Efficient Determinant Maximization for All Matroids

TL;DR

This paper presents a combinatorial algorithm for determinant maximization under matroid constraints, achieving a near-optimal approximation ratio for large matroids, advancing the understanding of algebraic combinatorial optimization.

Contribution

It introduces a new combinatorial algorithm that achieves a $O(d^{O(d)})$-approximation for determinant maximization under matroid constraints, extending previous geometric approaches.

Findings

Achieves $O(d^{O(d)})$-approximation for large matroids

Matches the best-known estimation algorithms in approximation quality

Combines combinatorial optimization with algebraic properties of determinants

Abstract

Determinant maximization provides an elegant generalization of problems in many areas, including convex geometry, statistics, machine learning, fair allocation of goods, and network design. In an instance of the determinant maximization problem, we are given a collection of vectors $v_1,\ldots, v_n \in \mathbb{R}^d$, and the goal is to pick a subset $S\subseteq [n]$ of given vectors to maximize the determinant of the matrix $\sum_{i \in S} v_iv_i^\top$, where the picked set of vectors $S$ must satisfy some combinatorial constraint such as cardinality constraint ($|S| \leq k$) or matroid constraint ($S$ is a basis of a matroid defined on $[n]$). In this work, we give a combinatorial algorithm for the determinant maximization problem under a matroid constraint that achieves $O(d^{O(d)})$-approximation for any matroid of rank $r\geq d$. This complements the recent result of~\cite{BrownLPST22} that achieves a similar bound for matroids of rank $r\leq d$, relying on a geometric interpretation of the determinant. Our result matches the best-known estimation algorithms~\cite{madan2020maximizing} for the problem, which could estimate the objective value but could not give an approximate solution with a similar guarantee. Our work follows the framework developed by~\cite{BrownLPST22} of using matroid intersection based algorithms for determinant maximization. To overcome the lack of a simple geometric interpretation of the objective when $r \geq d$, our approach combines ideas from combinatorial optimization with algebraic properties of the determinant. We also critically use the properties of a convex programming relaxation of the problem introduced by~\cite{madan2020maximizing}.