Enhancing neural combinatorial optimization by progressive training paradigm
Neural Combinatorial Optimization (NCO) methods have garnered considerable attention, due to their effectiveness in automatic algorithm design for solving combinatorial optimization problems. Current constructive NCO methods predominantly employ a one-stage training paradigm using either reinforceme...
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| Vydáno v: | Neurocomputing (Amsterdam) Ročník 659; s. 131707 |
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| Jazyk: | angličtina |
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Elsevier B.V
01.01.2026
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| ISSN: | 0925-2312 |
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| Abstract | Neural Combinatorial Optimization (NCO) methods have garnered considerable attention, due to their effectiveness in automatic algorithm design for solving combinatorial optimization problems. Current constructive NCO methods predominantly employ a one-stage training paradigm using either reinforcement learning (RL) or supervised learning (SL). The one-stage training inevitably entails the computation-intensive labeling (i.e., solving optimal solutions) in SL or less-informative sparse rewards in RL. In this work, we propose a progressive training paradigm that pre-trains a neural network on small-scale instances using SL and then fine-tunes it using RL. In the former stage, the optimal solutions as labels effectively guide the neural network training, thereby bypassing the sparse reward issue. In the latter, the neural network is trained using RL to solve large-scale problems, avoiding the labels of optimal solutions that are hard to obtain. Moreover, we propose a decomposition-based approach that enables RL training with larger problem scales, alleviating the issue of insufficient memory induced by the heavy neural network. The proposed paradigm advances existing NCO models to obtain near-optimal solutions for the Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) with up to 10,000 nodes. Additionally, it enhances the generalization performance across instances of different sizes and distributions, as well as real-world TSPLib and CVRPLib instances. |
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| AbstractList | Neural Combinatorial Optimization (NCO) methods have garnered considerable attention, due to their effectiveness in automatic algorithm design for solving combinatorial optimization problems. Current constructive NCO methods predominantly employ a one-stage training paradigm using either reinforcement learning (RL) or supervised learning (SL). The one-stage training inevitably entails the computation-intensive labeling (i.e., solving optimal solutions) in SL or less-informative sparse rewards in RL. In this work, we propose a progressive training paradigm that pre-trains a neural network on small-scale instances using SL and then fine-tunes it using RL. In the former stage, the optimal solutions as labels effectively guide the neural network training, thereby bypassing the sparse reward issue. In the latter, the neural network is trained using RL to solve large-scale problems, avoiding the labels of optimal solutions that are hard to obtain. Moreover, we propose a decomposition-based approach that enables RL training with larger problem scales, alleviating the issue of insufficient memory induced by the heavy neural network. The proposed paradigm advances existing NCO models to obtain near-optimal solutions for the Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) with up to 10,000 nodes. Additionally, it enhances the generalization performance across instances of different sizes and distributions, as well as real-world TSPLib and CVRPLib instances. |
| ArticleNumber | 131707 |
| Author | Ge, Hongwei Wu, Yaoxin Hou, Yaqing Cao, Zhi |
