A lower bound technique for triangulations of simplotopes

TL;DR

This paper introduces new lower bound techniques for triangulating simplotopes, providing the first bounds for products of three or more simplices and demonstrating their application through specific examples.

Contribution

It develops novel methods to derive lower bounds for simplotopes triangulations, including those with interior vertices, and establishes minimal triangulations for certain products.

Findings

Minimal triangulation of two simplices is vertex-only.

First known lower bounds for products of three or more simplices.

Constructed a size 10 triangulation of a triangle-square product.

Abstract

Products of simplices, called simplotopes, and their triangulations arise naturally in algorithmic applications in game theory and optimization. We develop techniques to derive lower bounds for the size of simplicial covers and triangulations of simplotopes, including those with interior vertices. We establish that a minimal triangulation of a product of two simplices is given by a vertex triangulation, i.e., one without interior vertices. For products of more than two simplices, we produce bounds for products of segments and triangles. Aside from cubes, these are the first known lower bounds for triangulations of simplotopes with three or more factors, and our techniques suggest extensions to products of other kinds of simplices. We also construct a minimal triangulation of size 10 for the product of a triangle and a square using our lower bound.