Source-linked AI summary
Directing to Diversify: Multi-Directional Search for Automatic Heuristic Design
Jianghan Zhu, Jianan Zhou, Fei Liu, Ni Zhang, Yaoxin Wu, Zhuoyi Lin, Zhiguang Cao
TL;DR
Existing LLM-based heuristic-design approaches often rely on evolutionary operators or candidate selection, without explicitly directing transformations during generation. PRS applies expansion, compression, and steering to heuristic ideas, then uses breadth-first search with adaptive operator control. Across diverse combinatorial optimization tasks, it consistently finds high-quality heuristics and outperforms existing LLM-based AHD methods in various settings.
Problem
Method
PRS perturbs heuristic ideas through expansion, compression, and steering, rewrites them into executable candidates, and combines breadth-first search with adaptive operator selection.
Source: §1 · p. 2, §3.3 · p. 6Results
Takeaways & Limitations
The experiments emphasize the importance of exploring directionally diverse regions of the heuristic search space.
Source: §6 · p. 9PRS's exploration schedule and restart behavior are manually specified, which may limit search-control flexibility across tasks and search stages.
Source: J LIMITATIONS · p. 46
Abstract
from arXiv · showhide
Large language models (LLMs) have demonstrated strong potential to automate heuristic design for NP-hard combinatorial optimization (CO) tasks without human intervention. Recent methods predominantly rely on evolutionary computation (EC) frameworks, where heuristic diversification is mainly induced through EC-like operators or downstream selection, while the direction of heuristic transformation remains largely unharnessed. In this paper, we propose a multi-directional Perturbation-Rewrite Search (PRS) framework for automatic heuristic design that factorizes heuristic transformation through directional search primitives at the generation level. Specifically, PRS operates on heuristic ideas through three complementary directions: expansion toward richer and more expressive designs, compression toward simpler and less specialized designs, and steering for fine-grained refinement, together enabling broader and more sustained exploration of diverse regions in the heuristic design space. PRS further organizes candidate exploration within a breadth-first search-based process with adaptive operator control, which ensures thorough candidate utilization and sustained exploration, and maintains strong search dynamics while preventing premature convergence. Extensive empirical results across diverse CO tasks show that PRS consistently discovers high-quality heuristics, demonstrating the effectiveness and generality of explicitly directional exploration design.
1 INTRODUCTION
Existing AHD approaches often diversify through candidate management or generic operators, without explicitly directing how heuristic ideas change during generation. PRS makes directional transformations explicit and reports high-quality heuristics across 11 combinatorial optimization settings.
1 INTRODUCTION
- Existing approaches mainly diversify through candidate management, selection, similarity measures, or search-space partitioning rather than generation primitives for directional change.Consequently, diversification typically emerges from generic mutation, crossover, reflection, or novelty pressure.Source: §1 · pp. 1–2
- PRS perturbs heuristic ideas through expansion, compression, and steering toward richer, simpler, and locally refined designs, then rewrites them as executable code.Its breadth-first search and adaptive exploration control support candidate utilization and sustained exploration.Source: §1 · p. 2
- Across 11 combinatorial optimization settings, PRS consistently discovers high-quality heuristics and performs strongly against state-of-the-art LLM-based AHD methods.The conclusion also reports that PRS outperforms existing LLM-based AHD methods in various settings.Source: §1 · p. 2
2 PRELIMINARY
The preliminaries define automatic heuristic design as finding a heuristic with strong expected performance, and describe LLM-based design and evolutionary search as approaches to navigating the heuristic space. They motivate explicit directional generation as a contrast to existing search operators.
2 PRELIMINARY
- Automatic heuristic design seeks a heuristic that maximizes expected performance over a validation set of problem instances.Each heuristic maps problem instances to feasible solutions.Source: §2 · p. 2
- LLM-based AHD uses language models to navigate the heuristic space, including by designing constructive choices or updating weighted distance matrices within Guided Local Search.The passage gives TSP as an example of constructive heuristic design.Source: §2 · p. 2
- Evolutionary AHD methods use crossover and mutation to explore heuristics, but do not explicitly harness distinct directions of heuristic change as search primitives.PRS contrasts this with direct structuring of generation through complementary transformation directions.Source: §2 · pp. 2–3
- Breadth-first search expands a frontier layer by layer, scores resulting states, and retains a subset for the next layer until a stopping criterion is reached.The retained frontier may be limited to a fixed breadth.Source: §2 · p. 3
3 METHODOLOGY
PRS perturbs heuristic ideas in complementary directions, rewrites them as executable candidates, and explores candidates through bounded breadth-first search with adaptive control. Its mechanisms address structural change, candidate utilization, exploration–exploitation balance, and search stagnation.