| Author_xml | – sequence: 1 givenname: Zhi surname: Cao fullname: Cao, Zhi organization: School of Computer Science and Technology, Dalian University of Technology, Dalian 116023, China – sequence: 2 givenname: Yaoxin surname: Wu fullname: Wu, Yaoxin organization: Department of Industrial Engineering and Innovation Sciences, Eindhoven University of Technology, 5600 MB Eindhoven, the Netherlands – sequence: 3 givenname: Yaqing surname: Hou fullname: Hou, Yaqing organization: School of Computer Science and Technology, Dalian University of Technology, Dalian 116023, China – sequence: 4 givenname: Hongwei orcidid: 0000-0002-8937-1515 surname: Ge fullname: Ge, Hongwei email: hwge@dlut.edu.cn organization: School of Computer Science and Technology, Dalian University of Technology, Dalian 116023, China |
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| Cites_doi | 10.1016/j.ejor.2009.10.002 10.1145/3694690 10.1023/A:1022672621406 10.1038/nmeth.2926 10.1109/TII.2022.3189725 10.1109/TNNLS.2021.3068828 10.1109/TEPM.2003.813002 10.1109/TIP.2024.3472494 10.1007/s11063-023-11364-4 10.1287/mnsc.6.1.80 10.1109/TITS.2023.3253552 10.1016/j.cor.2021.105643 10.1016/j.bulsci.2024.103482 |
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| Copyright | 2025 Elsevier B.V. |
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| Keywords | Attention model Neural combinatorial optimization Vehicle routing problem Reinforcement learning |
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| References | Xin, Song, Cao, Zhang (bib0185) 2021; vol. 35 Radhika, Chandrasekar, Vijayakumar (bib0035) 2024; 195 Williams (bib0270) 1992; 8 Wu, Song, Cao, Zhang, Lim (bib0070) 2021; 33 Drakulic, Michel, Mai, Sors, Andreoli (bib0110) 2023; 36 Joshi, Laurent, Bresson (bib0050) 2019 Kumar, Luo (bib0250) 2003; 26 Jiang, Wu, Li, Cao, Zhang (bib0290) 2025 Choo, Kwon, Kim, Jae, Hottung, Tierney, Gwon (bib0265) 2022; 35 Bello, Pham, Le, Norouzi, Bengio (bib0140) 2016 Dantzig, Ramser (bib0240) 1959; 6 Geisler, Sommer, Schuchardt, Bojchevski, Günnemann (bib0160) 2022 Luo, Lin, Wu, Wang, Xialiang, Yuan, Zhang (bib0285) 2025 Ma, Cao, Chee (bib0120) 2023; vol. 36 Helsgaun (bib0025) 2017; 12 Xin, Song, Cao, Zhang (bib0115) 2021; vol. 34 Yao, Lin, Wang, Zhang, Wang (bib0205) 2024 Jin, Ding, Pan, He, Zhao, Qin, Song, Bian (bib0195) 2023; vol. 37 Song, Chen, Li, Cao (bib0105) 2023; 19 Kool, Hoof, Welling (bib0055) 2019 Radhika, Chandrasekar, Vijayakumar, Zhu (bib0030) 2023; 55 Wu, Zhou, Xia, Zhang, Cao, Zhang (bib0255) 2023; 24 Prates, Avelar, Lemos, Lamb, Vardi (bib0075) 2019; vol. 33 Zhou, Wu, Song, Cao, Zhang (bib0090) 2023 Kwon, Choo, Kim, Yoon, Gwon, Min (bib0060) 2020; 33 Bi, Ma, Zhou, Song, Cao, Wu, Zhang (bib0100) 2024; vol. 37 Kwon, Choo, Yoon, Park, Park, Gwon (bib0180) 2021; 34 Wu, Wang, Wen, Xiao, Wu, Wu, Yu, Maskell, Zhou (bib0235) 2024 Tang, Ge, Sun, Hou, Zhao (bib0040) 2024 Konstantakopoulos, Gayialis, Kechagias (bib0245) 2022; 22 Goh, Cao, Ma, Dong, Dupty, Lee (bib0095) 2024 Zhang, Zhang, Wang, Zhu (bib0145) 2022; vol. 36 D. Applegate, R. Bixby, V. Chvatal, W. Cook, Concorde TSP solver (2006). Luo, Lin, Liu, Zhang, Wang (bib0080) 2023; vol. 36 Mandal, Kumar Deva Sarma (bib0260) 2022 Vinyals, Fortunato, Jaitly (bib0045) 2015; vol. 28 Hottung, Kwon, Tierney (bib0065) 2022 Ye, Wang, Liang, Cao, Li, Li (bib0215) 2024; vol. 38 Sanokowski, Hochreiter, Lehner (bib0155) 2024 Min, Bai, Gomes (bib0130) 2023; 36 Sun, Yang (bib0150) 2023; vol. 36 Gao, Shang, Xue, Li, Qian (bib0230) 2024 Brophy, Voigt (bib0010) 2014; 11 Fu, Qiu, Zha (bib0125) 2022 Grinsztajn, Furelos-Blanco, Surana, Bonnet, Barrett (bib0200) 2023; 36 Nazari, Oroojlooy, Takáč, Snyder (bib0135) 2018 Garaix, Artigues, Feillet, Josselin (bib0005) 2010 Lisicki, Afkanpour, Taylor (bib0085) 2020 Kim, Park (bib0190) 2021; 34 Vaswani, Shazeer, Parmar, Uszkoreit, Jones, Gomez, Kaiser, Polosukhin (bib0175) 2017 Vidal (bib0280) 2022; 140 Hou, Yang, Su, Wang, Deng (bib0210) 2023 Zhang, Song, Cao, Zhang, Tan, Chi (bib0015) 2020; 33 F. Didier, L. Perron, S. Mohajeri, S.A. Gay, T. Cuvelier, V. Furnon, Or-tools’ vehicle routing solver: a generic constraint-programming solver with heuristic search for routing problems (2023). L. Xin, W. Song, Z. Cao, J. Zhang, Generative adversarial training for neural combinatorial optimization models (2023). Zong, Wang, Wang, Zheng, Li (bib0225) 2022 Khalil, Dai, Zhang, Dilkina, Song (bib0170) 2017; 30 Pan, Jin, Ding, Feng, Zhao, Song, Bian (bib0220) 2023; vol. 37 Khalil (10.1016/j.neucom.2025.131707_bib0170) 2017; 30 Vaswani (10.1016/j.neucom.2025.131707_bib0175) 2017 Jiang (10.1016/j.neucom.2025.131707_bib0290) 2025 