Comparison of popular meta-heuristics.

The objective of the max-cut problem is to cut any graph in such a way that the total weight of the edges that are cut off is maximum in both subsets of vertices that are divided due to the cut of the edges. Although it is an elementary graph partitioning problem, it is one of the most challenging combinatorial optimization-based problems, and tons of application areas make this problem highly admissible. Due to its admissibility, the problem is solved using the Harris Hawk Optimization algorithm (HHO). Though HHO effectively solved some engineering optimization problems, is sensitive to parameter settings and may converge slowly, potentially getting trapped in local optima. Thus, HHO and some additional operators are used to solve the max-cut problem. Crossover and refinement operators are used to modify the fitness of the hawk in such a way that they can provide precise results. A mutation mechanism along with an adjustment operator has improvised the outcome obtained from the updated hawk. To accept the potential result, the acceptance criterion has been used, and then the repair operator is applied in the proposed approach. The proposed system provided comparatively better outcomes on the G-set dataset than other state-of-the-art algorithms. It obtained 533 cuts more than the discrete cuckoo search algorithm in 9 instances, 1036 cuts more than PSO-EDA in 14 instances, and 1021 cuts more than TSHEA in 9 instances. But for four instances, the cuts are lower than PSO-EDA and TSHEA. Besides, the statistical significance has also been tested using the Wilcoxon signed rank test to provide proof of the superior performance of the proposed method. In terms of solution quality, MC-HHO can produce outcomes that are quite competitive when compared to other related state-of-the-art algorithms.最大割问题（max-cut problem）的优化目标为对任意给定图进行边割分，使得因割分得到的两个顶点子集中，被切断边的总权重达到最大值。尽管该问题属于基础的图划分问题，但其却是极具挑战性的组合优化问题之一，且应用场景丰富，具备极高的研究价值。鉴于该问题的研究价值，本文采用哈里斯鹰优化算法（Harris Hawk Optimization, HHO）对其求解。虽然HHO已成功解决了部分工程优化问题，但该算法对参数设置较为敏感，且易出现收敛速度缓慢、陷入局部最优解的问题。为此，本文结合多种额外算子对HHO进行改进以求解最大割问题：通过交叉算子与细化算子调整鹰群的适应度，以获得更精准的求解结果；结合变异机制与调整算子对更新后的鹰群求解结果进行优化。在所提改进方法中，首先采用接受准则筛选潜在优质解，随后应用修复算子对解进行修正。所提改进算法在G集数据集（G-set dataset）上的求解效果相较于其他主流前沿算法更为优异：在9个测试实例中，其割分总权重较离散布谷鸟搜索算法高出533；在14个测试实例中，较PSO-EDA算法高出1036；在9个测试实例中，较TSHEA算法高出1021。仅在4个测试实例中，其割分总权重低于PSO-EDA与TSHEA算法。此外，本文采用威尔科克森符号秩检验（Wilcoxon signed rank test）对算法性能的统计显著性进行验证，进一步证明了所提方法的优越性。从求解质量层面来看，MC-HHO算法可获得与其他相关前沿算法极具竞争力的求解结果。