Reducing the Coefficients of a Two-Dimensional Integer Linear Constraint

Abstract

Let us consider a two-dimensional linear constraint C of the form ax + by ≤ c with integer coefficients and such that |a| ≤ |b|. A constraint C′ of the form a′x + b′y ≤ c′ is equivalent to C relative to a domain iff all the integer points in the domain satisfying C satisfy C′ and all the integer points in the domain not satisfying C do not satisfy C′. This paper introduces a new method to transform a constraint C into an equivalent constraint C′ relative to a domain defined by {(x,y) | h ≤ x ≤ h + D } such that the absolute values of a′ and b′ do not exceed D. Our method achieves a O(log(D)) time complexity and it can operate when the constraints coefficients are real values with the same time complexity. This transformation can be used to compute the convex hull of the integer points which satisfy a system of n two-dimensional linear constraints in O(n log(D)) time where D represents the size of the solution space. Our algorithm uses elementary statements from number theory and leads to a simple and efficient implementation.