Minimum-volume enclosing ellipsoids and core sets

We study the problem of computing a ( 1 + epsilon)-approximation to the minimum-volume enclosing ellipsoid of a given point set S = {p(1), p(2),..., p(n)} subset of R-d. Based on a simple, initial volume approximation method, we propose a modi. cation of the Khachiyan first-order algorithm. Our analysis leads to a slightly improved complexity bound of O(nd(3)/epsilon) operations for epsilon is an element of(0, 1). As a byproduct, our algorithm returns a core set X subset of S with the property that the minimum-volume enclosing ellipsoid of X provides a good approximation to that of S. Furthermore, the size of X depends on only the dimension d and epsilon, but not on the number of points n. In particular, our results imply that | X| = O(d(2)/epsilon) for epsilon is an element of(0, 1).