In mathematics, a matroidpolytope, also called a matroid basis polytope to distinguish it from other polytopes derived from a matroid, is a polytope constructed via the bases of a matroid. Given a matroid, the matroid polytope is the convex hull of the indicator vectors of the bases of.
Definition
Let be a matroid on elements. Given a basis of, the indicator vector of is where is the standard thunit vector in. The matroid polytope is the convex hull of the set
Examples
Let be the rank 2 matroid on 4 elements with bases
Let be the rank 2 matroid on 4 elements with bases that are all 2-element subsets of. The corresponding matroid polytope is the octahedron. Observe that the polytope from the previous example is contained in.
A matroid polytope is contained in the hypersimplex, where is the rank of the associated matroid and is the size of the ground set of the associated matroid. More precisely, the vertices of are a subset of the vertices of.
Every edge of a matroid polytope is a parallel translate of for some, the ground set of the associated matroid. In other words, the edges of correspond exactly to the pairs of bases that satisfy the basis exchange property: for some Because of this property, every edge length is the square root of two. More generally, the families of sets for which the convex hull of indicator vectors has edge lengths one or the square root of two are exactly the delta-matroids.
Let be the rank function of a matroid. The matroid polytope can be written uniquely as a signed Minkowski sum of simplices:
Related polytopes
Independence matroid polytope
The matroid independence polytope or independence matroid polytope is the convex hull of the set The matroid polytope is a face of the independence matroid polytope. Given the rank of a matroid, the independence matroid polytope is equal to the polymatroid determined by.
Flag matroid polytope
The flag matroid polytope is another polytope constructed from the bases of matroids. A flag is a strictly increasing sequence of finite sets. Let be the cardinality of the set. Two matroids and are said to be concordant if their rank functions satisfy Given pairwise concordant matroids on the ground set with ranks, consider the collection of flags where is a basis of the matroid and . Such a collection of flags is a flag matroid. The matroids are called the constituents of. For each flag in a flag matroid, let be the sum of the indicator vectors of each basis in Given a flag matroid, the flag matroid polytope is the convex hull of the set A flag matroid polytope can be written as a Minkowski sum of the matroid polytopes of the constituent matroids:
