Multi-objective Evolutionary Algorithms are Still Good: Maximizing Monotone Approximately Submodular Minus Modular Functions

As evolutionary algorithms (EAs) are general-purpose optimization algorithms, recent theoretical studies have tried to analyze their performance for solving general problem classes, with the goal of providing a general theoretical explanation of the behavior of EAs. Particularly, a simple multi-objective EA, i.e., GSEMO, has been shown to be able to achieve good polynomial-time approximation guarantees for submodular optimization, where the objective function is only required to satisfy some properties and its explicit formulation is not needed. Submodular optimization has wide applications in diverse areas, and previous studies have considered the cases where the objective functions are monotone submodular, monotone nonsubmodular, or non-monotone submodular. To complement this line of research, this paper studies the problem class of maximizing monotone approximately submodular minus modular functions (i.e., [Formula: see text]) with a size constraint, where g is a so-called non-negative monotone approximately submodular function and c is a socalled non-negative modular function, resulting in the objective function [Formula: see text] being non-monotone non-submodular in general. Different from previous analyses, we prove that by optimizing the original objective function [Formula: see text] and the size simultaneously, the GSEMO fails to achieve a good polynomial-time approximation guarantee. However, we also prove that by optimizing a distorted objective function and the size simultaneously, the GSEMO can still achieve the best-known polynomialtime approximation guarantee. Empirical studies on the applications of Bayesian experimental design and directed vertex cover show the excellent performance of the GSEMO.