In an instance of the minimum eigenvalue problem, we are given a collection of $n$ vectors $v_1,\ldots, v_n \subset {\mathbb{R}^d}$, and the goal is to pick a subset $B\subseteq [n]$ of given vectors to maximize the minimum eigenvalue of the matrix $\sum_{i\in B} v_i v_i^{\top} $. Often, additional combinatorial constraints such as cardinality constraint $\left(|B|\leq k\right)$ or matroid constraint ($B$ is a basis of a matroid defined on $[n]$) must be satisfied by the chosen set of vectors. The minimum eigenvalue problem with matroid constraints models a wide variety of problems including the Santa Clause problem, the E-design problem, and the constructive Kadison-Singer problem. In this paper, we give a randomized algorithm that finds a set $B\subseteq [n]$ subject to any matroid constraint whose minimum eigenvalue is at least $(1-\epsilon)$ times the optimum, with high probability. The running time of the algorithm is $O\left( n^{O(d\log(d)/\epsilon^2)}\right)$. In particular, our results give a polynomial time asymptotic scheme when the dimension of the vectors is constant. Our algorithm uses a convex programming relaxation of the problem after guessing a rescaling which allows us to apply pipage rounding and matrix Chernoff inequalities to round to a good solution. The key new component is a structural lemma which enables us to "guess'' the appropriate rescaling, which could be of independent interest. Our approach generalizes the approximation guarantee to monotone, homogeneous functions and as such we can maximize $\det(\sum_{i\in B} v_i v_i^\top)^{1/d}$, or minimize any norm of the eigenvalues of the matrix $\left(\sum_{i\in B} v_i v_i^\top\right)^{-1} $, with the same running time under some mild assumptions. As a byproduct, we also get a simple algorithm for an algorithmic version of Kadison-Singer problem.

翻译：在最小特征值问题的一个实例中，给定一组 $n$ 个向量 $v_1,\ldots, v_n \subset {\mathbb{R}^d}$，目标是选择一个子集 $B\subseteq [n]$，使得矩阵 $\sum_{i\in B} v_i v_i^{\top}$ 的最小特征值最大化。通常，所选向量集合还需满足额外的组合约束，例如基数约束 ($|B|\leq k$) 或拟阵约束（$B$ 是定义在 $[n]$ 上的拟阵的一个基）。带拟阵约束的最小特征值问题建模了广泛的应用，包括圣诞老人问题、E-设计问题以及构造性Kadison-Singer问题。本文提出一种随机算法，该算法能够在满足任意拟阵约束的条件下，以高概率找到一个子集 $B\subseteq [n]$，使其最小特征值至少为最优值的 $(1-\epsilon)$ 倍。算法的运行时间为 $O\left( n^{O(d\log(d)/\epsilon^2)}\right)$。特别地，当向量维度为常数时，我们的结果给出了一个多项式时间渐近方案。该算法在猜测一个缩放因子后使用问题的凸松弛，进而应用管道舍入和矩阵Chernoff不等式来舍入得到优质解。其中的关键新组件是一个结构性引理，它使我们能够“猜测”适当的缩放因子，该引理可能具有独立的研究意义。我们的方法将近似保证推广到单调齐次函数，从而可以在温和假设下，以相同运行时间最大化 $\det(\sum_{i\in B} v_i v_i^\top)^{1/d}$，或最小化矩阵 $\left(\sum_{i\in B} v_i v_i^\top\right)^{-1}$ 特征值的任意范数。作为副产品，我们还得到了Kadison-Singer问题算法版本的一个简单算法。