Diverse Approximations for Monotone Submodular Maximization Problems with a Matroid Constraint

TL;DR

This paper introduces two greedy algorithms for maximizing monotone submodular functions with diversity considerations under a matroid constraint, providing theoretical guarantees and empirical insights into their trade-offs.

Contribution

It proposes simple greedy algorithms with parameters balancing objective and diversity, along with their approximation guarantees for the first time.

Findings

Algorithms achieve provable approximation guarantees.

Empirical results show distinct objective-diversity trade-offs.

Algorithms perform effectively on maximum vertex coverage instances.

Abstract

Finding diverse solutions to optimization problems has been of practical interest for several decades, and recently enjoyed increasing attention in research. While submodular optimization has been rigorously studied in many fields, its diverse solutions extension has not. In this study, we consider the most basic variants of submodular optimization, and propose two simple greedy algorithms, which are known to be effective at maximizing monotone submodular functions. These are equipped with parameters that control the trade-off between objective and diversity. Our theoretical contribution shows their approximation guarantees in both objective value and diversity, as functions of their respective parameters. Our experimental investigation with maximum vertex coverage instances demonstrates their empirical differences in terms of objective-diversity trade-offs.