Enumeration of 2-level polytopes

Abstract

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 $d \leqslant 7$. 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.

Appendix: On the implementation of the Enumeration algorithm

This appendix is meant to provide a high level description of the implementation of the Enumeration algorithm, Algorithm 1.⁸ We implemented the Enumeration algorithm in C++.

It is possible to run the code by passing it two parameters:

$$\begin{aligned} \texttt {./2L\_enum}\, d \,\texttt {verbose\_flag} \end{aligned}$$

where:

• $d$ is an integer between 1 and 7, denoting the dimension $d$ of the list of combinatorial inequivalent $2$-level polytopes to enumerate. The algorithm creates a text file with the list of their slack matrices. The algorithm reads a file named $(d-1)$.txt in input, containing the list of all $(d-1)$-dimensional slack matrices of $2$-level polytopes.

• verbose_flag is an integer between 0 and 3.

  • verbose_flag = 0 : minimal output.

  • verbose_flag = 1 : the output includes all closed sets and tests for classes of $2$-level polytopes.

  • verbose_flag = 2 : the output includes all closed sets, $\mathcal {H}$-embedding of ground set and tests for classes of $2$-level polytopes (see Table 1).

  • verbose_flag = 3 : the output includes all closed sets, $\mathcal {V}$- and $\mathcal {H}$-embedding ground sets, the slabs vs ground set incidence matrix and tests for classes of $2$-level polytopes (see Table 1).

Our implementation of Algorithm 1 is organized as follows. Fix $d \ge 1$.

• We read the complete list $L_{d-1}$ of $d-1$-dimensional $2$-level polytopes through their slack matrices. Moreover for every $d-1$-dimensional $2$-level polytope $P_0$, we assume we are given the embedding transformation matrix $M_{d-1} :=M(\varGamma _0)$ for some simplicial core $\varGamma _0 :=(F_2',\dots ,F_{d+1};v_2,\dots ,v_{d+1})$ of $P_0$. In particular, the top left $d \times d$ submatrix of the slack matrix $S(P_0)$ that we pass in input is lower triangular, and corresponds to the rows and columns of $S(P_0)$ which are indexed by facets $F_2',\dots ,F_{d+1}$ and vertices $v_2,\dots ,v_{d+1}$, respectively.

• We construct the $\mathcal {H}$-embedding of $P_0$ in the hyperplane $\{x \in \mathbb {R}^d \mid x_1 = 0\}$ with respect to the simplicial core $\varGamma _0$.

• Using the definition of $\mathcal {H}$-embedding and the slack matrix $S(P_0)$, we build the list of all sets $E$ such that either $x(E) \ge 0$ or $x(E) \le 1$ is a facet of $P_0$ in the $\mathcal {H}$-embedding of $P_0$ in the hyperplane $\{x \in \mathbb {R}^d \mid x_1 = 0\}$ with respect to the simplicial core $\varGamma _0$.

• We construct the $\mathcal {H}$-embedding of the ground set $\mathcal {X}_{\mathrm {final}}$ in the hyperplane $\{x \in \mathbb {R}^d \mid x_1 = 1\}$ with respect to the simplicial core $\varGamma _0$, see (10).

• We build the list of all possible slabs (as in Lemma 15) that extend facets of $P_0$ to $\mathbb {R}^d$.

• We build incidence matrices between points of $\mathcal {X}_{\mathrm {final}}$ and slabs.

• At this point we are ready to start Ganter’s Next-Closure algorithm [13, 14] to enumerate all closed sets with respect to closure operator $\mathrm {cl}$ in Lemma 36.

• According to Lemma 15), for every $A \subseteq \mathcal {X}_{\mathrm {final}}$, the set $\mathrm {cl}_{\mathrm {dc}}(A)$ is the set of row indices that contains $A$ and corresponds to a maximal rectangle of ones in the ground set points vs slabs incidence matrix.

• For a set $A \subseteq \mathcal {X}_{\mathrm {final}}$, we compute $\mathrm {cl}_{\mathrm {comp}}$ by using the list of facets of $P_0$ in its $\mathcal {H}$-embedding and applying the definition of incompatible pair of points in Lemma 22.

• For every closed set $A \subseteq \mathcal {X}_{\mathrm {final}}$ with respect to $\mathrm {cl}$, we construct the incidence matrix $S$ of points of $\mathrm {vert}(P_0) \cup A$ and the subset of slabs constructed as in Lemma 15 and containing a maximal set of point in $\mathrm {vert}(P_0) \cup A$.

• We test every such candidate martix $S$ for $2$-levelness using nauty and Algorithm 2.

• If we obtain a positive answer, that is, $S$ is the slack matrix of some $d$-dimensional $2$-level polytope, we browse the list $L_d$ of slack matrices of $d$-dimensional $2$-level polytopes that we constructed until that moment and check whether an $S$ already exists there, up to permutation of rows and columns. This test is performed by nauty, as explained in Sect. 6.1. If $S$ is not listed, then it is added to $L_d$.