We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the Klee-Minty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the Klee-Minty cubes leads to (nearly) quadratic lower bounds for the complexity of linear programming with random pivots. Thus we disprove two bounds (for the expected running time of the random-edge simplex algorithm on Klee-Minty cubes) conjectured in the literature. At the same time, we establish quadratic upper bounds for the expected length of a path for a simplex algorithm with random pivots on the classes of linear programs under investigation. In contrast to this, we find that the average length of an increasing path in a Klee-Minty cube is exponential when all paths are taken with equal probability.

We consider a class A of generalized linear programs on the d-cube (due to Matousek) and prove that Kalai's subexponential simplex algorithm Random-Facet is polynomial on all actual linear programs in the class. In contrast, the subexponential analysis is known to be best possible for general instances in A. Thus, we identify a "geometric" property of linear programming that goes beyond all abstract notions previously employed in generalized linear programming frameworks, and that can be exploited by the simplex method in a nontrivial setting.

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Neighborly cubical polytopes exist: for any n ≥ d ≥ 2r + 2, there is a cubical whose r-skeleton is combinatorially equivalent to that of the convex d-polytope Cn d n-dimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary ∂Cn d of a neighborly cubical polytope Cn d maximizes the f-vector among all cubical (d − 1)-spheres with 2n vertices. While we show that this is true for polytopal spheres if n ≤ d+1, we also give a counter-example for d = 4 and n = 6. Further, the existence of neighborly cubical polytopes shows that the graph of the n-dimensional cube, where n ≥ 5, is “dimensionally ambiguous ” in the sense of Grünbaum. We also show that the graph of the 5-cube is “strongly 4-ambiguous”. In the special case d = 4, neighborly cubical polytopes have f3 = f0 4 log2 f0 4 vertices, so the facet-vertex ratio f3/f0 is not bounded; this solves a problem of Kalai, Perles and Stanley studied by Jockusch.

In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound theorem for the face numbers of convex polytopes and its many extensions is the second topic. Next are general properties of subsets of the vertices of the discrete n-dimensional cube and some relations with questions of extremal and probabilistic combinatorics. Our fourth topic is tree enumeration and random spanning trees, and finally, some combinatorial and geometrical aspects of the simplex method for linear programming are considered.

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

The Monotone Upper Bound Problem asks for the maximal number M(d, n) ofvertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d, n) ≤ Mubt(d, n) provided by McMullen’s (1970) Upper Bound Theorem is tight, where Mubt(d, n) is the number of vertices of a dual-to-cyclic d-polytope with n facets. It was recently shown that the upper bound M(d, n) ≤ Mubt(d, n) holdswith equality for small dimensions (d ≤ 4: Pfeifle, 2003) and for small corank (n ≤ d +2: Gärtner et al., 2001). Here we prove that it is not tight in general: In dimension d =6apolytopewithn =9facetscanhaveMubt(6, 9) = 30 vertices, but not more than 27 ≤ M(6, 9) ≤ 29 vertices can lie on a strictly-increasing edge-path. The proof involves classification results about neighborly polytopes, Kalai’s (1988) concept of abstract objective functions, the Holt-Klee conditions (1998), explicit enumeration, Welzl’s (2001) extended Gale diagrams, randomized generation of instances, as well as non-realizability proofs via a version of the Farkas lemma.

It is an open problem to characterize the cone of f-vectors of 4-dimensional convex polytopes. The question whether the “fatness ” of the fvector of a 4-polytope can be arbitrarily large is a key problem in this context. Here we construct a 2-parameter family of 4-dimensional polytopes π(P 2r n) with extreme combinatorial structure. In this family, the “fatness ” of the fvector gets arbitrarily close to 9; an analogous invariant of the flag vector, the “complexity, ” gets arbitrarily close to 16. The polytopes are obtained from suitable deformed products of even polygons by a projection to R4.

We investigate the worst-case behavior of the simplex algorithm on linear programs with 3 variables, that is, on 3-dimensional simple polytopes. Among the pivot rules that we consider, the “random edge” rule yields the best asymptotic behavior as well as the most complicated analysis. All other rules turn out to be much easier to study, but also produce worse results: Most of them show essentially worst-possible behavior; this includes both Kalai’s “random-facet” rule, which is known to be subexponential without dimension restriction, as well as Zadeh’s deterministic history-dependent rule, for which no non-polynomial instances in general dimensions have been found so far.

We investigate the worst-case behavior of the simplex algorithm on linear programs with three variables, that is, on 3-dimensional simple polytopes. Among the pivot rules that we consider, the “random edge ” rule yields the best asymptotic behavior as well as the most complicated analysis. All other rules turn out to be much easier to study, but also produce worse results: Most of them show essentially worst-possible behavior; this includes both Kalai’s “random-facet ” rule, which without dimension restriction is known to be subexponential, as well as Zadeh’s deterministic history-dependent rule, for which no non-polynomial instances in general dimensions have been found so far. 1