We show that the problem of finding the simplex of largest volume in the convex hull of n points in Q d can be approximated with a factor of O(log d) 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 c d , unless P = N P . Our hardness result holds even if n = O(d), in which case there exists ac d -approximation algorithm that relies on recent sampling techniques, wherec 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.
Their results imply an e approximation ratio for Eq. (1.1) in the special case of k = p. On the other hand, Summa et al. REF showed that there exists a constant c > 1 such that polynomial-time c-approximation of the D-optimality is impossible for the p = k case, unless P = NP.
On largest volume simplices and sub-determinants
ArXiv
ArXiv
Mathematics Computer Science