Combinatorial Polytope Enumeration

TL;DR

This paper introduces a provably complete algorithm for generating all combinatorially distinct simple polytopes in R^d, providing valuable tools for solving longstanding problems in polytope theory.

Contribution

It presents a novel algorithm that generates a tight superset of simple polytopes using cutting planes and planar sweeps, advancing polytope enumeration methods.

Findings

Algorithm is provably complete and exact.

Enables exploration of polytope enumeration conjectures.

Impacts understanding of polytope graph properties.

Abstract

We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and face lattices. The technique applies repeated cutting planes and planar sweeps to a d-simplex. Our generator has implications for several outstanding problems in polytope theory, including conjectures about the number of distinct polytopes, the edge expansion of polytopal graphs, and the d-step conjecture.