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2008-18

Integer Knapsacks: Average behavior of the frobenius numbers

by Henk, M., Aliev, I..


Series: 2008-18, Preprints

MSC:
11D04 Linear equations
11D85 Representation problems
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