Results 1  10
of
15
The Many Facets of Linear Programming
, 2000
"... . We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interiorpoint, and other methods. Key words. linear programming  history  simplex method  ellipsoid method  interiorpoint methods 1. Introduction A ..."
Abstract

Cited by 25 (1 self)
 Add to MetaCart
. We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interiorpoint, and other methods. Key words. linear programming  history  simplex method  ellipsoid method  interiorpoint 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 fortyfifth of Frisch's suggestion of the logarithmic barrier function for linear programming [37]; the twentyfifth of the awarding of the 1975 Nobe...
Linear Programming, the Simplex Algorithm and Simple Polytopes
 Math. Programming
, 1997
"... In the first part of the paper we survey some farreaching 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 ru ..."
Abstract

Cited by 22 (1 self)
 Add to MetaCart
In the first part of the paper we survey some farreaching 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 ddimensional polyhedron (briefly, a dpolyhedron) if the points in P affinely span R d . A convex ddimensional polytope (briefly, a dpolytope) is a bounded convex dpolyhedron. Alternatively, a convex dpolytope is the convex hull of a finite set of points which affinely span R d . A (nontrivial) face F of a dpolyhedron 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 kface of P . The empty set and P itself are...
More polytopes meeting the conjectured Hirsch bound
 Discrete Math
, 1999
"... In 1957 W.M. Hirsch conjectured that every dpolytope with n facets has edgediameter 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, w ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
In 1957 W.M. Hirsch conjectured that every dpolytope with n facets has edgediameter 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 edgediameter 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 edgediameter 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...
Onepoint suspensions and wreath products of polytopes and spheres
"... 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 ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
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 nonPL spheres as well as contractible simplicial complexes with a vertextransitive group of automorphisms can be obtained in this way. 1.
Polytope Skeletons And Paths
 Handbook of Discrete and Computational Geometry (Second Edition ), chapter 20
"... INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton 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 i ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton 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
Incremental Construction Properties in Dimension Two  Shellability, Extendable Shellability and Vertex Decomposability
, 2000
"... 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 ext ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
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.
Diameter of Polyhedra: Limits of Abstraction
, 2009
"... We investigate the diameter of a natural abstraction of the 1skeleton 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 ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
We investigate the diameter of a natural abstraction of the 1skeleton 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.
A CONSTRUCTIVE PROOF OF THE HIRSCH CONJECTURE
, 2009
"... 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 conjectur ..."
Abstract
 Add to MetaCart
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.