Geodesic
Integer points in knapsack polytopes and \(s\)-covering radius
The electronic journal of combinatorics, Tome 20 (2013) no. 2
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Given a matrix $A\in \mathbb{Z}^{m\times n}$ satisfying certain regularity assumptions, we consider for a positive integer $s$ the set ${\mathcal F}_s(A)\subset \mathbb{Z}^m$ of all vectors $b\in \mathbb{Z}^m$ such that the associated knapsack polytope\begin{equation*}P(A, b)=\{ x \in \mathbb{R}^n_{\ge 0}: A x= b\}\end{equation*}contains at least $s$ integer points. We present lower and upper bounds on the so called diagonal $s$-Frobenius number associated to the set ${\mathcal F}_s(A)$. In the case $m=1$ we prove an optimal lower bound for the $s$-Frobenius number, which is the largest integer $b$ such that $P(A,b)$ contains less than $s$ integer points.
DOI : 10.37236/2887
Classification : 90C10, 52C07, 11H06
Mots-clés : Knapsack polytope, (diagonal) Frobenius numbers, inhomogeneous minimum, covering radius, successive minima 
@article{10_37236_2887,
     author = {Iskander Aliev and Martin Henk and Eva Linke},
     title = {Integer points in knapsack polytopes and \(s\)-covering radius},
     journal = {The electronic journal of combinatorics},
     year = {2013},
     volume = {20},
     number = {2},
     doi = {10.37236/2887},
     zbl = {1267.90071},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/2887/}
}
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Iskander Aliev; Martin Henk; Eva Linke. Integer points in knapsack polytopes and \(s\)-covering radius. The electronic journal of combinatorics, Tome 20 (2013) no. 2. doi: 10.37236/2887

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