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 examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interior-point, and other methods. Key words. linear programming -- history -- simplex method -- ellipsoid method -- interior-point methods 1. Introduction At the last Mathematical Programming Symposium in Lausanne, we celebrated the 50th anniversary of the simplex method. Here, we are at or close to several other anniversaries relating to linear programming: the sixtieth of Kantorovich's 1939 paper on "Mathematical Methods in the Organization and Planning of Production" (and the fortieth of its appearance in the Western literature) [55]; the fiftieth of the historic 0th Mathematical Programming Symposium that took place in Chicago in 1949 on Activity Analysis of Production and Allocation [64]; the forty-fifth of Frisch's suggestion of the logarithmic barrier function for linear programming [37]; the twenty-fifth of the awarding of the 1975 Nobe...

In 1957 W.M. Hirsch conjectured that every d-polytope with n facets has edge-diameter at most n \Gamma d. Recently Holt and Klee constructed polytopes which meet this bound for a number of (d; n) pairs with d 13 and for all pairs (14; n). These constructions involve a judicious use of truncation, wedging, and blending on polytopes which already meet the Hirsch bound. In this paper we extend these techniques to construct polytopes of edge-diameter n \Gamma 8 for all (8; n). The improvement from d = 14 to d = 8 follows from identifying circumstances in which the results for wedging when n ? 2d can be extended to the cases n 2d, our lemma 2.2. 1 Introduction For two vertices x and y of a polytope P , the distance ffi P (x; y) is defined as the smallest number of edges of P that can be used to form a path from x to y. The edge-diameter ffi(P ) of P is the maximum over all pairs (x; y) of P 's vertices. An undirected edge [u; v] in a polytope P is said to be slow toward a vertex x of P...

Abstract. It is known that the suspension of a simplicial complex can be realized with only one additional point. Suitable iterations of this construction generate highly symmetric simplicial complexes with various interesting combinatorial and topological properties. In particular, infinitely many non-PL spheres as well as contractible simplicial complexes with a vertex-transitive group of automorphisms can be obtained in this way. 1.

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

We give new examples of shellable but not extendably shellable two dimensional simplicial complexes. They include minimal examples, which are smaller than those previously known. We also give examples of shellable but not vertex decomposable two dimensional simplicial complexes. Among them are extendably shellable ones.

We investigate the diameter of a natural abstraction of the 1-skeleton of polyhedra. Although this abstraction is simpler than other abstractions that were previously studied in the literature, the best upper bounds on the diameter of polyhedra continue to hold here. On the other hand, we show that this abstraction has its limits by providing a superlinear lower bound. Categories and Subject Descriptors G.1.6 [Numerical Analysis]: Optimization—linear programming;

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Warren M. Hirsch posed the conjecture which bears his name in a letter of 1957 to George B. Dantzig. Simply stated in geometric terms, Hirsch proposed that a convex polyhedron in dimension d with n facets admits a path of at most (n − d) edges connecting any two vertices. Hirsch posed his conjecture in the context of a linear program in d variables and n constraints as requiring a maximum of (n − d) pivots — steps of the simplex algorithm — on the shortest path to achieve an optimum. While the conjecture is known to be false for some unbounded polyhedra, over the years it has attracted much research attention for polytopes, and has been proved in special cases. This article contributes a general proof for polytopes.