Zong (10.1016/j.neucom.2025.131707_bib0225) 2022 Gao (10.1016/j.neucom.2025.131707_bib0230) 2024 Radhika (10.1016/j.neucom.2025.131707_bib0035) 2024; 195 Ma (10.1016/j.neucom.2025.131707_bib0120) 2023; vol. 36 Kwon (10.1016/j.neucom.2025.131707_bib0060) 2020; 33 Radhika (10.1016/j.neucom.2025.131707_bib0030) 2023; 55 Wu (10.1016/j.neucom.2025.131707_bib0070) 2021; 33 Geisler (10.1016/j.neucom.2025.131707_bib0160) 2022 Xin (10.1016/j.neucom.2025.131707_bib0115) 2021; vol. 34 Song (10.1016/j.neucom.2025.131707_bib0105) 2023; 19 Sun (10.1016/j.neucom.2025.131707_bib0150) 2023; vol. 36 Tang (10.1016/j.neucom.2025.131707_bib0040) 2024 Garaix (10.1016/j.neucom.2025.131707_bib0005) 2010 Kim (10.1016/j.neucom.2025.131707_bib0190) 2021; 34 Min (10.1016/j.neucom.2025.131707_bib0130) 2023; 36 10.1016/j.neucom.2025.131707_bib0020 Bello (10.1016/j.neucom.2025.131707_bib0140) 2016 Grinsztajn (10.1016/j.neucom.2025.131707_bib0200) 2023; 36 Sanokowski (10.1016/j.neucom.2025.131707_bib0155) 2024 Luo (10.1016/j.neucom.2025.131707_bib0285) 2025 Wu (10.1016/j.neucom.2025.131707_bib0255) 2023; 24 Vinyals (10.1016/j.neucom.2025.131707_bib0045) 2015; vol. 28 Zhang (10.1016/j.neucom.2025.131707_bib0015) 2020; 33 Luo (10.1016/j.neucom.2025.131707_bib0080) 2023; vol. 36 Helsgaun (10.1016/j.neucom.2025.131707_bib0025) 2017; 12 Wu (10.1016/j.neucom.2025.131707_bib0235) Kool (10.1016/j.neucom.2025.131707_bib0055) 2019 Choo (10.1016/j.neucom.2025.131707_bib0265) 2022; 35 Ye (10.1016/j.neucom.2025.131707_bib0215) 2024; vol. 38 Dantzig (10.1016/j.neucom.2025.131707_bib0240) 1959; 6 10.1016/j.neucom.2025.131707_bib0275 Goh (10.1016/j.neucom.2025.131707_bib0095) 2024 Yao (10.1016/j.neucom.2025.131707_bib0205) 2024 Nazari (10.1016/j.neucom.2025.131707_bib0135) 2018 Prates (10.1016/j.neucom.2025.131707_bib0075) 2019; vol. 33 Zhang (10.1016/j.neucom.2025.131707_bib0145) 2022; vol. 36 Joshi (10.1016/j.neucom.2025.131707_bib0050) Konstantakopoulos (10.1016/j.neucom.2025.131707_bib0245) 2022; 22 Kwon (10.1016/j.neucom.2025.131707_bib0180) 2021; 34 Jin (10.1016/j.neucom.2025.131707_bib0195) 2023; vol. 37 Kumar (10.1016/j.neucom.2025.131707_bib0250) 2003; 26 Brophy (10.1016/j.neucom.2025.131707_bib0010) 2014; 11 Lisicki (10.1016/j.neucom.2025.131707_bib0085) 2020 Bi (10.1016/j.neucom.2025.131707_bib0100) 2024; vol. 37 Hottung (10.1016/j.neucom.2025.131707_bib0065) 2022 10.1016/j.neucom.2025.131707_bib0165 Mandal (10.1016/j.neucom.2025.131707_bib0260) 2022 Xin (10.1016/j.neucom.2025.131707_bib0185) 2021; vol. 35 Hou (10.1016/j.neucom.2025.131707_bib0210) 2023 Drakulic (10.1016/j.neucom.2025.131707_bib0110) 2023; 36 Pan (10.1016/j.neucom.2025.131707_bib0220) 2023; vol. 37 Williams (10.1016/j.neucom.2025.131707_bib0270) 1992; 8 Vidal (10.1016/j.neucom.2025.131707_bib0280) 2022; 140 Zhou (10.1016/j.neucom.2025.131707_bib0090) 2023 Fu (10.1016/j.neucom.2025.131707_bib0125) 2022 |
| References_xml | – volume: vol. 28 year: 2015 ident: bib0045 article-title: Pointer networks publication-title: Advances in Neural Information Processing Systems – start-page: 6914 year: 2024 end-page: 6922 ident: bib0230 article-title: Towards generalizable neural solvers for vehicle routing problems via ensemble with transferrable local policy publication-title: Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence, IJCAI-24 – year: 2019 ident: bib0050 article-title: An efficient graph convolutional network technique for the travelling salesman problem – volume: 30 year: 2017 ident: bib0170 article-title: Learning combinatorial optimization algorithms over graphs publication-title: Adv. Neural Inf. Process. Syst. – volume: 33 start-page: 21188 year: 2020 end-page: 21198 ident: bib0060 article-title: Pomo: policy optimization with multiple optima for reinforcement learning publication-title: Neural Inf. Process. Syst. – year: 2024 ident: bib0040 article-title: A virtual-sensor construction network based on physical imaging for image super-resolution publication-title: IEEE Trans. Image Process. – year: 2019 ident: bib0055 article-title: Attention, learn to solve routing problems! publication-title: International Conference on Learning Representations – volume: vol. 35 start-page: 12042 year: 2021 end-page: 12049 ident: bib0185 article-title: Multi-decoder attention