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156
Combinatorial Polytope Enumeration
, 2009
"... We describe a provably complete algorithm for the generation of a tight, possibly exact, superset of all combinatorially distinct simple nfacet polytopes in R d, along with their graphs, fvectors, 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 nfacet polytopes in R d, along with their graphs, fvectors, and face lattices. The technique applies repeated cutting planes and planar sweeps to a d
Approximations of Polytope Enumerators using Linear Expansions
, 2007
"... Several scientific problems are represented as sets of linear (or affine) constraints over a set of variables and symbolic constants. When solutions of interest 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) constraints over a set of variables and symbolic constants. When solutions of interest 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 structure (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
"... What should the ddimensional analogues of polygons be? The short answer is “convex dpolytopes”. 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 ddimensional analogues of polygons be? The short answer is “convex dpolytopes”. 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
FLAG ENUMERATIONS OF MATROID BASE POLYTOPES
, 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 cdindex of a polytope can be expressed when a polytope is split by a h ..."
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Cited by 4 (0 self)
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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 cdindex of a polytope can be expressed when a polytope is split by a
Enumeration of neighborly polytopes and oriented matroids
, 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
"... 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
, 2004
"... Abstract. Many compiler optimization techniques depend on the ability to calculate the number of integer values that satisfy a given set of linear constraints. This count (the enumerator of a parametric polytope) is a function of the symbolic parameters that may appear in the constraints. In an ex ..."
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Cited by 20 (7 self)
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Abstract. Many compiler optimization techniques depend on the ability to calculate the number of integer values that satisfy a given set of linear constraints. This count (the enumerator of a parametric polytope) is a function of the symbolic parameters that may appear in the constraints
Enumerating foldings and unfoldings between polygons and polytopes
 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 ..."
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Cited by 13 (8 self)
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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
, 2005
"... We construct CW spheres from the lattices that arise as the closed sets of a convex closure, the meetdistributive 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 ..."
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Cited by 5 (0 self)
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We construct CW spheres from the lattices that arise as the closed sets of a convex closure, the meetdistributive 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
Results 1  10
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156