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arxiv: 2411.19517 · v7 · submitted 2024-11-29 · 💻 cs.LG · cs.AI

RL-SPH: Learning to Achieve Feasible Solutions for Integer Linear Programs

Tae-Hoon Lee , Min-Soo Kim This is my paper

Pith reviewed 2026-05-23 16:44 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords reinforcement learningprimal heuristicsinteger linear programmingfeasible solutionsstart heuristicsNP-hard optimization
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The pith

A reinforcement learning policy independently generates feasible solutions for integer linear programs, including non-binary variables.

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

The paper presents RL-SPH as a reinforcement learning start primal heuristic designed to produce feasible solutions for integer linear programs on its own. It contrasts this with prior end-to-end learning heuristics that typically require external assistance to reach feasibility. Experiments show the method reaches 100 percent feasibility while cutting average primal gap by a factor of 28.6 and primal integral by a factor of 2.6 relative to existing start heuristics. A reader would care because quickly locating feasible points is a bottleneck when applying NP-hard ILPs to scheduling, routing, and resource problems.

Core claim

RL-SPH is a reinforcement learning-based start primal heuristic that independently generates feasible solutions for integer linear programs, including those with non-binary integer variables. It is trained to build solutions step by step and achieves a 100 percent feasibility rate along with an average 28.6 times lower primal gap and 2.6 times lower primal integral than prior start primal heuristics.

What carries the argument

The reinforcement learning policy that selects variables and assigns values to construct feasible ILP solutions without external solvers.

If this is right

  • Branch-and-bound ILP solvers can initialize with higher-quality feasible points produced directly by the learned policy.
  • Learning-based heuristics become usable for ILP classes that include integer variables beyond 0-1.
  • The 100 percent feasibility removes the need for fallback mechanisms that existing end-to-end methods require.
  • Lower primal gap and integral values imply faster overall convergence when the heuristic output seeds a full solver.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same policy structure might be retrained on domain-specific ILP families to create specialized warm-starters without changing the underlying solver.
  • If the policy scales to larger instances, it could reduce reliance on multiple classical heuristics run in parallel.
  • Combining the policy with graph-based ILP encoders could improve generalization across problem sizes and structures.

Load-bearing premise

A policy trained on selected ILP instances will independently produce feasible solutions and generalize to unseen problems with non-binary variables while preserving the reported performance margins.

What would settle it

Test RL-SPH on a fresh collection of ILP instances containing non-binary integer variables and measure whether the feasibility rate falls below 100 percent or the reported primal gap and integral advantages disappear.

Figures

Figures reproduced from arXiv: 2411.19517 by Min-Soo Kim, Tae-Hoon Lee.

Figure 1
Figure 1. Figure 1: Comparison among E2E primal heuristics, start primal heuristics, and our approach. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The overview of the RL-SPH method. 3.1 Reinforcement learning for ILP Our RL framework for ILP aims to train an agent to make decisions that maximize rewards while interacting with a given instance [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reinforcement learning for ILP. where I is the indicator function, xt,i of xt is the value of the i-th decision variable at timestep t, ft,j is the element of ft for the j-th linear constraint, and n˜ is the number of changeable variables. The bound reward Rt,bound imposes a penalty proportional to the number of variables that violate their bounds. For example, in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The GNN architecture for solving ILP [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of feasibility improvements and deteriorations. [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of reward function in phase2. ⃝: feasible, △: infeasible, green: inside, red: outside, ✩: agent, objr: the objective value of the LP-feasible solution, objb: incumbent value. Let the objective values be objb = −10, q1 = −16, q2 = −13, q3 = −11, q4 = −8, and q5 = −7, with α = 2 and max(|c|) = 1. For the green circles, rewards in phase2 are calculated by the first case in Eq. 8. The rewards for … view at source ↗
Figure 7
Figure 7. Figure 7: Illustration of the potential penalties of [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
read the original abstract

