In the first part of the paper we survey some far-reaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes. In the second part we discuss some recent developments concerning the simplex algorithm. We describe subexponential randomized pivot rules and upper bounds on the diameter of graphs of polytopes. 1 Introduction: A convex polyhedron is the intersection P of a finite number of closed halfspaces in R d . P is a d-dimensional polyhedron (briefly, a d-polyhedron) if the points in P affinely span R d . A convex d-dimensional polytope (briefly, a d-polytope) is a bounded convex d-polyhedron. Alternatively, a convex d-polytope is the convex hull of a finite set of points which affinely span R d . A (nontrivial) face F of a d-polyhedron P is the intersection of P with a supporting hyperplane. F itself is a polyhedron of some lower dimension. If the dimension of F is k we call F a k-face of P . The empty set and P itself are...

. We start with a theorem of Perles on the k-skeleton, Skel k (P ) (faces of dimension k) of d- polytopes P with d+b vertices for large d. The theorem says that for fixed b and d, if d is sufficiently large, then Skel k (P ) is the k-skeleton of a pyramid over a (d \Gamma 1)-dimensional polytope. Therefore the number of combinatorially distinct k-skeleta of d-polytopes with d + b vertices is bounded by a function of k and b alone. Next we replace b (the number of vertices minus the dimension) by related but deeper invariants of P , the g-numbers. For a d-polytope P there are [d=2] invariants g1 (P ); g2 (P ); :::; g [d=2] (P ) which are of great importance in the combinatorial theory of polytopes. We study polytopes for which g k is small and carried away to related and slightly related problems. Key words: Convex polytopes, skeleton, simplicial sphere, simplicial manifold, f-vector, g- theorem, ranked atomic lattices, stress, rigidity, sunflower, lower bound theorem, elementary poly...

Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...

Abstract. We give a unified explanation of the geometric and algebraic properties of two well-known maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of pattern-avoidance. Applying these results, we build the Malvenuto-Reutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of non-commutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations. 1.

A k-system of the graph GP of a simple polytope P is a set of induced subgraphs of GP that shares certain properties with the set of subgraphs induced by the k-faces of P . This new concept leads to polynomial-size certicates in terms of GP for both the set of vertex sets of facets as well as for abstract objective functions (AOF) in the sense of Kalai. Moreover, it is proved that an acyclic orientation yields an AOF if and only if it induces a unique sink on every 2-face.

We introduce unique sink orientations of grids as digraph models for many well-studied problems, including linear programming over products of simplices, generalized linear complementarity problems over P-matrices (PGLCP), and simple stochastic games. We investigate the combinatorial structure of such orientations and develop randomized algorithms for finding the sink. We show that the orientations arising from PGLCP satisfy the Holt-Klee condition known to hold for polytope digraphs, and we give the first expected linear-time algorithms for solving PGLCP with a fixed number of blocks.

Abstract. We describe many different realizations with integer coordinates for the associahedron (i.e. the Stasheff polytope) and for the cyclohedron (i.e. the Bott-Taubes polytope) and compare them to the permutahedron of type A and B respectively. The coordinates are obtained by an algorithm which uses an oriented Coxeter graph of type An or Bn respectively as only input and which specialises to a procedure presented by J.-L. Loday for a certain orientation of An. 1.

We show that the problem to decide whether two (convex) polytopes, given by their vertex-facet incidences, are combinatorially isomorphic is graph isomorphism complete, even for simple or simplicial polytopes. On the other hand, we give a polynomial time algorithm for the combinatorial polytope isomorphism problem in bounded dimensions. Furthermore, we derive that the problems to decide whether two polytopes, given either by vertex or by facet descriptions, are projectively or affinely isomorphic are graph isomorphism hard.

The combinatorial structure of a d-dimensional simple convex polytope -- as given, for example, by the set of the (d 1)-regular subgraphs of the facets -- can be reconstructed from its abstract graph. However, no polynomial/efficient algorithm is known for this task, although a polynomially checkable certificate for the correct reconstruction exists. A much stronger certicate would be given by the following characterization of the facet subgraphs, conjectured by M. Perles: "The facet subgraphs of a simple d-polytope are exactly all the (d 1)-regular, connected, induced, non-separating subgraphs." We present non-trivial classes of examples for the validity of the Perles conjecture: In particular, it holds for the duals of cyclic polytopes, and for the duals of stacked polytopes. On the other hand, we observe that for any 4-dimensional counterexample, the boundary of the (simplicial) dual polytope P contains a 2-complex without a free edge, and without 2-dimensional homology. Examples of such complexes are known; we use a modification of "Bing's house" (two walls removed) to construct explicit 4-dimensional counterexamples to the Perles conjecture.

INTRODUCTION The k-dimensional skeleton of a d-polytope P is the set of all faces of the polytope of dimension at most k. The 1-skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs--- subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t