Counting vertices of integral polytopes defined by facets

TL;DR

This paper investigates the computational complexity of counting vertices in integral polytopes, especially those with small integer coordinates like 0/1 and half-integral polytopes, revealing various complexity results.

Contribution

It provides new complexity results for counting vertices in integral polytopes with small integer vertices, expanding understanding of these computational problems.

Findings

Complexity results vary depending on polytope type

Counting vertices in 0/1 polytopes has specific computational complexity

Results highlight challenges in counting vertices of half-integral polytopes

Abstract

We present a number of complexity results concerning the problem of counting vertices of an integral polytope defined by a system of linear inequalities. The focus is on polytopes with small integer vertices, particularly 0/1 polytopes and half-integral polytopes.