We show that the problem of finding the simplex of largest volume in the
convex hull of n points in Qd can be approximated with a factor
of O(logd)d/2 in polynomial time. This improves upon the previously best
known approximation guarantee of d(d−1)/2 by Khachiyan. On the other hand,
we show that there exists a constant c>1 such that this problem cannot be
approximated with a factor of cd, unless P=NP. % This improves over the
1.09 inapproximability that was previously known. Our hardness result holds
even if n=O(d), in which case there exists a \bar c\,^{d}-approximation
algorithm that relies on recent sampling techniques, where cˉ is again a
constant. We show that similar results hold for the problem of finding the
largest absolute value of a subdeterminant of a d×n matrix