model with embedding glimpse for solving vehicle routing problems publication-title: Proceedings of the AAAI Conference on Artificial Intelligence – year: 2022 ident: bib0160 article-title: Generalization of neural combinatorial solvers through the lens of adversarial robustness publication-title: ICLR – volume: vol. 37 start-page: 93479 year: 2024 end-page: 93509 ident: bib0100 article-title: Learning to handle complex constraints for vehicle routing problems publication-title: Advances in Neural Information Processing Systems – volume: 55 start-page: 11055 year: 2023 end-page: 11072 ident: bib0030 article-title: Analysis of Markovian jump stochastic Cohen–Grossberg BAM neural networks with time delays for exponential input-to-state stability publication-title: Neural Process. Lett. – volume: vol. 34 start-page: 7472 year: 2021 end-page: 7483 ident: bib0115 article-title: NeuroLKH: combining deep learning model with Lin-Kernighan-Helsgaun heuristic for solving the traveling salesman problem publication-title: NEURIPS – volume: 195 year: 2024 ident: bib0035 article-title: Finite-time h publication-title: Bull. Sci. Math. – year: 2022 ident: bib0065 article-title: Efficient active search for combinatorial optimization problems publication-title: International Conference on Learning Representations – reference: L. Xin, W. Song, Z. Cao, J. Zhang, Generative adversarial training for neural combinatorial optimization models (2023). – year: 2020 ident: bib0085 article-title: Evaluating curriculum learning strategies in neural combinatorial optimization publication-title: NEURIPS 2020 Workshop on Learning Meets Combinatorial Algorithms – volume: vol. 36 start-page: 49555 year: 2023 end-page: 49578 ident: bib0120 article-title: Learning to search feasible and infeasible regions of routing problems with flexible neural k-opt publication-title: NEURIPS – volume: 33 start-page: 1621 year: 2020 end-page: 1632 ident: bib0015 article-title: Learning to dispatch for job shop scheduling via deep reinforcement learning publication-title: Adv. Neural Inf. Process. Syst. – year: 2025 ident: bib0285 article-title: Boosting neural combinatorial optimization for large-scale vehicle routing problems publication-title: The Thirteenth International Conference on Learning Representations – volume: vol. 36 start-page: 9136 year: 2022 end-page: 9144 ident: bib0145 article-title: Learning to solve travelling salesman problem with hardness-adaptive curriculum publication-title: AAAI – volume: 35 start-page: 8760 year: 2022 end-page: 8772 ident: bib0265 article-title: Simulation-guided beam search for neural combinatorial optimization publication-title: Adv. Neural Inf. Process. Syst. – year: 2023 ident: bib0210 article-title: Generalize learned heuristics to solve large-scale vehicle routing problems in real-time publication-title: The Eleventh International Conference on Learning Representations – volume: 26 start-page: 14 year: 2003 end-page: 21 ident: bib0250 article-title: Optimizing the operation sequence of a chip placement machine using TSP model publication-title: IEEE Trans. Electron. Packag. Manuf. – start-page: 42769 year: 2023 end-page: 42789 ident: bib0090 article-title: Towards omni-generalizable neural methods for vehicle routing problems publication-title: International Conference on Machine Learning – start-page: 62 year: 2010 end-page: 75 ident: bib0005 article-title: Vehicle routing problems with alternative paths: an application to on-demand transportation publication-title: Eur. J. Oper. Res. – start-page: 43346 year: 2024 end-page: 43367 ident: bib0155 article-title: A diffusion model framework for unsupervised neural combinatorial optimization publication-title: International Conference on Machine Learning – volume: 36 start-page: 48485 year: 2023 end-page: 48509 ident: bib0200 article-title: Winner takes it all: training performant RL populations for combinatorial optimization publication-title: Adv. Neural Inf. Process. Syst. – volume: vol. 33 start-page: 4731 year: 2019 end-page: 4738 ident: bib0075 article-title: Learning to solve NP-complete problems: a graph neural network for decision TSP publication-title: Proceedings of the AAAI Conference on Artificial Intelligence – volume: vol. 37 start-page: 