A Basis for Slicing Birkhoff Polytopes

TL;DR

This paper introduces a new basis for Birkhoff polytopes that facilitates more efficient volume calculations via slicing, with a method to construct this basis for any Birkhoff polytope and potential future research directions.

Contribution

It proposes a novel basis derived from special matrices to improve volume computation of Birkhoff polytopes, enhancing existing slicing methods.

Findings

New basis simplifies volume calculations

Method for constructing basis for any Birkhoff polytope

Examples demonstrating the basis's application

Abstract

We present a change of basis that may allow more efficient calculation of the volumes of Birkhoff polytopes using a slicing method. We construct the basis from a special set of square matrices. We explain how to construct this basis easily for any Birkhoff polytope, and give examples of its use. We also discuss possible directions for future work.