Efficient Algorithms for Subdeterminant Maximization under Partition Matroids

2026-09-16

AI Summary

本文解决了在分划拟阵约束下子行列式最大化问题，提出了一种基于几何最大最小松弛的e^O(k)近似算法。

Abstract

We consider the determinant maximization problem under partition constraints: Given an $n\times n$ PSD matrix A and a partition matroid $M$ on $[n]$, find a base $S$ of $M$ that maximizes $\det(A_{S,S})$. We give an $e^{O(k)}$-approximation algorithm to find such a set $S$, where $k$ is the rank of $M$. This improves upon the current $k^{O(k)}$-approximation, and matches the current $e^k$-estimation guarantee, up to $O(1)$ factors in the exponent. Our algorithm is based on rounding the geometric max-min relaxation due to Nikolov-Singh'2016, using a continuous potential-driven process, and several new structural and analytic properties of this relaxation.

Problem

• determinant maximization
• partition matroids
• approximation algorithm

Innovation

• geometric max-min relaxation
• continuous potential-driven process
• partition matroids
• subdeterminant maximization