Computing the Löwner-John ellipsoid of a polyhedron The Löwner-John ellipsoid of a convex set $C$ is the minimum-volume ellipsoid (MVE) that encloses it. The ellipsoid can be computed using Khachiyan's method, and there are a number of approximations available if $C$ is (the convex hull of) a set of points. 

Are there fast (i.e non-ellipsoid-method based) approximations to the MVE of a bounded polyhedron presented only in terms of the halfplanes whose intersections define it ? In particular, I'd be interested in methods that run in time polynomial in the dimension and the inverse error $1/\varepsilon$.
According to Boyd, it's NP-hard: http://youtu.be/mNzu42FrlHo?t=41m3s