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The discovery of Markov boundaries is highly effective at identifying features that are causally related to the target variable, providing strong interpretability and robustness. While there are numerous methods for discovering Markov boundaries in real-world applications, no single method is universally applicable to all datasets. Therefore, in order to balance precision and recall, we propose an ensemble framework of divide-and-conquer Markov boundary discovery algorithms based on U-I selection strategy. We put three divide-and-conquer Markov boundary methods into the framework to obtain an ensemble algorithm, focusing on judging controversial parent–child variables to further balance precision and recall. By combining multiple algorithms, the ensemble algorithm can leverage their respective strengths and more thoroughly analyze the cause-and-effect relationships of target variables through various perspectives. Furthermore, it can enhance the robustness of the algorithm and reduce dependence on a single algorithm. In the experiment, we select four advanced Markov boundary discovery algorithms as comparison algorithms and compare them on nine benchmark Bayesian networks and three real-world datasets. The results show that EDMB ranks first in the overall ranking, which illustrates the superiority of the integrated algorithm and the effectiveness of the adopted U-I selection strategy. The main contribution of this paper lies in proposing an ensemble framework for divide-and-conquer Markov boundary discovery algorithms, balancing precision and recall through the U-I selection strategy, and judging controversial parent–child variables to enhance algorithm performance and robustness. The advantage of the U-I selection strategy and its difference from existing methods is the ability to independently obtain the maximum precision and recall of multiple algorithms within the ensemble framework. By assessing controversial parent–child variables, it further balances precision and recall, leading to results that are closer to the true Markov boundary. [ABSTRACT FROM AUTHOR]

Various metaheuristic algorithms inspired by nature have been designed to deal with a variety of practical optimization problems. As an excellent metaheuristic algorithm, the improved particle swarm optimization algorithm based on grouping (IPSO) has strong global search capabilities. However, it lacks a strong local search ability and the ability to solve constrained discrete optimization problems. This paper focuses on improving these two aspects of the IPSO algorithm. Based on IPSO, we propose an improved particle swarm optimization algorithm based on variable neighborhood search (VN-IPSO) and design a 0-1 integer programming solution with constraints. In the experiment, the performance of the VN-IPSO algorithm is fully tested and analyzed using 23 classic benchmark functions (continuous optimization), 6 knapsack problems (discrete optimization), and 10 CEC2017 composite functions (complex functions). The results show that the VN-IPSO algorithm wins 18 first places in the classic benchmark function test set, including 6 first places in the solutions for seven unimodal test functions, indicating a good local search ability. In solving the six knapsack problems, it wins four first places, demonstrating the effectiveness of the 0-1 integer programming constraint solution and the excellent solution ability of VN-IPSO in discrete optimization problems. In the test of 10 composite functions, VN-IPSO wins first place four times and ranks the first in the comprehensive ranking, demonstrating its excellent solving ability for complex functions. [ABSTRACT FROM AUTHOR]