9345 year: 2023 end-page: 9353 ident: bib0220 article-title: H-tsp: Hierarchically solving the large-scale traveling salesman problem publication-title: Proceedings of the AAAI Conference on Artificial Intelligence – volume: 19 start-page: 1600 year: 2023 end-page: 1610 ident: bib0105 article-title: Flexible job-shop scheduling via graph neural network and deep reinforcement learning publication-title: IEEE Trans. Ind. Inf. – start-page: 9861 year: 2018 end-page: 9871 ident: bib0135 article-title: Reinforcement learning for solving the vehicle routing problem publication-title: Proceedings of the 32nd International Conference on Neural Information Processing Systems – volume: 36 start-page: 47264 year: 2023 end-page: 47278 ident: bib0130 article-title: Unsupervised learning for solving the travelling salesman problem publication-title: Adv. Neural Inf. Process. Syst. – reference: D. Applegate, R. Bixby, V. Chvatal, W. Cook, Concorde TSP solver (2006). – start-page: 4648 year: 2022 end-page: 4658 ident: bib0225 article-title: Rbg: hierarchically solving large-scale routing problems in logistic systems via reinforcement learning publication-title: Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining – start-page: 1 year: 2022 end-page: 6 ident: bib0260 article-title: Novel applications of ant colony optimization with the traveling salesman problem in DNA sequence optimization publication-title: 2022 IEEE 2nd International Symposium on Sustainable Energy, Signal Processing and Cyber Security (ISSSC) – volume: 6 start-page: 80 year: 1959 end-page: 91 ident: bib0240 article-title: The truck dispatching problem publication-title: Manag. Sci. – volume: 8 start-page: 229 year: 1992 end-page: 256 ident: bib0270 article-title: Simple statistical gradient-following algorithms for connectionist reinforcement learning publication-title: Mach. Learn. – volume: vol. 36 start-page: 3706 year: 2023 end-page: 3731 ident: bib0150 article-title: Difusco: graph-based diffusion solvers for combinatorial optimization publication-title: Advances in Neural Information Processing Systems – volume: 36 start-page: 77416 year: 2023 end-page: 77429 ident: bib0110 article-title: Bq-NCO: bisimulation quotienting for efficient neural combinatorial optimization publication-title: Adv. Neural Inf. Process. Syst. – volume: 11 start-page: 508 year: 2014 end-page: 520 ident: bib0010 article-title: Principles of genetic circuit design publication-title: Nat. Methods – volume: 140 year: 2022 ident: bib0280 article-title: Hybrid genetic search for the CVRP: open-source implementation and swap* neighborhood publication-title: Comput. Oper. Res. – volume: 12 start-page: 966 year: 2017 end-page: 980 ident: bib0025 article-title: An extension of the Lin-Kernighan-Helsgaun TSP solver for constrained traveling salesman and vehicle routing problems publication-title: Roskilde: Roskilde University – start-page: 884 year: 2024 end-page: 895 ident: bib0095 article-title: Hierarchical neural constructive solver for real-world TSP scenarios publication-title: Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining – volume: 22 start-page: 2033 year: 2022 end-page: 2062 ident: bib0245 article-title: Vehicle routing problem and related algorithms for logistics distribution: a literature review and classification publication-title: Oper. Res. – volume: vol. 36 start-page: 8845 year: 2023 end-page: 8864 ident: bib0080 article-title: Neural combinatorial optimization with heavy decoder: toward large scale generalization publication-title: Advances in Neural Information Processing Systems – volume: 34 start-page: 5138 year: 2021 end-page: 5149 ident: bib0180 article-title: Matrix encoding networks for neural combinatorial optimization publication-title: Adv. Neural Inf. Process. Syst. – year: 2024 ident: bib0235 article-title: Neural combinatorial optimization algorithms for solving vehicle routing problems: a comprehensive survey with perspectives – volume: vol. 37 start-page: 8132 year: 2023 end-page: 8140 ident: bib0195 article-title: Pointerformer: deep reinforced multi-pointer transformer for the traveling salesman problem publication-title: Proceedings of the AAAI Conference on Artificial Intelligence – year: 2025 ident: bib0290 article-title: