A linear optimization oracle for zonotope computation

TL;DR

This paper introduces efficient algorithms utilizing a linear optimization oracle to compute and analyze zonotopes, including vertex enumeration, generator recovery, and zonotope recognition, advancing computational geometry methods.

Contribution

The paper presents novel algorithms for zonotope vertex computation and generator recovery using a linear optimization oracle, improving efficiency in geometric enumeration tasks.

Findings

Algorithms successfully compute zonotope vertices from generators.

Methods recover generators from zonotope vertices.

Approach can identify and extract zonotopal parts of polytopes.

Abstract

A class of counting problems ask for the number of regions of a central hyperplane arrangement. By duality, this is the same as counting the vertices of a zonotope. We give several efficient algorithms, based on a linear optimization oracle, that solve this and related enumeration problems. More precisely, our algorithms compute the vertices of a zonotope from the set of its generators and inversely, recover the generators of a zonotope from its vertices. A variation of the latter algorithm also allows to decide whether a polytope, given as its vertex set, is a zonotope and when it is not, to compute its greatest zonotopal summand.