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 multiobjective EA, that is, 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 non-submodular, or non-monotone submodular. To complement this line of research, this article studies the problem class of maximizing monotone approximately submodular minus modular functions (i.e., $g-c$⁠) with a size constraint, where $g$ is a so-called non-negative monotone approximately submodular function and $c$ is a so-called non-negative modular function, resulting in the objective function $(g-c)$ being non-monotone non-submodular in general. Different from previous analyses, we prove that by optimizing the original objective function $(g-c)$ 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 polynomial-time approximation guarantee. Empirical studies on the applications of Bayesian experimental design and directed vertex cover show the excellent performance of the GSEMO.

由于进化算法 (EA) 是通用优化算法，最近的理论研究试图分析它们在解决一般问题类方面的性能，目的是提供对 EA 行为的一般理论解释。特别是，一个简单的多目标 EA，即 GSEMO，已被证明能够为子模优化实现良好的多项式时间近似保证，其中目标函数只需要满足某些属性，不需要其显式公式。子模优化在不同领域有着广泛的应用，之前的研究已经考虑了目标函数是单调子模、单调非子模或非单调子模的情况。为了补充这方面的研究，$G——C$⁠ ) 具有大小约束，其中$G$ 是一个所谓的非负单调近似子模函数和 $C$ 是所谓的非负模函数，导致目标函数 $(G——C)$通常是非单调非子模块的。与之前的分析不同，我们证明通过优化原始目标函数$(G——C)$和大小同时，GSEMO 无法实现良好的多项式时间近似保证。然而，我们也证明，通过同时优化失真的目标函数和大小，GSEMO 仍然可以实现最著名的多项式时间近似保证。对贝叶斯实验设计和定向顶点覆盖应用的实证研究表明 GSEMO 具有出色的性能。