Maximizing Submodular or Monotone Approximately Submodular Functions by Multi-objective Evolutionary Algorithms

TL;DR

This paper analyzes the performance of multi-objective evolutionary algorithms, specifically GSEMO-C, in maximizing submodular and approximately submodular functions, providing theoretical approximation guarantees for these classes.

Contribution

It extends the theoretical analysis of EAs to broader classes of submodular and approximately submodular maximization problems, beyond monotone cases.

Findings

GSEMO-C achieves good approximation guarantees.

The algorithm runs in polynomial expected time.

Theoretical analysis applies to non-monotone and non-submodular functions.

Abstract

Evolutionary algorithms (EAs) are a kind of nature-inspired general-purpose optimization algorithm, and have shown empirically good performance in solving various real-word optimization problems. During the past two decades, promising results on the running time analysis (one essential theoretical aspect) of EAs have been obtained, while most of them focused on isolated combinatorial optimization problems, which do not reflect the general-purpose nature of EAs. To provide a general theoretical explanation of the behavior of EAs, it is desirable to study their performance on general classes of combinatorial optimization problems. To the best of our knowledge, the only result towards this direction is the provably good approximation guarantees of EAs for the problem class of maximizing monotone submodular functions with matroid constraints. The aim of this work is to contribute to this line of research. Considering that many combinatorial optimization problems involve non-monotone or non-submodular objective functions, we study the general problem classes, maximizing submodular functions with/without a size constraint and maximizing monotone approximately submodular functions with a size constraint. We prove that a simple multi-objective EA called GSEMO-C can generally achieve good approximation guarantees in polynomial expected running time.