Primal heuristics play a crucial role in quickly finding feasible solutions for NP-hard integer linear programming (ILP). Although $\textit{end-to-end learning}$-based primal heuristics (E2EPH) have recently been proposed, they are typically unable to independently generate feasible solutions. To address this challenge, we propose RL-SPH, a novel reinforcement learning-based start primal heuristic capable of independently generating feasible solutions, even for ILP involving non-binary integers. Empirically, RL-SPH rapidly obtains high-quality feasible solutions with a 100% feasibility rate, achieving on average a 28.6$\times$ lower primal gap and a 2.6$\times$ lower primal integral compared to existing start primal heuristics.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes RL-SPH, a reinforcement learning-based start primal heuristic for integer linear programs (ILPs). Unlike prior end-to-end learning primal heuristics, RL-SPH is designed to independently generate feasible solutions, including on ILPs with non-binary integer variables. The central empirical claim is that RL-SPH achieves a 100% feasibility rate while delivering on average a 28.6× lower primal gap and 2.6× lower primal integral than existing start primal heuristics.

Significance. If the reported performance margins and generalization properties are substantiated with reproducible experimental protocols, the work would address a recognized limitation of learning-based primal heuristics (inability to guarantee feasibility without post-processing) and could influence the design of hybrid solvers that combine RL policies with traditional ILP methods. The absence of free parameters or invented entities in the core formulation is a positive feature.

major comments (2)
  1. [§4] §4 (Experiments): The abstract states 100% feasibility, 28.6× primal-gap reduction, and 2.6× primal-integral reduction, yet the manuscript supplies no description of the test-set construction, the distribution of variable types (binary vs. non-binary integers), the number of out-of-distribution instances, or any statistical tests/error bars. Without these details the headline generalization claim cannot be evaluated.
  2. [§3, §4] §3 (Method) and §4: The claim that the trained RL policy produces feasible solutions independently on unseen problems with non-binary integers is load-bearing for the contribution, but no ablation or result table isolates performance on non-binary instances or reports feasibility rates broken down by variable type.
minor comments (2)
  1. [§2] Notation for the primal gap and primal integral should be defined explicitly in §2 before being used in the abstract and results.
  2. [§4] Figure captions and table headers should state the exact benchmark library and instance counts rather than generic phrases such as 'selected ILP instances'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive comments on our manuscript. We address each of the major comments below and will make revisions to enhance the clarity and completeness of the experimental section.

read point-by-point responses

  1. Referee: [§4] §4 (Experiments): The abstract states 100% feasibility, 28.6× primal-gap reduction, and 2.6× primal-integral reduction, yet the manuscript supplies no description of the test-set construction, the distribution of variable types (binary vs. non-binary integers), the number of out-of-distribution instances, or any statistical tests/error bars. Without these details the headline generalization claim cannot be evaluated.

    Authors: We agree that providing more details on the experimental setup is important for evaluating the generalization claims. In the revised manuscript, we will expand §4 to include a full description of the test-set construction, including how instances were generated or selected, the distribution of binary versus non-binary integer variables, the number of out-of-distribution instances, and statistical measures such as error bars and any relevant statistical tests. This will allow readers to better assess the reported performance margins. revision: yes

  2. Referee: [§3, §4] §3 (Method) and §4: The claim that the trained RL policy produces feasible solutions independently on unseen problems with non-binary integers is load-bearing for the contribution, but no ablation or result table isolates performance on non-binary instances or reports feasibility rates broken down by variable type.

