An old linear programming algorithm runs in polynomial time

Abstract

The Ellipsoid Algorithm (EA) for linear programming attracted recently great attention. EA was proposed in [N76] and developed in [K79, G81] and other works. It is a modification of Method of Centralized Splitting presented in [L65], which differs from EA in two essential respects. Firstly, [L65] uses simplexes instead of ellipsoids; it is admitted, secondly, that, several (q(n))splittings of the n-dimensional simplex may be needed before the remaining polyhedron can be enclosed into a simplex of a smaller volume. Only a very rough upper bound q(n) ≪ nlog(n)follows from the reasoning of [L65]. This does not imply polynomiality of the computation time, since n, log(n) splittings may make the simplex very complex. We prove below that, q(n)= 1. Let the problem be to find x∈Rn such that Ax ≫ 0, where A is an m × n matrix of rank n. We normalize solutions by a restriction (e - Ax) = 1 where e ≫ 0. On every step the algorithm considers a simplex BAx ≥ 0 containing all solutions, where B is a non-negative n × m matrix with det(BA) ≠ 0. Let us denote this simplex by ΔB, its volume by VB and its center by CB. Initially we take an arbitrary B and e = BT(1,..,1).