相关论文: Lectures on 0/1-polytopes

We present slight refinements of known general lower and upper bounds on sizes of extended formulations for polytopes. With these observations we are able to compute the extension complexities of all 0/1-polytopes up to dimension 4. We…

In Rothvo\ss{} it was shown that there exists a 0/1 polytope (a polytope whose vertices are in \{0,1\}^{n}) such that any higher-dimensional polytope projecting to it must have 2^{\Omega(n)} facets, i.e., its linear extension complexity is…

We present a number of complexity results concerning the problem of counting vertices of an integral polytope defined by a system of linear inequalities. The focus is on polytopes with small integer vertices, particularly 0/1 polytopes and…

We provide a characterization of the compressed lattice polytopes in terms of their facet defining inequalities and we show that every compressed lattice polytope is affinely isomorphic to a 0/1-polytope. As an application, we characterize…

We prove that every 0/1-polytope has a unique Minkowski decomposition into indecomposable polytopes, up to translation of summands. The summands lie in pairwise orthogonal subspaces. Thus, every 0/1-polytope is the Cartesian product of…

Let P be a random $d$-dimensional 0/1-polytope with $n(d)$ vertices, and denote by $\phi_k(P)$ the \emph{$k$-face density} of $P$, i.e., the quotient of the number of $k$-dimensional faces of $P$ and $\binom{n(d)}{k+1}$. For each $k\ge 2$,…

We give a complete enumeration of all 2-neighborly 0/1-polytopes of dimension 7. There are 13 959 358 918 different 0/1-equivalence classes of such polytopes. They form 5 850 402 014 combinatorial classes and 1 274 089 different f-vectors.…

We introduce the simple extension complexity of a polytope P as the smallest number of facets of any simple (i.e., non-degenerate in the sense of linear programming) polytope which can be projected onto P. We devise a combinatorial method…

Let $Q_n$ denote the $n$-dimensional hypercube with the vertex set $V_n=\{0,1}^n$. A 0/1-polytope of $Q_n$ is a convex hull of a subset of $V_n$. This paper is concerned with the enumeration of equivalence classes of full-dimensional…

We introduce revlex-initial 0/1-polytopes as the convex hulls of reverse-lexicographically initial subsets of 0/1-vectors. These polytopes are special knapsack-polytopes. It turns out that they have remarkable extremal properties. In…

We give compact extended formulations for the packing and partitioning orbitopes (with respect to the full symmetric group) described and analyzed in (Kaibel and Pfetsch, 2008). These polytopes are the convex hulls of all 0/1-matrices with…

We prove that there are 0/1 polytopes P that do not admit a compact LP formulation. More precisely we show that for every n there is a sets X \subseteq {0,1}^n such that conv(X) must have extension complexity at least 2^{n/2 * (1-o(1))}. In…

We present new pivot rules for the Simplex method for LPs over 0/1 polytopes. We show that the number of non-degenerate steps taken using these rules is strongly polynomial and even linear in the dimension or in the number of variables. Our…

An \emph{interval vector} is a $(0,1)$-vector in $\mathbb{R}^n$ for which all the 1's appear consecutively, and an \emph{interval-vector polytope} is the convex hull of a set of interval vectors in $\mathbb{R}^n$. We study three particular…

We characterize the edges of two classes of $0/1$-polytopes. The first class corresponds to the stable set polytope of a graph $G$ and includes chain polytopes of posets, some instances of matroid independence polytopes, as well as…

We relate the maximum semidefinite and linear extension complexity of a family of polytopes to the cardinality of this family and the minimum pairwise Hausdorff distance of its members. This result directly implies a known lower bound on…

The extension complexity of a polytope measures its amenability to succinct representations via lifts. There are several versions of extension complexity, including linear, real semidefinite, and complex semidefinite. We focus on the last…

This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The…

The lectures are devoted to a remarkable class of $3$-dimensional polytopes, which are mathematical models of the important object of quantum physics, quantum chemistry and nanotechnology -- fullerenes. The main goal is to show how results…

Over the past twenty years, lecture hall partitions have emerged as fundamental combinatorial structures, leading to new generalizations and interpretations of classical theorems and new results. In recent years, geometric approaches to…