Maximizing the Minimum Eigenvalue in Constant Dimension

TL;DR

This work addresses selecting a subset under matroid constraints to maximize the minimum eigenvalue of the sum of outer products $\sum_{i\in B} v_i v_i^{\top}$. It introduces a randomized pipeline built on a convex relaxation over the matroid base polytope, a rescaling/structural lemma to bound leverage, and pipage rounding with matrix Chernoff bounds to achieve a $(1-\varepsilon)$-approximation with high probability in time $O(n^{O(d\log(d)/\varepsilon^2)})$, yielding a PTAS for constant dimension and an algorithmic Kadison-Singer result. The framework extends to monotone, homogeneous objective functions (e.g., determinant maximization or inverse-eigenvalue norms), broadening applicability to experimental design and spectral sparsification problems. By solving the problem for partition matroids first and then general matroids via pipage rounding and renormalized concentration, the paper unifies several spectral-optimization paradigms (max-min allocation, E-design, Kadison-Singer) under a single, efficient algorithmic approach in constant dimension. These results provide practical tools for constant-dimension spectral design and deepen the algorithmic understanding of KS-type guarantees.

Abstract

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 $(1-ε)$ times the optimum, with high probability. The running time of the algorithm is $O\left( n^{O(d\log(d)/ε^2)}\right)$. 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.

Theorems & Definitions (18)

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• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 4