Polytopes - Combinations and Computation
Questions that arose from linear programming and combinatorial optimization have been a driving force for modern polytope theory, such as the diameter questions motivated by the desire to understand the complexity of the simplex algorithm, or the need to study facets for use in cutting plane procedures. In addition, algorithms now provide the means to computationally study polytopes, to compute their parameters such as flag vectors, graphs and volumes, and to construct examples of large complexity. The papers of this volume thus display a wide panorama of connections of polytope theory with other fields. Areas such as discrete and computational geometry, linear and combinatorial optimization, and scientific computing have contributed a combination of questions, ideas, results, algorithms and, finally, computer programs.
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Lectures on 01Polytopes
a Framework for Analyzing Convex Polytopes
Flag Numbers and FLAGTOOL
A Census of Flagvectors of 4Polytopes
Extremal Properties of 01Polytopes of Dimension 5
A Practical Study
Reconstructing a Simple Polytope from its Graph
Reconstructing a Nonsimple Polytope from its Graph
A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm
The Complexity of Yamnitsky and Levins Simplices Algorithm
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affine algorithm appear apply assume bases basic basis bound called classes coefficients column combinatorial complexity computation conjecture consider construction contains convex hull coordinates corresponding cube cubical d-dimensional d-polytope define denote described determinant dictionary dimension Discrete dual edges enumeration equations equivalent example exists extreme face facets fact feasible Figure flag numbers FLAGTOOL function Geometry give given graph hyperplane implemented inequalities input integer known Lemma lex-positive lexicographic linear linear inequalities lower bound Math matrix maximal method Note obtained optimal original pivot poly polyhedron polymake polytopes positive possible problem projection Proof properties Proposition proved random relations represents respect result reverse rule satisfies signed simple simplex simplices solution solve Step Table Theorem transformation triangulation unique vector vertex set vertices volume yields