منابع مشابه
Finding Short Paths on Polytopes by the Shadow Vertex Algorithm
We show that the shadow vertex algorithm can be used to compute a short path between a given pair of vertices of a polytope P = {x ∈ R : Ax ≤ b} along the edges of P , where A ∈ Rm×n. Both, the length of the path and the running time of the algorithm, are polynomial in m, n, and a parameter 1/δ that is a measure for the flatness of the vertices of P . For integer matrices A ∈ Zm×n we show a con...
متن کاملLattice Points in Lattice Polytopes
We show that, for any lattice polytope P ⊂ R, the set int(P ) ∩lZ (provided it is non-empty) contains a point whose coefficient ofasymmetry with respect to P is at most 8d · (8l+7)2d+1. If, moreover,P is a simplex, then this bound can be improved to 8 · (8l+ 7)d+1.As an application, we deduce new upper bounds on the volume ofa lattice polytope, given its ...
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We investigate the vertex-connectivity of the graph of f -monotone paths on a d-polytope P with respect to a generic functional f . The third author has conjectured that this graph is always (d − 1)-connected. We resolve this conjecture positively for simple polytopes and show that the graph is 2-connected for any d-polytope with d ≥ 3. However, we disprove the conjecture in general by exhibiti...
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[1] Victor V. Batyrev and Benjamin Nill. Multiples of lattice polytopes without interior lattice points. Moscow Mathematical Journal 7:195–207, 2007. [2] Victor V. Batyrev, Benjamin Nill. Combinatorial aspects of mirror symmetry. Contemporary Mathematics, 452:35–66, 2008. [3] Barbara Baumeister, Christian Haase, Benjamin Nill and Andreas Paffenholz. On permutation polytopes. Advances in Mathema...
متن کاملLattice Points inside Lattice Polytopes
We show that, for any lattice polytope P ⊂ R, the set int(P ) ∩ lZ (provided it is non-empty) contains a point whose coefficient of asymmetry with respect to P is at most 8d · (8l+7) 2d+1 . If, moreover, P is a simplex, then this bound can be improved to 9 · (8l+ 7) d+1 . This implies that the maximum volume of a lattice polytope P ⊂ R d containing exactly k ≥ 1 points of lZ in its interior, is...
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ژورنال
عنوان ژورنال: Discrete & Computational Geometry
سال: 2021
ISSN: 0179-5376,1432-0444
DOI: 10.1007/s00454-020-00268-y