Enumeration of 2-level polytopes

摘要

A(convex) polytope P is said to be 2-level if for each hyperplane H that supports a facet of P, the vertices of P can be covered with H and exactly one other translate of H. 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 ≤ 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.