pith. sign in

arxiv: 2606.00718 · v1 · pith:QQR3NP77new · submitted 2026-05-30 · 💻 cs.AI · math.OC

LLM-Driven Co-Evolutionary Automated Heuristic Design for Bi-Component Coupled Combinatorial Optimization

Mingen Kuang , Xudong Deng , Xi Lin , Ye Fan , Jianyong Sun , Jialong Shi This is my paper

Pith reviewed 2026-06-28 18:31 UTC · model grok-4.3

classification 💻 cs.AI math.OC
keywords automated heuristic designco-evolutionary frameworklarge language modelscombinatorial optimizationTraveling Thief ProblemTraveling Purchaser Problemdual-population evolution
0
0 comments X

The pith

CoEvo-AHD co-evolves two LLM populations to design cooperative heuristics for problems with coupled decisions.

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

The paper proposes CoEvo-AHD, a framework where large language models co-evolve separate populations of route operators and selection operators for bi-component problems. It introduces a cooperative evaluation that scores pairs of operators to capture how they interact in coupled subspaces. A tool library allows generated heuristics to call standard functions instead of writing custom code. Experiments on the Traveling Thief Problem and Traveling Purchaser Problem show the method finds effective combinations that compete with traditional hand-designed heuristics. This matters because many optimization problems involve intertwined decisions that are hard to handle with isolated heuristic design.

Core claim

CoEvo-AHD is an LLM-driven dual-population co-evolutionary framework for automated heuristic design in coupled combinatorial optimization. It co-evolves two closely related operator populations, using a cooperative evaluation mechanism with pairwise scoring and synergistic joint crossover to discover complementary operator logic for joint improvement across coupled decision subspaces, enabled by a tool-invocation environment library for standardized interfaces.

What carries the argument

The dual-population co-evolution with cooperative evaluation mechanism that explicitly captures interactions between route and selection operators through pairwise scoring.

If this is right

  • CoEvo-AHD automatically discovers cooperative heuristic combinations for TTP and TPP.
  • It achieves competitive solution quality against traditional heuristics.
  • The tool-invocation environment library enables LLM-generated operators to use standardized interfaces.
  • Pairwise scoring and synergistic joint crossover help identify complementary operator logic.

Where Pith is reading between the lines

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

  • The method could extend to other problems with multiple coupled decision components beyond TTP and TPP.
  • Standardized tool libraries might improve reliability of LLM-generated code in optimization tasks more generally.
  • Co-evolutionary approaches may reduce the need for problem-specific manual tuning in heuristic design.

Load-bearing premise

The cooperative evaluation mechanism and pairwise scoring can accurately capture and exploit interactions between route and selection operators in coupled decision subspaces.

What would settle it

Running CoEvo-AHD on TTP or TPP instances and finding that the evolved heuristics do not match or exceed the solution quality of established traditional heuristics would challenge the claim of effective automatic discovery.

Figures

Figures reproduced from arXiv: 2606.00718 by Jialong Shi, Jianyong Sun, Mingen Kuang, Xi Lin, Xudong Deng, Ye Fan.

Figure 1
Figure 1. Figure 1: Overview of the CoEvo-AHD framework. The framework consists of four stages: ini [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Evolutionary trajectory and representative operator milestones of CoEvo-AHD on TTP. [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolutionary trajectory and representative operator milestones of CoEvo-AHD on TPP. [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
read the original abstract

While Large Language Models (LLMs) have recently shown promise in Automated Heuristic Design (AHD), existing methods typically generate and evolve heuristics as a single operator or search strategy, limiting their ability to model strong coupling among multiple decision substructures in problems such as the Traveling Thief Problem (TTP) and the Traveling Purchaser Problem (TPP). In this work, we propose CoEvo-AHD, an LLM-driven dual-population co-evolutionary framework for automated heuristic design in coupled combinatorial optimization. Unlike prior methods that evolve individual heuristics in isolation, CoEvo-AHD leverages LLMs to co-evolve two closely related operator populations. A cooperative evaluation mechanism explicitly captures interactions between route and selection operators, while pairwise scoring and synergistic joint crossover help discover complementary operator logic for joint improvement across coupled decision subspaces. We further design a tool-invocation environment library that encapsulates frequently used core operations, such as local-search delta computation, into callable functions, enabling LLM-generated operators to use standardized interfaces instead of reimplementing inefficient and error-prone problem-specific loops. Experiments on TTP and TPP show that CoEvo-AHD automatically discovers cooperative heuristic combinations and achieves competitive solution quality against traditional 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 / 1 minor