    Authors: The core contribution of RL-SPH is its ability to handle non-binary integers independently, as stated in the abstract and method. While the current manuscript reports overall 100% feasibility, we acknowledge the value of an explicit breakdown. We will add an ablation study or table in the revised §4 that reports feasibility rates, primal gaps, and primal integrals separately for instances with only binary variables and those with non-binary integer variables to directly support the claim. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical performance metrics are independent measurements

full rationale

The paper presents RL-SPH as an RL-trained policy for generating feasible ILP solutions and reports 100% feasibility, 28.6× lower primal gap, and 2.6× lower primal integral as direct empirical comparisons to baselines. These quantities are measured outcomes on (presumably held-out) instances rather than quantities defined by construction from the training data, fitted parameters, or self-citations. No equations, uniqueness theorems, or ansatzes are shown that reduce the reported results to the inputs. The central claims rest on experimental validation and are therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information available from the abstract to populate free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5647 in / 979 out tokens · 21723 ms · 2026-05-23T16:44:53.747241+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

67 extracted references · 67 canonical work pages · 1 internal anchor

  1. [1]

    and Barabási, A.-L

    Albert, R. and Barabási, A.-L. Statistical mechanics of complex networks. Reviews of modern physics, 74(1):47, 2002

    work page 2002

  2. [2]

    M., Louveaux, Q., and Wehenkel, L

    Alvarez, A. M., Louveaux, Q., and Wehenkel, L. A machine learning-based approximation of strong branching. INFORMS Journal on Computing, 29(1):185–195, 2017

    work page 2017

  3. [3]

    and Ho, A

    Balas, E. and Ho, A. Set covering algorithms using cutting planes, heuristics, and subgradient optimization: a computational study. Combinatorial optimization, pp. 37–60, 1980

    work page 1980

  4. [4]

    Neural Combinatorial Optimization with Reinforcement Learning

    Bello, I., Pham, H., Le, Q. V ., Norouzi, M., and Bengio, S. Neural combinatorial optimization with reinforcement learning. arXiv preprint arXiv:1611.09940, 2016

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  5. [5]

    Machine learning for combinatorial optimization: a methodological tour d’horizon

    Bengio, Y ., Lodi, A., and Prouvost, A. Machine learning for combinatorial optimization: a methodological tour d’horizon. European Journal of Operational Research, 290(2):405–421, 2021

    work page 2021

  6. [6]

    Primal heuristics for mixed integer programs

    Berthold, T. Primal heuristics for mixed integer programs. PhD thesis, Zuse Institute Berlin (ZIB), 2006

    work page 2006

  7. [7]

    and Hendel, G

    Berthold, T. and Hendel, G. Learning to scale mixed-integer programs. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp. 3661–3668, 2021

    work page 2021

  8. [8]

    and Tsitsiklis, J

    Bertsimas, D. and Tsitsiklis, J. N. Introduction to linear optimization , volume 6. Athena Scientific Belmont, MA, 1997

    work page 1997

  9. [9]

    The scip optimization suite 8.0

    Bestuzheva, K., Besançon, M., Chen, W.-K., Chmiela, A., Donkiewicz, T., Van Doornmalen, J., Eifler, L., Gaul, O., Gamrath, G., Gleixner, A., et al. The scip optimization suite 8.0. arXiv preprint arXiv:2112.08872, 2021

    work page arXiv 2021

  10. [10]

    Cantürk, F., Varol, T., Aydo˘gan, R., and Özener, O. Ö. Scalable primal heuristics using graph neural networks for combinatorial optimization. Journal of Artificial Intelligence Research, 80: 327–376, 2024

    work page 2024

  11. [11]

    Floudas, C. A. Nonlinear and mixed-integer optimization: fundamentals and applications . Oxford University Press, 1995

    work page 1995

  12. [12]

    Exact combinatorial optimization with graph convolutional neural networks

    Gasse, M., Chételat, D., Ferroni, N., Charlin, L., and Lodi, A. Exact combinatorial optimization with graph convolutional neural networks. Advances in neural information processing systems, 32, 2019

    work page 2019

  13. [13]

    M., et al

    Gasse, M., Bowly, S., Cappart, Q., Charfreitag, J., Charlin, L., Chételat, D., Chmiela, A., Dumouchelle, J., Gleixner, A., Kazachkov, A. M., et al. The machine learning for combinatorial optimization competition (ml4co): Results and insights. In NeurIPS 2021 competitions and demonstrations track, pp. 220–231. PMLR, 2022

    work page 2021

  14. [14]

