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Combinatorial Polytope Enumeration

by Sandeep Koranne, Anand Kulkarni , 2009
"... We describe a provably complete algorithm for the generation of a tight, possibly exact, superset of all combinatorially distinct simple n-facet polytopes in R d, along with their graphs, f-vectors, and face lattices. The technique applies repeated cutting planes and planar sweeps to a d−simplex. Ou ..."
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We describe a provably complete algorithm for the generation of a tight, possibly exact, superset of all combinatorially distinct simple n-facet polytopes in R d, along with their graphs, f-vectors, and face lattices. The technique applies repeated cutting planes and planar sweeps to a d

Approximations of Polytope Enumerators using Linear Expansions

by Benoît Meister , 2007
"... Several scientific problems are represented as sets of linear (or affine) con-straints over a set of variables and symbolic constants. When solutions of inter-est are integers, the number of such integer solutions is generally a meaningful information. Ehrhart polynomials are functions of the symbol ..."
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Several scientific problems are represented as sets of linear (or affine) con-straints over a set of variables and symbolic constants. When solutions of inter-est are integers, the number of such integer solutions is generally a meaningful information. Ehrhart polynomials are functions of the symbolic constants that count these solutions. Unfortunately, they have a complex mathematical struc-ture (resembling polynomials, hence the name), making it hard for other tools to manipulate them. Furthermore, their use may imply exponential computational complexity. This paper presents two contributions towards the useability of Ehrhart polynomials, by showing how to compute the following polynomial functions: an approximation and an upper (and a lower) bound of an Ehrhart polynomial. The computational complexity of this polynomial is less than or equal to that of

FROM POLYTOPES TO ENUMERATION

by Ed Swartz
"... What should the d-dimensional analogues of polygons be? The short answer is “convex d-polytopes”. We may not know what these are yet, but that does not stop us from asking a number of questions about them. Example 1.1. The cube and the octahedron. The cube has 8 vertices, 12 edges and 6 faces, while ..."
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What should the d-dimensional analogues of polygons be? The short answer is “convex d-polytopes”. We may not know what these are yet, but that does not stop us from asking a number of questions about them. Example 1.1. The cube and the octahedron. The cube has 8 vertices, 12 edges and 6 faces

Enumerating Triangulations of Convex Polytopes

by Sergei Bespamyatnikh , 2001
"... ..."
Abstract - Cited by 2 (1 self) - Add to MetaCart
Abstract not found

FLAG ENUMERATIONS OF MATROID BASE POLYTOPES

by Sangwook Kim , 2009
"... In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the cd-index of a polytope can be expressed when a polytope is split by a h ..."
Abstract - Cited by 4 (0 self) - Add to MetaCart
In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the cd-index of a polytope can be expressed when a polytope is split by a

Enumeration of neighborly polytopes and oriented matroids

by Hiroyuki Miyata, Arnau Padrol , 2014
"... Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties they form a surprisingly rich class of polytopes, which has been widely studied and is the subject of many open problems and conj ..."
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and conjectures. In this paper, we study the enumeration of neighborly polytopes beyond the cases that have been computed so far. To this end, we enumerate neighborly oriented matroids — a combinatorial abstraction of neighborly polytopes — of small rank and corank. In particular, if we denote by OM(n, r) the set

Enumeration of Integer Projections of Parametric Polytopes

by Sven Verdoolaege, Maurice Bruynooghe, Francky Catthoor, Gerda Janssens
"... Although Barvinok’s algorithm for counting lattice points in a rational polytope easily extends to linearly parametrized polytopes, it is not immediately obvious whether the same applies to Barvinok and Woods ’ algorithm for counting the number of points in the integer projection of a polytope. We t ..."
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Although Barvinok’s algorithm for counting lattice points in a rational polytope easily extends to linearly parametrized polytopes, it is not immediately obvious whether the same applies to Barvinok and Woods ’ algorithm for counting the number of points in the integer projection of a polytope. We

Experiences with enumeration of integer projections of parametric polytopes

by Sven Verdoolaege, Kristof Beyls, Maurice Bruynooghe, Francky Catthoor , 2004
"... Abstract. Many compiler optimization techniques depend on the abil-ity to calculate the number of integer values that satisfy a given set of linear constraints. This count (the enumerator of a parametric poly-tope) is a function of the symbolic parameters that may appear in the constraints. In an ex ..."
Abstract - Cited by 20 (7 self) - Add to MetaCart
Abstract. Many compiler optimization techniques depend on the abil-ity to calculate the number of integer values that satisfy a given set of linear constraints. This count (the enumerator of a parametric poly-tope) is a function of the symbolic parameters that may appear in the constraints

Enumerating foldings and unfoldings between polygons and polytopes

by Erik D. Demaine, Martin L. Demaine, Anna Lubiw - Graphs Comb
"... Abstract. We pose and answer several questions concerning the number of ways to fold a polygon to a polytope, and how many polytopes can be obtained from one polygon; and the analogous questions for unfolding polytopes to polygons. Our answers are, roughly: exponentially many, or nondenumerably infi ..."
Abstract - Cited by 13 (8 self) - Add to MetaCart
Abstract. We pose and answer several questions concerning the number of ways to fold a polygon to a polytope, and how many polytopes can be obtained from one polygon; and the analogous questions for unfolding polytopes to polygons. Our answers are, roughly: exponentially many, or nondenumerably

Enumeration in convex geometries and associated polytopal subdivisions of spheres

by Louis J. Billera, Samuel K. Hsiao, J. Scott Provan , 2005
"... We construct CW spheres from the lattices that arise as the closed sets of a convex closure, the meet-distributive lattices. These spheres are nearly polytopal, in the sense that their barycentric subdivisions are simplicial polytopes. The complete information on the numbers of faces and chains of ..."
Abstract - Cited by 5 (0 self) - Add to MetaCart
We construct CW spheres from the lattices that arise as the closed sets of a convex closure, the meet-distributive lattices. These spheres are nearly polytopal, in the sense that their barycentric subdivisions are simplicial polytopes. The complete information on the numbers of faces and chains
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