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arxiv: 2604.16581 · v1 · submitted 2026-04-17 · 💻 cs.LG · cs.AI

NCO4CVRP: Neural Combinatorial Optimization for the Capacitated Vehicle Routing Problem

Mahir Labib Dihan , Md. Ashrafur Rahman Khan , Wasif Jalal , Md. Roqunuzzaman Sojib , Mashroor Hasan Bhuiyan This is my paper

Pith reviewed 2026-05-10 08:50 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords optimizationcombinatorialsearchinferencemodelsolutiontechniquesacross
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The pith

Adding simulated annealing to random reconstruction and beam search to POMO in neural CVRP solvers reduces optimality gaps on standard benchmarks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The work starts from two prior neural models that learn to build routes for trucks with capacity limits. One model uses a light encoder and heavy decoder; the other uses multiple starting points for policy learning. The authors change how these models pick final routes at test time. Instead of always taking the best immediate fix, they let the model sometimes accept a slightly worse fix with a probability that decreases over time, like cooling metal in simulated annealing. They also let the model keep several good partial routes at once and expand them systematically with beam search. They try different ways to sample next steps, such as softmax, greedy, or epsilon-greedy, and they flip or rotate the input maps to make the model more robust. Experiments on common CVRP test sets show smaller gaps to the best known solutions when these search tricks are used together.

Core claim

Our extensive experiments demonstrate that these modifications significantly reduce the optimality gap across various Capacitated Vehicle Routing Problem (CVRP) benchmarks, with Beam Search and SA-based RRC consistently yielding superior performance.

Load-bearing premise

That the probabilistic acceptance in SA-RRC and the beam expansion in POMO produce genuinely better generalization rather than merely fitting the chosen benchmark distributions and augmentation set.

Figures

Figures reproduced from arXiv: 2604.16581 by Mahir Labib Dihan, Mashroor Hasan Bhuiyan, Md. Ashrafur Rahman Khan, Md. Roqunuzzaman Sojib, Wasif Jalal.

Figure 1
Figure 1. Figure 1: The structure of our proposed LEHD model, which has a single-layer light encoder and [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Experiments using different temperatures [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Experiments using different values of epsilon [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of different inference techniques and augmentation types. [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
read the original abstract

Neural Combinatorial Optimization (NCO) has emerged as a powerful framework for solving combinatorial optimization problems by integrating deep learning-based models. This work focuses on improving existing inference techniques to enhance solution quality and generalization. Specifically, we modify the Random Re-Construct (RRC) approach of the Light Encoder Heavy Decoder (LEHD) model by incorporating Simulated Annealing (SA). Unlike the conventional RRC, which greedily replaces suboptimal segments, our SA-based modification introduces a probabilistic acceptance mechanism that allows the model to escape local optima and explore a more diverse solution space. Additionally, we enhance the Policy Optimization with Multiple Optima (POMO) approach by integrating Beam Search, enabling systematic exploration of multiple promising solutions while maintaining diversity in the search space. We further investigate different inference strategies, including Softmax Sampling, Greedy, Gumbel-Softmax, and Epsilon-Greedy, analyzing their impact on solution quality. Furthermore, we explore instance augmentation techniques, such as horizontal and vertical flipping and rotation-based augmentations, to improve model generalization across different CVRP instances. Our extensive experiments demonstrate that these modifications significantly reduce the optimality gap across various Capacitated Vehicle Routing Problem (CVRP) benchmarks, with Beam Search and SA-based RRC consistently yielding superior performance. By refining inference techniques and leveraging enhanced search strategies, our work contributes to the broader applicability of NCO models in real-world combinatorial optimization tasks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the base LEHD and POMO models already capture useful CVRP structure and that the added search procedures improve upon them without introducing new biases. Because only the abstract is available, a complete ledger of fitted parameters cannot be extracted; the work implicitly depends on many neural-network weights and several inference hyperparameters whose values are not stated.

free parameters (2)
  • Beam width
    Controls how many partial solutions are kept during decoding; value chosen to trade off quality and compute time.
  • Simulated annealing temperature schedule
    Determines probability of accepting worse routes; schedule parameters are tuned for the reported benchmarks.
axioms (1)
  • domain assumption The LEHD and POMO architectures provide a strong starting point whose inference can be improved by classical search heuristics.
    Invoked when the authors state they modify existing models rather than train new ones from scratch.