Summary. The manuscript proposes CoEvo-AHD, an LLM-driven dual-population co-evolutionary framework for automated heuristic design in bi-component coupled combinatorial optimization problems such as the Traveling Thief Problem (TTP) and Traveling Purchaser Problem (TPP). It co-evolves route and selection operator populations using a cooperative evaluation mechanism with pairwise scoring and synergistic joint crossover to capture interactions across coupled decision subspaces, supported by a tool-invocation environment library that provides standardized callable functions for core operations. The central claim is that this framework automatically discovers cooperative heuristic combinations and achieves competitive solution quality against traditional heuristics.

Significance. If the results hold with proper validation, the work would advance automated heuristic design by extending LLM-based methods from isolated operators to explicitly coupled multi-component problems, potentially improving solution quality where decision substructures interact strongly. The tool-invocation library addresses a practical barrier in LLM-generated code by reducing reimplementation of error-prone loops.

major comments (2)
  1. [Abstract] Abstract: The claim that 'Experiments on TTP and TPP show that CoEvo-AHD automatically discovers cooperative heuristic combinations and achieves competitive solution quality against traditional heuristics' is presented without any reported experimental details, instance sets, baselines, quantitative metrics, statistical tests, or run counts. This absence prevents assessment of the central empirical claim.
  2. [Framework description] Framework description: The core claim that the dual-population mechanism, cooperative evaluation, pairwise scoring, and joint crossover exploit operator interactions rests on an untested assumption. No ablation is described that replaces pairwise cooperative scoring with independent fitness evaluation for each population (while holding LLM generation and the tool library fixed) to isolate whether the coupling mechanism, rather than general evolutionary search or the LLM environment, drives any observed gains.
minor comments (1)
  1. [Tool-invocation environment library] The description of the tool-invocation environment library would benefit from an explicit list of the encapsulated core operations and their interfaces to support reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight opportunities to strengthen the presentation of empirical claims and the validation of the co-evolutionary mechanism. We address each major comment below and commit to revisions that directly respond to the concerns raised.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that 'Experiments on TTP and TPP show that CoEvo-AHD automatically discovers cooperative heuristic combinations and achieves competitive solution quality against traditional heuristics' is presented without any reported experimental details, instance sets, baselines, quantitative metrics, statistical tests, or run counts. This absence prevents assessment of the central empirical claim.

    Authors: We agree that the abstract, as currently written, is too concise and omits the specific experimental details needed to substantiate the central claim. In the revised manuscript we will expand the abstract to explicitly reference the instance sets (standard TTP and TPP benchmark suites from the literature), the traditional heuristic baselines, the primary quantitative metrics (solution quality and runtime), the number of independent runs, and the statistical tests employed. This change will allow readers to evaluate the empirical support without needing to consult the full experimental section. revision: yes

  2. Referee: [Framework description] Framework description: The core claim that the dual-population mechanism, cooperative evaluation, pairwise scoring, and joint crossover exploit operator interactions rests on an untested assumption. No ablation is described that replaces pairwise cooperative scoring with independent fitness evaluation for each population (while holding LLM generation and the tool library fixed) to isolate whether the coupling mechanism, rather than general evolutionary search or the LLM environment, drives any observed gains.

    Authors: We acknowledge that the current manuscript does not contain an ablation isolating the contribution of the cooperative evaluation components. While the overall performance results are reported, an explicit comparison that replaces pairwise cooperative scoring and joint crossover with independent fitness evaluation (keeping LLM generation and the tool-invocation library unchanged) would provide clearer evidence that the coupling mechanism is responsible for the observed gains. We will design, execute, and report this ablation study in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity; empirical framework with external benchmarks

full rationale

The paper proposes an LLM-driven co-evolutionary framework (CoEvo-AHD) for heuristic design on coupled problems like TTP and TPP. Its central claims rest on experimental results showing competitive solution quality against traditional heuristics, with the method described directly via dual populations, cooperative evaluation, pairwise scoring, and a tool library. No equations, fitted parameters, or predictions are presented that reduce by construction to inputs. No self-citation chains or uniqueness theorems are invoked as load-bearing. The work is self-contained against external problem benchmarks and does not rely on renaming known results or smuggling ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim depends on the unverified assumption that LLMs can reliably generate functional operators and that the co-evolutionary mechanisms capture real interactions, with no independent evidence provided in the abstract.

