The cutting plane method is a key technique for successful branch-and-cut and branch-price-and-cut algorithms that find the exact optimal solutions for various vehicle routing problems (VRPs). Among various cuts, the rounded capacity inequalities (RCIs) are the most fundamental. To generate RCIs, we need to solve the separation problem, whose exact solution takes a long time to obtain; therefore, heuristic methods are widely used. We design a learning-based separation heuristic algorithm with graph coarsening that learns the solutions of the exact separation problem with a graph neural network (GNN), which is trained with small instances of 50 to 100 customers. We embed our separation algorithm within the cutting plane method to find a lower bound for the capacitated VRP (CVRP) with up to 1,000 customers. We compare the performance of our approach with CVRPSEP, a popular separation software package for various cuts used in solving VRPs. Our computational results show that our approach finds better lower bounds than CVRPSEP for large-scale problems with 400 or more customers, while CVRPSEP shows strong competency for problems with less than 400 customers.

翻译：切割平面法是成功实现分支切割和分支定价切割算法的关键技术，这些算法能够精确求解各种车辆路径问题（VRPs）的最优解。在各种割平面中，舍入容量不等式（RCIs）是最基础的。为了生成RCIs，我们需要解决分离问题，而该问题的精确求解需要很长时间；因此，启发式方法被广泛使用。我们设计了一种基于学习的分离启发式算法，该算法采用图粗化技术，通过图神经网络（GNN）学习精确分离问题的解，并使用包含50至100个客户的小规模实例进行训练。我们将此分离算法嵌入切割平面法中，用于求解最多包含1000个客户的容量受限VRP（CVRP）的下界。我们将我们的方法与CVRPSEP（一种用于求解VRP中多种割平面的流行分离软件包）进行了性能对比。计算结果表明，对于具有400个或更多客户的大规模问题，我们的方法能找到比CVRPSEP更优的下界；而对于客户数少于400的问题，CVRPSEP则表现出较强的竞争力。