On the Complexity of Recognizing the Hilbert Basis of a Linear Diophantine System

  • Arnaud Durand
  • Miki Hermann
  • Laurent Juban
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1672)


The problem of computing the Hilbert basis of a linear Diophantine system over nonnegative integers is often considered in automated deduction and integer programming. In automated deduction, the Hilbert basis of a corresponding system serves to compute the minimal complete set of associative-commutative unifiers, whereas in integer programming the Hilbert bases are tightly connected to integer polyhedra and to the notion of total dual integrality. In this paper, we sharpen the previously known result that the problem, asking whether a given solution belongs to the Hilbert basis of a given system, is coNP-complete. We show that the problem has a pseudopolynomial algorithm if the number of equations in the system is fixed, but it is coNP-complete in the strong sense if the given system is unbounded. This result is important in the scope of automated deduction, where the input is given in unary and therefore the previously known coNP-completeness result was unusable. Moreover, we prove that, given a linear Diophantine system and a set of solutions, asking whether this set constitutes the Hilbert basis of the system, is also coNP-complete in the strong sense, answering this way an open problem formulated by Henk and Weismantel in 1996. Our result also allows us to solve another open problem, formulated by Edmonds and Giles in 1982, where we prove that asking whether a given set of vectors constitutes the Hilbert basis of an unknown linear Diophantine system, is coNP-complete in the strong sense.


Integer Programming Nonnegative Integer Minimal Solution Integral Vector Integral Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [CD94]
    E. Contejean and H. Devie. An efficient incremental algorithm for solving systems of linear Diophantine equations. Information and Computation, 113(1):143–172, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [CF89]
    M. Clausen and A. Fortenbacher. Efficient solution of linear Diophantine equations. Journal of Symbolic Computation, 8(1–2):201–216, 1989.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [Dom91]
    E. Domenjoud. Solving systems of linear Diophantine equations: An algebraic approach. In A. Tarlecki, ed., Proceedings 16th MFCS, Kazimierz Dolny (Poland), LNCS 520, pages 141–150. Springer, 1991.Google Scholar
  4. [EG82]
    J. Edmonds and R. Giles. Total dual integrality of linear inequality systems. In W. R. Pulleyblank, ed., Proceedings Progress in Combinatorial Optimization, Waterloo (Ontario, Canada), pages 324–333. Academic Press, 1982.Google Scholar
  5. [GJ79]
    M. R. Garey and D. S. Johnson. Computers and intractability: A guide to the theory of NP-completeness. W.H. Freeman and Co, 1979.Google Scholar
  6. [Gor73]
    P. Gordan. Ueber die Auflösung linearen Gleichungen mit reellen Coefficienten. Mathematische Annalen, 6:23–28, 1873.CrossRefMathSciNetGoogle Scholar
  7. [Hil90]
    D. Hilbert. Ueber die Theorie der algebraischen Formen. Mathematische Annalen, 36:473–534, 1890.CrossRefMathSciNetGoogle Scholar
  8. [Hue78]
    G. Huet. An algorithm to generate the basis of solutions to homogeneous linear Diophantine equations. Inf. Proc. Letters, 7(3):144–147, 1978.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [HW96]
    M. Henk and R. Weismantel. On Hilbert bases of polyhedral cones. Preprint SC 96-12, Konrad-Zuse-Zentrum für Informationstechnik, Berlin, April 1996. URL =
  10. [KB79]
    R. Kannan and A. Bachem. Algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM J. Computing, 8(4):499–507, 1979.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [Lam87]
    J.-L. Lambert. Une borne pour les générateurs des solutions entières positives d’une équation diophantienne linéaire. Compte-rendus de l’Académie des Sciences de Paris, 305(1):39–40, 1987.zbMATHMathSciNetGoogle Scholar
  12. [Lan89]
    D. Lankford. Non-negative integer basis algorithms for linear equations with integer coefficients. Journal of Automated Reasoning, 5(1):25–35, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [Pap81]
    C. H. Papadimitriou. On the complexity of integer programming. Journal of the Association for Computing Machinery, 28(4):765–768, 1981.zbMATHMathSciNetGoogle Scholar
  14. [Pap94]
    C. H. Papadimitriou. Computational complexity. Addison-Wesley, 1994.Google Scholar
  15. [Sch86]
    A. Schrijver. Theory of linear and integer programming. Wiley, 1986.Google Scholar
  16. [Seb90]
    A. Sebő. Hilbert bases, Carathéodory’s theorem and combinatorial optimization. In R. Kannan and W. R. Pulleyblank, eds., Proc. 1st IPCO, Waterloo (Ontario, Canada), pages 431–455. University of Waterloo Press, May 1990.Google Scholar
  17. [Sti81]
    M. Stickel. A unification algorithm for associative-commutative functions. Journal of the Association for Computing Machinery, 28(3):423–434, 1981.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Arnaud Durand
    • 1
  • Miki Hermann
    • 2
  • Laurent Juban
    • 2
  1. 1.LACL, Department of Computer ScienceUniversité Paris 12CréteilFrance
  2. 2.LORIA (CNRS and Université Henri Poincaré Nancy 1Vandceuvre-lès-NancyFrance

Personalised recommendations