Cycle algebras and polytopes of matroids

TL;DR

This paper explores the algebraic and geometric properties of cycle polytopes of matroids by analyzing their associated toric algebras, revealing new insights into their structure and generators.

Contribution

It introduces the study of cycle algebras of matroid polytopes, examining their ideals, minors, and algebraic retracts, advancing understanding of their algebraic structure.

Findings

Analysis of matroid minors yields algebra retracts.

Determination of degrees of generators of cycle algebra ideals.

Application to cut polytopes and Eulerian subgraph polytopes.

Abstract

Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes associated to graphs. Here we start an algebraic and geometric investigation of these polytopes by studying their toric algebras, called cycle algebras, and their defining ideals. Several matroid operations are considered which determine faces of cycle polytopes that belong again to this class of polyhedral objects. As a key technique used in this paper, we study certain minors of given matroids which yield algebra retracts on the level of cycle algebras. In particular, that allows us to use a powerful algebraic machinery. As an application, we study highest possible degrees in minimal homogeneous systems of generators of defining ideals of cycle algebras as well as interesting cases of cut polytopes and Eulerian subgraph polytopes.