallel-element structure; it also discusses limits for broader unions via distributional hardness results. Overall, the work advances online matroid optimization by generalizing secretary results to redundant constraints and linking combinatorial structure to competitive performance.
Abstract
In the matroid secretary problem, elements $N := [n]$ of a matroid $\mathcal{M} \subseteq 2^N$ arrive in random order. When an element arrives, its weight is revealed and a choice must be made to accept or reject the element, subject to the constraint that the accepted set $S \in \mathcal{M}$. Kleinberg'05 gives a $(1-O(1/\sqrt{k}))$-competitive algorithm when $\mathcal{M}$ is a $k$-uniform matroid. We generalize their result, giving a $(1-O(\sqrt{\log(n)/k}))$-competitive algorithm when $\mathcal{M}$ is a $k$-fold matroid union.