# Lattice translates of a polytope and the Frobenius problem

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## Abstract

This paper considers the “Frobenius problem”: Given*n* natural numbers*a*_{1},*a*_{2},...*a*_{n} such that their greatest common divisor is 1, find the largest natural number that is not expressible as a nonnegative integer combination of them. This problem can be seen to be NP-hard. For the cases*n*=2,3 polynomial time algorithms, are known to solve it. Here a polynomial time algorithm is given for every fixed*n*. This is done by first proving an exact relation between the Frobenius problem and a geometric concept called the “covering radius”. Then a polynomial time algorithm is developed for finding the covering radius of any polytope in a fixed number of dimensions. The last algorithm relies on a structural theorem proved here that describes for any polytope*K*, the set*K*+ℤ^{h}={*x*∶*x*∈ℝ^{n};*x=y+z*;*y*∈*K*;*z*∈ℤ^{n}} which is the portion of space covered by all lattice translates of*K*. The proof of the structural theorem relies on some recent developments in the Geometry of Numbers. In particular, it uses a theorem of Kannan and Lovász [11], bounding the width of lattice-point-free convex bodies and the techniques of Kannan, Lovász and Scarf [12] to study the shapes of a polyhedron obtained by translating each facet parallel, to itself. The concepts involved are defined from first principles. In a companion paper [10], I extend the structural result and use that to solve a general problem of which the Frobenius problem is a special case.

### AMS subject classification code (1991)

11 H 31 52 C 07 52 C 17 90 C 10## Preview

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