The Minimum Bisection Problem is a well-known NP-hard problem in combinatorial optimization, with practical applications in areas such as parallel computing, network design, and machine learning. In this paper, we examine the potential of using D-Wave Systems' quantum annealing solvers to solve the Minimum Bisection Problem, which we formulate as a Quadratic Unconstrained Binary Optimization model. A key challenge in this formulation lies in choosing an appropriate penalty parameter, as it plays a crucial role in ensuring both the quality of the solution and the satisfaction of the problem's constraints. To address this, we introduce a novel machine learning-based approach for adaptive tuning of the penalty parameter. Specifically, we use a Gradient Boosting Regressor model trained to predict suitable penalty parameter values based on structural properties of the input graph, the number of nodes and the graph's density. This method enables the penalty parameter to be adjusted dynamically for each specific problem instance, improving the solver's ability to balance the competing goals of minimizing the cut size and maintaining equally sized partitions. We test our approach on a large dataset of randomly generated Erd\H{o}s-R\'enyi graphs with up to 4,000 nodes, and we compare the results with classical partitioning algorithms, Metis and Kernighan-Lin. Experimental findings demonstrate that our adaptive tuning strategy significantly improves the performance of the quantum annealing hybrid solver and consistently outperforms the classical methods used, indicating its potential as an alternative for the graph partitioning problem.

翻译：最小二分问题是组合优化中一个著名的NP难问题，在并行计算、网络设计和机器学习等领域具有实际应用。本文研究了利用D-Wave Systems的量子退火求解器解决最小二分问题的潜力，我们将该问题建模为二次无约束二元优化模型。该建模的一个关键挑战在于选择合适的惩罚参数，因为它对确保解的质量和满足问题约束条件都起着至关重要的作用。为此，我们提出了一种基于机器学习的自适应惩罚参数调优新方法。具体而言，我们使用梯度提升回归模型，该模型经过训练能够根据输入图的结构特性（节点数量和图的密度）预测合适的惩罚参数值。这种方法使得惩罚参数能够针对每个具体问题实例进行动态调整，从而提升求解器在最小化割集规模与保持均衡划分这两个竞争目标之间的平衡能力。我们在包含多达4,000个节点的随机生成Erdős-Rényi图的大型数据集上测试了该方法，并将结果与经典划分算法Metis和Kernighan-Lin进行了比较。实验结果表明，我们的自适应调优策略显著提升了量子退火混合求解器的性能，并持续优于所使用的经典方法，这显示了其作为图划分问题替代方案的潜力。