Lectures on 0/1-Polytopes

Abstract

These lectures on the combinatorics and geometry of 0/1-polytopes are meant as an introduction and invitation. Rather than heading for an extensive survey on 0/1polytopes I present some interesting aspects of these objects; all of them are related to some quite recent work and progress. 0/1-polytopes have a very simple definition and explicit descriptions; we can enumerate and analyze small examples explicitly in the computer (e. g. using polymake). However, any intuition that is derived from the analysis of examples in "low dimensions" will miss the true complexity of 0/1-polytopes. Thus, in the following we will study several aspects of the complexity of higher-dimensional 0/1-polytopes: the doubly-exponential number of combinatorial types, the number of facets which can be huge, and the coefficients of defining inequalities which sometimes turn out to be extremely large. Some of the effects and results will be backed by proofs in the course of these lectures; we will also be able to verify some of them on explicit examples, which are accessible as a polymake database.

An Efficient Algorithm to Compute Subsets of Points in ℤ n

“…This new algorithm computes the group of isometries of the ndimensional unit hypercube in R n and uses it to ignore the subsets of points of it that differ by isometries of the n-dimensional space. Proposition 7 in [13] proves that an algorithm of this type returns the subsets of points non-congruent that differ by isometries of the n-dimensional space. A scheme of this algorithm is shown in Figure 3.…”

Finding a Simple Polytope from Its Graph in Polynomial Time

“…Thus, we are not concerned with actual realization of the polytope in Euclidean space; however the existence of such an embedding is necessary for the analysis. The following definitions are all standard and are discussed in more detail in standard texts, e.g., [11].…”

“…1 Our discussion in the remainder of the paper will be self contained but compact. See the book [11] for more details on the geometry of polytopes, [3] for more technical details on some of the specific techniques we use, and [4] for an excellent review, including the history, of the problem studied in this paper.…”

What is known about unit cubes

“…It is easy to get 0/1 polytopes that are affinely equivalent but not congruent. To show the other cases we have the following examples, both from Ziegler [115]. (1, 1, 1, 1, 1).…”

“…Let ς(n, k) denote the maximal number of the k-dimensional faces of an n-dimensional 0/1 polytope, and especially abbreviate ς(n, n − 1) to ς(n). The known exact values of ς(n) are listed in Table 5 (see Ziegler [115]). Let e 1 , e 2 , · · · , e n denote the n vectors of an orthonormal basis of E n and write e = (1, 1, · · · , 1).…”