axioms (2)
  • domain assumption LLMs can generate functional and effective heuristic operators for combinatorial optimization problems
    The framework relies on LLMs to produce usable code for operators via the tool library.
  • domain assumption The tool-invocation environment library provides sufficient standardized interfaces without requiring problem-specific reimplementation
    Described as enabling LLM-generated operators to avoid inefficient loops.
invented entities (1)
  • CoEvo-AHD dual-population co-evolutionary framework no independent evidence
    purpose: To co-evolve two operator populations with cooperative evaluation for coupled problems
    New method introduced in the paper with no external validation mentioned.

pith-pipeline@v0.9.1-grok · 5758 in / 1276 out tokens · 26864 ms · 2026-06-28T18:31:26.811084+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

58 extracted references · 16 canonical work pages

  1. [1]

    URL https: //doi.org/10.1038/s41586-023-06924-6

    Bernardino Romera-Paredes, Mohammadamin Barekatain, Alexander Novikov, Matej Balog, M. Pawan Kumar, Emilien Dupont, Francisco J. R. Ruiz, Jordan S. Ellenberg, Pengming Wang, Omar Fawzi, et al. Mathematical discoveries from program search with large language models. Nature, 625(7995):468–475, 2024. doi: 10.1038/s41586-023-06924-6

    work page doi:10.1038/s41586-023-06924-6 2024

  2. [2]

    Le, Denny Zhou, and Xinyun Chen

    Chengrun Yang, Xuezhi Wang, Yifeng Lu, Hanxiao Liu, Quoc V . Le, Denny Zhou, and Xinyun Chen. Large Language Models as Optimizers. InThe Twelfth International Conference on Learning Representations, 2024

    2024

  3. [3]

    Evolution of heuristics: Towards efficient automatic algorithm design using large language model

    Fei Liu, Xialiang Tong, Mingxuan Yuan, Xi Lin, Fu Luo, Zhenkun Wang, Zhichao Lu, and Qingfu Zhang. Evolution of heuristics: Towards efficient automatic algorithm design using large language model. InProceedings of the 41st International Conference on Machine Learning, volume 235 ofProceedings of Machine Learning Research, pages 32201–32223. PMLR, 2024

    2024

  4. [4]

    ReEvo: Large language models as hyper-heuristics with reflective evolution

    Haoran Ye, Jiarui Wang, Zhiguang Cao, Federico Berto, Chuanbo Hua, Haeyeon Kim, Jinkyoo Park, and Guojie Song. ReEvo: Large language models as hyper-heuristics with reflective evolution. InAdvances in Neural Information Processing Systems, vol- ume 37, 2024. URL https://papers.nips.cc/paper_files/paper/2024/hash/ 4ced59d480e07d290b6f29fc8798f195-Abstract-...

    2024

  5. [5]

    The travelling thief problem: The first step in the transition from theoretical problems to realistic problems

    Mohammad Reza Bonyadi, Zbigniew Michalewicz, and Luigi Barone. The travelling thief problem: The first step in the transition from theoretical problems to realistic problems. In 2013 IEEE Congress on Evolutionary Computation, pages 1037–1044. IEEE, 2013. doi: 10.1109/CEC.2013.6557681

    work page doi:10.1109/cec.2013.6557681 2013

  6. [6]

    A comprehensive benchmark set and heuristics for the traveling thief problem

    Sergey Polyakovskiy, Mohammad Reza Bonyadi, Markus Wagner, Zbigniew Michalewicz, and Frank Neumann. A comprehensive benchmark set and heuristics for the traveling thief problem. InGECCO 2014: Proceedings of the 2014 Genetic and Evolutionary Computation Conference, pages 477–484. Association for Computing Machinery, 2014. ISBN 9781450326629. doi: 10.1145/2...

    work page doi:10.1145/2576768.2598249 2014

  7. [7]

