Polynomial algorithms for linear programming over the algebraic numbers

Abstract

We derive a bound on the computational complexity of linear programs whose coefficients are real algebraic numbers. Key to this result is a notion of problem size that is analogous in function to the binary size of a rational-number problem. We also view the coefficients of a linear program as members of a finite algebraic extension of the rational numbers. The degree of this extension is an upper bound on the degree of any algebraic number that can occur during the course of the algorithm, and in this sense can be viewed as a supplementary measure of problem dimension. Working under an arithmetic model of computation, and making use of a tool for obtaining upper and lower bounds on polynomial functions of algebraic numbers, we derive an algorithm based on the ellipsoid method that runs in time bounded by a polynomial in the dimension, degree, and size of the linear program. Similar results hold under a rational number model of computation, given a suitable binary encoding of the problem input.