On Computing the Shadows and Slices of Polytopes

TL;DR

This paper investigates the computational complexity of projecting polytopes along multiple directions, revealing equivalences to vertex enumeration and NP-hardness in various forms, and introduces new complexity classes related to vertex enumeration.

Contribution

It characterizes the complexity of polytope projection problems in different input and output scenarios, and introduces new complexity classes based on vertex enumeration.

Findings

Projection is equivalent to vertex enumeration in most forms.

In some cases, projection problems are NP-hard or trivial.

New complexity classes related to vertex enumeration are proposed.

Abstract

We study the complexity of computing the projection of an arbitrary $d$-polytope along $k$ orthogonal vectors for various input and output forms. We show that if $d$ and $k$ are part of the input (i.e. not a constant) and we are interested in output-sensitive algorithms, then in most forms the problem is equivalent to enumerating vertices of polytopes, except in two where it is NP-hard. In two other forms the problem is trivial. We also review the complexity of computing projections when the projection directions are in some sense non-degenerate. For full-dimensional polytopes containing origin in the interior, projection is an operation dual to intersecting the polytope with a suitable linear subspace and so the results in this paper can be dualized by interchanging vertices with facets and projection with intersection. To compare the complexity of projection and vertex enumeration, we define new complexity classes based on the complexity of Vertex Enumeration.