A note on embracing exchange sequences in oriented matroids

TL;DR

This paper generalizes a convex geometry problem to oriented matroids, proposing a conjecture on exchange sequences and proving it for a specific class, linking it to longstanding open problems.

Contribution

It introduces a matroidal framework for exchange sequences, formulates a new conjecture, and proves it for oriented graphic matroids, connecting to open problems in matroid theory.

Findings

Proposed a conjecture on minimum length of exchange sequences in oriented matroids.

Proved the conjecture for oriented graphic matroids of directed graphs.

Explored connections to open problems in exchange sequences in unoriented matroids.

Abstract

An open problem in convex geometry asks whether two simplices $A,B\subseteq\mathbb{R}^d$, both containing the origin in their convex hulls, admit a polynomial-length sequence of vertex exchanges transforming $A$ into $B$ while maintaining the origin in the convex hull throughout. We propose a matroidal generalization of the problem to oriented matroids, concerning exchange sequences between bases under sign constraints on elements appearing in certain fundamental circuits. We formulate a conjecture on the minimum length of such a sequence, and prove it for oriented graphic matroids of directed graphs. We also study connections between our conjecture and several long-standing open problems on exchange sequences between pairs of bases in unoriented matroids.