Matroid Partition Property and the Secretary Problem

TL;DR

This paper investigates the matroid partition property in relation to the secretary problem, demonstrating limitations of the complete binary matroid and refuting a recent conjecture about its colorability and partition properties.

Contribution

It proves that the complete binary matroid does not satisfy the alpha-partition property for any constant alpha and refutes a conjecture regarding its colorability and partition matroid reduction.

Findings

Complete binary matroid lacks alpha-partition property for any constant alpha.

Refutes conjecture that the matroid can be reduced to a sublinear alpha-colorable partition matroid.

Shows the matroid is 2^d/d-colorable but not reducible to a sublinear alpha 2^d/d-colorable partition matroid.

Abstract

A matroid $\mathcal{M}$ on a set $E$ of elements has the $\alpha$-partition property, for some $\alpha>0$, if it is possible to (randomly) construct a partition matroid $\mathcal{P}$ on (a subset of) elements of $\mathcal{M}$ such that every independent set of $\mathcal{P}$ is independent in $\mathcal{M}$ and for any weight function $w:E\to\mathbb{R}_{\geq 0}$, the expected value of the optimum of the matroid secretary problem on $\mathcal{P}$ is at least an $\alpha$-fraction of the optimum on $\mathcal{M}$. We show that the complete binary matroid, ${\cal B}_d$ on $\mathbb{F}_2^d$ does not satisfy the $\alpha$-partition property for any constant $\alpha>0$ (independent of $d$). Furthermore, we refute a recent conjecture of B\'erczi, Schwarcz, and Yamaguchi by showing the same matroid is $2^d/d$-colorable but cannot be reduced to an $\alpha 2^d/d$-colorable partition matroid for any $\alpha$ that is sublinear in $d$.