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25
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...
Realizations of the associahedron and cyclohedron
, 2005
"... We describe many different realizations with integer coordinates for the associahedron (i.e. the Stasheff polytope) and for the cyclohedron (i.e. the BottTaubes polytope) and compare them to the permutahedron of type A and B respectively. The coordinates are obtained by an algorithm which uses an o ..."
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Cited by 18 (4 self)
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We describe many different realizations with integer coordinates for the associahedron (i.e. the Stasheff polytope) and for the cyclohedron (i.e. the BottTaubes polytope) and compare them to the permutahedron of type A and B respectively. The coordinates are obtained by an algorithm which uses an oriented Coxeter graph of type An or Bn respectively as only input and which specialises to a procedure presented by J.L. Loday for a certain orientation of An.
BASIC PROPERTIES OF CONVEX POLYTOPES
, 1997
"... Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry (toric varieties) ..."
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Cited by 14 (2 self)
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Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry (toric varieties) to linear and combinatorial
Some Algorithmic Problems in Polytope Theory
 IN ALGEBRA, GEOMETRY, AND SOFTWARE SYSTEMS
, 2003
"... Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact... ..."
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Cited by 11 (1 self)
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Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...
On the kSystems of a Simple Polytope
 ISRAEL J. MATH
, 2001
"... A ksystem of the graph GP of a simple polytope P is a set of induced subgraphs of GP that shares certain properties with the set of subgraphs induced by the kfaces of P . This new concept leads to polynomialsize certicates in terms of GP for both the set of vertex sets of facets as well as fo ..."
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Cited by 9 (3 self)
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A ksystem of the graph GP of a simple polytope P is a set of induced subgraphs of GP that shares certain properties with the set of subgraphs induced by the kfaces of P . This new concept leads to polynomialsize certicates in terms of GP for both the set of vertex sets of facets as well as for abstract objective functions (AOF) in the sense of Kalai. Moreover, it is proved that an acyclic orientation yields an AOF if and only if it induces a unique sink on every 2face.
On the Complexity of Polytope Isomorphism Problems
 Graphs and Combinatorics
, 2003
"... We show that the problem to decide whether two (convex) polytopes, given by their vertexfacet incidences, are combinatorially isomorphic is graph isomorphism complete, even for simple or simplicial polytopes. On the other hand, we give a polynomial time algorithm for the combinatorial polytope isom ..."
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Cited by 8 (0 self)
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We show that the problem to decide whether two (convex) polytopes, given by their vertexfacet incidences, are combinatorially isomorphic is graph isomorphism complete, even for simple or simplicial polytopes. On the other hand, we give a polynomial time algorithm for the combinatorial polytope isomorphism problem in bounded dimensions. Furthermore, we derive that the problems to decide whether two polytopes, given either by vertex or by facet descriptions, are projectively or affinely isomorphic are graph isomorphism hard.
Unique sink orientations of grids
 Proc. 11th Conference on Integer Programming and Combinatorial Optimization (IPCO
, 2005
"... We introduce unique sink orientations of grids as digraph models for many wellstudied problems, including linear programming over products of simplices, generalized linear complementarity problems over Pmatrices (PGLCP), and simple stochastic games. We investigate the combinatorial structure of su ..."
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Cited by 7 (4 self)
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We introduce unique sink orientations of grids as digraph models for many wellstudied problems, including linear programming over products of simplices, generalized linear complementarity problems over Pmatrices (PGLCP), and simple stochastic games. We investigate the combinatorial structure of such orientations and develop randomized algorithms for finding the sink. We show that the orientations arising from PGLCP satisfy the HoltKlee condition known to hold for polytope digraphs, and we give the first expected lineartime algorithms for solving PGLCP with a fixed number of blocks.
Examples and counterexamples for the Perles' conjecture
, 2000
"... The combinatorial structure of a ddimensional 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 check ..."
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Cited by 7 (3 self)
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The combinatorial structure of a ddimensional 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 dpolytope are exactly all the (d 1)regular, connected, induced, nonseparating subgraphs." We present nontrivial 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 4dimensional counterexample, the boundary of the (simplicial) dual polytope P contains a 2complex without a free edge, and without 2dimensional homology. Examples of such complexes are known; we use a modification of "Bing's house" (two walls removed) to construct explicit 4dimensional counterexamples to the Perles conjecture.
Polytope Skeletons And Paths
 HANDBOOK OF DISCRETE AND COMPUTATIONAL GEOMETRY (SECOND EDITION ), CHAPTER 20
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