On integer points in polyhedra

Abstract

We give an upper bound on the number of vertices ofP I , 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(m n ϕ n−). 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.

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Supported by Sonderfschungsbereich 303 (DFG) and NSF grant ECS-8611841.

Partially supported by NSF grant DMS-8905645.

Supported by NSF grants ECS-8418392 and CCR-8805199.

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Cook, W., Hartmann, M., Kannan, R. et al. On integer points in polyhedra. Combinatorica 12, 27–37 (1992). https://doi.org/10.1007/BF01191202

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AMS subject classification code (1991)

  • 52 A 25
  • 90 C 10