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

  • Miki Hermann
  • Laurent Juban
  • Phokion G. Kolaitis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1705)


We investigate the computational complexity of counting the Hilbert basis of a homogeneous system of linear Diophantine equations. We establish lower and upper bounds on the complexity of this problem by showing that counting the Hilbert basis is #P-hard and belongs to the class #NP. Moreover, we investigate the complexity of variants obtained by restricting the number of occurrences of the variables in the system.


Bipartite Graph Minimal Solution Truth Assignment Disjunctive Normal Form Satisfying Assignment 
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  1. [BCD90]
    A. Boudet, E. Contejean, and H. Devie. A new AC unification algorithm with a new algorithm for solving Diophantine equations. In Proceedings 5th LICS, Philadelphia (PA, USA), pages 289–299, June 1990.Google Scholar
  2. [BKN87]
    D. Benanav, D. Kapur, and P. Narendran. Complexity of matching problems. Journal of Symbolic Computation, 3:203–216, 1987.zbMATHMathSciNetGoogle Scholar
  3. [BN98]
    F. Baader and T. Nipkow. Term rewriting and all that. Cambridge University Press, 1998.Google Scholar
  4. [Bou93]
    A. Boudet. Competing for the AC-unification race. Journal of Automated Reasoning, 11(2):185–212, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [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
  6. [CF89]
    M. Clausen and A. Fortenbacher. Efficient solution of linear Diophantine equations. Journal of Symbolic Computation, 8(1–2):201–216, 1989.MathSciNetzbMATHGoogle Scholar
  7. [DHJ99]
    A. Durand, M. Hermann, and L. Juban. On the complexity of recognizing the Hilbert basis of a linear Diophantine system. In L. Pacholski, editor, Proceedings 24th MFCS, Szklarska Poreba (Poland), LNCS, Springer-Verlag, September 1999.Google Scholar
  8. [Dom91a]
    E. Domenjoud. Outils pour la déduction automatique dans les théories associative-commutatives. PhD thesis, Université Henri Poincaré, Nancy, France, September 1991.Google Scholar
  9. [Dom91b]
    E. Domenjoud. Solving systems of linear Diophantine equations: An algebraic approach. In A. Tarlecki, editor, Proceedings 16th MFCS, Kazimierz Dolny (Poland), LNCS 520, pages 141–150. Springer-Verlag, September 1991.Google Scholar
  10. [Dur99]
    A. Durand. Personal communication, June 1999.Google Scholar
  11. [EE85]
    A. A. Elimam and S. E. Elmaghraby. On the reduction method for integer linear programs II. Discrete Applied Mathematics, 12(3):241–260, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [Fag87]
    F. Fages. Associative commutative unification. Journal of Symbolic Computation, 3(3):257–275, 1987.zbMATHMathSciNetGoogle Scholar
  13. [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
  14. [Gor73]
    P. Gordan. Ueber die Auflösung linearen Gleichungen mit reellen Coefficienten. Mathematische Annalen, 6:23–28, 1873.CrossRefMathSciNetGoogle Scholar
  15. [Hil90]
    D. Hilbert. Ueber die Theorie der algebraischen Formen. Mathematische Annalen, 36:473–534, 1890.CrossRefMathSciNetGoogle Scholar
  16. [HK95a]
    M. Hermann and P. G. Kolaitis. The complexity of counting problems in equational matching. Journal of Symbolic Computation, 20(3):343–362, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [HK95b]
    M. Hermann and P. G. Kolaitis. Computational complexity of simultaneous elementary matching problems. In J. Wiedermann and P. Hájek, editors, Proceedings 20th MFCS, Prague (Czech Republic), LNCS 969, pages 359–370. Springer-Verlag, August 1995.Google Scholar
  18. [HS87]
    A. Herold and J. H. Siekmann. Unification in Abelian semigroups. Journal of Automated Reasoning, 3(3):247–283, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [Hue78]
