AGDN: Learning to Solve Traveling Salesman Problem with Anisotropic Graph Diffusion Network

Problem
The Traveling Salesman Problem (TSP) is a fundamental challenge in combinatorial optimization, with significant implications in logistics and routing. Existing graph-based learning approaches have not fully leveraged the graph structure inherent in TSP, particularly in fully connected graphs where informative topological priors are lacking. Additionally, traditional graph sparsification techniques often result in the loss of connected nodes critical to optimal solutions. This paper addresses these gaps by proposing a novel framework, the Anisotropic Graph Diffusion Network (AGDN), to improve TSP solution quality. The work is presented as a preprint and has not undergone peer review.

Method
AGDN introduces a MixScore transition matrix that integrates node similarity with pairwise distances, enhancing the representation of the graph’s topology. The architecture employs an anisotropic graph diffusion strategy, which facilitates efficient information propagation across multiple hops in the graph. This approach allows the model to maintain connectivity among nodes that are crucial for finding optimal solutions. The authors conducted extensive experiments on various TSP instances, varying in size and node distribution, to validate the effectiveness of AGDN. The training process and computational resources utilized are not explicitly detailed in the paper.

Results
AGDN demonstrates superior performance compared to several baseline methods on standard TSP benchmarks. Specifically, it achieves a reduction in solution cost by up to 15% compared to the best-performing baseline, which includes traditional heuristics and other graph neural network approaches. The model maintains competitive computation times, suggesting that the improvements in solution quality do not come at the expense of efficiency. AGDN also exhibits strong generalization capabilities, effectively solving TSP instances that differ in size and distribution from those encountered during training.

Limitations
The authors acknowledge that while AGDN shows promising results, it may still struggle with extremely large TSP instances due to inherent computational complexity. Additionally, the reliance on the MixScore transition matrix may limit the model’s adaptability to other combinatorial optimization problems without further modifications. The paper does not address potential overfitting issues or the impact of hyperparameter tuning on performance.

Why it matters
The introduction of AGDN represents a significant advancement in the application of graph neural networks to combinatorial optimization problems, particularly TSP. By effectively utilizing graph structure and enhancing information exchange, AGDN sets a new benchmark for future research in this domain. The implications of this work extend beyond TSP, as the methodologies developed could be adapted for other optimization problems that benefit from graph-based representations. This research contributes to the growing body of literature on graph neural networks and their applications in solving complex combinatorial challenges, as published in arXiv.