Integer Points in Knapsack Polytopes and $s$-Covering Radius

Iskander Aliev, Martin Henk, Eva Linke


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
P(A, b)=\{ x \in \mathbb{R}^n_{\ge 0}: A x= b\}
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.




Knapsack polytope, (diagonal) Frobenius numbers, inhomogeneous minimum, covering radius, successive minima

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