Computing the Integer Points of a Polyhedron, II: Complexity Estimates

  • Rui-Juan JingEmail author
  • Marc Moreno Maza
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10490)


Let K be a polyhedron in \({{{\mathbb R}}}^d\), given by a system of m linear inequalities, with rational number coefficients bounded over in absolute value by L. In this series of two papers, we propose an algorithm for computing an irredundant representation of the integer points of K, in terms of “simpler” polyhedra, each of them having at least one integer point. Using the terminology of W. Pugh: for any such polyhedron P, no integer point of its grey shadow extends to an integer point of P. We show that, under mild assumptions, our algorithm runs in exponential time w.r.t. d and in polynomial w.r.t m and L. We report on a software experimentation. In this series of two papers, the first one presents our algorithm and the second one discusses our complexity estimates.



The authors would like to thank IBM Canada Ltd (CAS project 880) and NSERC of Canada (CRD grant CRDPJ500717-16), as well as the University of Chinese Academy of Sciences, UCAS Joint PhD Training Program, for supporting their work.


  1. 1.
    4ti2 team. 4ti2–a software package for algebraic, geometric and combinatorial problems on linear spaces.
  2. 2.
    Aubry, P., Lazard, D., Moreno Maza, M.: On the theories of triangular sets. J. Symb. Comput. 28, 105–124 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bruns, W., Ichim, B., Römer, T., Sieg, R., Söger, C.: Normaliz. Algorithms for rational cones and affine monoids.
  4. 4.
    Chen, C., Davenport, J.H., May, J.P., Moreno Maza, M., Xia, B., Xiao, R.: Triangular decomposition of semi-algebraic systems. J. Symb. Comput. 49, 3–26 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Imbert, J.-L.: Fourier’s elimination: which to choose? pp. 117–129 (1993)Google Scholar
  6. 6.
    Karmarkar, N.: A new polynomial-time algorithm for linear programming. In: Proceedings of the Sixteenth Annual ACM Aymposium on Theory of Computing. STOC 1984, pp. 302–311. ACM, New York, NY, USA (1984)Google Scholar
  7. 7.
    Khachiyan, L.: Fourier-motzkin elimination method. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, pp. 1074–1077. Springer, Heidelberg (2009). doi: 10.1007/978-0-387-74759-0_187 CrossRefGoogle Scholar
  8. 8.
    Motzkin, T.S.: Beiträge zur Theorie der linearen Ungleichungen. Azriel Press, Jerusalem (1936)zbMATHGoogle Scholar
  9. 9.
    Pugh, W.: The omega test: a fast and practical integer programming algorithm for dependence analysis. In: Martin, J.L. (ed.), Proceedings Supercomputing 1991, Albuquerque, NM, USA, 18–22 November 1991, pp. 4–13. ACM (1991)Google Scholar
  10. 10.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.KLMM, UCAS, Academy of Mathematics and Systems Science, Chinese Academy of SciencesBeijingChina
  2. 2.University of Western OntarioLondonCanada

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