The subset selection problem with a monotone and submodular objective function under a linear cost constraint has wide applications, such as maximum coverage, influence maximization, and feature selection, just to name a few. Various greedy algorithms have been proposed with good performance both theoretically and empirically. Recently, evolutionary algorithms (EAs), inspired by Darwin's evolution theory, have emerged as a prominent methodology, offering both empirical advantages and theoretical guarantees. Among these, the multi-objective EA, POMC, has demonstrated the best empirical performance to date, achieving an approximation guarantee of $(1/2)(1-1/e)$. However, there remains a gap in the approximation bounds of EAs compared to greedy algorithms, and their full theoretical potential is yet to be realized. In this paper, we re-analyze the approximation performance of POMC theoretically, and derive an improved guarantee of $1/2$, which thus provides theoretical justification for its encouraging empirical performance. Furthermore, we propose a novel multi-objective EA, EPOL, which not only achieves the best-known practical approximation guarantee of $0.6174$, but also delivers superior empirical performance in applications of maximum coverage and influence maximization. We hope this work can help better solving the subset selection problem, but also enhance our theoretical understanding of EAs.

Machine learning models have liberated manpower greatly in many real-world tasks, but their predictions are still worse than humans on some specific instances. To improve the performance, it is natural to optimize machine learning models to take decisions for most instances while delivering a few tricky instances to humans, resulting in the problem of Human Assisted Learning (HAL). Previous works mainly formulated HAL as a constrained optimization problem that tries to find a limited subset of instances for human decision such that the sum of model and human errors can be minimized; and employed the greedy algorithms, whose performance, however, may be limited due to the greedy nature. In this paper, we propose a new framework HAL-EMO based on Evolutionary Multi-objective Optimization, which reformulates HAL as a bi-objective optimization problem that minimizes the number of selected instances for human decision and the total errors simultaneously, and employs a Multi-Objective Evolutionary Algorithm (MOEA) to solve it. We implement HAL-EMO using two MOEAs, the popular NSGA-II as well as the theoretically grounded GSEMO. We also propose a specific MOEA, called BSEMO, with biased selection and balanced mutation for HAL-EMO, and prove that for human assisted regression and classification, HAL-EMO using BSEMO can achieve better and same theoretical guarantees than previous greedy algorithms, respectively. Experiments on the tasks of medical diagnosis and content moderation show the superiority of HAL-EMO (with either NSGA-II, GSEMO or BSEMO) over previous algorithms, and that using BSEMO leads to the best performance of HAL-EMO.