  Recently, Transformer has become a prevailing deep architecture for solving
vehicle routing problems (VRPs). However, it is less effective in learning
improvement models for VRP because its positional encoding (PE) method is not
suitable in representing VRP solutions. This paper presents a novel Dual-Aspect
Collaborative Transformer (DACT) to learn embeddings for the node and
positional features separately, instead of fusing them together as done in
existing ones, so as to avoid potential noises and incompatible correlations.
Moreover, the positional features are embedded through a novel cyclic
positional encoding (CPE) method to allow Transformer to effectively capture
the circularity and symmetry of VRP solutions (i.e., cyclic sequences). We
train DACT using Proximal Policy Optimization and design a curriculum learning
strategy for better sample efficiency. We apply DACT to solve the traveling
salesman problem (TSP) and capacitated vehicle routing problem (CVRP). Results
show that our DACT outperforms existing Transformer based improvement models,
and exhibits much better generalization performance across different problem
sizes on synthetic and benchmark instances, respectively.