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Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    An improved hybrid firefly algorithm for capacitated vehicle routing problem.Applied Soft Computing, 84:105728, 2019

    Asma M Altabeeb, Abdulqader M Mohsen, and Abdullatif Ghallab. An improved hybrid firefly algorithm for capacitated vehicle routing problem.Applied Soft Computing, 84:105728, 2019

    work page 2019

  2. [2]

    Heuristics for unequal weight delivery problems with a fixed error guarantee.Operations Research Letters, 6(4):149–158, 1987

    Kemal Altinkemer and Bezalel Gavish. Heuristics for unequal weight delivery problems with a fixed error guarantee.Operations Research Letters, 6(4):149–158, 1987

    work page 1987

  3. [3]

    An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts.Mathematical Programming, 115:351–385, 2008

    Roberto Baldacci, Nicos Christofides, and Aristide Mingozzi. An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts.Mathematical Programming, 115:351–385, 2008

    work page 2008

  4. [4]

    New route relaxation and pricing strategies for the vehicle routing problem.Operations research, 59(5):1269–1283, 2011

    Roberto Baldacci, Aristide Mingozzi, and Roberto Roberti. New route relaxation and pricing strategies for the vehicle routing problem.Operations research, 59(5):1269–1283, 2011

    work page 2011

  5. [5]

    A quasi-polynomial-time approximation scheme for vehicle routing on planar and bounded-genus graphs

    Amariah Becker, Philip N Klein, and David Saulpic. A quasi-polynomial-time approximation scheme for vehicle routing on planar and bounded-genus graphs. In25th Annual European Sym- posium on Algorithms (ESA 2017). Schloss-Dagstuhl-Leibniz Zentrum f¨ ur Informatik, 2017

    work page 2017

  6. [6]

    Polynomial-time approximation schemes for k-center, k-median, and capacitated vehicle routing in bounded highway dimension

    Amariah Becker, Philip N Klein, and David Saulpic. Polynomial-time approximation schemes for k-center, k-median, and capacitated vehicle routing in bounded highway dimension. In26th Annual European Symposium on Algorithms (ESA 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018

    work page 2018

  7. [7]

    A ptas for bounded-capacity vehicle routing in planar graphs

    Amariah Becker, Philip N Klein, and Aaron Schild. A ptas for bounded-capacity vehicle routing in planar graphs. InAlgorithms and Data Structures: 16th International Symposium, WADS 2019, Edmonton, AB, Canada, August 5–7, 2019, Proceedings 16, pages 99–111. Springer, 2019

    work page 2019

  8. [8]

    Improving the approximation ratio for capacitated vehicle routing.Mathematical Programming, 197(2):451–497, 2023

    Jannis Blauth, Vera Traub, and Jens Vygen. Improving the approximation ratio for capacitated vehicle routing.Mathematical Programming, 197(2):451–497, 2023

    work page 2023

  9. [9]

    On light spanners, low- treewidth embeddings and efficient traversing in minor-free graphs

    Vincent Cohen-Addad, Arnold Filtser, Philip N Klein, and Hung Le. On light spanners, low- treewidth embeddings and efficient traversing in minor-free graphs. In2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 589–600. IEEE, 2020

    work page 2020

  10. [10]

    A new exact algorithm for the multi-depot vehicle routing problem under capacity and route length constraints

    Claudio Contardo. A new exact algorithm for the multi-depot vehicle routing problem under capacity and route length constraints. 2012

    work page 2012

  11. [11]

    A new exact algorithm for the multi-depot vehicle routing problem under capacity and route length constraints.Discrete Optimization, 12:129– 146, 2014

    Claudio Contardo and Rafael Martinelli. A new exact algorithm for the multi-depot vehicle routing problem under capacity and route length constraints.Discrete Optimization, 12:129– 146, 2014

    work page 2014

  12. [12]

    A taxonomic review of metaheuristic algorithms for solving the vehicle routing problem and its variants.Computers & Industrial Engineering, 140:106242, 2020

    Raafat Elshaer and Hadeer Awad. A taxonomic review of metaheuristic algorithms for solving the vehicle routing problem and its variants.Computers & Industrial Engineering, 140:106242, 2020

    work page 2020

  13. [13]

    An efficient meta-heuristic algo- rithm for solving capacitated vehicle routing problem.International Journal of Advances in Intelligent Informatics, 4(3):212–225, 2018

