On largest volume simplices and sub-determinants

TL;DR

This paper presents improved approximation algorithms for finding the largest volume simplex in a convex hull and establishes hardness results, showing the limits of approximability unless P=NP.

Contribution

It introduces a polynomial-time approximation algorithm with a factor of O( log d)^{d/2} for the largest volume simplex problem, improving previous bounds, and proves exponential inapproximability bounds under P≠NP.

Findings

Approximation factor improved to O(log d)^{d/2}.

Hardness of approximation established with a factor c^d unless P=NP.

Results extend to subdeterminant maximization problems.

Abstract

We show that the problem of finding the simplex of largest volume in the convex hull of $n$ points in $\mathbb{Q}^d$ 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 $P=NP$. % 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 $\bar c$ is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a $d\times n$ matrix.