Determinant maximization subject to a partition matroid constraint via stable distributions

RESUMO

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 $det ( \sum_{i \in I} v_{i} v_{i}^T) \geq e^{-O(d)} OPT$, where $OPT = \max_{I^*} det ( \sum_{i \in I^*} v_{i} v_{i}^T)$. For partition matroids of rank $m \leq d$, 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.