Exponential Inapproximability of Selecting a Maximum Volume Sub-matrix

TL;DR

This paper proves that selecting a maximum volume submatrix is computationally hard to approximate within an exponential factor, indicating significant complexity barriers for this problem.

Contribution

The authors establish exponential inapproximability bounds for the maximum volume submatrix selection problem, a novel hardness result in matrix subset selection.

Findings

No polynomial-time approximation within 2^{-ck} for certain parameters

Hardness result holds unless P=NP

Demonstrates fundamental computational difficulty of the problem

Abstract

Given a matrix $A \in \mathbb{R}^{m \times n}$ ($n$ vectors in $m$ dimensions), and a positive integer $k < n$, we consider the problem of selecting $k$ column vectors from $A$ such that the volume of the parallelepiped they define is maximum over all possible choices. We prove that there exists $\delta<1$ and $c>0$ such that this problem is not approximable within $2^{-ck}$ for $k = \delta n$, unless $P=NP$.