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14
A quasipolynomial bound for the diameter of graphs of polyhedra
 Bulletin Amer. Math. Soc
, 1992
"... Abstract. The diameter of the graph of a ddimensional polyhedron with n facets is at most nlog d+2 graph whose vertices are the extreme points of P and two vertices u and v are adjacent if the interval [v, u] is an extreme edge ( = 1dimensional face) of P. The diameter of the graph of P is denoted ..."
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Cited by 52 (4 self)
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Abstract. The diameter of the graph of a ddimensional polyhedron with n facets is at most nlog d+2 graph whose vertices are the extreme points of P and two vertices u and v are adjacent if the interval [v, u] is an extreme edge ( = 1dimensional face) of P. The diameter of the graph of P is denoted by δ(P). Let ∆(d, n) be the maximal diameter of the graphs of ddimensional polyhedra P with n facets. (A facet is a (d − 1)dimensional face.) Thus, P is the set of solutions of n linear inequalities in d variables. It is an old standing problem to determine the behavior of the function ∆(d, n). The value of ∆(d, n) is a lower bound for the number of iterations needed for Dantzig’s simplex algorithm for linear programming with any pivot rule. In 1957 Hirsch conjectured [2] that ∆(d, n) ≤ n−d. Klee and Walkup [6] showed that the Hirsch conjecture is false for unbounded polyhedra. They proved that for n ≥ 2d, ∆(d, n) ≥ n − d + [d/5]. This is the best known lower bound for ∆(d, n). The statement of the Hirsch conjecture for bounded polyhedra is still open. For a recent survey on the Hirsch conjecture and its relatives, see [5]. In 1967 Barnette proved [1, 3] that ∆(d, n) ≤ n3 d−3. An improved upper bound, ∆(d, n) ≤ n2 d−3, was proved in 1970 by Larman [7]. Barnette’s and Larman’s bounds are linear in n but exponential in the dimension √ d. In 1990 the first author [4] proved a subexponential bound ∆(d, n) ≤ 2 (n−d)log(n−d). The purpose of this paper is to announce and to give a complete proof of a quasipolynomial upper bound for ∆(d, n). Such a bound was proved by the first author in March 1991. The proof presented here is a substantial simplification that was subsequently found by the second author. See [4] for the original proof and related results. The existence of a polynomial (or even linear) upper bound for ∆(d, n) is still open. Recently, the first author found a randomized pivot rule for linear programming which requires an expected n 4 √ d (or less) arithmetic operations for every linear programming problem with d variables and n constraints.
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 ..."
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Cited by 25 (1 self)
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. 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 ..."
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Cited by 22 (1 self)
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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...
Randomized Simplex Algorithms on KleeMinty Cubes
 COMBINATORICA
, 1994
"... We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the KleeMinty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the KleeMinty cubes ..."
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Cited by 19 (6 self)
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We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the KleeMinty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the KleeMinty 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 randomedge simplex algorithm on KleeMinty 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 KleeMinty cube is exponential when all paths are taken with equal probability.
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 ..."
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Cited by 6 (0 self)
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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
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 ..."
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Cited by 5 (1 self)
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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.
LatticeFree Polytopes and Their Diameter
 Discrete and Computational Geometry
, 1995
"... A convex polytope in real Euclidean space is latticefree if it intersects some lattice in space exactly in its vertex set. Latticefree polytopes form a large and computationally hard class, and arise in many combinatorial and algorithmic contexts. In this article, affine and combinatorial properti ..."
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Cited by 3 (3 self)
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A convex polytope in real Euclidean space is latticefree if it intersects some lattice in space exactly in its vertex set. Latticefree polytopes form a large and computationally hard class, and arise in many combinatorial and algorithmic contexts. In this article, affine and combinatorial properties of such polytopes are studied. First, bounds on some invariants, such as the diameter and layernumber, are given. It is shown that the diameter of a ddimensional latticefree polytope is O(d 3 ). A bound of O(nd+d 3 ) on the diameter of a dpolytope with n facets is deduced for a large class of integer polytopes. Second, Delaunay polytopes and [0; 1]polytopes, which form major subclasses of latticefree polytopes, are considered. It is shown that, up to affine equivalence, for any d 3 there are infinitely many ddimensional latticefree polytopes but only finitely many Delaunay and [0; 1]polytopes. Combinatorialtypes of latticefree polytopes are discussed, and the inclusion rel...