Volumes and Tangent Cones of Matroid Polytopes

TL;DR

This paper investigates the complexity of computing Ehrhart polynomials of matroid polytopes, showing that for uniform matroids with varying rank, the number of simplicial cones grows exponentially, indicating computational challenges.

Contribution

It demonstrates that the number of simplicial cones in tangent cone triangulations grows exponentially when the rank varies, limiting the efficiency of certain algorithms.

Findings

Number of simplicial cones is polynomial in fixed rank

Exponential growth of cones when rank varies

Triangulation relates to spanning thrackles of bipartite graphs

Abstract

De Loera et al. 2009, showed that when the rank is fixed the Ehrhart polynomial of a matroid polytope can be computed in polynomial time when the number of elements varies. A key to proving this is the fact that the number of simplicial cones in any triangulation of a tangent cone is bounded polynomially in the number of elements when the rank is fixed. The authors speculated whether or not the Ehrhart polynomial could be computed in polynomial time in terms of the number of bases, where the number of elements and rank are allowed to vary. We show here that for the uniform matroid of rank $r$ on $n$ elements, the number of simplicial cones in any triangulation of a tangent cone is $n-2 \choose r-1$. Therefore, if the rank is allowed to vary, the number of simplicial cones grows exponentially in $n$. Thus, it is unlikely that a Brion-Lawrence type of approach, such as Barvinok's Algorithm, can compute the Ehrhart polynomial efficiently when the rank varies with the number of elements. To prove this result, we provide a triangulation in which the maximal simplicies are in bijection with the spanning thrackles of the complete bipartite graph $K_{r,n-r}$.