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REVIEW 4 major objections 6 minor 62 references

Efficient Heuristics Generation for Solving Combinatorial Optimization Problems Using Large Language Models

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read By abstracting elite heuristics' core components as prompt prior knowledge, Hercules steers LLMs to specific search directions, and its confidence-gated fitness predictor cuts search time by up to 59%.

desk verdict The PPP performance predictor is a genuinely new and useful idea with broad experiments, but the universal SOTA claim overshoots the evidence and the 'proof' of CAP is an identity plus an assumption. read the letter →

arxiv 2505.12627 v2 pith:HSI7Z3T6 submitted 2025-05-19 cs.NE

Xuan Wu , Di Wang , Chunguo Wu , Lijie Wen , Chunyan Miao , Yubin Xiao , You Zhou This is my paper · ORCID
classification cs.NE
keywords largelanguagemodelsheuristicgenerationcombinatorialoptimizationcoreabstractionpromptingperformancepredictionevolutionarycomputationresourceefficiency
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the main weakness of LLM-based heuristic discovery for combinatorial optimization is not the LLM's coding ability but the vagueness of the search directions it is prompted to give. To fix this, the authors introduce Hercules, a heuristic-generation loop whose Core Abstraction Prompting (CAP) first asks the LLM to extract the core components of the current elite heuristics and then uses those components as prior knowledge when asking for new directions. They prove, via an information-gain argument, that such abstraction reduces the entropy (unspecificity) of the produced directions. They also add Performance Prediction Prompting (PPP) to let the LLM estimate the fitness of new heuristics from semantic similarity to evaluated ones, with confidence-gated acceptance, yielding a variant, Hercules-P, that cuts search time by 7%–59% while matching Hercules's quality. If the claims hold, LLM-based heuristics become both stronger and much cheaper to produce.

What carries the argument

The central object is the prompting pipeline wrapped around a generational loop. Core Abstraction Prompting (CAP) is a zero-shot prompt that summarizes the top-k heuristics' core components, e.g., 'normalize penalties relative to overall distance', and feeds them as prior knowledge for search directions. Performance Prediction Prompting (PPP) is a few-shot prompt that presents a set of previously evaluated heuristics with distinct fitness values and asks the LLM to predict the fitness of new code from semantic similarity, returning a confidence score. Two small mechanisms make PPP usable: EXEMPLAR selects the historical best/worst plus top parents with distinct fitness values as examples, and Confidence Stratification (ConS) decides per heuristic whether to accept the LLM's predicted fitness or pay for a real COP-instance evaluation, using the acceptance threshold schedule m_t = ⌊α·β^t·N_o⌋. Together they carry the paper's twin claims: CAP reduces unspecificity (with a proven entropy bound) and PPP reduces evaluation cost without degrading population quality.

What would settle it

Measure prediction error separately for heuristics that ConS trusts (confidence at least 1−δ) and for those it re-evaluates; if the trusted set's errors are not significantly smaller in absolute value than the untrusted set's, then the confidence signal carries no information and Hercules-P's speedup is not the reliable filtering the paper claims, so rerunning Hercules-P with shuffled confidence values should leave search time and gain essentially unchanged.

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Extended reading notes

Core claim

The paper's central claim is that feeding an LLM the abstracted core components of its best-so-far heuristics, rather than asking for reflections on performance, shifts the distribution of generated search directions toward actionable task-specific ideas, and that this shift can be quantified as an information gain. In formal terms, partitioning the space of possible directions by the k core components reduces entropy from H(Ω_t) to ∑ p_j H(Ω_j), giving IG = −∑ p_j log p_j in (0, log(k+1)]; the authors prove this identity and present it as the mechanism behind CAP's improvement. The companion claim is that an LLM can act as a fitness predictor: given a few well-chosen evaluated heuristics (best, worst, and top parents with distinct fitness values), the LLM predicts a new heuristic's performance from semantic similarity and outputs a confidence value; the ConS rule then decides which predictions to trust. Empirically, Hercules outperforms previous LLM-based heuristic generators across four tasks, five combinatorial optimization problems, and eight LLMs, while Hercules-P delivers similar gains in less compute.

