The Flatness Theorem for Some Class of Polytopes and Searching an Integer Point

  • Dmitry V. GribanovEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 104)


Let A be an m × n integral matrix of the rank n, we say that A has bounded minors if the maximum of the absolute values of the n × n minors is at most k, we will call these matrices as k-modular. We investigate an integer program \(\mathit{max}\{c^{\top }x: Ax \leq b,x \in \mathbb{Z}^{n}\}\), where A is k-modular. We say that A is almost unimodular if it is 2-modular and the absolute values of its \((n - 1) \times (n - 1)\) minors are at most 1. We also refer 2-modular matrices to as bimodular. We say that A is strict k-modular if the absolute values of its n × n minors are from set {0, k, −k}. We prove that the width of an empty lattice polytope is less than \((k - 1)\,(n + 1)\) if it is induced by a system of inequalities with a strict k-modular matrix. Furthermore, we can give a polynomial-time algorithm for searching an integer point in a strict k-modular polytope if its width is grater than \((k - 1)\,(n + 1)\).


Convex Hull Interior Point Integer Program Convex Body Normal Cone 
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The author wishes to express special thanks for the invaluable assistance to A.J. Chirkov, S.I. Veselov, D.S. Malyshev, V.N. Shevchenko, and S.V. Sorochan. The work is partly supported by National Research University Higher School of Economics, Russian Federation Government grant, N. 11.G34.31.0057.


  1. 1.
    Alekseev, V.E., Zakharova, D.V.: Independent sets in graphs with bounded minors of the extended incidence matrix. Discrete Anal. Oper. Res. 17(1), 3–10 (2010) [in russian]MathSciNetzbMATHGoogle Scholar
  2. 2.
    Banaszczyk, W., Litvak, A.E., Pajor, A., Szarek, S.J.: The flatness theorem for non-symmetric convex bodies via the local theory of Banach spaces. Math. Oper. Res. 24(3), 728–750 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Banaszczyk, W.: Inequalities for convex bodies and polar reciprocal lattices in \(\mathbb{R}^{n}\) II: Application of K-convexity. Discrete Comput. Geom. 16(3), 305–311 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cornuéjols, G., Zuluaga, L.F.: On Padberg’s conjecture about almost totally unimodular matrices. Oper. Res. Lett. 27(3), 97–99 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dadush, D.: Transference Theorems in the Geometry of Numbers.
  6. 6.
    Gribanov, D.V.: On Integer Programing With Almost Unimodular Matrices and The Flatness Theorem for Simplexes. Preprint (2014)Google Scholar
  7. 7.
    Grossman, J.W., Kilkarni, D.M., Schochetman, I.E.: On the minors of an incidence matrix and its Smith normal form. Linear Algebra Appl. 218, 213–224 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Haase, C., Ziegler, G.: On the maximal width of empty lattice simplices. Eur. J. Combinatorics 21, 111–119 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kannan, R., Lovász, L.: Covering minima and lattice-point-free convex bodies. Ann. Math. 128, 577–602 (1988)CrossRefzbMATHGoogle Scholar
  10. 10.
    Kantor, J.M.: On the width of lattice-free simplexes. Cornell University Library (1997)
  11. 11.
    Khachiyan, L.G.: Polynomial algorithms in the linear programming. Comput. Math. Math. Phys. 20(1), 53–72 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Khinchine, A.: A quantitative formulation of Kronecker’s theory of approximation. Izvestiya Akademii Nauk SSR Seriya Matematika 12, 113–122 (1948) [in russian]Google Scholar
  13. 13.
    Rudelson, M.: Distances between non-symmetric convex bodies and the MM -estimate. Positivity 4(2), 161–178 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Schrijver, A.: Theory of Linear and Integer Programming. WileyInterscience Series in Discrete Mathematics. Wiley (1998)zbMATHGoogle Scholar
  15. 15.
    Sebö, A.: An introduction to empty lattice simplexes. In: Cornuéjols, G., Burkard, R.R., Woeginger, R.E. (eds.) LNCS, vol. 1610, 400–414 (1999)Google Scholar
  16. 16.
    Shevchenko, V.N.: Qualitative Topics in Integer Linear Programming (Translations of Mathematical Monographs). AMS (1996)Google Scholar
  17. 17.
    Veselov, S.I., Chirkov, A.J.: Integer program with bimodular matrix. Discrete Optim. 6(2), 220–222 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ziegler, G.: Lectures on Polytopes, vol. 152. GTM, Springer,New York/Berlin/Heidelberg (1996)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Nizhny Novgorod Lobachevsky State UniversityNational Research University Higher School of EconomicsNizhny NovgorodRussian Federation

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