, Volume 12, Issue 1, pp 27-37

On integer points in polyhedra

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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.