Online Matroid Embeddings

TL;DR

This paper introduces online matroid embeddings to relate different variants of the matroid secretary problem, establishing equivalences and extending understanding of online matroid algorithms.

Contribution

It proves the existence of online embeddings for binary matroids and links various matroid secretary problem variants through these embeddings.

Findings

Established the existence of online embeddings for binary matroids

Showed equivalence between different matroid secretary problem variants

Connected known and unknown matroid structures in online settings

Abstract

We introduce the notion of an online matroid embedding, which is an algorithm for mapping an unknown matroid that is revealed in an online fashion to a larger-but-known matroid. We establish the existence of such an embedding for binary matroids, and use it to relate variants of the binary matroid secretary problem to each other, showing that seemingly simpler problems are in fact equivalent to seemingly harder ones (up to constant-factors). Specifically, we show this to be the case for the version of the matroid secretary problem in which the matroid is not known in advance, and where it is known in advance. We also show that the version with known matroid structure, is equivalent to the problem where weights are not fully adversarial but drawn from a known pairwise-independent distribution.