, Volume 12, Issue 2, pp 161177
Lattice translates of a polytope and the Frobenius problem
 Ravi KannanAffiliated withSchool of Computer Science, Carnegie Mellon University
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This paper considers the “Frobenius problem”: Givenn natural numbersa _{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 NPhard. For the casesn=2,3 polynomial time algorithms, are known to solve it. Here a polynomial time algorithm is given for every fixedn. 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 polytopeK, the setK+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 ofK. 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 latticepointfree 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 Title
 Lattice translates of a polytope and the Frobenius problem
 Journal

Combinatorica
Volume 12, Issue 2 , pp 161177
 Cover Date
 199206
 DOI
 10.1007/BF01204720
 Print ISSN
 02099683
 Online ISSN
 14396912
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 11 H 31
 52 C 07
 52 C 17
 90 C 10
 Industry Sectors
 Authors

 Ravi Kannan ^{(1)}
 Author Affiliations

 1. School of Computer Science, Carnegie Mellon University, 16213, Pittsburgh, Pennsylvania