Integer Linear Systems: Projection and Inverse Projection

Abstract

In this chapter we address the problem of finding an x ∈ ℜn which is a solution to the system Ax = b or Ax ≥ b with some or all of the x j required to be integer. This problem has a very long history. Integer equalities were studied by the Greek mathematician Diophantos during the third century A.D. In fact, when all the variables are required to be integer the equations in the system Ax = b are called linear Diophantine equations. We use the terms Diophantine and integer interchangeably. The study of linear and nonlinear Diophantine equations is an important part of number theory. Although linear Diophantine equations were studied in the third century A.D. it was not until the 20th century A.D. (1976 to be precise) that a polynomial algorithm was given for finding an integer solution the system Ax = b.