    Alfian Faiz, Subiyanto Subiyanto, and Ulfah Mediaty Arief. An efficient meta-heuristic algo- rithm for solving capacitated vehicle routing problem.International Journal of Advances in Intelligent Informatics, 4(3):212–225, 2018. 23

    work page 2018

  14. [14]

    Improved approximations for capacitated vehicle routing with unsplittable client demands

    Zachary Friggstad, Ramin Mousavi, Mirmahdi Rahgoshay, and Mohammad R Salavatipour. Improved approximations for capacitated vehicle routing with unsplittable client demands. InInternational Conference on Integer Programming and Combinatorial Optimization, pages 251–261. Springer, 2022

    work page 2022

  15. [15]

    Robust branch-and-cut-and-price for the capacitated vehicle routing problem.Mathematical programming, 106:491–511, 2006

    Ricardo Fukasawa, Humberto Longo, Jens Lysgaard, Marcus Poggi de Arag˜ ao, Marcelo Reis, Eduardo Uchoa, and Renato F Werneck. Robust branch-and-cut-and-price for the capacitated vehicle routing problem.Mathematical programming, 106:491–511, 2006

    work page 2006

  16. [16]

    Bounds and heuristics for capacitated routing problems.Mathematics of operations Research, 10(4):527–542, 1985

    Mordecai Haimovich and Alexander HG Rinnooy Kan. Bounds and heuristics for capacitated routing problems.Mathematics of operations Research, 10(4):527–542, 1985

    work page 1985

  17. [17]

    An extension of the lin-kernighan-helsgaun tsp solver for constrained traveling salesman and vehicle routing problems.Roskilde: Roskilde University, 12:966–980, 2017

    Keld Helsgaun. An extension of the lin-kernighan-helsgaun tsp solver for constrained traveling salesman and vehicle routing problems.Roskilde: Roskilde University, 12:966–980, 2017

    work page 2017

  18. [18]

    Ali Asghar Rahmani Hosseinabadi, Najmeh Sadat Hosseini Rostami, Maryam Kardgar, Seyed- saeid Mirkamali, and Ajith Abraham. A new efficient approach for solving the capacitated vehicle routing problem using the gravitational emulation local search algorithm.Applied Mathematical Modelling, 49:663–679, 2017

    work page 2017

  19. [19]

    Path-reduced costs for eliminating arcs in routing and scheduling.INFORMS Journal on Computing, 22(2):297–313, 2010

    Stefan Irnich, Guy Desaulniers, Jacques Desrosiers, and Ahmed Hadjar. Path-reduced costs for eliminating arcs in routing and scheduling.INFORMS Journal on Computing, 22(2):297–313, 2010

    work page 2010

  20. [20]

    An evolutionary algorithm for solving capacitated vehicle routing problems by using local information.Applied Soft Computing, 117:108431, 2022

    Hao Jiang, Mengxin Lu, Ye Tian, Jianfeng Qiu, and Xingyi Zhang. An evolutionary algorithm for solving capacitated vehicle routing problems by using local information.Applied Soft Computing, 117:108431, 2022

    work page 2022

  21. [21]

    Ptas for the euclidean capacitated vehicle routing problem in

    Michael Khachay and Roman Dubinin. Ptas for the euclidean capacitated vehicle routing problem in. InInternational Conference on Discrete Optimization and Operations Research, pages 193–205. Springer, 2016

    work page 2016

  22. [22]

    Thau Soon Khoo and Babrdel Bonab Mohammad. The parallelization of a two-phase dis- tributed hybrid ruin-and-recreate genetic algorithm for solving multi-objective vehicle routing problem with time windows.Expert Systems with Applications, 168:114408, 2021

    work page 2021

  23. [23]

    Attention, Learn to Solve Routing Problems!

    Wouter Kool, Herke Van Hoof, and Max Welling. Attention, learn to solve routing problems! arXiv preprint arXiv:1803.08475, 2018

    work page Pith review arXiv 2018

  24. [24]

    Neural combinatorial optimiza- tion with heavy decoder: Toward large scale generalization.Advances in Neural Information Processing Systems, 36:8845–8864, 2023

    Fu Luo, Xi Lin, Fei Liu, Qingfu Zhang, and Zhenkun Wang. Neural combinatorial optimiza- tion with heavy decoder: Toward large scale generalization.Advances in Neural Information Processing Systems, 36:8845–8864, 2023

    work page 2023

  25. [25]

