Lattice translates of a polytope and the Frobenius problem

Abstract

This paper considers the “Frobenius problem”: Givenn natural numbersa1,a2,...an 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.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    D. E. Bell: A theorem concerning the integer lattice,Studies in Applied Mathematics56 (1976/77) 187–188.

    MathSciNet  Article  Google Scholar 

  2. [2]

    A. Brauer, andJ. E. Shockley: On a problem of Frobenius,Journal für reine und angewandte Mathematik211 (1962) 399–408.

    MATH  Google Scholar 

  3. [3]

    W. Cook, A. M. H. Gerards., A. Schrijver, andE. Tardos: Sensitivity theorems in integer linear programming,Mathematical Programming34 (1986) 251–264

    MathSciNet  Article  Google Scholar 

  4. [4]

    P. Erdős, andR. Graham: On a linear diophantine problem of Frobenius,Acta Arithmetica21 (1972).

  5. [5]

    H. Greenberg:, Solution to a linear diophantine equation for nonnegative integers,Journal of Algorithms9 (1988) 343–353.

    MathSciNet  Article  Google 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.

    MathSciNet  MATH  Google Scholar 

  7. [7]

    M. Grötschel, L. Lovász, andA Schrijver:Geometric algorithms and combinatorial optimization, Springer-Verlag, 1988.

  8. [8]

    J. Incerpi, andR. Sedgwick: Improved upper bounds on ShellSort,Journal of Computer and Systems Sciences31 (1985), 210–224.

    MathSciNet  Article  Google Scholar 

  9. [9]

    R. Kannan: Minkowski's Convex body theorem and integer programming,Mathematics of Operations Research12 (1987), 415–440.

    MathSciNet  Article  Google 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 39–47

  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.

  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.

  13. [13]

    H. Krawczyk, andA. Paz: The diophantine problem of Frobenius: A close bound,Discrete Applied Mathematics23 (1989) 289–291.

    MathSciNet  Article  Google Scholar 

  14. [14]

    H. W. Lenstra: Integer programming with a fixed number of variables,Mathematics of Operations Research8 (1983) 538–548.

    MathSciNet  Article  Google 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.

  16. [16]

    O. J. Rödseth: On a linear diophantine problem of Frobenius,Journal für reine und angewandte Mathematik301 (1978), 171–178.

    MathSciNet  MATH  Google 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.

    MathSciNet  Article  Google Scholar 

  18. [18]

    H. E. Scarf, andD. Shallcross: The Frobenius problem and maximal lattice free bodies, Manuscript (1989).

  19. [19]

    R. Sedgwick: A new upper bound for ShellSort,Journal of Algorithms7 (1986), 159–173.

    MathSciNet  Article  Google Scholar 

  20. [20]

    E. S. Selmer: On the linear diophantine problem of FrobeniusJournal für reine und angewandte Mathematik293/294 (1977) 1–17.

    MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google Scholar 

  22. [22]

    A. Schrijver:Theory of Linear and Integer Programming, Wiley, 1986.

  23. [23]

    B. Vizvári: An application of Gomory cuts in number theory,Periodica Mathematica Hungarica18 (1987) 213–228.

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

Supported by NSF-Grant CCR 8805199

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kannan, R. Lattice translates of a polytope and the Frobenius problem. Combinatorica 12, 161–177 (1992). https://doi.org/10.1007/BF01204720

Download citation

AMS subject classification code (1991)

  • 11 H 31
  • 52 C 07
  • 52 C 17
  • 90 C 10