Leveraging Structural Constraints for Diffusion-based Neural TSP Solvers

Researchers introduce Projected Consistency Inference (PCI), a neural optimization method that solves the Traveling Salesman Problem more efficiently than gradient-based approaches by using structure-aware projections and local search instead of computationally expensive refinement. PCI achieves better optimality gaps (0.17% for 500 cities, 0.31% for 1000 cities) while reducing inference time by 30-40% compared to state-of-the-art FT2T methods.

This research addresses a fundamental challenge in neural combinatorial optimization: bridging the gap between continuous neural network outputs and discrete solution requirements. While diffusion and consistency models have demonstrated promise for solving NP-hard problems like TSP, previous approaches relied on gradient-based refinement during inference, which introduces computational bottlenecks and may push solutions away from structurally valid configurations. The research community has increasingly recognized that forcing neural models to align with discrete constraints through gradient descent is suboptimal, motivating development of constraint-aware decoding methods.

PCI's core innovation lies in its simplicity and practicality. Rather than requiring model retraining or complex optimization procedures, it applies Hamiltonian cycle decoding followed by lightweight 2-opt local search. This structure-aware approach respects the problem's inherent constraints while leveraging the neural model's learned representations. The 30-40% inference speedup carries significant implications for real-world applications where solution quality and speed are both critical.

For the AI and optimization communities, this work validates an important principle: incorporating domain knowledge at inference time can outperform end-to-end learned refinement. The results suggest that neural combinatorial optimizers benefit from hybrid approaches combining learning with classical algorithmic techniques. For practitioners and researchers developing optimization systems, PCI provides a plug-and-play methodology that doesn't require model retraining, making adoption straightforward.

Future work likely explores similar structure-aware inference strategies across other combinatorial problems, potentially extending this paradigm beyond TSP to vehicle routing, scheduling, and constraint satisfaction domains.

• →PCI achieves 0.17% optimality gap on TSP-500 and 0.31% on TSP-1000, outperforming gradient-based FT2T approaches while reducing inference time by 30-40%
• →The method eliminates need for model retraining by applying structure-aware projections and 2-opt local search at inference time
• →Constraint-aware decoding respects discrete problem structure better than gradient refinement, reducing variance and memory usage
• →Hybrid neural-classical approaches outperform pure end-to-end learned optimization for combinatorial problems
• →PCI enables competitive performance with classical heuristics like LKH3 while offering faster solution generation