3 METHODOLOGY
- PRS combines directional perturbation and rewriting with a heuristic pool, breadth-first expansion, selective pruning, and adaptive exploration control.After each completed BFS tree, random pool sampling promotes diversity while preserving the current best heuristic.Source: §3 · p. 3
3.1 PERTURBATION-REWRITE OPERATOR
- Expansion increases expressiveness, compression reduces complexity and over-specialization, and steering locally refines promising structures while approximately preserving their overall form.Perturbed ideas are rewritten by an LLM into executable candidates; the same idea can yield multiple implementations according to beam size κ.Source: §3.1 · p. 4
3.2 BFS-BASED HEURISTIC SEARCH
- The bounded BFS process expands frontier parents, evaluates offspring, and selectively retains candidates to continue at the next depth.Search stops at the maximum tree depth or when the evaluation budget is exhausted; surviving candidates are merged into the pool, preserving its best heuristic.Source: §3.2 · p. 5, Figure 2 · p. 5
3.3 ADAPTIVE EXPLORATION CONTROLLER
- A UCB-based controller selects among perturbation operators according to recent search utility, while a stagnation-aware restart responds when the global best stops improving.The restart retains the global best, resets the operator reward window, and restores the exploration coefficient to its initial value before it gradually returns to reward-guided selection.Source: §3.3 · p. 6
4 EXPERIMENTS
PRS is evaluated across combinatorial optimization problems using ACO, constructive, and GLS frameworks, with comparisons against handcrafted, solver, neural, and LLM-based methods. The reported results favor PRS on ACO out-of-distribution tests, KP and online BPP settings, and nearly all FSSP test sets.
4 EXPERIMENTS
- Evaluation spans ACO, step-by-step construction, and GLS across CVRP, OP, MKP, KP, OBPP, TSP, ASP, and FSSP, with handcrafted, solver, neural, and LLM-based baselines.Unless otherwise stated, LLM-based methods use GPT-5-nano, share initial seed heuristics and validation datasets, and are averaged over three runs.Source: §4 · pp. 6–7
4.1 COMPARISON ON ACO FRAMEWORK
- PRS consistently outperforms compared LLM-based AHD baselines on ACO out-of-distribution CVRP and OP, and leads on in-distribution CVRP while remaining competitive on OP.The comparison uses three test sets of 64 randomly generated instances each.Source: §4.1 · p. 7, Table 1 · p. 7
4.2 COMPARISON ON KP AND OBPP
- PRS achieves the lowest or near-lowest KP optimality gaps across problem sizes and superior average performance across six online BPP test settings.Table 3 reports KP performance; Table 2 covers online BPP results, including held-out problem scales.Source: §4.2 · p. 8, Table 3 · p. 8, Table 2 · p. 8
4.3 COMPARISON ON FSSP
- PRS shows a consistent advantage over other methods in nearly all FSSP test sets and on average.The comparison reports relative makespan improvements on Taillard instances.Source: §4.3 · p. 8
5 DISCUSSIONS
Mechanism ablations indicate contributions from all three perturbation operators and adaptive search controls. Across two training scenarios, PRS has smoother, more continuous performance gains than baselines that often converge early; further analyses cover diversity, costs, significance, sensitivity, and additional settings.
Ablations on Mechanisms
- Removing any perturbation operator, adaptive UCB selection, or stagnation-aware restart weakens performance; replacing expansion and compression with EC-style operators also degrades it.The reported restart ablation uses C = 105; results are means and standard deviations over five independent runs.Source: §5 · p. 8, Table 4 · p. 9
Evolution Curves of Different AHD methods
- Across two scenarios, PRS gains performance more smoothly and continuously, while most baselines improve early then converge; the plotted curves average five runs with standard deviation shaded.The paper attributes PRS's search dynamics to multi-directional perturbation and a stagnation-aware controller.Source: §5 · p. 9, Figure 3 · p. 9
Additional Experimental Results
- Additional analyses examine idea- and implementation-level diversity, significance, LLM cost, hyperparameter sensitivity, ID/OOD performance, operator contributions, TSP and TSPLib, black-box settings, and different LLMs.Source: §5 · p. 9
6 CONCLUSION
PRS makes heuristic transformation direction an explicit generation-level search primitive, combining complementary transformations with breadth-first search and adaptive operator control. The paper positions this approach against prior AHD methods and evaluates it across diverse combinatorial optimization tasks.