Load-bearing premise

Hercules-P's time savings rest on the assumption that the LLM's self-reported confidence in its fitness prediction reliably flags predictions worth trusting; the paper reports only a 0.39 correlation between predicted and true fitness, so a miscalibrated confidence could let poor heuristics slip through while the claimed on-par performance is not guaranteed.

Editorial extensions

If this is right

  • Hercules sets a new performance bar across four heuristic-generation tasks, five combinatorial optimization problems, and eight LLMs, so the specificity gain from CAP transfers across model families and problem types.
  • Hercules-P cuts search time by 7% to 59% while matching Hercules's gain, making LLM-driven heuristic design practical in settings with limited compute, such as near-real-time resource allocation.
  • The theoretical bound IG = −∑ p_j log p_j ∈ (0, log(k+1)] implies that any partitioning of the direction space by core components reduces entropy, so the benefit of CAP is structural rather than accidental to one model.
  • The rank-based selection mechanism and the early-iteration use of elite components accelerate convergence, as shown by the ablation where replacing rank-based selection with random selection drops gain from 11.10% to 8.49%.
  • PPP's EXEMPLAR mechanism improves predictive accuracy by 26% to 37% in median accuracy relative to using duplicate-fitness or random examples, indicating that example quality matters more than example quantity for LLM fitness prediction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If CAP's entropy-reduction argument applies as generally as stated, the same abstraction trick could be applied to other LLM-driven program synthesis tasks beyond combinatorial optimization, such as circuit design or scheduling policies, which the paper does not explore.
  • The paper documents that LLMs frequently produce semantically equivalent heuristics; this suggests an explicit deduplication step based on code embedding similarity could complement PPP and further cut evaluation counts, a step the authors do not take.
  • ConS currently thresholds confidence into three discrete bands; smoothing the acceptance probability as a continuous function of the confidence value could recover more of PPP's time savings while avoiding the bias-variance cliff at band boundaries.
  • A stress test for PPP would be to run it on a search space where semantic novelty is high; in such a regime the semantic-similarity signal may weaken, and the claimed 7%–59% speedup could shrink or require more real evaluations.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes two components for LLM-based heuristic generation. Core Abstraction Prompting (CAP) extracts core components from elite heuristics and feeds them into prompts so that LLM-generated search directions are more specific; a theoretical claim is made that CAP reduces unspecificity in the sense of information gain. The second component, Performance Prediction Prompting (PPP), uses a few-shot LLM prompt to predict fitness values of newly generated heuristics, together with an EXEMPLAR selection mechanism and a Confidence Stratification (ConS) acceptance rule, yielding a resource-efficient variant Hercules-P. The authors report extensive experiments across four heuristic-generation tasks, five combinatorial optimization problems, and eight LLMs, and claim state-of-the-art gains for Hercules and 7%--59% search-time reductions for Hercules-P at on-par quality.

Significance. If the empirical claims hold, the paper makes a useful practical contribution: CAP is a simple, zero-shot mechanism that plausibly improves the specificity of LLM search directions, and PPP is, to my knowledge, a novel use of LLMs as fitness predictors in heuristic generation. The manuscript is also strong in breadth: it covers multiple LLMs, seed algorithms, and problem types, and the authors release code and prompts. The ablation studies, especially the statistical comparison of EXEMPLAR variants in Figure 6, are a good-faith attempt to isolate design choices. However, the current evidence does not support the universal SOTA claim in the abstract and introduction, and the theoretical argument in Section 3.1/Appendix A is better described as a formalization of an assumption than a proof of CAP's benefit. These are load-bearing issues for the paper's central claims.

