Theta rank, levelness, and matroid minors

  title={Theta rank, levelness, and matroid minors},
  author={Francesco Grande and Raman Sanyal},
  journal={J. Comb. Theory, Ser. B},
Enumeration of 2-level polytopes
The approach is based on the notion of a simplicial core, that allows the problem to be reduced to the enumeration of the closed sets of a discrete closure operator, along with some convex hull computations and isomorphism tests.
Equivariant Semidefinite Lifts of Regular Polygons
This paper shows that one can construct an equivariant psd lift of the regular 2^n-gon of size 2n-1, which is exponentially smaller than the psd Lift of the sum-of-squares hierarchy, and proves that the construction is essentially optimal.
Many 2-Level Polytopes from Matroids
The counting results are used to show that the number of combinatorially non-equivalent 2-level polytopes is bounded from below by c·n-5/2·ρ-n, which brings to light some structural properties of 2- level matroids and exploits them for enumerative purposes.
Two-Level Polytopes with a Prescribed Facet
It is obtained, for the first time, the complete list of d-dimensional 2-level polytopes up to affine equivalence for dimension d = 7, and geometric constructions that the authors call suspensions play a prominent role in both theoretical and experimental results.
Small Shadows of Lattice Polytopes
. The diameter of the graph of a d -dimensional lattice polytope P ⊆ [0 ,k ] n is known to be at most dk due to work by Kleinschmidt and Onn. However, it is an open question whether the monotone
On 2-Level Polytopes Arising in Combinatorial Settings
A trade-off formula for the number of cliques and stable sets in a graph; a description of stable matching poly topes as affine projections of certain order polytopes; and a linear-size description of the base polytope of matroids that are 2-level in terms of cuts of an associated tree are presented.
Extended Formulations for Independence Polytopes of Regular Matroids
We show that the independence polytope of every regular matroid has an extended formulation of size quadratic in the size of its ground set. This generalizes a similar statement for (co-)graphic
Binary scalar products
The Slack Realization Space of a Polytope
The slack ideal offers an effective computational framework for several classical questions about polytopes such as rational realizability, non-prescribability of faces, and realizing the combinatorics of combinatorial poly topes.


Theta Bodies for Polynomial Ideals
A hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal called theta bodies of the ideal is introduced and a geometric description of the first theta body for all ideals is given.
Matroid polytopes, nested sets and Bergman fans
The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present
Lifts of Convex Sets and Cone Factorizations
This paper addresses the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone and shows that the existence of a lift of a conveX set to a cone is equivalent to theexistence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual.
A new semidefinite programming hierarchy for cycles in binary matroids and cuts in graphs
The theta bodies of the vanishing ideal of cycles in a binary matroid are constructed and applied to cuts in graphs, this yields a new hierarchy of semidefinite programming relaxations of the cut polytope of the graph.
A combinatorial model for series-parallel networks
The category of pregeometries with basepoint is defined and explored. In this category two important operations are extensively characterized: the series connection S(G, H), and the parallel
Polytopes of Minimum Positive Semidefinite Rank
This paper shows that the psd rank of a polytope is at least the dimension of the polytopes plus one, and characterize those polytopes whose pSD rank equals this lower bound.
Graph Minors: XV. Giant Steps
Abstract Let G be a graph with a subgraph H drawn with high representativity on a surface Σ . When can the drawing of H be extended “up to 3-separations” to a drawing of G in Σ if we permit a bounded
Submodular Functions, Matroids, and Certain Polyhedra
  • J. Edmonds
  • Mathematics
    Combinatorial Optimization
  • 2001
The viewpoint of the subject of matroids, and related areas of lattice theory, has always been, in one way or another, abstraction of algebraic dependence or, equivalently, abstraction of the
The regular matroids with no 5-wheel minor
  • J. Oxley
  • Mathematics
    J. Comb. Theory, Ser. B
  • 1989