    Dong, Y.-S

    Daniele Manerba, Renata Mansini, and Jorge Riera-Ledesma. The traveling purchaser problem and its variants.European Journal of Operational Research, 259(1):1–18, 2017. doi: 10.1016/j. ejor.2016.12.017

    work page doi:10.1016/j 2017

  8. [8]

    EoH-S: Evolution of heuristic set using LLMs for automated heuristic design

    Fei Liu, Yilu Liu, Qingfu Zhang, Xialiang Tong, and Mingxuan Yuan. EoH-S: Evolution of heuristic set using LLMs for automated heuristic design. InProceedings of the AAAI Conference on Artificial Intelligence, volume 40, pages 37090–37098, 2026. doi: 10.1609/aaai.v40i43. 41038

    work page doi:10.1609/aaai.v40i43 2026

  9. [9]

    G-LNS: Generative large neighborhood search for LLM-based automatic heuristic design.arXiv preprint arXiv:2602.08253, 2026

    Baoyun Zhao, He Wang, and Liang Zeng. G-LNS: Generative large neighborhood search for LLM-based automatic heuristic design.arXiv preprint arXiv:2602.08253, 2026

    work page arXiv 2026

  10. [10]

    Evolving inter- dependent operators with large language models for multi-objective combinatorial optimization

    Junhao Qiu, Xin Chen, Liang Ge, Liyong Lin, Zhichao Lu, and Qingfu Zhang. Evolving inter- dependent operators with large language models for multi-objective combinatorial optimization. arXiv preprint arXiv:2601.17899, 2026. 10

    work page arXiv 2026

  11. [11]

    Anomalygpt: Detecting industrial anomalies using large vision-language models

    Nguyen Viet Tuan Kiet, Tung Dao, Cong Dao Tran, and Huynh Thi Thanh Binh. Motif: Multi-strategy optimization via turn-based interactive framework. InProceedings of the AAAI Conference on Artificial Intelligence, volume 40, pages 37000–37008, 2026. doi: 10.1609/aaai. v40i43.41028

    work page doi:10.1609/aaai 2026

  12. [12]

    HSEvo: Elevating automatic heuristic design with diversity-driven harmony search and genetic algorithm using LLMs

    Pham Vu Tuan Dat, Long Doan, and Huynh Thi Thanh Binh. HSEvo: Elevating automatic heuristic design with diversity-driven harmony search and genetic algorithm using LLMs. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 39, pages 26931–26938,

  13. [13]

    doi: 10.1609/aaai.v39i25.34898

    work page doi:10.1609/aaai.v39i25.34898

  14. [14]

    Monte carlo tree search for comprehensive exploration in LLM-based automatic heuristic design

    Zhi Zheng, Zhuoliang Xie, Zhenkun Wang, and Bryan Hooi. Monte carlo tree search for comprehensive exploration in LLM-based automatic heuristic design. InProceedings of the 42nd International Conference on Machine Learning, volume 267 ofProceedings of Machine Learning Research, pages 78338–78373. PMLR, 2025. URL https://proceedings.mlr. press/v267/zheng25o.html

    2025

  15. [15]

    Multi-objective evolution of heuristic using large language model

    Shunyu Yao, Fei Liu, Xi Lin, Zhichao Lu, Zhenkun Wang, and Qingfu Zhang. Multi-objective evolution of heuristic using large language model. InProceedings of the AAAI Conference on Artificial Intelligence, volume 39, pages 27144–27152, 2025. doi: 10.1609/aaai.v39i25.34922

    work page doi:10.1609/aaai.v39i25.34922 2025

  16. [16]

    Approximate approaches to the traveling thief problem

    Hayden Faulkner, Sergey Polyakovskiy, Tom Schultz, and Markus Wagner. Approximate approaches to the traveling thief problem. InProceedings of the Genetic and Evolutionary Computation Conference, pages 385–392, 2015

    2015

  17. [17]

    Improving efficiency of heuristics for the large scale traveling thief problem

    Yi Mei, Xiaodong Li, Flora Salim, and Xin Yao. Improving efficiency of heuristics for the large scale traveling thief problem. InProceedings of the Asia-Pacific Conference on Simulated Evolution and Learning, pages 631–643. Springer, 2014

    2014

  18. [18]

