Combinatorica

, Volume 12, Issue 2, pp 135–142

On integer points in polyhedra: A lower bound

Article

Abstract

Given a polyhedronP⊂ℝ we writePI for the convex hull of the integral points inP. It is known thatPI can have at most135-2 vertices ifP is a rational polyhedron with size φ. Here we give an example showing thatPI can have as many as Ω(ϕn−1) vertices. The construction uses the Dirichlet unit theorem.

AMS subject classification code (1991)

52 C 07 11 H 06 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Z. I. Borevich, andI. R. Safarevich:Number theory, Academic Press, New York and London, 1966.Google Scholar
  2. [2]
    W. Cook, M. Hartmann, R. Kannan, andC. McDiarmid: On integer points in polyhedra,Combinatorica 12 (1992), 27–37.Google Scholar
  3. [3]
    A. C. Hayes, andD. G. Larman: The vertices of the knapsack polytope,Discrete Applied Math. 6 (1983), 135–138.Google Scholar
  4. [4]
    S. Lang:Algebraic number theory, Graduate Texts in Mathematics 110, Springer Verlag, New York etc., 1986.Google Scholar
  5. [5]
    D. Morgan: Personal communication, 1989.Google Scholar
  6. [6]
    D. S. Rubin: On the unlimited number of faces in integer hulls of linear programs with a single constraint,Operations Research 18 (1970), 940–946.Google Scholar
  7. [7]
    A. Schrijver:Theory of linear and integer programming, Wiley, Chichester, 1987.Google Scholar
  8. [8]
    V. N. Shevchenko: On the number of extreme points in integer programming,Kibernetika 2 (1981), 133–134.Google Scholar

Copyright information

© Akadémiai Kiadó 1992

Authors and Affiliations

  1. 1.Mathematical InstituteBudapestHungary
  2. 2.Department of MathematicsYale UniversityNew HavenU. S. A.
  3. 3.Department of Computer ScienceEötvös UniversityBudapestHungary
  4. 4.Princeton UniversityPrincetonU. S. A.

Personalised recommendations