Towards Efficient Constraint Handling in Neural Solvers for Routing Problems

TL;DR

The paper addresses constraint handling in neural solvers for routing problems by introducing Construct-and-Refine (CaR), a framework that explicitly refines infeasible or suboptimal constructions through a lightweight neural improvement step. CaR jointly trains a construction module and a refinement module, guided by tailored losses and a short refinement horizon, enabling efficient constraint satisfaction even for hard VRPs. A shared encoder across construction and refinement enables cross-paradigm representation learning, improving feasibility awareness and enabling knowledge transfer. Empirical results on hard VRPs (e.g., TSPTW, CVRPBLTW) show significant speedups and superior feasibility and quality compared to classical and neural baselines, with strong generalization across scales and constraint types, highlighting CaR’s practical impact for real-world constrained routing tasks.

Abstract

Neural solvers have achieved impressive progress in addressing simple routing problems, particularly excelling in computational efficiency. However, their advantages under complex constraints remain nascent, for which current constraint-handling schemes via feasibility masking or implicit feasibility awareness can be inefficient or inapplicable for hard constraints. In this paper, we present Construct-and-Refine (CaR), the first general and efficient constraint-handling framework for neural routing solvers based on explicit learning-based feasibility refinement. Unlike prior construction-search hybrids that target reducing optimality gaps through heavy improvements yet still struggle with hard constraints, CaR achieves efficient constraint handling by designing a joint training framework that guides the construction module to generate diverse and high-quality solutions well-suited for a lightweight improvement process, e.g., 10 steps versus 5k steps in prior work. Moreover, CaR presents the first use of construction-improvement-shared representation, enabling potential knowledge sharing across paradigms by unifying the encoder, especially in more complex constrained scenarios. We evaluate CaR on typical hard routing constraints to showcase its broader applicability. Results demonstrate that CaR achieves superior feasibility, solution quality, and efficiency compared to both classical and neural state-of-the-art solvers.

Figures (14)

• Figure 1: Overall framework of CaR, where the construction module provides diverse and high-quality solutions for the refinement module to generate better solutions.
• Figure 2: Overview of the unified network architecture in CaR. Blue dashed arrows indicate information flow specific to refinement, while purple dashed arrows indicate flow exclusive to construction.
• Figure 3: Performance over time on TSPTW-100 (left) and CVRPBLTW-100 (right). For CVRPBLTW, POMO+EAS+SGBS and CaR always achieve $0\%$ infeasibility due to feasibility guarantee by masking. Dots with black circles represent results reported in Table \ref{['tab:combined']}.
• Figure 3: Results on TSPDL-50.
• Figure 4: CVRPLIB results.