    Population-based vs

    Mohamed El Yafrani and Belaïd Ahiod. Population-based vs. single-solution heuristics for the travelling thief problem. InProceedings of the Genetic and Evolutionary Computation Conference 2016, GECCO ’16, pages 317–324, New York, NY , USA, 2016. Association for Computing Machinery. ISBN 9781450342063. doi: 10.1145/2908812.2908847

    work page doi:10.1145/2908812.2908847 2016

  19. [19]

    Stealing items more efficiently with ants: A swarm intelligence approach to the travelling thief problem

    Markus Wagner. Stealing items more efficiently with ants: A swarm intelligence approach to the travelling thief problem. InProceedings of the International Conference on Swarm Intelligence, pages 273–281. Springer, 2016

    2016

  20. [20]

    Exact approaches for the travelling thief problem.Algorithmica, 82:2090–2111, 2020

    Junhua Wu, Sergey Polyakovskiy, Markus Wagner, and Frank Neumann. Exact approaches for the travelling thief problem.Algorithmica, 82:2090–2111, 2020

    2090

  21. [21]

    Golden, Lawrence Levy, and Robert Dahl

    Bruce L. Golden, Lawrence Levy, and Robert Dahl. Two generalizations of the traveling salesman problem.Omega, 9(4):439–441, 1981

    1981

  22. [22]

    H. L. Ong. Approximate algorithms for the traveling purchaser problem.Operations Research Letters, 1(5):201–205, 1982

    1982

  23. [23]

    W. L. Pearn and R. C. Chien. Improved solutions for the traveling purchaser problem.Computers & Operations Research, 25(11):879–885, 1998

    1998

  24. [24]

    Boctor, Gilbert Laporte, and Jacques Renaud

    Fayez F. Boctor, Gilbert Laporte, and Jacques Renaud. Heuristics for the traveling pur- chaser problem.Computers & Operations Research, 30(4):491–504, 2003. doi: 10.1016/ S0305-0548(02)00020-5

    2003

  25. [25]

    Dynamic tabu search strategies for the traveling purchaser problem.Annals of Operations Research, 63:253–275, 1996

    Stefan V oß. Dynamic tabu search strategies for the traveling purchaser problem.Annals of Operations Research, 63:253–275, 1996

    1996

  26. [26]

    Ant colony optimization for the traveling purchaser problem.Computers & Operations Research, 35(2):628–637, 2008

    Boris Bontoux and Dominique Feillet. Ant colony optimization for the traveling purchaser problem.Computers & Operations Research, 35(2):628–637, 2008. doi: 10.1016/j.cor.2006.03. 023

    work page doi:10.1016/j.cor.2006.03 2008

  27. [27]

    Goldbarg, Ligia B

    Marco C. Goldbarg, Ligia B. Bagi, and Elizabeth F. G. Goldbarg. Transgenetic algorithm for the traveling purchaser problem.European Journal of Operational Research, 199(1):36–45,

  28. [28]

    doi: 10.1016/j.ejor.2008.10.027. 11

    work page doi:10.1016/j.ejor.2008.10.027 2008

  29. [29]

    A branch-and-cut algorithm for the undirected traveling purchaser problem.Operations Research, 51(6):940–951, 2003

    Gilbert Laporte, Jorge Riera-Ledesma, and Juan-José Salazar-González. A branch-and-cut algorithm for the undirected traveling purchaser problem.Operations Research, 51(6):940–951, 2003

    2003

  30. [30]

    Haofeng Yuan, Rongping Zhu, Wanlu Yang, Shiji Song, Keyou You, Wei Fan, and C. L. Philip Chen. Deep reinforcement learning for traveling purchaser problems.IEEE Transactions on Emerging Topics in Computational Intelligence, 10(1):425–439, 2026. doi: 10.1109/TETCI. 2025.3581113

    work page doi:10.1109/tetci 2026

  31. [31]

    An iterated local search algorithm for the traveling purchaser problem.European Journal of Operational Research, 324(3):759–772, 2025

    Tomás Kapancioglu and Raquel Bernardino. An iterated local search algorithm for the traveling purchaser problem.European Journal of Operational Research, 324(3):759–772, 2025. doi: 10.1016/j.ejor.2025.02.024

    work page doi:10.1016/j.ejor.2025.02.024 2025

  32. [32]