6 CONCLUSION
- PRS explicitly directs heuristic generation through expansion, compression, and steering, and reports high-quality heuristics across diverse tasks that outperform existing LLM-based AHD methods in various settings.Its conclusion describes these directions as exploring complementary regions of the heuristic design space and links them with breadth-first search and adaptive operator control.Source: §6 · p. 9
A.1 AUTOMATED HEURISTIC DESIGN
- Traditional AHD methods, including genetic programming, depend on manually designed components and expert knowledge, limiting scalability and portability.GP mutation and crossover effectiveness is also sensitive to representation choices and domain knowledge.Source: §A.1 · p. 15
A.2 LLM-BASED AHD
- LLM-based AHD methods commonly use population-based evolutionary loops, while recent work also manages diversity through population structure, similarity, novelty, or search-space partitioning.PRS instead selects a transformation direction at generation time.Source: §A.2 · p. 15
A.3 LLM FOR CO
- LLMs for combinatorial optimization also appear as iterative optimizers or end-to-end solvers, distinct from designing reusable or instance-specific heuristics.Optimizer approaches refine candidate solutions through repeated prompting and evaluation; end-to-end solvers map problem descriptions to complete solutions.Source: §A.3 · pp. 15–16
A.4 ADAPTIVE CONTROL IN EC AND LLM-BASED AHD.
- Prior adaptive control adjusts operator rates or probabilities, while PRS combines utility-based operator adjustment with stagnation-triggered elite restart and temporary reheating.The related-work passage notes that low-level numerical credit assignment does not reason about dynamics such as stagnation or loss of diversity.Source: §A.4 · p. 16
B.1 DEFINITION OF TASKS
- The evaluation spans routing, packing, scheduling, and set-construction tasks, including TSP, CVRP, OP, MKP, BPP, KP, ASP, and FSSP.BPP is tested only in its online setting.Source: §B.1 · p. 16
- The task set also includes constructive selection formulations for MKP and ASP, alongside routing objectives for TSP, CVRP, and OP.MKP selects items sequentially under multiple resource constraints; ASP constructs a large admissible set by prioritizing candidate vectors.Source: §B.1 · pp. 16–18
- In the task formulations, online BPP assigns each arriving item immediately, while FSSP schedules jobs across machines to minimize makespan.For FSSP, the described evaluation uses 50-job instances with 2–20 machines and average makespan as heuristic fitness.Source: §B.1 · p. 17
B.2 EVALUATED FRAMEWORK FOR TASKS
The evaluation applies PRS across eight NP-hard combinatorial optimization problems using constructive, GLS, and ACO frameworks. It specifies the heuristic component designed in each framework and gives the PRS search procedure used to generate candidates.
B.2 EVALUATED FRAMEWORK FOR TASKS
- PRS is evaluated on eight problems across step-by-step construction, ACO, and GLS, with minimization tasks measured by relative optimality gap and maximization tasks by comparison with a reference value.The task frameworks cover TSP, KP, ASP, Online-BPP, MKP, CVRP, OP, and FSSP.Source: §B.2 · p. 18
B.2.1 STEP-BY-STEP CONSTRUCTIVE FRAMEWORK
- In step-by-step construction, heuristics repeatedly score feasible candidates and select the highest-priority option; the Greedy Construct baseline uses nearest-neighbor for TSP and value-to-weight ratio for KP.The designed component is a scoring function for selecting the next node, item, bin, or ASP vector.Source: §B.2.1 · p. 18
B.2.2 GLS-BASED PROBLEMS
- For GLS, PRS-designed heuristics guide edge penalties in TSP and control which operations to penalize and which schedule parts to perturb in FSSP.The GLS evaluation uses Swap and Relocate operators, while the TSP setup also uses 2-opt and relocate within KGLS.Source: §B.2.2 · pp. 18–19
B.2.3 ACO-BASED PROBLEMS
- In ACO, the designed heuristic information defines edge desirability used with pheromone values to select feasible next nodes; only this heuristic component changes across compared methods.The framework is used for CVRP, OP, and MKP, while pheromone updates and other ACO hyperparameters remain identical.Source: §B.2.3 · p. 19
C COMPLETE ALGORITHM
- Algorithm 1 initializes and evaluates a heuristic pool, then runs bounded BFS with perturbation and rewrite operators, adaptive UCB operator selection, and candidate selection.After a stagnation threshold, it resets the pool to the elite heuristic and reheats operator exploration; the algorithm returns the best valid heuristic found.Source: C COMPLETE ALGORITHM · p. 20
D DETAILED METHODOLOGY
This section details how PRS guides directional changes in heuristic ideas and realizes them as candidate heuristics, with examples for constructive bin packing and ant-colony orienteering. It also describes scoring candidate edges using edge, route, depot, and structural signals, alongside a separate constraint-sensitive black-box heuristic.
