Test-Time Search in Neural Graph Coarsening Procedures for the Capacitated Vehicle Routing Problem

The identification of valid inequalities, such as the rounded capacity inequalities (RCIs), is a key component of cutting plane methods for the Capacitated Vehicle Routing Problem (CVRP). While a deep learning-based separation method can learn to find high-quality cuts, our analysis reveals that the model produces fewer cuts than expected because it is insufficiently sensitive to generate a diverse set of generated subsets. This paper proposes an alternative: enhancing the performance of a trained model at inference time through a new test-time search with stochasticity. First, we introduce stochastic edge selection into the graph coarsening procedure, replacing the previously proposed greedy approach. Second, we propose the Graph Coarsening History-based Partitioning (GraphCHiP) algorithm, which leverages coarsening history to identify not only RCIs but also, for the first time, the Framed capacity inequalities (FCIs). Experiments on randomly generated CVRP instances demonstrate the effectiveness of our approach in reducing the dual gap compared to the existing neural separation method. Additionally, our method discovers effective FCIs on a specific instance, despite the challenging nature of identifying such cuts.