Simple Randomized Rounding for Max-Min Eigenvalue Augmentation

Jourdain Lamperski, Haeseong Yang, Oleg Prokopyev

TL;DR: The paper derives a new matrix concentration inequality to establish an approximation guarantee for a simple algorithm for max-min eigenvalue augmentation in a practically motivated regime that has yet to be studied from a theoretical standpoint.

Abstract: We consider the *max-min eigenvalue augmentation* problem: given $n \times n$ symmetric positive semidefinite matrices $M,A_1,\ldots, A_m$ and a positive integer $k < m$, the goal is to choose a subset $I \subset \{1,\ldots,m\}$ of cardinality at most $k$ that maximizes the minimum eigenvalue of the matrix $M + \sum_{i \in I} A_i$. The problem captures both the *Bayesian E-optimal design* and *maximum algebraic connectivity augmentation* problems. In contrast to the existing work, we do not assume that the *augmentation matrices* are rank-one matrices, and we focus on the setting in which $k < n$. We show that a *simple* randomized rounding method provides a constant-factor approximation if the *optimal increase* is sufficiently large, specifically, if $\mathrm{OPT} - \lambda_{\mathrm{min}}(M) = \Omega(R \ln k)$, where $\mathrm{OPT}$ is the optimal value, and $R$ is the maximum trace of an augmentation matrix. To establish the guarantee, we derive a matrix concentration inequality that is of independent interest. The inequality can be interpreted as an *intrinsic dimension* analog of the matrix Chernoff inequality for the minimum eigenvalue of a sum of independent random positive semidefinite matrices; such an inequality has already been established for the maximum eigenvalue, but not for the minimum eigenvalue.

Lay Summary: We consider a class of optimization problems that finds applications in statistics and network design. We study a simple algorithm that returns approximate solutions to the problem. We establish theoretical properties of the algorithm. We expect that the techniques that we develop towards this end can be applied to other problem classes.

Primary Area: Optimization->Discrete and Combinatorial Optimization

Keywords: Approximation algorithms, Bayesian E-optimal design, matrix concentration inequalities, maximum algebraic connectivity augmentation

Submission Number: 11220