Primitive Zonotopes

TL;DR

This paper introduces a new family of polytopes called primitive zonotopes, exploring their properties and connections to combinatorial optimization and polytope diameter problems.

Contribution

It generalizes the permutahedron of type B_d and links these polytopes to diameter bounds and multicriteria matroid optimization complexity.

Findings

Primitive zonotopes extend known polytopes like permutahedra.

Connections established between these polytopes and diameter bounds.

Implications for multicriteria matroid optimization complexity.

Abstract

We introduce and study a family of polytopes which can be seen as a generalization of the permutahedron of type $B_d$. We highlight connections with the largest possible diameter of the convex hull of a set of points in dimension $d$ whose coordinates are integers between $0$ and $k$, and with the computational complexity of multicriteria matroid optimization.