Approximation of a Maximum-Submodular-Coverage problem involving spectral functions, with application to Experimental Design

TL;DR

This paper investigates a spectral function-based combinatorial optimization problem related to experimental design, proving submodularity for all parameters and analyzing approximation guarantees of greedy and rounding algorithms.

Contribution

It establishes submodularity of the objective for all p in [0,1], enabling approximation guarantees for greedy algorithms and analyzing rounding methods.

Findings

Greedy approach achieves 1-1/e approximation for all p.

Rounding method's approximation factor approaches 1 as p approaches 1.

The problem generalizes maximum coverage and relates to the knapsack problem.

Abstract

We study a family of combinatorial optimization problems defined by a parameter $p\in[0,1]$, which involves spectral functions applied to positive semidefinite matrices, and has some application in the theory of optimal experimental design. This family of problems tends to a generalization of the classical maximum coverage problem as $p$ goes to 0, and to a trivial instance of the knapsack problem as $p$ goes to 1. In this article, we establish a matrix inequality which shows that the objective function is submodular for all $p\in[0,1]$, from which it follows that the greedy approach, which has often been used for this problem, always gives a design within $1-1/e$ of the optimum. We next study the design found by rounding the solution of the continuous relaxed problem, an approach which has been applied by several authors. We prove an inequality which generalizes a classical result from the theory of optimal designs, and allows us to give a rounding procedure with an approximation factor which tends to 1 as $p$ goes to 1.