The $k$-Fold Matroid Secretary Problem

TL;DR

This paper extends the matroid secretary problem to $k$-fold matroid unions, providing a new competitive algorithm that improves upon previous results for uniform matroids.

Contribution

It generalizes Kleinberg's algorithm from $k$-uniform to $k$-fold matroid unions, achieving a better competitive ratio.

Findings

Achieves a $(1-O(\sqrt{rac{\log(n)}{k}}))$-competitive ratio.

Extends the matroid secretary problem to more complex matroid structures.

Provides a new algorithm with improved performance for $k$-fold matroid unions.

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.