Günter M. Ziegler

Lectures on 0/1-Polytopes

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.

Matroid Applications: Introduction to Greedoids

Introduction Greedoids were invented around 1980 by B. Korte and L. Lovasz. Originally, the main motivation for proposing this generalization of the matroid concept came from combinatorial optimization. Korte and Lovasz had observed that the optimality of a ‘greedy’ algorithm could in several instances be traced back to an underlying combinatorial structure that was not a matroid – but (as they named it) a ‘greedoid’. In subsequent research greedoids have been shown to be interesting also from various non-algorithmic points of view. The basic distinction between greedoids and matroids is that greedoids are modeled on the algorithmic construction of certain sets, which means that the ordering of elements in a set plays an important role. Viewing such ordered sets as words, and the collection of words as a formal language, we arrive at the general definition of a greedoid as a finite language that is closed under the operation of taking initial substrings and satisfies a matroid-type exchange axiom. It is a pleasant feature that greedoids can also be characterized in terms of set systems (the unordered version), but the language formulation (the ordered version) seems more fundamental. Consider, for instance, the algorithmic construction of a spanning tree in a connected graph. Two simple strategies are: (1) pick one edge at a time, making sure that the current edge does not form a circuit with those already chosen; (2) pick one edge at a time, starting at some given node, so that the current edge connects a visited node with an unvisited node.

Homotopy types of subspace arrangements via diagrams of spaces.

We prove combinatorial formulas for the homotopy type of the union of the subspaces in an (affine, compactified affine, spherical or projective) subspace arrangement. From these formulas we derive results of Goresky & MacPherson on the homology of the arrangement and the cohomology of its complement. The union of an arrangement can be interpreted as the direct limit of a diagram of spaces over the intersection poset. A closely related space is obtained by taking the homotopy direct limit of this diagram. Our method consists in constructing a combinatorial model diagram over the same poset, whose homotopy limit can be compared to the original one by usual homotopy comparison results for diagrams of spaces.

Polytopes : combinatorics and computation

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Higher bruhat orders and cyclic hyperplane arrangements

We study the higher Bruhat orders

Hyperplane arrangements with a lattice of regions

B(n,k)

Homotopy colimits – comparison lemmas for combinatorial applications

of Manin & Schechtman [MaS] and - characterize them in terms of inversion sets, - identify them with the posets

Broken circuit complexes: factorization and generalizations

U(C^{n+1,r},n+1)

Many non-equivalent realizations of the associahedron

of uniform extensions of the alternating oriented matroids

0/1-Integer Programming: Optimization and Augmentation are Equivalent

C^{n,r}