Wilcoxon signed-rank test for G-set dataset.

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）的目标是对任意无向图进行割分，使得因边割而划分得到的两个顶点子集之间，被割去的边的总权重达到最大值。尽管它属于基础的图划分问题，却是最具挑战性的组合优化问题之一，其丰富的应用场景更使其极具研究与应用价值。鉴于其广泛的应用价值，研究人员采用哈里斯鹰优化算法（Harris Hawk Optimization, HHO）求解该问题。虽然HHO已成功求解了诸多工程优化问题，但该算法对参数设置较为敏感，且可能收敛速度缓慢，甚至易陷入局部最优。因此，本文结合HHO与若干附加算子来求解最大割问题。交叉算子与细化算子被用于修正鹰群个体的适应度，以实现更精准的求解效果；变异机制与调整算子则进一步优化了更新后的鹰群所得到的求解结果。在所提方法中，首先通过接受准则筛选潜在最优解，随后应用修复算子对候选解进行修正完善。在所构建的G-set数据集上，所提方法相较其他前沿算法取得了更优的求解效果。在9个测试实例中，其割边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。但在4个测试实例中，其割边总权重低于PSO-EDA与TSHEA算法。此外，本文采用Wilcoxon符号秩检验（Wilcoxon signed rank test）对实验结果进行了显著性统计检验，以验证所提方法的性能优越性。在求解质量方面，MC-HHO算法的求解结果相较其他同类前沿算法仍具有较强的竞争力。