Ebook: Polytopes — Combinatorics 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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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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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.
Content:
Front Matter....Pages i-vi
Lectures on 0/1-Polytopes....Pages 1-41
polymake: a Framework for Analyzing Convex Polytopes....Pages 43-73
Flag Numbers and FLAGTOOL....Pages 75-103
A Census of Flag-vectors of 4-Polytopes....Pages 105-110
Extremal Properties of 0/1-Polytopes of Dimension 5....Pages 111-130
Exact Volume Computation for Polytopes: A Practical Study....Pages 131-154
Reconstructing a Simple Polytope from its Graph....Pages 155-165
Reconstructing a Non-simple Polytope from its Graph....Pages 167-176
A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm....Pages 177-198
The Complexity of Yamnitsky and Levin’s Simplices Algorithm....Pages 199-225

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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.
Content:
Front Matter....Pages i-vi
Lectures on 0/1-Polytopes....Pages 1-41
polymake: a Framework for Analyzing Convex Polytopes....Pages 43-73
Flag Numbers and FLAGTOOL....Pages 75-103
A Census of Flag-vectors of 4-Polytopes....Pages 105-110
Extremal Properties of 0/1-Polytopes of Dimension 5....Pages 111-130
Exact Volume Computation for Polytopes: A Practical Study....Pages 131-154
Reconstructing a Simple Polytope from its Graph....Pages 155-165
Reconstructing a Non-simple Polytope from its Graph....Pages 167-176
A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm....Pages 177-198
The Complexity of Yamnitsky and Levin’s Simplices Algorithm....Pages 199-225
....