Minimum-Volume Enclosing Ellipsoids and Core Sets1

We study the problem of computing a (1 + )-approximation to the minimum volume enclosing ellipsoid of a given point set S = {p, p, . . . , p} ⊆ R. Based on a simple, initial volume approximation method, we propose a modification of Khachiyan’s first-order algorithm. Our analysis leads to a slightly improved complexity bound of O(nd/ ) operations for ∈ (0, 1). As a byproduct, our algorithm returns a core set X ⊆ 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 only on the dimension d and , but not on the number of points n. In particular, our results imply that |X | = O(d/ ) for ∈ (0, 1).