The Complexity of Yamnitsky and Levin’s Simplices Algorithm

Abstract

In 1982 Yamnitsky and Levin gave a variant of the ellipsoid method which uses simplices instead of ellipsoids. Unlike the ellipsoid method this simplices method can be implemented in rational arithmetic. We show, however, that this results in a non-polynomial method since the storage requirement may grow exponentially with the size of the input. Neverthe-less, by introducing a rounding procedure we can guarantee polynomiality for both a central-cut and a shallow-cut version. Thus in most applications the simplices method can serve as a substitute for the ellipsoid method. In particular, it performs better than the ellipsoid method if it is used to ob-tain bounds for the volume of a convex body. Furthermore it can be used to estimate the optimal function value of total approximation problems.