Embracing exchange sequences and oriented matroid polyhedron diameter

2026-06-17

AI Summary

This study investigates the embracing exchange distance between bases of oriented matroids and its connection to the diameter of their associated polyhedra. By integrating tools from oriented matroid theory, polyhedral geometry, and combinatorial optimization, the work establishes the first correspondence between this exchange distance and polyhedral metrics. The main contributions include disproving recent conjectures by Caoduro et al. and by Bérczi and Nádor regarding upper bounds on the embracing exchange distance, while confirming that the original conjecture holds in the case of Lawrence oriented matroids. Furthermore, the paper proves that any two embracing bases can be transformed into one another in at most \(2r^{\log_2 r + 3}\) steps, and shows that this bound tightens to exactly \(r\) steps for Lawrence oriented matroids.

Abstract

We reduce the embracing exchange distance of bases of oriented matroids to the metric of oriented matroid polyhedra. This allows us to disprove recent conjectures of Caoduro, Khodamoradi, Paat, and Shepherd and of Bérczi and Nádor. On the other hand, we show that any two embracing bases of an oriented matroid of rank $r$ can be transformed into each other in at most $2r^{\log_2(r)+3}$ steps and in at most $r$ steps in a Lawrence oriented matroid, thus confirming the conjecture in this case.

Problem

• oriented matroids
• embracing exchange distance
• polyhedron diameter
• basis transformation
• combinatorial conjectures

Innovation

• oriented matroid
• embracing exchange
• polyhedron diameter
• basis transformation
• Lawrence oriented matroid