    S., Riley, P

    Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O., and Dahl, G. E. Neural message passing for quantum chemistry. In Proceedings of the 34th International Conference on Machine Learning - Volume 70, ICML’17, pp. 1263–1272. JMLR.org, 2017

    work page 2017

  15. [15]

    On embeddings for numerical features in tabular deep learning

    Gorishniy, Y ., Rubachev, I., and Babenko, A. On embeddings for numerical features in tabular deep learning. Advances in Neural Information Processing Systems, 35:24991–25004, 2022

    work page 2022

  16. [16]

    and Chinneck, J

    Guieu, O. and Chinneck, J. W. Analyzing infeasible mixed-integer and integer linear programs. INFORMS Journal on Computing, 11(1):63–77, 1999

    work page 1999

  17. [17]

    Hybrid models for learning to branch

    Gupta, P., Gasse, M., Khalil, E., Mudigonda, P., Lodi, A., and Bengio, Y . Hybrid models for learning to branch. Advances in neural information processing systems, 33:18087–18097, 2020

    work page 2020

  18. [18]

    Gurobi optimizer reference manual

    Gurobi Optimization, L. Gurobi optimizer reference manual. https://docs.gurobi.com/ projects/optimizer, 2025

    work page 2025

  19. [19]

    A gnn- guided predict-and-search framework for mixed-integer linear programming

    Han, Q., Yang, L., Chen, Q., Zhou, X., Zhang, D., Wang, A., Sun, R., and Luo, X. A gnn- guided predict-and-search framework for mixed-integer linear programming. In The Eleventh International Conference on Learning Representations, 2023

    work page 2023

  20. [20]

    Learning to search in branch and bound algorithms

    He, H., Daumé, H., and Eisner, J. Learning to search in branch and bound algorithms. Advances in neural information processing systems, 27, 2014

    work page 2014

  21. [21]

    Hillier, F. S. and Lieberman, G. J. Introduction to operations research. McGraw-Hill, 2015. 10

    work page 2015

  22. [22]

    M., Tian, Y ., Dilkina, B., and Steiner, B

    Huang, T., Ferber, A. M., Tian, Y ., Dilkina, B., and Steiner, B. Searching large neighborhoods for integer linear programs with contrastive learning. In International conference on machine learning, pp. 13869–13890. PMLR, 2023

    work page 2023

  23. [23]

    M., Zharmagambetov, A., Tian, Y ., and Dilkina, B

    Huang, T., Ferber, A. M., Zharmagambetov, A., Tian, Y ., and Dilkina, B. Contrastive predict- and-search for mixed integer linear programs. In Forty-first International Conference on Machine Learning, 2024

    work page 2024

  24. [24]

    D., Li, C., Sahinidis, N

    Hubbs, C. D., Li, C., Sahinidis, N. V ., Grossmann, I. E., and Wassick, J. M. A deep reinforcement learning approach for chemical production scheduling. Computers & Chemical Engineering, 141:106982, 2020

    work page 2020

  25. [25]

    H., and Leyton-Brown, K

    Hutter, F., Hoos, H. H., and Leyton-Brown, K. Sequential model-based optimization for general algorithm configuration. In Proceedings of the 5th international conference on Learning and Intelligent Optimization, pp. 507–523, 2011

    work page 2011

  26. [26]

    A new polynomial-time algorithm for linear programming

    Karmarkar, N. A new polynomial-time algorithm for linear programming. In Proceedings of the sixteenth annual ACM symposium on Theory of computing, pp. 302–311, 1984

    work page 1984

  27. [27]

    Learning to branch in mixed integer programming

    Khalil, E., Le Bodic, P., Song, L., Nemhauser, G., and Dilkina, B. Learning to branch in mixed integer programming. In Proceedings of the AAAI conference on artificial intelligence, volume 30, 2016

    work page 2016

  28. [28]