D.1 PRS PROMPT
- Separate system prompts direct the LLM to describe heuristic decision logic and translate an idea into code, while initialization first generates a distinct idea and then implements it.The idea prompt requests concrete decision principles and behavior-affecting parameters; the code prompt translates the supplied idea into an executable heuristic.Source: §D.1 · pp. 22–23
- Expansion enriches heuristic ideas, Compression removes or consolidates mechanisms, and Steering makes localized refinements; a separate rewrite realizes the target idea in code.The rewrite treats the target idea as the specification and preserves parent mechanisms only when consistent with it.Source: §D.1 · pp. 24–25
D.2 GENERATED HEURISTICS
- Generated examples score OBPP bin choices using fill and exhaustion signals, and OP transitions using prize, distance, neighborhood density, and connectivity signals.The OP example describes favoring profitable, nearby, well-connected transitions while suppressing moves that are difficult to complete feasibly.Source: §D.2 · pp. 27–28
- Route depth reflects distance to and from the depot, infeasible edges are those unable to return within maxlen, and depot affinity determines a depot bonus.Depot distances are normalized by maxlen plus epsilon, and the bonus scales with depot_weight.Source: §D.2 · p. 28
- The edge-scoring matrix combines edge prize and inverse distance with structural, route-depth, depot-bonus, and ring-penalty factors, then zeros infeasible edges.The implementation also clears the diagonal, replaces non-finite values, and clips scores to nonnegative values.Source: §D.2 · p. 28
- The OP ACO black-box heuristic scores positive-cost edges by endpoint attributes divided by edge cost, halving the score when combined endpoint cost exceeds node_constraint.Nonpositive-cost edges receive -1 before the score matrix is clipped at -1.Source: §D.2 · p. 29
- The black-box heuristic standardizes its score matrix by subtracting its mean and dividing by its standard deviation plus 1e-10.This is the final normalization step before returning the matrix.Source: §D.2 · p. 29
E DETAILS OF EVALUATION CONFIGURATIONS
The evaluation configurations set common comparison budgets and task-specific validation and testing protocols. The diversity analysis measures how heuristic archives spread during search and finds that PRS sustains particularly high diversity in later stages.
E DETAILS OF EVALUATION CONFIGURATIONS
- Most experiments use GPT-5-nano-2025-08-08; CVRP heuristic design under ACO averages 3.09M input tokens, 1.55M output tokens, 1353 queries, and 6h per run.Some settings additionally use GPT-4.1-nano, and the framework is also evaluated with closed- and open-source LLMs.Source: E DETAILS OF EVALUATION CONFIGURATIONS · p. 29
- T = 2000 for online BPP and T = 1000 for other tasks provide a shared maximum number of evaluated heuristics across methods.Baseline methods otherwise use their original reported hyperparameters where possible.Source: E DETAILS OF EVALUATION CONFIGURATIONS · p. 30
F.1.1 DIVERSITY ANALYSIS
- CDI is computed from Euclidean edge lengths in a minimum spanning tree over cumulative archives of top-20 candidate embeddings added every 80 evaluations.The analysis samples early, middle, and final search stages; higher CDI indicates more dispersed heuristic structures.Source: §F.1.1 · p. 31
- Across OBPP and OP, PRS reaches the largest final-stage CDI and has the highest or near-highest CDI at middle and final stages.All four frameworks’ cumulative archives become more diverse as evaluation steps grow; PRS’s growth is especially pronounced from 0.8T to T.Source: §F.1.1 · pp. 31–32, Figure 4 · p. 31
F.1.2 IDEA-SPACE DIVERSITY ANALYSIS
The idea-space analysis measures semantic distance between retained heuristic ideas, separately from diversity among realized heuristic implementations. PRS generally maintains less locally redundant idea archives than EoH and MCTS-AHD across both tasks.
