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45
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 ..."
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Cited by 16 (0 self)
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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...
Complexity results for InfiniteHorizon Markov Decision Processes
, 2000
"... Markov decision processes (MDPs) are models of dynamic decision making under uncertainty. These models arise in diverse applications and have been developed extensively in fields such as operations research, control engineering, and the decision sciences in general. Recent research, especially in a ..."
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Cited by 15 (3 self)
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Markov decision processes (MDPs) are models of dynamic decision making under uncertainty. These models arise in diverse applications and have been developed extensively in fields such as operations research, control engineering, and the decision sciences in general. Recent research, especially in artificial intelligence, has highlighted the significance of studying the computational properties of MDP problems. We address
Convex Combinatorial Optimization
, 2004
"... We introduce the convex combinatorial optimization problem, a farreaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edgeguaranteed family, and discuss several applications. ..."
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Cited by 14 (7 self)
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We introduce the convex combinatorial optimization problem, a farreaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edgeguaranteed family, and discuss several applications.
Small examples of nonconstructible simplicial balls and spheres
 SIAM J. Discrete Math
, 2004
"... We construct nonconstructible simplicial dspheres with d + 10 vertices and nonconstructible, nonrealizable simplicial dballs with d + 9 vertices for d≥3. 1 ..."
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Cited by 10 (4 self)
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We construct nonconstructible simplicial dspheres with d + 10 vertices and nonconstructible, nonrealizable simplicial dballs with d + 9 vertices for d≥3. 1
A Survey on Pivot Rules for Linear Programming
 ANNALS OF OPERATIONS RESEARCH. (SUBMITTED
, 1991
"... The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the finiteness property of simplex type pivot rules. Th ..."
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Cited by 9 (1 self)
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The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the finiteness property of simplex type pivot rules. There are some other important topics in linear programming, e.g. complexity theory or implementations, that are not included in the scope of this paper. We do not discuss ellipsoid methods nor interior point methods. Well known classical results concerning the simplex method are also not particularly discussed in this survey, but the connection between the new methods and the classical ones are discussed if there is any. In this paper we discuss three classes of recently developed pivot rules for linear programming. The first class (the largest one) of the pivot rules we discuss is the class of essentially combinatorial pivot rules. Namely these rules only use labeling and signs of the variab...
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 ..."
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Cited by 7 (2 self)
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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 ..."
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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
The Simplex Algorithm in Dimension Three
, 2004
"... We investigate the worstcase behavior of the simplex algorithm on linear programs with three variables, that is, on 3dimensional 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 ..."
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Cited by 5 (1 self)
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We investigate the worstcase behavior of the simplex algorithm on linear programs with three variables, that is, on 3dimensional 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 worstpossible behavior; this includes both Kalai’s “randomfacet” rule, which without dimension restriction is known to be subexponential, as well as Zadeh’s deterministic historydependent rule, for which no nonpolynomial instances in general dimensions have been found so far.