    G. Huet. An algorithm to generate the basis of solutions to homogeneous linear Diophantine equations. Information Processing Letters, 7(3):144–147, 1978.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [HV95]
    L. A. Hemaspaandra and H. Vollmer. The satanic notations: Counting classes beyond #P and other definitional adventures. SIGACT News, 26(1):2–13, March 1995.CrossRefGoogle Scholar
  21. [JK91]
    J.-P. Jouannaud and C. Kirchner. Solving equations in abstract algebras: A rule-based survey of unification. In J.-L. Lassez and G. Plotkin, editors, Computational Logic. Essays in honor of Alan Robinson, chapter 8, pages 257–321. MIT Press, Cambridge (MA, USA), 1991.Google Scholar
  22. [Joh90]
    D. S. Johnson. A catalog of complexity classes. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity, chapter 2, pages 67–161. North-Holland, Amsterdam, 1990.Google Scholar
  23. [Jub98]
    L. Juban. Comptage de l’ensemble des éléments de la base de Hilbert d’un système d’équations diophantiennes linéaires. Technical Report 98-R-066, LORIA, 1998.Google Scholar
  24. [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
  25. [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
  26. [LC89]
    P. Lincoln and J. Christian. Adventures in associative-commutative unification. Journal of Symbolic Computation, 8(1–2):217–240, 1989.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [Pap81]
    C. H. Papadimitriou. On the complexity of integer programming. Journal of the Association for Computing Machinery, 28(4):765–768, 1981.zbMATHMathSciNetGoogle Scholar
  28. [Pap94]
    C. H. Papadimitriou. Computational complexity. Addison-Wesley, 1994.Google Scholar
  29. [Pot91]
    L. Pottier. Minimal solutions of linear Diophantine systems: bounds and algorithms. In R.V. Book, editor, Proceedings 4th RTA, Como (Italy), LNCS 488, pages 162–173. Springer-Verlag, April 1991.Google Scholar
  30. [Sch86]
    A. Schrijver. Theory of linear and integer programming. John Wiley & Sons, 1986.Google Scholar
  31. [Sti75]
    M. Stickel. A complete unification algorithm for associative-commutative functions. In Proceedings 4th IJCAI, Tbilisi (USSR), pages 71–82, 1975.Google Scholar
  32. [Sti81]
    M. Stickel. A unification algorithm for associative-commutative functions. Journal of the Association for Computing Machinery, 28(3):423–434, 1981.zbMATHMathSciNetGoogle Scholar
  33. [Tod89]
    S. Toda. On the computational power of PP and ⊕P. In Proceedings 30th FOCS, Research Triangle Park (NC, USA), pages 514–519, 1989.Google Scholar
  34. [Tod91]
    S. Toda. Computational complexity of counting complexity classes. PhD thesis, Tokyo Institute of Technology, Department of Computer Science, Tokyo, Japan, 1991.Google Scholar
  35. [TW92]
    S. Toda and O. Watanabe. Polynomial-time 1-Turing reductions from #PH to #P. Theoretical Computer Science, 100(1):205–221, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  36. [Val79a]
    L. G. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8(2):189–201, 1979.zbMATHCrossRefMathSciNetGoogle Scholar
  37. [Val79b]
    L. G. Valiant. The complexity of enumeration and reliability problems. SIAM Journal on Computing, 8(3):410–421, 1979.zbMATHCrossRefMathSciNetGoogle Scholar
  38. [Zan91]
    V. Zankó. #P-completeness via many-one reductions. International Journal of Foundations of Computer Science, 2(1):77–82, 1991.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Miki Hermann
    • 1
  • Laurent Juban
    • 1
  • Phokion G. Kolaitis
    • 2
  1. 1.LORIA (CNRS and Université Henri Poincaré Nancy 1)Vandœuvre-lès-NancyFrance
  2. 2.Computer Science DepartmentUniversity of CaliforniaSanta CruzUSA

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