Parameterized complexity and improved inapproximability for computing the largest j-simplex in a V-polytope

We consider the problem of computing the squared volume of the largest j-simplex contained in an n-dimensional polytope presented by its vertices (a V -polytope). We show that the related decision problem is W [1]-complete, with respect to the parameter j. We also improve the constant inapproximability factor given in [Packer, 2004, Discrete Applied Mathematics, 134], by showing that there are constants μ < 1, c > 1, such that it is NP-hard to approximate within a factor of c the volume of the largest bμnc-simplex contained in an n-dimensional polytope with O(n) vertices.