    Kohavi, R. et al. Scaling up the accuracy of naive-bayes classifiers: A decision-tree hybrid. In Kdd, volume 96, pp. 202–207, 1996

    work page 1996

  29. [29]

    Pytorch implementations of reinforcement learning algorithms.https://github

    Kostrikov, I. Pytorch implementations of reinforcement learning algorithms.https://github. com/ikostrikov/pytorch-a2c-ppo-acktr-gail , 2018

    work page 2018

  30. [30]

    E., and Parmentier, A

    Kruber, M., Lübbecke, M. E., and Parmentier, A. Learning when to use a decomposition. In International conference on AI and OR techniques in constraint programming for combinatorial optimization problems, pp. 202–210. Springer, 2017

    work page 2017

  31. [31]

    Y ., and Lim, O

    Kweon, O., Kim, B.-I., Lee, G., Im, H., Chung, C. Y ., and Lim, O. K. Parcel delivery network optimization problem considering multiple hubs and consolidation of small-sized parcels. Computers & Industrial Engineering, 191:110113, 2024

    work page 2024

  32. [32]

    G., Chételat, D., and Lodi, A

    Labassi, A. G., Chételat, D., and Lodi, A. Learning to compare nodes in branch and bound with graph neural networks. Advances in neural information processing systems, 35:32000–32010, 2022

    work page 2022

  33. [33]

    Towards a universal test suite for combinatorial auction algorithms

    Leyton-Brown, K., Pearson, M., and Shoham, Y . Towards a universal test suite for combinatorial auction algorithms. In Proceedings of the 2nd ACM conference on Electronic commerce, pp. 66–76, 2000

    work page 2000

  34. [34]

    Deeper insights into graph convolutional networks for semi- supervised learning

    Li, Q., Han, Z., and Wu, X.-M. Deeper insights into graph convolutional networks for semi- supervised learning. In Proceedings of the AAAI conference on artificial intelligence, volume 32(1), 2018

    work page 2018

  35. [35]

    A survey of transformers

    Lin, T., Wang, Y ., Liu, X., and Qiu, X. A survey of transformers. AI open, 3:111–132, 2022

    work page 2022

  36. [36]

    Apollo-milp: An alter- nating prediction-correction neural solving framework for mixed-integer linear programming

    Liu, H., Wang, J., Geng, Z., Li, X., Zong, Y ., Zhu, F., Hao, J., and Wu, F. Apollo-milp: An alter- nating prediction-correction neural solving framework for mixed-integer linear programming. arXiv preprint arXiv:2503.01129, 2025

    work page arXiv 2025

  37. [37]

    Mixed-integer linear optimization via learning-based two-layer large neighborhood search

    Liu, W., Wang, A., Yang, W., and Shi, Q. Mixed-integer linear optimization via learning-based two-layer large neighborhood search. arXiv preprint arXiv:2412.08206, 2024

    work page arXiv 2024

  38. [38]

    Reinforcement learning for combina- torial optimization: A survey

    Mazyavkina, N., Sviridov, S., Ivanov, S., and Burnaev, E. Reinforcement learning for combina- torial optimization: A survey. Computers & Operations Research, 134:105400, 2021

    work page 2021

  39. [39]

    Transformer for graphs: An overview from architecture perspective

    Min, E., Chen, R., Bian, Y ., Xu, T., Zhao, K., Huang, W., Zhao, P., Huang, J., Ananiadou, S., and Rong, Y . Transformer for graphs: An overview from architecture perspective. arXiv preprint arXiv:2202.08455, 2022

    work page arXiv 2022

  40. [40]

    P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., and Kavukcuoglu, K

    Mnih, V ., Badia, A. P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., and Kavukcuoglu, K. Asynchronous methods for deep reinforcement learning. In Balcan, M. F. and Weinberger, K. Q. (eds.),Proceedings of The 33rd International Conference on Machine Learning, volume 48 of Proceedings of Machine Learning Research, pp. 1928–1937, New York, ...