    The traveling purchaser problem for perishable foods.Computers & Industrial Engineering, 195:110424, 2024

    Ilker Kucukoglu, Pieter Vansteenwegen, and Dirk Cattrysse. The traveling purchaser problem for perishable foods.Computers & Industrial Engineering, 195:110424, 2024

    2024

  33. [33]

    Majid Namazi, M. A. Hakim Newton, Conrad Sanderson, and Abdul Sattar. Solving travelling thief problems using coordination based methods.Journal of Heuristics, 29(4–6):487–544,

  34. [34]

    doi: 10.1007/s10732-023-09518-7

    work page doi:10.1007/s10732-023-09518-7

  35. [35]

    Jorge Riera-Ledesma. TPPLIB. https://jriera.webs.ull.es/TPP.htm, 2012. Bench- mark instances for the Traveling Purchaser Problem, accessed 2026-05-07

    2012

  36. [36]

    Efficiently solving the traveling thief problem using hill climbing and simulated annealing.Information Sciences, 432:231–244, 2018

    Mohamed El Yafrani and Belaïd Ahiod. Efficiently solving the traveling thief problem using hill climbing and simulated annealing.Information Sciences, 432:231–244, 2018. doi: 10.1016/ j.ins.2017.12.011. A Additional Related Work and Positioning This appendix provides additional discussion on how CoEvo-AHD differs from existing LLM-driven automated heurist...

    2018

  37. [38]

    The logic should be different from the provided reference operators to improve population diversity

  38. [39]

    Implement it in Python using the required function signature:{function_signature}

  39. [42]

    If helper functions are needed, define them inside the main function

  40. [43]

    Use the exposed environment tools when available

  41. [45]

    Prompt for Mutation You are an algorithm optimizer

    Do not provide additional explanations outside the required output. Prompt for Mutation You are an algorithm optimizer. We have a{component_type} Operatorfor{problem_type}. Problem Description:{task_description} Strategy:{advice} Current Code:{operator_code} Optional Execution Profile:{profiling_report} Task:Refine and improve this operator code according...

  42. [46]

    Keep the required function signature unchanged:{function_signature}

  43. [52]

    Prompt for Homogeneous Crossover You are an expert in heuristic optimization

    Do not provide additional explanations outside the required output. Prompt for Homogeneous Crossover You are an expert in heuristic optimization. Create a new{component_type} Operatorfor{prob- lem_type}by combining ideas from two parent operators of the same component type. Problem Description:{task_description} Parent Operator 1:{parent1_code} Parent Ope...

  44. [53]

    The description must be inside braces{...}

    First, describe the new algorithm and its main steps in one sentence. The description must be inside braces{...}

  45. [54]

    Implement the new operator using the required function signature:{function_signature}

  46. [55]

    The function must return a feasible output for the corresponding component

  47. [57]

    Define helper functions inside the main function if needed

  48. [58]

    Use exposed environment tools when available

  49. [60]

    Prompt for Cross-Component Joint Crossover You are an expert in heuristic optimization

    Do not provide additional explanations outside the required output. Prompt for Cross-Component Joint Crossover You are an expert in heuristic optimization. We use cross-component joint crossover to co-evolve two heterogeneous operators for{problem_type}. Component A Description:{task_description_A} Component B Description:{task_description_B} Best-perform...

  50. [61]

    The description must be inside braces{...}

    First, describe the joint design idea and its main steps in one sentence. The description must be inside braces{...}

  51. [62]

    Return one code block containing both generated operators

  52. [63]

    Use the required function signatures: {function_signature_A} and{function_signature_B}

  53. [64]

    Each operator must return a feasible output for its corresponding component

  54. [65]

    Only use standard Python libraries and NumPy

  55. [66]

    Define helper functions inside each main function if needed

  56. [67]

    Use shared information and exposed environment tools when useful

  57. [68]

    Wrap code in“‘python ... “‘

  58. [69]

    D.2 Validation Pipeline for LLM-Generated Operators CoEvo-AHD does not directly trust raw LLM-generated code

    Do not provide additional explanations outside the required output. D.2 Validation Pipeline for LLM-Generated Operators CoEvo-AHD does not directly trust raw LLM-generated code. The implementation validates gener- ated code before insertion and again during per-instance evaluation. Code extraction and registration.Generated text must contain a Python code...

    2000