The minimum weight dominating set (MWDS) problem is an important generalization of the minimum dominating set problem with various applications. In this work, we develop an efficient local search scheme that can dynamically adjust the number of added and removed vertices according to the information of the candidate solution. Based on this scheme, we further develop three novel ideas to improve performance, resulting in our so-called DeepOpt-MWDS algorithm. First, we use a new construction method with five reduction rules to significantly reduce massive graphs and construct an initial solution efficiently. Second, an improved configuration checking strategy called CC²V3+ is designed to reduce the cycling phenomenon in local search. Third, a general perturbation framework called deep optimization mechanism (DeepOpt) is proposed to help the algorithm avoid local optima and to converge to a new solution quickly.

Extensive experiments based on eight popular benchmarks of different scales are carried out to evaluate the proposed algorithm. Compared to seven state-of-the-art heuristic algorithms, DeepOpt-MWDS performs better on random and classic benchmarks and obtains the best solutions on almost all massive graphs. We investigate three main algorithmic ingredients to understand their impacts on the performance of the proposed algorithm. Moreover, we adapt the proposed general framework DeepOpt to another NP-hard problem to verify its generality and achieve good performance.

最小权重支配集（MWDS）问题是最小支配集问题的重要推广，具有各种应用。在这项工作中，我们开发了一种高效的局部搜索方案，可以根据候选解的信息动态调整添加和删除的顶点数量。基于这个方案，我们进一步开发了三个新的想法来提高性能，从而产生了我们所谓的 DeepOpt-MWDS 算法。首先，我们使用了一种新的构造方法，具有五个归约规则，以显着减少海量图并有效地构造初始解决方案。其次，一种改进的配置检查策略，称为 CC ²V3+ 旨在减少局部搜索中的循环现象。第三，提出了一种称为深度优化机制（DeepOpt）的通用扰动框架，以帮助算法避免局部最优并快速收敛到新的解决方案。

基于八个不同尺度的流行基准进行了广泛的实验来评估所提出的算法。与七种最先进的启发式算法相比，DeepOpt-MWDS 在随机和经典基准上表现更好，并且在几乎所有海量图上都获得了最佳解决方案。我们研究了三个主要的算法成分，以了解它们对所提出算法性能的影响。此外，我们将提出的通用框架 DeepOpt 应用于另一个 NP-hard 问题，以验证其通用性并取得良好的性能。