Diophantine Approximations and Integer Points of Cones

“…The first example is given by n − 1 standard unit vectors and one additional integral vector r. This setting has been studied with respect to the integer Carathéodory rank and only special instances with the (ICP) are known; see Henk and Weismantel (2002) for some of them. We assume r n = and that the first n − 1 coordinates of r admit at most two different values apart from 0 and − 1, e.g., r = (0, .…”

Section: Special Cones With the (Icp)mentioning

confidence: 99%

Integer Carathéodory results with bounded multiplicity

Kuhlmann

2024

Beitr Algebra Geom

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The integer Carathéodory rank of a pointed rational cone C is the smallest number k such that every integer vector contained in C is an integral non-negative combination of at most k Hilbert basis elements. We investigate the integer Carathéodory rank of simplicial cones with respect to their multiplicity, i.e., the determinant of the integral generators of the cone. One of the main results states that simplicial cones with multiplicity bounded by five have the integral Carathéodory property, that is, the integer Carathéodory rank equals the dimension. Furthermore, we give a novel upper bound on the integer Carathéodory rank that depends on the dimension and the multiplicity. This bound improves upon the best known upper bound on the integer Carathéodory rank if the dimension exceeds the multiplicity. At last, we present special cones that have the integral Carathéodory property such as certain dual cones of Gorenstein cones.

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“…Pochet and Weismantel [13] provided a linear inequality description of the knapsack set where all variables are bounded. Other hard problems studied under the assumption of divisibility of the coefficients include network design [14], lot-sizing problems [4] and the integer Carathéodory property for rational cones [10].…”

Section: Introductionmentioning

confidence: 99%