Extension of Simple Algorithms to the Matroid Secretary Problem

TL;DR

This paper explores extending simple algorithms to the Matroid Secretary Problem, analyzing their effectiveness and limitations, including competitive ratios and structural constraints on algorithms with forbidden sets.

Contribution

It generalizes simple algorithms from classical secretary problems to matroids, analyzing their performance and limitations, including competitive ratios and forbidden set constraints.

Findings

A generalized Virtual Algorithm achieves a constant competitive ratio on the Hat Graph.

One sample-based algorithm is an instance of the Greedy Algorithm, while the other is not.

No algorithm with Strong Forbidden Sets of size 1 exists for all graphic matroids.

Abstract

Whereas there are simple algorithms that are proven to be optimal for the Classical and the Multiple Choice Secretary Problem, the Matroid Secretary Problem is less thoroughly understood. This paper proposes the generalization of some simple algorithms from the Classical and Multiple Choice versions on the Matroid Secretary Problem. Out of two algorithms that make decisions based on samples, like the Dynkin's algorithm, one is proven to be an instance of Greedy Algorithm (Bahrani et al., 2022), while the other is not. A generalized version of the Virtual Algorithm (Babaioff et al., 2018) obtains a constant competitive ratio for the Hat Graph, the adversarial example for Greedy Algorithms, but fails to do so when a slight modificiation is introduced to the graph. We show that there is no algorithm with Strong Forbidden Sets (Soto et al., 2021) of size 1 on all graphic matroids.