Cycle algebras and polytopes of matroids

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.

Classes of cut ideals and their Betti numbers

“…In our particular case of a monomial cut ideal I, the fiber ring F (I) is just the cut algebra of I which was introduced by Sturmfels and Sullivant [21] and has been further studied, in particular, by Römer-Saeedi Madani [16,17] and Koley-Römer [13].…”

“…Sturmfels and Sullivant [21] started an interesting connection to algebraic geometry and commutative algebra by introducing toric cut ideals and -algebras, which have been intensively studied in the last decade. See, e.g., [7,14,16,17,18,19,20].…”