    Optimised crossover genetic algorithm for capacitated vehicle routing problem.Applied Mathematical Modelling, 36(5):2110–2117, 2012

    Habibeh Nazif and Lai Soon Lee. Optimised crossover genetic algorithm for capacitated vehicle routing problem.Applied Mathematical Modelling, 36(5):2110–2117, 2012

    work page 2012

  26. [26]

    Improved branch-cut-and-price for capacitated vehicle routing.Mathematical Programming Computation, 9:61–100, 2017

    Diego Pecin, Artur Pessoa, Marcus Poggi, and Eduardo Uchoa. Improved branch-cut-and-price for capacitated vehicle routing.Mathematical Programming Computation, 9:61–100, 2017

    work page 2017

  27. [27]

    Robust branch-cut-and-price algorithms for vehicle routing problems.The Vehicle Routing Problem/springer, 2008

    A Pessoa. Robust branch-cut-and-price algorithms for vehicle routing problems.The Vehicle Routing Problem/springer, 2008. 24

    work page 2008

  28. [28]

    In-out separation and column generation stabilization by dual price smoothing

    Artur Pessoa, Ruslan Sadykov, Eduardo Uchoa, and Francois Vanderbeck. In-out separation and column generation stabilization by dual price smoothing. InExperimental Algorithms: 12th International Symposium, SEA 2013, Rome, Italy, June 5-7, 2013. Proceedings 12, pages 354–365. Springer, 2013

    work page 2013

  29. [29]

    A simple and effective evolutionary algorithm for the vehicle routing problem

    Christian Prins. A simple and effective evolutionary algorithm for the vehicle routing problem. Computers & operations research, 31(12):1985–2002, 2004

    work page 1985

  30. [30]

    Symmetry helps: Bounded bi-directional dynamic pro- gramming for the elementary shortest path problem with resource constraints.Discrete Opti- mization, 3(3):255–273, 2006

    Giovanni Righini and Matteo Salani. Symmetry helps: Bounded bi-directional dynamic pro- gramming for the elementary shortest path problem with resource constraints.Discrete Opti- mization, 3(3):255–273, 2006

    work page 2006

  31. [31]

    New dynamic programming algorithms for the re- source constrained elementary shortest path problem.Networks: An International Journal, 51(3):155–170, 2008

    Giovanni Righini and Matteo Salani. New dynamic programming algorithms for the re- source constrained elementary shortest path problem.Networks: An International Journal, 51(3):155–170, 2008

    work page 2008

  32. [32]

    Branching decisions in branch-and-cut-and-price algorithms for vehicle routing problems.Presentation in Column Generation, 2012, 2012

    Stefan Røpke. Branching decisions in branch-and-cut-and-price algorithms for vehicle routing problems.Presentation in Column Generation, 2012, 2012

    work page 2012

  33. [33]

    A novel algorithm for capacitated vehicle routing problem for smart cities.Symmetry, 13(10):1923, 2021

    Mohammad Sajid, Jagendra Singh, Raza Abbas Haidri, Mukesh Prasad, Vijayakumar Varadarajan, Ketan Kotecha, and Deepak Garg. A novel algorithm for capacitated vehicle routing problem for smart cities.Symmetry, 13(10):1923, 2021

    work page 1923

  34. [34]

    Two meta-heuristics for solving the capacitated vehicle routing problem: the case of the tunisian post office.Operational Research, pages 1–43, 2022

    Ines Sbai, Saoussen Krichen, and Olfa Limam. Two meta-heuristics for solving the capacitated vehicle routing problem: the case of the tunisian post office.Operational Research, pages 1–43, 2022

    work page 2022

  35. [35]

    Differential evolu- tion algorithm with local search for capacitated vehicle routing problem.International Journal of Bio-Inspired Computation, 7(5):321–342, 2015

    Boon Ean Teoh, Sivalinga Govinda Ponnambalam, and Ganesan Kanagaraj. Differential evolu- tion algorithm with local search for capacitated vehicle routing problem.International Journal of Bio-Inspired Computation, 7(5):321–342, 2015

    work page 2015

  36. [36]

    Hybrid genetic search for the cvrp: Open-source implementation and swap* neighborhood.Computers & Operations Research, 140:105643, 2022

    Thibaut Vidal. Hybrid genetic search for the cvrp: Open-source implementation and swap* neighborhood.Computers & Operations Research, 140:105643, 2022. 25

    work page 2022