Randomized Rounding for the Largest Simplex Problem

TL;DR

This paper presents a deterministic approximation algorithm for the maximum volume j-simplex problem, achieving an exponential approximation ratio, and introduces a novel rounding technique related to D-optimal design.

Contribution

It introduces a new deterministic approximation algorithm with exponential ratio for the largest simplex problem and connects it to D-optimal design and invertibility principles.

Findings

Achieves an approximation ratio of e^{j/2 + o(j)} for the maximum volume j-simplex problem.

Provides a simple proof of a restricted invertibility principle for determinants.

Extends the approach to find principal submatrices with maximum determinant.

Abstract

The maximum volume $j$-simplex problem asks to compute the $j$-dimensional simplex of maximum volume inside the convex hull of a given set of $n$ points in $\mathbb{Q}^d$. We give a deterministic approximation algorithm for this problem which achieves an approximation ratio of $e^{j/2 + o(j)}$. The problem is known to be $\mathrm{NP}$-hard to approximate within a factor of $c^{j}$ for some constant $c > 1$. Our algorithm also gives a factor $e^{j + o(j)}$ approximation for the problem of finding the principal $j\times j$ submatrix of a rank $d$ positive semidefinite matrix with the largest determinant. We achieve our approximation by rounding solutions to a generalization of the $D$-optimal design problem, or, equivalently, the dual of an appropriate smallest enclosing ellipsoid problem. Our arguments give a short and simple proof of a restricted invertibility principle for determinants.