F.1.2 IDEA-SPACE DIVERSITY ANALYSIS
- MNND measures the semantic distance from each retained heuristic idea to its nearest neighbor, with higher values indicating less local redundancy.Ideas are embedded with sentence-transformers/all-mpnet-base-v2, and exact duplicate ideas are counted once.Source: §F.1.2 · p. 32
- PRS generally explores less locally redundant idea-space regions than EoH and MCTS-AHD across OBPP-Constructive and OP-ACO.On OBPP-Constructive, PRS finishes at approximately 0.117 MNND, versus 0.116 for MCTS-AHD and 0.109 for EoH; on OP-ACO, PRS has the highest final mean.Source: §F.1.2 · p. 32, Figure 5 · p. 33
- Together, CDI and MNND indicate that PRS diversity appears both among realized implementations and among semantic ideas before code realization.The reported trends associate the broader exploration with PRS’s expansion, compression, and steering primitives.Source: §F.1.2 · p. 32
F.1.3 OPERATOR CONTRIBUTION ANALYSIS
Across tasks, PRS operators contribute differently by problem and search stage, and the reported evaluations show competitive performance across multiple settings, including larger and black-box tasks. The results also indicate that no single perturbation direction or search strategy is uniformly best.
F.1.3 OPERATOR CONTRIBUTION ANALYSIS
- Operator contributions vary by task: compression accounts for approximately 46% of FSSP-GLS improvements and 47% in CVRP-ACO, while expansion contributes approximately 42% in KP-Constructive.In KP-Constructive, steering also contributes around 37%; stage-wise utility differences across tasks and search stages support adaptive operator allocation.Source: §F.1.3 · pp. 33–34
F.1.4 OPERATOR-WISE GENERALIZATION
- For the strongest operator-generated candidates, expansion performs best on ID instances and compression on larger OOD instances in both KP-Constructive and CVRP-ACO.Steering remains competitive across both regimes, but the results do not establish a general ID or OOD advantage for any direction.Source: §F.1.4 · pp. 34–35
F.2 COMPARISON ON BLACK-BOX SETTINGS
- In black-box CVRP, MKP, and OP settings, PRS remains highly competitive or consistently exceeds other AHD frameworks.The evaluations require executable programs under strict interfaces and rely on outcome-driven trial and error.Source: §F.2 · p. 35, Table 8 · p. 35
F.3 TSP STEP-BY-STEP CONSTRUCTIVE FRAMEWORK
- Across TSP sizes and both LLM settings, PRS outperforms compared LLM-based AHD methods, with the advantage persisting as problem scale increases.With GPT-5-nano, it also achieves lower gaps across all four test sizes and generalizes to substantially larger instances.Source: §F.3 · pp. 35–36, Table 9 · p. 36
F.4 TSP HEURISTIC UNDER GUIDED LOCAL SEARCH FRAMEWORK
- PRS improves KGLS across TSP scales, reaching zero gaps on TSP50 and TSP100 and a 1.649% gap on TSP500.It has the lowest gap on TSP500 and the second-lowest on TSP200, while maintaining competitive or lowest runtime.Source: §F.4 · p. 36, Table 10 · p. 37
F.5 FLOW SHOP SCHEDULING PROBLEM UNDER GUIDED LOCAL SEARCH FRAMEWORK
- PRS achieves lower Taillard-instance gaps than EoH in most test sets, including 0.27% vs 0.62% on j100_m20, and outperforms it on average.It also exceeds non-AHD baselines including NEH-based heuristics and iterated local search methods.Source: §F.5 · p. 38, Table 11 · p. 38
F.6 RESULTS ON TSPLIB INSTANCES
- On TSPLib, PRS has a lower overall optimality gap than the reported manual heuristics and GHPP, and leads all methods on 8 of 15 instances.These results support competitive generalization to this out-of-distribution benchmark.Source: §F.6 · p. 39, Table 12 · p. 39
F.7 STATISTICAL SIGNIFICANCE; F.8 SEARCH STRATEGY VARIANTS
- PRS gains over MCTS-AHD are statistically significant in CVRP-ACO, KP-Constructive, and TSP-GLS, while PRS-BFS performs best overall on KP and CVRP.PRS-MCTS and PRS-CoT-SC remain competitive, so effectiveness does not rely solely on BFS.Source: §F.7 · p. 39, Table 13 · p. 39, §F.8 · p. 40
G COST ANALYSIS
PRS uses fewer LLM queries than MCTS-AHD across the evaluated tasks, but token usage and runtime comparisons depend on the baseline and setting. Heuristic evaluation, rather than LLM inference, primarily governs total runtime.
