Learning to Handle Constraints in Routing Problems via a Construct-and-Refine Framework

Neural solvers have achieved impressive progress on simple routing problems via data-driven training, but often struggle with complex constraints. We rethink the popular single-paradigm neural solvers and identify paradigm-inherent limitations: construction solvers suffer from inflexible stepwise feasibility, and improvement solvers easily get stuck in infeasible searches with long runtimes. However, these paradigms are naturally complementary: construction efficiently provides strong initial solutions that help improvement rapidly reach feasible, high-quality solutions. Motivated by this, we propose Construct-and-Refine (CaR), the first generic neural framework for efficient constraint handling, compatible with existing construction and improvement solvers. To promote synergistic paradigm integration, we introduce a joint training framework with bespoke losses to generate diverse, high-quality, (near)-feasible solutions that are refined by a light improvement process (e.g., only 10 steps down from 5k). We also present the first study of a shared encoder for cross-paradigm representation learning in handling complex constraints. Extensive experiments on hard-constrained TSPTW and CVRPBLTW demonstrate that CaR achieves superior feasibility, solution quality, and efficiency compared to both traditional and neural state-of-the-art solvers.