, Volume 12, Issue 2, pp 161–177 | Cite as

Lattice translates of a polytope and the Frobenius problem

  • Ravi Kannan


This paper considers the “Frobenius problem”: Givenn natural numbersa1,a2, 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 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+ℤh={xx∈ℝn;x=y+z;yK;z∈ℤn} 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 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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D. E. Bell: A theorem concerning the integer lattice,Studies in Applied Mathematics56 (1976/77) 187–188.MathSciNetCrossRefGoogle Scholar
  2. [2]
    A. Brauer, andJ. E. Shockley: On a problem of Frobenius,Journal für reine und angewandte Mathematik211 (1962) 399–408.zbMATHGoogle Scholar
  3. [3]
    W. Cook, A. M. H. Gerards., A. Schrijver, andE. Tardos: Sensitivity theorems in integer linear programming,Mathematical Programming34 (1986) 251–264MathSciNetCrossRefGoogle Scholar
  4. [4]
    P. Erdős, andR. Graham: On a linear diophantine problem of Frobenius,Acta Arithmetica21 (1972).Google Scholar
  5. [5]
    H. Greenberg:, Solution to a linear diophantine equation for nonnegative integers,Journal of Algorithms9 (1988) 343–353.MathSciNetCrossRefGoogle Scholar
  6. [6]
    M. Hujter, andB. Vizvári: The exact solution to the Frobenius problem with three variables.,Journal of the Ramanujan Math. Soc.2 (1987) 117–143.MathSciNetzbMATHGoogle Scholar
  7. [7]
    M. Grötschel, L. Lovász, andA Schrijver:Geometric algorithms and combinatorial optimization, Springer-Verlag, 1988.Google Scholar
  8. [8]
    J. Incerpi, andR. Sedgwick: Improved upper bounds on ShellSort,Journal of Computer and Systems Sciences31 (1985), 210–224.MathSciNetCrossRefGoogle Scholar
  9. [9]
    R. Kannan: Minkowski's Convex body theorem and integer programming,Mathematics of Operations Research12 (1987), 415–440.MathSciNetCrossRefGoogle Scholar
  10. [10]
    R. Kannan: Test sets for integer programs, Å∀ sentences, in:DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 1,Polyhedral Combinatorics, (eds., W. Cook, P. D. Seymour), 1990, American Mathematical Society 39–47Google Scholar
  11. [11]
    R. Kannan, andL. Lovász,: Covering minima and lattice point free convex bodies, in: Lecture Notes in Computer Science 241, ed. (K. V. Nori), Springer-Verlag (1986) 193–213. Final version inAnnals of Mathematics128 (1988) 577–602.Google Scholar
  12. [12]
    R. Kannan, L. Lovász, andH. E. Scarf,: The shapes of polyhedra, Cowles Foundation Discussion paper No. 883, September (1988), to appear inMathematics of Operations Research.Google Scholar
  13. [13]
    H. Krawczyk, andA. Paz: The diophantine problem of Frobenius: A close bound,Discrete Applied Mathematics23 (1989) 289–291.MathSciNetCrossRefGoogle Scholar
  14. [14]
    H. W. Lenstra: Integer programming with a fixed number of variables,Mathematics of Operations Research8 (1983) 538–548.MathSciNetCrossRefGoogle Scholar
  15. [15]
    L. Lovász: Geometry of Numbers and Integer Programming, Proceedings of the 13th International Symposium onMathematical Programming, (M. Iri and K. Tanabe eds.),Mathematical Programming (1989) 177–201.Google Scholar
  16. [16]
    O. J. Rödseth: On a linear diophantine problem of Frobenius,Journal für reine und angewandte Mathematik301 (1978), 171–178.MathSciNetzbMATHGoogle Scholar
  17. [17]
    H. E. Scarf: An observation on the structure of production sets with indivisibilities,Proceedings of the National Academy of Sciences USA74 (1977) 3637–3641.MathSciNetCrossRefGoogle Scholar
  18. [18]
    H. E. Scarf, andD. Shallcross: The Frobenius problem and maximal lattice free bodies, Manuscript (1989).Google Scholar
  19. [19]
    R. Sedgwick: A new upper bound for ShellSort,Journal of Algorithms7 (1986), 159–173.MathSciNetCrossRefGoogle Scholar
  20. [20]
    E. S. Selmer: On the linear diophantine problem of FrobeniusJournal für reine und angewandte Mathematik293/294 (1977) 1–17.MathSciNetzbMATHGoogle Scholar
  21. [21]
    E. S. Selmer, andO. Beyer: On the linear diophantine problem of Frobenius in three variables,Journal für reine und angewandte Mathematik301 (1978), 161–170.MathSciNetzbMATHGoogle Scholar
  22. [22]
    A. Schrijver:Theory of Linear and Integer Programming, Wiley, 1986.Google Scholar
  23. [23]
    B. Vizvári: An application of Gomory cuts in number theory,Periodica Mathematica Hungarica18 (1987) 213–228.MathSciNetCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó 1992

Authors and Affiliations

  • Ravi Kannan
    • 1
  1. 1.School of Computer ScienceCarnegie Mellon UniversityPittsburgh

Personalised recommendations