Theory of linear and integer programming
Theory of linear and integer programming
Solving a System of Linear Diophantine Equations with Lower and Upper Bounds on the Variables
Mathematics of Operations Research
Non-standard approaches to integer programming
Discrete Applied Mathematics
Hard Equality Constrained Integer Knapsacks
Mathematics of Operations Research
Frobenius Problem and the Covering Radius of a Lattice
Discrete & Computational Geometry
Nonlinear Optimization over a Weighted Independence System
AAIM '09 Proceedings of the 5th International Conference on Algorithmic Aspects in Information and Management
Integer Knapsacks: Average Behavior of the Frobenius Numbers
Mathematics of Operations Research
Journal of Combinatorial Theory Series A
Discrete Optimization
Bounds on generalized Frobenius numbers
European Journal of Combinatorics
LLL-reduction for integer knapsacks
Journal of Combinatorial Optimization
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Given a matrix $A\in\mathbb{Z}^{m\times n}$ satisfying certain regularity assumptions, we consider the set $\mathcal{F}(A)$ of all vectors $\boldsymbol{b}\in\mathbb{Z}^m$ such that the associated knapsack polytope $P(A,\boldsymbol{b})=\{\boldsymbol{x}\in\mathbb{R}^n_{\geq0}:A\boldsymbol{x}=\boldsymbol{b}\}$ contains an integer point. When $m=1$ the set $\mathcal{F}(A)$ is known to contain all consecutive integers greater than the Frobenius number associated with $A$. In this paper we introduce the diagonal Frobenius number $\mathrm{g}(A)$ which reflects in an analogous way feasibility properties of the problem and the structure of $\mathcal{F}(A)$ in the general case. We give an optimal upper bound for $\mathrm{g}(A)$ and also estimate the asymptotic growth of the diagonal Frobenius number on average.