Large language models as end-to-end combinatorial optimization solvers publication-title: Adv. Neural Inf. Process. Syst. – year: 2017 ident: bib0175 article-title: Attention is all you need publication-title: Adv. Neural Inf. Process. Syst. – year: 2024 ident: bib0205 article-title: Rethinking supervised learning based neural combinatorial optimization for routing problem publication-title: ACM Trans. Evol. Learn. – year: 2016 ident: bib0140 article-title: Neural combinatorial optimization with reinforcement learning – volume: 33 start-page: 5057 year: 2021 end-page: 5069 ident: bib0070 article-title: Learning improvement heuristics for solving routing problems publication-title: IEEE Trans. Neural Netw. Learn. Syst. – volume: 24 start-page: 15652 year: 2023 end-page: 15666 ident: bib0255 article-title: Neural airport ground handling publication-title: IEEE Trans. Intell. Transp. Syst. – start-page: 7474 year: 2022 end-page: 7482 ident: bib0125 article-title: Generalize a small pre-trained model to arbitrarily large TSP instances publication-title: Proc. AAAI Conf. Artif. Intell. – volume: vol. 38 start-page: 20284 year: 2024 end-page: 20292 ident: bib0215 article-title: Glop: learning global partition and local construction for solving large-scale routing problems in real-time publication-title: Proceedings of the AAAI Conference on Artificial Intelligence – reference: F. Didier, L. Perron, S. Mohajeri, S.A. Gay, T. Cuvelier, V. Furnon, Or-tools’ vehicle routing solver: a generic constraint-programming solver with heuristic search for routing problems (2023). – volume: 34 start-page: 10418 year: 2021 end-page: 10430 ident: bib0190 article-title: Learning collaborative policies to solve NP-hard routing problems publication-title: Adv. Neural Inf. Process. Syst. – start-page: 62 year: 2010 ident: 10.1016/j.neucom.2025.131707_bib0005 article-title: Vehicle routing problems with alternative paths: an application to on-demand transportation publication-title: Eur. J. Oper. Res. doi: 10.1016/j.ejor.2009.10.002 – volume: 36 start-page: 48485 year: 2023 ident: 10.1016/j.neucom.2025.131707_bib0200 article-title: Winner takes it all: training performant RL populations for combinatorial optimization publication-title: Adv. Neural Inf. Process. Syst. – year: 2024 ident: 10.1016/j.neucom.2025.131707_bib0205 article-title: Rethinking supervised learning based neural combinatorial optimization for routing problem publication-title: ACM Trans. Evol. Learn. doi: 10.1145/3694690 – volume: 35 start-page: 8760 year: 2022 ident: 10.1016/j.neucom.2025.131707_bib0265 article-title: Simulation-guided beam search for neural combinatorial optimization publication-title: Adv. Neural Inf. Process. Syst. – volume: 8 start-page: 229 issue: 3 year: 1992 ident: 10.1016/j.neucom.2025.131707_bib0270 article-title: Simple statistical gradient-following algorithms for connectionist reinforcement learning publication-title: Mach. Learn. doi: 10.1023/A:1022672621406 – start-page: 1 year: 2022 ident: 10.1016/j.neucom.2025.131707_bib0260 article-title: Novel applications of ant colony optimization with the traveling salesman problem in DNA sequence optimization – volume: vol. 34 start-page: 7472 year: 2021 ident: 10.1016/j.neucom.2025.131707_bib0115 article-title: NeuroLKH: combining deep learning model with Lin-Kernighan-Helsgaun heuristic for solving the traveling salesman problem – volume: 12 start-page: 966 year: 2017 ident: 10.1016/j.neucom.2025.131707_bib0025 article-title: An extension of the Lin-Kernighan-Helsgaun TSP solver for constrained traveling salesman and vehicle routing problems publication-title: Roskilde: Roskilde University – volume: 11 start-page: 508 issue: 5 year: 2014 ident: 10.1016/j.neucom.2025.131707_bib0010 article-title: Principles of genetic circuit design publication-title: Nat. Methods doi: 10.1038/nmeth.2926 – start-page: 7474 year: 2022 ident: 10.1016/j.neucom.2025.131707_bib0125 article-title: Generalize a small pre-trained model to arbitrarily large TSP instances publication-title: Proc. AAAI Conf. Artif. Intell. – start-page: 9861 year: 2018 ident: 10.1016/j.neucom.2025.131707_bib0135 article-title: Reinforcement learning for solving the vehicle routing problem – ident: 10.1016/j.neucom.2025.131707_bib0050 – volume: 19 start-page: 1600 issue: 2 year: 2023 ident: 10.1016/j.neucom.2025.131707_bib0105 article-title: Flexible job-shop scheduling via graph neural network and deep reinforcement learning publication-title: IEEE Trans. Ind. Inf. doi: 10.1109/TII.2022.3189725 – volume: 33 start-page: 5057 issue: 9 year: 2021 ident: 10.1016/j.neucom.2025.131707_bib0070 article-title: Learning improvement heuristics for solving routing problems publication-title: IEEE Trans. Neural Netw. Learn. Syst. doi: 10.1109/TNNLS.2021.3068828 – year: 2022 ident: 10.1016/j.neucom.2025.131707_bib0065 article-title: Efficient active search for combinatorial optimization problems – start-page: 43346 year: 2024 ident: 10.1016/j.neucom.2025.131707_bib0155 article-title: A diffusion model framework for unsupervised neural combinatorial optimization – volume: 26 start-page: 14 issue: 1 year: 2003 ident: 10.1016/j.neucom.2025.131707_bib0250 article-title: Optimizing the operation sequence of a chip placement machine using TSP model publication-title: IEEE Trans. Electron. Packag. Manuf. doi: 10.1109/TEPM.2003.813002 – ident: 10.1016/j.neucom.2025.131707_bib0020 – year: 2017 ident: 10.1016/j.neucom.2025.131707_bib0175 article-title: Attention is all you need publication-title: Adv. Neural Inf. Process. Syst. – volume: 33 start-page: 1621 year: 2020 ident: 10.1016/j.neucom.2025.131707_bib0015 article-title: Learning to dispatch for job shop scheduling via deep reinforcement learning publication-title: Adv. Neural Inf. Process. Syst. – start-page: 42769 year: 2023 ident: 10.1016/j.neucom.2025.131707_bib0090 article-title: Towards omni-generalizable neural methods for vehicle routing problems – volume: vol. 37 start-page: 9345 year: 2023 ident: 10.1016/j.neucom.2025.131707_bib0220 article-title: H-tsp: Hierarchically solving the large-scale traveling salesman problem – year: 2024 ident: 10.1016/j.neucom.2025.131707_bib0040 article-title: A virtual-sensor construction network based on physical imaging for image super-resolution publication-title: IEEE Trans. Image Process. doi: 10.1109/TIP.2024.3472494 – volume: vol. 36 start-page: 9136 year: 2022 ident: 10.1016/j.neucom.2025.131707_bib0145 article-title: Learning to solve travelling salesman problem with hardness-adaptive curriculum – year: 2023 ident: 10.1016/j.neucom.2025.131707_bib0210 article-title: Generalize learned heuristics to solve large-scale vehicle routing problems in real-time – volume: vol. 35 start-page: 12042 year: 2021 ident: 10.1016/j.neucom.2025.131707_bib0185 article-title: Multi-decoder attention model with embedding glimpse for solving vehicle routing problems – year: 2025 ident: 10.1016/j.neucom.2025.131707_bib0290 article-title: Large language models as end-to-end combinatorial optimization solvers publication-title: Adv. Neural Inf. Process. Syst. – year: 2016 ident: 10.1016/j.neucom.2025.131707_bib0140 article-title: Neural combinatorial optimization with reinforcement learning – ident: 10.1016/j.neucom.2025.131707_bib0235 – volume: vol. 36 start-page: 49555 year: 2023 ident: 10.1016/j.neucom.2025.131707_bib0120 article-title: Learning to search feasible and infeasible regions of routing problems with flexible neural k-opt – volume: vol. 36 start-page: 3706 year: 2023 ident: 10.1016/j.neucom.2025.131707_bib0150 article-title: Difusco: graph-based diffusion solvers for combinatorial optimization – volume: 36 start-page: 77416 year: 2023 ident: 10.1016/j.neucom.2025.131707_bib0110 article-title: Bq-NCO: bisimulation quotienting for efficient neural combinatorial optimization publication-title: Adv. Neural Inf. Process. Syst. – volume: 55 start-page: 11055 issue: 8 year: 2023 ident: 10.1016/j.neucom.2025.131707_bib0030 article-title: Analysis of Markovian jump stochastic Cohen–Grossberg BAM neural networks with time delays for exponential input-to-state stability publication-title: Neural Process. Lett. doi: 10.1007/s11063-023-11364-4 – volume: vol. 38 start-page: 20284 year: 2024 ident: 10.1016/j.neucom.2025.131707_bib0215 article-title: Glop: learning global partition and local construction for solving large-scale routing problems in real-time – volume: vol. 37 start-page: 93479 year: 2024 ident: 10.1016/j.neucom.2025.131707_bib0100 article-title: Learning