08-18
Integer Knapsacks: Average behavior of the frobenius numbers
Preprint series: 08-18, Preprints
- MSC:
- 11D04 Linear equations
- 11D85 Representation problems, See also {11P55}
- 68Q25 Analysis of algorithms and problem complexity
Abstract: Given a primitive integer vector $a\in \mathbb{Z}^N_{>0}$, the largest integer $b$ such that the knapsack polytope $P=\{x\in \mathbb{R}_{\geq 0}^N:\langle a,x\rangle=b\}$ contains no integer point is called the Frobenius number of $a$. We show that the asymptotic growth of the Frobenius number in average is significantly slower than the growth of the maximum Frobenius number. More precisely, we prove that it does not essentially exceed $\|a\|_{\infty}^{1+1/(N-1)}$, where $\|\cdot \|_{\infty}$ denotes the maximum norm.
Keywords: Frobenius number, successive minima, inhomogeneous minimum, distribution of sublattices
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