Constraint Optimization Problems (COP) pose intricate challenges in combinatorial problems usually addressed through Branch and Bound (B\&B) methods, which involve maintaining priority queues and iteratively selecting branches to search for solutions. However, conventional approaches take a considerable amount of time to find optimal solutions, and it is also crucial to quickly identify a near-optimal feasible solution in a shorter time. In this paper, we aim to investigate the effectiveness of employing a depth-first search algorithm for solving COP, specifically focusing on identifying optimal or near-optimal solutions within top $n$ solutions. Hence, we propose a novel heuristic neural network algorithm based on MCTS, which, by simultaneously conducting search and training, enables the neural network to effectively serve as a heuristic during Backtracking. Furthermore, our approach incorporates encoding COP problems and utilizing graph neural networks to aggregate information about variables and constraints, offering more appropriate variables for assignments. Experimental results on stochastic COP instances demonstrate that our method identifies feasible solutions with a gap of less than 17.63% within the initial 5 feasible solutions. Moreover, when applied to attendant Constraint Satisfaction Problem (CSP) instances, our method exhibits a remarkable reduction of less than 5% in searching nodes compared to state-of-the-art approaches.

翻译：约束优化问题（COP）在组合问题中构成复杂挑战，通常通过分支定界（B&B）方法解决，该方法涉及维护优先队列并迭代选择分支以搜索解。然而，传统方法寻找最优解需耗费大量时间，而在较短时间内快速找到接近最优的可行解同样至关重要。本文旨在探究采用深度优先搜索算法求解COP的有效性，重点聚焦于在前n个解中识别最优解或接近最优解。为此，我们提出一种基于MCTS的新型启发式神经网络算法，通过同步进行搜索与训练，使神经网络在回溯过程中能够有效充当启发式策略。此外，我们的方法对COP问题进行编码，并利用图神经网络聚合变量与约束信息，从而为变量赋值提供更合适的候选变量。针对随机COP实例的实验结果表明，在前5个可行解中，我们的方法能够找到间隙小于17.63%的可行解。同时，在应用于伴随的约束满足问题（CSP）实例时，与现有最优方法相比，我们的方法在搜索节点数量上实现了低于5%的显著减少。