Fast algorithms for maximizing the minimum eigenvalue in fixed dimension

Abstract

In the minimum eigenvalue problem, we are given a collection of vectors in $R^d$, and the goal is to pick a subset B to maximize the minimum eigenvalue of the matrix ${\sum }_{i\in B}{v}_i{v}_i^{\top }$. We give a $O\left(n^{O(d\log (d)/{ϵ}^2)}\right)$-time randomized algorithm that finds an assignment subject to a partition constraint whose minimum eigenvalue is at least $(1-ϵ)$ times the optimum, with high probability. As a byproduct, we also get a simple algorithm for an algorithmic version of Kadison-Singer problem.