The Partitioning Min-Max Weighted Matching (PMMWM) problem, being a practical NP-hard problem, integrates the task of partitioning the vertices of a bipartite graph into disjoint sets of limited size with the classical Maximum-Weight Perfect Matching (MPWM) problem. Initially introduced in 2015, the state-of-the-art method for addressing PMMWM is the MP$_{\text{LS}}$. In this paper, we present a novel approach, the Fast Iterative Match-Partition Hybrid Genetic Algorithm (FIMP-HGA), for addressing PMMWM. Similar to MP$_{\text{LS}}$, FIMP-HGA divides the solving into match and partition stages, iteratively refining the solution. In the match stage, we propose the KM-M algorithm, which reduces matching complexity through incremental adjustments, significantly enhancing runtime efficiency. For the partition stage, we introduce a Hybrid Genetic Algorithm (HGA) incorporating an elite strategy and design a Greedy Partition Crossover (GPX) operator alongside a Multilevel Local Search (MLS) to optimize individuals in the population. Population initialization employs various methods, including the multi-way Karmarkar-Karp (KK) algorithm, ensuring both quality and diversity. At each iteration, the bipartite graph is adjusted based on the current solution, aiming for continuous improvement. To conduct comprehensive experiments, we develop a new instance generation method compatible with existing approaches, resulting in four benchmark groups. Extensive experiments evaluate various algorithm modules, accurately assessing each module's impact on improvement. Evaluation results on our benchmarks demonstrate that the proposed FIMP-HGA significantly enhances solution quality compared to MP$_{\text{LS}}$, meanwhile reducing runtime by 3 to 20 times.

翻译：划分最小-最大加权匹配问题（PMMWM）作为一个实际NP-难问题，将二分图顶点划分为有限大小不相交集合的任务与经典的最大权完美匹配问题（MPWM）相结合。该问题于2015年首次提出，当前最先进的求解方法是MP$_{\text{LS}}$。本文提出了一种新方法——快速迭代匹配-划分混合遗传算法（FIMP-HGA）用于求解PMMWM。与MP$_{\text{LS}}$类似，FIMP-HGA将求解过程分为匹配阶段和划分阶段，并通过迭代逐步优化解。在匹配阶段，我们提出KM-M算法，通过增量调整降低匹配复杂度，显著提升运行效率。在划分阶段，我们引入采用精英策略的混合遗传算法（HGA），并设计了贪婪划分交叉算子（GPX）以及多层次局部搜索（MLS）来优化种群中的个体。种群初始化采用多种方法，包括多维Karmarkar-Karp（KK）算法，确保了种群的质量与多样性。在每次迭代中，基于当前解调整二分图，以实现持续改进。为进行全面的实验，我们开发了一种与现有方法兼容的实例生成方法，形成了四个基准组。大量实验评估了不同算法模块的贡献，准确衡量了每个模块对性能提升的影响。在我们的基准测试上的评估结果表明，与MP$_{\text{LS}}$相比，所提出的FIMP-HGA显著提高了解的质量，同时将运行时间缩短了3到20倍。