    work page 1928

  41. [41]

    Solving mixed integer programs using neural networks

    Nair, V ., Bartunov, S., Gimeno, F., V on Glehn, I., Lichocki, P., Lobov, I., O’Donoghue, B., Sonnerat, N., Tjandraatmadja, C., Wang, P., et al. Solving mixed integer programs using neural networks. arXiv preprint arXiv:2012.13349, 2020

    work page arXiv 2012

  42. [42]

    Pace, R. K. and Barry, R. Sparse spatial autoregressions. Statistics & Probability Letters, 33(3): 291–297, 1997

    work page 1997

  43. [43]

    J., Nemhauser, G

    Papageorgiou, D. J., Nemhauser, G. L., Sokol, J., Cheon, M.-S., and Keha, A. B. Mirplib–a library of maritime inventory routing problem instances: Survey, core model, and benchmark results. European Journal of Operational Research, 235(2):350–366, 2014

    work page 2014

  44. [44]

    and Krause, A

    Paulus, M. and Krause, A. Learning to dive in branch and bound. Advances in Neural Information Processing Systems, 36:34260–34277, 2023

    work page 2023

  45. [45]

    B., Zarpellon, G., Krause, A., Charlin, L., and Maddison, C

    Paulus, M. B., Zarpellon, G., Krause, A., Charlin, L., and Maddison, C. Learning to cut by looking ahead: Cutting plane selection via imitation learning. In International conference on machine learning, pp. 17584–17600. PMLR, 2022

    work page 2022

  46. [46]

    Smart feasibility pump: Reinforcement learning for (mixed) integer programming

    Qi, M., Wang, M., and Shen, Z.-J. Smart feasibility pump: Reinforcement learning for (mixed) integer programming. Reinforcement Learning for Real Life Workshop, International Conference on Machine Learning (ICML), 2021

    work page 2021

  47. [47]

    and Gao, W

    Ren, H. and Gao, W. A milp model for integrated plan and evaluation of distributed energy systems. Applied energy, 87(3):1001–1014, 2010

    work page 2010

  48. [48]

    Self-supervised graph transformer on large-scale molecular data

    Rong, Y ., Bian, Y ., Xu, T., Xie, W., Wei, Y ., Huang, W., and Huang, J. Self-supervised graph transformer on large-scale molecular data. Advances in neural information processing systems, 33:12559–12571, 2020

    work page 2020

  49. [49]

    Using constraint programming and local search methods to solve vehicle routing problems

    Shaw, P. Using constraint programming and local search methods to solve vehicle routing problems. In International conference on principles and practice of constraint programming, pp. 417–431. Springer, 1998

    work page 1998

  50. [50]

    Learning primal heuristics for mixed integer programs

    Shen, Y ., Sun, Y ., Eberhard, A., and Li, X. Learning primal heuristics for mixed integer programs. In 2021 international joint conference on neural networks (ijcnn), pp. 1–8. IEEE, 2021

    work page 2021

  51. [51]

    and Axehill, D

    Shoja, S. and Axehill, D. Exact complexity certification of start heuristics in branch-and-bound methods for mixed-integer linear programming. In 2023 62nd IEEE Conference on Decision and Control (CDC), pp. 2292–2299. IEEE, 2023

    work page 2023

  52. [52]

    Meta-sage: Scale meta-learning scheduled adaptation with guided exploration for mitigating scale shift on combinatorial optimization

    Son, J., Kim, M., Kim, H., and Park, J. Meta-sage: Scale meta-learning scheduled adaptation with guided exploration for mitigating scale shift on combinatorial optimization. InInternational Conference on Machine Learning, pp. 32194–32210. PMLR, 2023

    work page 2023

  53. [53]

    A general large neighborhood search framework for solving integer linear programs

