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

TL;DR

This paper tackles cutting-plane strength in CVRP by augmenting a learned RCI/FCI separation framework with test-time search. It introduces stochastic edge selection (pi-greedy) during neural graph coarsening to increase diversity of candidate subsets, and proposes GraphCHiP to exploit coarsening history for discovering RCIs and FCIs without retraining. Empirical results show reduced dual gaps and more effective RCIs, with a demonstrated FCI on a challenging instance, highlighting practical gains for neural-enhanced solvers. The approach is general to other learning-based graph-coarsening methods and offers a practical path toward stronger root-node bounds in BC/BPC without additional training.

Abstract

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.

Figures (14)

• Figure 1: A Hybrid approach for CVRP using cutting planes with NeuralSEP and test-time search. The cutting plane method alternates between solving the current LP relaxation and employing NeuralSEP as a separation algorithm to generate valid cuts. Our proposed test-time search technique enhances NeuralSEP's performance.
• Figure 2: The details of the NeuralSEP framework. $p_i$ is the predicted probability that vertex $i \in V_C$ is included in the subset $S$. $q_{ij}$ is the contraction probability that vertices $i \in V_C$ and $j \in V_C$ are contracted into a single vertex.
• Figure 3: The comparison of the adjacent graphs $G^{(m)}$ and $G^{(m+1)}$ in the NeuralSEP framework.
• Figure 4: Box plots of three metrics over graph sizes
• Figure 5: Illustration of $\pi$-greedy selection method in the Neural Graph Coarsening procedure

Theorems & Definitions (4)

• Proposition 1
• Proposition 2
• proof
• proof