Computing the Volume of Polytropes

TL;DR

The paper tackles exact volume computation for polytropes by marrying Jim Lawrence's polytope volume algorithm with tropical geometry tools. It leverages a tropical Cramer rule and Kleene-star representations to efficiently locate pseudovertices, which are then summed via $N_v$ to obtain the volume through $\mathrm{vol}(P)=\sum_v N_v$. A key theoretical contribution is that polytrope facet normals form a totally unimodular incidence structure, yielding $\delta_v=1$ and enabling a streamlined, faster algorithm with asymptotic complexity $O(d^2 4^d)$. The authors validate the approach with 2D and 3D examples, confirm agreement with Polymake, and provide extensive appendices with higher-dimensional calculations implemented in Python. This work offers a practical pathway to exact polytrope volumes and demonstrates how tropical methods can reduce computational burden in polyhedral volume problems.

Abstract

We apply an algorithm for measuring the volume of polytopes described by Jim Lawrence to polytropes. By using a tropical form of Cramer's rule, we found an efficient way to find all pseudovertices which are necessary for computing the volume. Due to the limited possibilities for hyperplanes of polytropes, this led to a simplification of the algorithm, decreasing the time complexity significantly.

Figures (4)

• Figure 1: Tropical convex hulls
• Figure 2: Kleene Star
• Figure 3: Stable intersection of hyperplanes
• Figure 4:

Theorems & Definitions (11)

• Definition 1.1
• Definition 1.2
• Definition 2.1
• Theorem 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• proof : Proof of Theorem 3.1