The ellipsoid algorithm for linear inequalities in exact arithmetic

Abstract

A modification of the ellipsoid algorithm is shown to be capable of testing for satisfiability a system of linear equations and inequalities with integer coefficients of the form Ax = b, x ≥ 0. All the rational arithmetic is performed exactly, without losing polynomiality of the computing time. In case of satisfiability, the approach always provides a rational feasible point. The bulk of the computations consists of a sequence of linear least squares problems, each a rank one modification of the preceding one. The continued fractions jump is used to compute some of the coordinates of a feasible point.