A&I-ED-TSP: Association and Integration Encoder–Decoder for Traveling Shortest Path Planning

Traveling shortest path planning, encompassing the Traveling Salesman Problem (TSP) in graph theory, holds profound significance. The motivation for addressing the TSP stems from its critical application in real-world scenarios, such as logistics, where optimal routing can substantially reduce costs and improve service efficiency. Furthermore, TSP-like challenges play a pivotal role in assisting travelers to chart the optimal itinerary, encompassing all landmarks in the least distance or time and concluding at the departure site. This optimization not only streamlines travel routes but also economizes time and energy, ensuring maximal sightseeing within a confined timeframe. Recognizing the limitations of current solutions in achieving high efficiency and accuracy simultaneously, we propose an innovative Association & Integration-based Encoder–Decoder structure tailored for solving the Traveling Salesman Problem, i.e., A&I-ED-TSP.The proposed structure comprises four blocks: the information linkage space, dual-path integration encoder, node encoder, and representation decoder. Specifically, the information linkage space constructs associations among hidden information between input sequence samples. The dual-path integration encoder extracts and merges the original representations of the sequence with associated representations. The node encoder extracts current sequence representations, while the representation decoder block computes the probability distribution of sequence samples, completing the combinatorial optimization of the entire sequence. In the experimental evaluation, we utilized three different metrics: Average Tour Length (ATL), Optimality Gap (OG), and Evaluation Time (ET). We compared the proposed method with classical approximation methods and various state-of-the-art deep learning approaches. The experimental results show that our A&I-ED-TSP structure achieved the best ATLs of 5.704, 12.770, 17.981, 21.979, and 25.293 for TSP instances of TSP50, TSP250, TSP500, TSP750, and TSP1000, respectively. The corresponding best OGs were 0.06%, 8.09%, 8.57%, 9.17%, and 9.41%, respectively, representing improvements of 2.83%, 0.68%, 2.47%, 2.91%, and 3.63% over the best previous comprehensive method. The best ETs were 2.15 seconds, 19.24 seconds, 45.81 seconds, 97.03 seconds, and 213.97 seconds, respectively. Experimental results demonstrate that the proposed A&I-ED-TSP structure significantly enhances the model’s solution accuracy and efficiency, offering a promising approach to overcoming the limitations of traditional algorithms and recent machine learning models in solving complex combinatorial problems such as TSP.