Abstract
Existing graph convolutional network (GCN) models for the traveling salesman problem (TSP) cannot generalize well to TSP instances with larger number of cities than training samples, and the NP-Hard nature of the TSP renders it impractical to use large-scale instances for training. This paper proposes a novel approach that generalizes well a pre-trained GCN model for a fixed small TSP size to large scale instances with the help of Lin-Kernighan-Helsgaun (LKH) heuristics. This is realized by first devising a Sierpinski partition scheme to partition a large TSP into sub-problems that can be efficiently solved by the pre-trained GCN, and then developing an attention-based merging mechanism to integrate the sub-solutions as a whole solution to the original TSP instance. Specifically, we train a GCN model by supervised learning to produce edge prediction heat maps of small-scale TSP instances, then apply it to the sub-problems of a large TSP instance generated by partition strategies. Controlled by an attention mechanism, all the heat maps of the sub-problems are merged into a complete one to construct the edge candidate set for LKH. Experiments show that this new approach significantly enhances the generalization ability of the pre-trained GCN model without using labeled large-scale TSP instances in the training process and also outperforms LKH in the same time limit.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Applegate, D., Bixby, R., Chvatal, V., Cook, W.: Concorde TSP solver (2006). www.math.uwaterloo.ca/tsp/concorde
Bello, I., Pham, H., Le, Q.V., Norouzi, M., Bengio, S.: Neural combinatorial optimization with reinforcement learning. In: Workshop Track of the International Conference on Learning Representations (2017)
Bengio, Y., Lodi, A., Prouvost, A.: Machine learning for combinatorial optimization: a methodological tour d’horizon. Eur. J. Oper. Res. 290(2), 405–421 (2021)
Emami, P., Ranka, S.: Learning permutations with sinkhorn policy gradient. arXiv preprint arXiv:1805.07010 (2018)
Fu, Z.H., Qiu, K.B., Zha, H.: Generalize a small pre-trained model to arbitrarily large tsp instances. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 35, pp. 7474–7482 (2021)
Helsgaun, K.: An effective implementation of the lin-kernighan traveling salesman heuristic. Eur. J. Oper. Res. 126(1), 106–130 (2000)
Helsgaun, K.: General k-opt submoves for the lin-kernighan tsp heuristic. Math. Program. Comput. 1(2), 119–163 (2009)
Helsgaun, K.: An extension of the lin-kernighan-helsgaun tsp solver for constrained traveling salesman and vehicle routing problems. Roskilde: Roskilde University, pp. 24–50 (2017)
Johnson, D.S., McGeoch, L.A.: The traveling salesman problem: a case study in local optimization. Local Search Comb. Optim. 1(1), 215–310 (1997)
Joshi, C.K., Laurent, T., Bresson, X.: An efficient graph convolutional network technique for the travelling salesman problem. arXiv preprint arXiv:1906.01227 (2019)
Karp, R.M.: Probabilistic analysis of partitioning algorithms for the traveling-salesman problem in the plane. Math. Oper. Res. 2(3), 209–224 (1977)
Khalil, E., Dai, H., Zhang, Y., Dilkina, B., Song, L.: Learning combinatorial optimization algorithms over graphs. In: Advances in Neural Information Processing Systems 30 (2017)
Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization. In: International Conference on Learning Representations (2015)
Kool, W., Hoof, H.V., Welling, M.: Attention, learn to solve routing problems! In: International Conference on Learning Representations (2019)
Lin, S., Kernighan, B.W.: An effective heuristic algorithm for the traveling-salesman problem. Oper. Res. 21(2), 498–516 (1973)
Ma, H., Tu, S., Xu, L.: IA-CL: A deep bidirectional competitive learning method for traveling salesman problem. In: Tanveer, M., Agarwal, S., Ozawa, S., Ekbal, A., Jatowt, A. (eds.) Neural Information Processing. ICONIP 2022. Lecture Notes in Computer Science, vol. 13623, pp. 525–536. Springer International Publishing, Cham (2023). https://doi.org/10.1007/978-3-031-30105-6_44
Nowak, A., Villar, S., Bandeira, A.S., Bruna, J.: Revised note on learning quadratic assignment with graph neural networks. In: 2018 IEEE Data Science Workshop, DSW 2018, Lausanne, Switzerland, June 4–6, 2018, pp. 229–233. IEEE (2018)
Platzman, L.K., Bartholdi, J.J., III.: Spacefilling curves and the planar travelling salesman problem. J. ACM (JACM) 36(4), 719–737 (1989)
Rohe, A.: Parallele heuristiken für sehr große travelling salesman probleme. diplom. de (1998)
Vinyals, O., Fortunato, M., Jaitly, N.: Pointer networks. In: Advances in Neural Information Processing Systems 28 (2015)
Xin, L., Song, W., Cao, Z., Zhang, J.: Neurolkh: combining deep learning model with lin-kernighan-helsgaun heuristic for solving the traveling salesman problem. Adv. Neural. Inf. Process. Syst. 34, 7472–7483 (2021)
Xing, Z., Tu, S.: A graph neural network assisted monte Carlo tree search approach to traveling salesman problem. IEEE Access 8, 108418–108428 (2020)
Xu, L.: Deep bidirectional intelligence: AlphaZero, deep IA-search, deep IA-infer, and TPC causal learning. Appl. Inform. 5(1), 1–38 (2018). https://doi.org/10.1186/s40535-018-0052-y
Xu, L.: Deep IA-BI and five actions in circling. In: International Conference on Intelligent Science and Big Data Engineering, pp. 1–21. Springer, New York, NY (2019). https://doi.org/10.1007/0-387-23081-5_11
Zheng, J., He, K., Zhou, J., Jin, Y., Li, C.M.: Combining reinforcement learning with lin-kernighan-helsgaun algorithm for the traveling salesman problem. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 35, pp. 12445–12452 (2021)
Acknowledgement.
This work was supported by the Shanghai Municipal Science and Technology Major Project (2021SHZDZX0102). Shikui Tu and Lei Xu are corresponding authors.
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Li, M., Tu, S., Xu, L. (2024). Generalizing Graph Network Models for the Traveling Salesman Problem with Lin-Kernighan-Helsgaun Heuristics. In: Luo, B., Cheng, L., Wu, ZG., Li, H., Li, C. (eds) Neural Information Processing. ICONIP 2023. Lecture Notes in Computer Science, vol 14447. Springer, Singapore. https://doi.org/10.1007/978-981-99-8079-6_41
Download citation
DOI: https://doi.org/10.1007/978-981-99-8079-6_41
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-99-8078-9
Online ISBN: 978-981-99-8079-6
eBook Packages: Computer ScienceComputer Science (R0)