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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0/1-equivalent 2-neighbourly 0/1-polytope acyclic orientation affine algorithm bases basic feasible solution Bayer Billera cobasic coefficients column combinatorial complexity conjecture contains convex polytopes coordinates corresponding cross polytope cube cubical polytope cut polytopes CUT(n d-cube d-polytope denote determinant dictionary dimension dimensional dual dual graph ellipsoid method example extreme rays face numbers Figure flag numbers flag-vectors FLAGTOOL Fourier-Motzkin elimination Fukuda full-dimensional g-numbers geometric ray Geometry given graph hypercubes hyperplane implemented induced subgraph input integer Joswig Kalai Lasserre's Lemma lex-positive basis lexicographic linear inequalities linear program lower bound Math matrix maximal number number of edges number of facets number of vertices obtained optimal pivot polyhedra polyhedron polymake polynomial problem Proof properties Proposition quasifacet random recursion regular polytopes result reverse search Schlegel diagrams Section signed decomposition simple polytopes simplex simplices method simplicial polytopes Theorem triangulation upper bound vertex enumeration vertex set volume computation Ziegler zonotopes