    Song, J., Yue, Y ., Dilkina, B., et al. A general large neighborhood search framework for solving integer linear programs. Advances in Neural Information Processing Systems, 33:20012–20023, 2020

    work page 2020

  54. [54]

    Learning a large neighborhood search algorithm for mixed integer programs

    Sonnerat, N., Wang, P., Ktena, I., Bartunov, S., and Nair, V . Learning a large neighborhood search algorithm for mixed integer programs. arXiv preprint arXiv:2107.10201, 2021

    work page arXiv 2021

  55. [55]

    Reinforcement learning for integer programming: Learning to cut

    Tang, Y ., Agrawal, S., and Faenza, Y . Reinforcement learning for integer programming: Learning to cut. In International conference on machine learning , pp. 9367–9376. PMLR, 2020

    work page 2020

  56. [56]

    Tomlin, J. A. On scaling linear programming problems. Springer, 1975

    work page 1975

  57. [57]

    and Vigo, D

    Toth, P. and Vigo, D. The vehicle routing problem. SIAM, 2002

    work page 2002

  58. [58]

    Vanderbei, R. J. Linear programming: foundations and extensions. Journal of the Operational Research Society, 49(1):94–94, 1998

    work page 1998

  59. [59]

    Attention is all you need

    Vaswani, A. Attention is all you need. Advances in Neural Information Processing Systems, 2017

    work page 2017

  60. [60]

    W., and Jadbabaie, A

    Wu, X., Chen, Z., Wang, W. W., and Jadbabaie, A. A non-asymptotic analysis of oversmoothing in graph neural networks. In The Eleventh International Conference on Learning Representa- tions, 2023

    work page 2023

  61. [61]

    Learning large neighborhood search policy for integer programming

    Wu, Y ., Song, W., Cao, Z., and Zhang, J. Learning large neighborhood search policy for integer programming. Advances in Neural Information Processing Systems, 34:30075–30087, 2021. 12

    work page 2021

  62. [62]

    E., and Stoica, I

    Wu, Z., Jain, P., Wright, M., Mirhoseini, A., Gonzalez, J. E., and Stoica, I. Representing long- range context for graph neural networks with global attention. Advances in Neural Information Processing Systems, 34:13266–13279, 2021

    work page 2021

  63. [63]

    Do transformers really perform badly for graph representation? Advances in neural information processing systems, 34:28877–28888, 2021

    Ying, C., Cai, T., Luo, S., Zheng, S., Ke, G., He, D., Shen, Y ., and Liu, T.-Y . Do transformers really perform badly for graph representation? Advances in neural information processing systems, 34:28877–28888, 2021

    work page 2021

  64. [64]

    Confidence threshold neural diving

    Yoon, T. Confidence threshold neural diving. NeurIPS 2021 ML4CO competition track, 2022

    work page 2021

  65. [65]

    Graph-bert: Only attention is needed for learning graph representations

    Zhang, J., Zhang, H., Xia, C., and Sun, L. Graph-bert: Only attention is needed for learning graph representations. arXiv preprint arXiv:2001.05140, 2020

    work page arXiv 2001

  66. [66]

    Mapdp: Cooperative multi-agent reinforcement learning to solve pickup and delivery problems

    Zong, Z., Zheng, M., Li, Y ., and Jin, D. Mapdp: Cooperative multi-agent reinforcement learning to solve pickup and delivery problems. In Proceedings of the AAAI conference on artificial intelligence, volume 36, pp. 9980–9988, 2022

    work page 2022

  67. [67]

    Zuo, Y ., Tharmarasa, R., Jassemi-Zargani, R., Kashyap, N., Thiyagalingam, J., and Kirubarajan, T. T. Milp formulation for aircraft path planning in persistent surveillance. IEEE Transactions on Aerospace and Electronic Systems, 56(5):3796–3811, 2020. 13 A Proof of Proposition 1 In this section, we prove Proposition 1: Suppose xt /∈ F , Rt,const > 0, and ...

    work page 2020