# Enumeration of $2$-level polytopes

## TL;DR

This paper introduces the first algorithm to enumerate all combinatorial types of 2-level polytopes in a given dimension, providing complete results up to dimension 7 using an inductive approach based on facet enumeration.

## Contribution

The paper presents a novel inductive algorithm for enumerating all 2-level polytopes of a fixed dimension, advancing understanding of their combinatorial structure.

## Key findings

- Complete enumeration of 2-level polytopes up to dimension 7
- Development of an inductive enumeration algorithm
- Insights into the structure of 2-level polytopes

## Abstract

A (convex) polytope $P$ is said to be $2$-level if for every direction of hyperplanes which is facet-defining for $P$, the vertices of $P$ can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of $2$-level polytopes of a given dimension $d$, and provide complete experimental results for $d \leqslant 7$. Our approach is inductive: for each fixed $(d-1)$-dimensional $2$-level polytope $P_0$, we enumerate all $d$-dimensional $2$-level polytopes $P$ that have $P_0$ as a facet. This relies on the enumeration of the closed sets of a closure operator over a finite ground set. By varying the prescribed facet $P_0$, we obtain all $2$-level polytopes in dimension $d$.