Determinant maximization subject to a partition matroid constraint via stable distributions

Abstract

Given vectors $v_i\in {\mathbb{R}}^d$, we consider the problem of choosing a set $I$ independent in a partition matroid in order to maximize the determinant $\det ({\sum }_{i\in I}{v}_i{v}_i^T)$. Our main result is a polynomial-time approximation algorithm that finds a solution of value $det({\sum }_{i\in I}{v}_i{v}_i^T)\ge e^{-O(d)}OPT$, where $OPT={\max }_{I^{\ast }}det({\sum }_{i\in I^{\ast }}{v}_i{v}_i^T)$. For partition matroids of rank $m\le d$, we give a similar result for approximating the $m$-dimensional volume spanned by the chosen vectors, within a factor of $e^{O(m)}$. This matches earlier known algorithms that estimate the optimal value but do not find the corresponding solution, up to a constant in the exponent. Similar to these estimation algorithms, our algorithm is based on the saddle-point relaxation proposed by Nikolov and Singh. A new ingredient is a randomized transformation based on $1/2$-stable distributions, which converts the saddle-point relaxation into a more convenient multilinear relaxation.

The key results were obtained with ChatGPT-5.6 Sol