, Volume 12, Issue 1, pp 27–37

On integer points in polyhedra

  • W. Cook
  • M. Hartmann
  • R. Kannan
  • C. McDiarmid

DOI: 10.1007/BF01191202

Cite this article as:
Cook, W., Hartmann, M., Kannan, R. et al. Combinatorica (1992) 12: 27. doi:10.1007/BF01191202


We give an upper bound on the number of vertices ofPI, the integer hull of a polyhedronP, in terms of the dimensionn of the space, the numberm of inequalities required to describeP, and the size ϕ of these inequalities. For fixedn the bound isO(mnϕn−). We also describe an algorithm which determines the number of integer points in a polyhedron to within a multiplicative factor of 1+ε in time polynomial inm, ϕ and 1/ε when the dimensionn is fixed.

AMS subject classification code (1991)

52 A 25 90 C 10 

Copyright information

© Akadémiai Kiadó 1992

Authors and Affiliations

  • W. Cook
    • 1
  • M. Hartmann
    • 2
  • R. Kannan
    • 3
  • C. McDiarmid
    • 4
  1. 1.Bell Communications Research, U.S.A. and Institut für Ökonometrie und Operations ResearchUniversität BonnGermany
  2. 2.Department of Operations ResearchUniversity of North CarolinaUSA
  3. 3.Computer Science DepartmentCarnegie-Mellon UniversityUSA
  4. 4.Institute of Economics and StatisticsOxfordUK

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