to handle complex constraints for vehicle routing problems – year: 2020 ident: 10.1016/j.neucom.2025.131707_bib0085 article-title: Evaluating curriculum learning strategies in neural combinatorial optimization – year: 2025 ident: 10.1016/j.neucom.2025.131707_bib0285 article-title: Boosting neural combinatorial optimization for large-scale vehicle routing problems – start-page: 6914 year: 2024 ident: 10.1016/j.neucom.2025.131707_bib0230 article-title: Towards generalizable neural solvers for vehicle routing problems via ensemble with transferrable local policy – ident: 10.1016/j.neucom.2025.131707_bib0165 – volume: 34 start-page: 5138 year: 2021 ident: 10.1016/j.neucom.2025.131707_bib0180 article-title: Matrix encoding networks for neural combinatorial optimization publication-title: Adv. Neural Inf. Process. Syst. – volume: 6 start-page: 80 issue: 1 year: 1959 ident: 10.1016/j.neucom.2025.131707_bib0240 article-title: The truck dispatching problem publication-title: Manag. Sci. doi: 10.1287/mnsc.6.1.80 – volume: 33 start-page: 21188 year: 2020 ident: 10.1016/j.neucom.2025.131707_bib0060 article-title: Pomo: policy optimization with multiple optima for reinforcement learning publication-title: Neural Inf. Process. Syst. – start-page: 884 year: 2024 ident: 10.1016/j.neucom.2025.131707_bib0095 article-title: Hierarchical neural constructive solver for real-world TSP scenarios – volume: vol. 37 start-page: 8132 year: 2023 ident: 10.1016/j.neucom.2025.131707_bib0195 article-title: Pointerformer: deep reinforced multi-pointer transformer for the traveling salesman problem – volume: 24 start-page: 15652 issue: 12 year: 2023 ident: 10.1016/j.neucom.2025.131707_bib0255 article-title: Neural airport ground handling publication-title: IEEE Trans. Intell. Transp. Syst. doi: 10.1109/TITS.2023.3253552 – volume: 30 year: 2017 ident: 10.1016/j.neucom.2025.131707_bib0170 article-title: Learning combinatorial optimization algorithms over graphs publication-title: Adv. Neural Inf. Process. Syst. – volume: 22 start-page: 2033 issue: 3 year: 2022 ident: 10.1016/j.neucom.2025.131707_bib0245 article-title: Vehicle routing problem and related algorithms for logistics distribution: a literature review and classification publication-title: Oper. Res. – volume: vol. 36 start-page: 8845 year: 2023 ident: 10.1016/j.neucom.2025.131707_bib0080 article-title: Neural combinatorial optimization with heavy decoder: toward large scale generalization – volume: 140 year: 2022 ident: 10.1016/j.neucom.2025.131707_bib0280 article-title: Hybrid genetic search for the CVRP: open-source implementation and swap* neighborhood publication-title: Comput. Oper. Res. doi: 10.1016/j.cor.2021.105643 – volume: vol. 28 year: 2015 ident: 10.1016/j.neucom.2025.131707_bib0045 article-title: Pointer networks – volume: vol. 33 start-page: 4731 year: 2019 ident: 10.1016/j.neucom.2025.131707_bib0075 article-title: Learning to solve NP-complete problems: a graph neural network for decision TSP – volume: 195 year: 2024 ident: 10.1016/j.neucom.2025.131707_bib0035 article-title: Finite-time h∞ synchronization of semi-Markov jump neural networks with two delay components with stochastic sampled-data control publication-title: Bull. Sci. Math. doi: 10.1016/j.bulsci.2024.103482 – ident: 10.1016/j.neucom.2025.131707_bib0275 – volume: 36 start-page: 47264 year: 2023 ident: 10.1016/j.neucom.2025.131707_bib0130 article-title: Unsupervised learning for solving the travelling salesman problem publication-title: Adv. Neural Inf. Process. Syst. – volume: 34 start-page: 10418 year: 2021 ident: 10.1016/j.neucom.2025.131707_bib0190 article-title: Learning collaborative policies to solve NP-hard routing problems publication-title: Adv. Neural Inf. Process. Syst. – year: 2022 ident: 10.1016/j.neucom.2025.131707_bib0160 article-title: Generalization of neural combinatorial solvers through the lens of adversarial robustness – year: 2019 ident: 10.1016/j.neucom.2025.131707_bib0055 article-title: Attention, learn to solve routing problems! – start-page: 4648 year: 2022 ident: 10.1016/j.neucom.2025.131707_bib0225 article-title: Rbg: hierarchically solving large-scale routing problems in logistic systems via reinforcement learning |
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