LLM queries and token consumption
- PRS uses approximately 32–33% fewer LLM queries than MCTS-AHD across the evaluated settings, but does not uniformly use fewer queries than EoH.For example, it uses 2706 versus 4012 queries on OBPP; its query cost can exceed EoH's.Source: G COST ANALYSIS · p. 42
- PRS uses more tokens than lightweight baselines in several settings, but fewer than MCTS-AHD on CVRP and OP.Its token use reflects the idea-generation and idea-guided rewrite stages.Source: G COST ANALYSIS · p. 42
Runtime Cost and efficiency
- PRS has runtime comparable to lightweight evolutionary methods and is faster than search-heavy MCTS-AHD; heuristic evaluation primarily drives total runtime.Per-heuristic evaluation time remains controlled, and runtime scales with heuristic complexity rather than branching search structure.Source: G COST ANALYSIS · p. 42
H SENSITIVITY ANALYSIS
The sensitivity analysis finds that PRS performs robustly under moderate changes to several search-control settings, while its examples illustrate how expansion and compression rewrite heuristic structure.
H SENSITIVITY ANALYSIS
- Increasing beam size and depth from 3 to 4 yields comparable performance with lower token cost, while reducing them to 2 moderately degrades solution quality and increases token cost.The authors retain κ = d = 3 as a balance between exploration and cost.Source: H SENSITIVITY ANALYSIS · p. 43, Table 18 · p. 43
- Changing ∆/T from 0.1 to 0.05 or 0.2 caused only minor performance and computational-cost changes; increasing |H| from 50 to 100 caused minor performance and token-cost changes on KP and OBPP.Across the two tasks, both Tsel = 0.5 and Tsel = 2.0 remain competitive with the default Tsel = 1.0, and token usage remains relatively stable across these settings.Source: H SENSITIVITY ANALYSIS · p. 43
I EXAMPLE RESPONSE FOR MULTI-DIRECTIONAL PERTURBATIONS
- In the OP example, expansion retains the parent heuristic’s backbone while adding reward-aware structures, improving validation score from g = 13.560 to g = 14.194 (+4.68%).In the OBPP example, compression retains fit, disparity, and smoothness mechanisms while simplifying other logic, improving the score from g = 2.170 to g = 1.846; lower is preferred for OBPP.Source: Figure 8 · p. 44, I EXAMPLE RESPONSE FOR… · p. 44, Figure 9 · p. 45
J LIMITATIONS
PRS has limitations in exploration control and evaluation cost, and the paper identifies LLM-driven control and surrogate models as future directions. The baselines and licenses appendix describes evaluation references and implemented baselines.
J LIMITATIONS
- PRS’s controller uses predefined scheduling rules and stagnation thresholds; manual scheduling and restarts may limit flexibility across tasks and search stages.UCB-based operator selection adapts to recent search outcomes, but does not remove those manually specified controls.Source: J LIMITATIONS · p. 46
- PRS realizes each perturbed heuristic idea as executable code before evaluation, which can make costly evaluations especially burdensome on large instances.Not every proposed idea is necessarily worth implementation and downstream evaluation.Source: J LIMITATIONS · p. 46
- Future work proposes LLM-driven operator selection, scheduling, and stagnation response, plus surrogate models to estimate idea utility before implementation.The proposed surrogates could prioritize promising ideas and filter less useful ones, reducing unnecessary implementation and evaluation costs.Source: J LIMITATIONS · p. 46
K BASELINES & LICENSES
- The appendix uses Lin–Kernighan references for TSP, OR-Tools for KP optimal values, and implements POMO and DeepACO as representative baselines.The TSP solver returns proven optimal solutions up to N = 315; larger-instance solutions are described as high-quality approximations. Table 19 summarizes external codebases and datasets.Source: K BASELINES & LICENSES · p. 46, Table 19 · p. 46