Reconfiguring Ordered Bases of a Matroid

TL;DR

This paper characterizes when one ordered basis of a matroid can be transformed into another through basis exchange steps, providing bounds on the number of steps needed, with improvements for graphic matroids.

Contribution

It establishes necessary and sufficient conditions for reconfiguring ordered bases and derives bounds on the number of exchange steps, including improved bounds for graphic matroids.

Findings

Reconfiguration is possible iff initial and final elements with the same label are in the same component.

Number of steps needed is $O(r^{1.5})$ for general matroids.

Improved bound of $O(r \, \log r)$ for graphic matroids.

Abstract

For a matroid with an ordered (or "labelled") basis, a basis exchange step removes one element with label $l$ and replaces it by a new element that results in a new basis, and with the new element assigned label $l$. We prove that one labelled basis can be reconfigured to another if and only if for every label, the initial and final elements with that label lie in the same connected component of the matroid. Furthermore, we prove that when the reconfiguration is possible, the number of basis exchange steps required is $O(r^{1.5})$ for a rank $r$ matroid. For a graphic matroid we improve the bound to $O(r \log r)$.