major comments (4)
  1. [Section 3.1 and Appendix A] The proof of Proposition 1 is internally correct only after assuming the conditional-probability identity p_j * p(omega_i | Omega_j) = p(omega_i | Omega_t) for every subset j and every direction omega_i. This identity is not a consequence of probability calculus; it is an assumption that the abstraction partitions the direction space without loss. Furthermore, the claimed positivity range (0, log(k+1)] in Eq. (4) requires that no subset has probability 1, and this non-degeneracy is exactly the behavioral conclusion the theorem is supposed to establish. Please state these assumptions explicitly and either prove them from the CAP mechanism or reframe the result as a formalization of the intended effect rather than a proof of empirical benefit.
  2. [Section 1, Tables 1, 3, and 5] The headline claim that Hercules outperforms state-of-the-art LLM-based HG algorithms across four tasks, five COPs, and eight LLMs is not supported by the paper's own tables. Table 1 shows that with Llama3-70b at n=200, Hercules obtains gain 3.42 versus EoH's 5.82. Table 3 shows average gains of 4.87 for Hercules versus 4.80 for EoH, a near tie with no reported variance. In Table 5, POMO TSP n=200 gives Hercules 2.49 versus Random 3.05 and EoH 2.19, and LEHD TSP n=1000 gives Hercules 5.14 versus EoH 6.09. A systematic win/loss count across all cells and statistical significance tests are needed before the universal SOTA statement can be made.
  3. [Tables 5 and 6, Section 4.5] The on-par performance claim for Hercules-P is not demonstrated. Table 5 contains several cells where Hercules-P is substantially worse than Hercules or even negative, e.g., POMO TSP n=200 (-0.10 vs 2.49) and POMO CVRP n=200 (-0.57 vs 1.53). Table 6 reports only search time, not quality, so the reported 7%--59% savings do not by themselves establish resource efficiency at on-par performance. In addition, the reliance of ConS on the LLM's self-reported confidence phi_i is weakly supported: the paper itself reports a Pearson correlation of only 0.39 between predicted and true fitness values (Section 4.5), and no calibration analysis for the confidence values is provided. The authors should report quality and time jointly, and provide evidence that the confidence threshold in Eq. (7) is well-calibrated.
  4. [Experimental methodology, Tables 1--5] Most gain tables report only the mean over three independent runs, without standard deviations or tests, while the few reported standard deviations (Table 4) show overlapping intervals for several key comparisons, e.g., BPP n=1000 Hercules 2.19 +/- 0.20 versus ReEvo 2.01 +/- 0.34. Given that the universal SOTA claim rests on small numeric gaps, the paper should report per-cell standard deviations or confidence intervals and run paired significance tests (or a systematic win/loss summary) across all tasks and LLMs. Without this, the claim that Hercules outperforms SOTA is not verifiable from the presented data.
minor comments (6)
  1. [Abstract and Introduction] The phrase "we theoretically prove the effectiveness of CAP in reducing unspecificity" overstates what Eq. (4) establishes; consider replacing it with "formalize the intended reduction" or similar.
  2. [Section 4.1] There is a typo in "real-wolrd case study" that should be corrected to "real-world".
  3. [Section 4.4, Table 5] The text notes that Random outperforms the other methods on some NCO tasks, but the discussion does not reconcile this with the universal SOTA claim; please add an explicit discussion of these counterexamples.
  4. [Section 4.5, Figure 6] The figure label contains garbled text ("w w/o w/o1") and the figure is difficult to read; the axis labels and legend should be clarified.
  5. [Appendix G] There is a typo "repsectively" that should be "respectively".
  6. [Appendix A] The paragraph after the proof argues that the one-to-one mapping assumption is reasonable, but the argument would be more convincing if it stated that the information-gain formula changes only by replacing k with k_c; the notation should be made explicit.

Circularity Check

1 steps flagged · score 6.0 of 10

Information-gain 'proof' of CAP reduces to the non-degeneracy assumption it needs to establish; empirical SOTA claims remain independent.

  1. self definitional [Section 3.1, Eqs. (1)-(4); Appendix A, Proposition 1 and its proof]
    "As proven in Appendix A, Eq. (3) simplifies to the following expression, whose value ranges within the (0,log(k+1)] interval: IG(Ωt)=−∑_{j=0}^k p_j log p_j. Therefore, in theory, providing the core components as prior knowledge in prompts can reduce unspecificity in LLM responses and yield more specific search directions. ... However, due to the diverse nature of LLM training corpora, the LLM will not consistently provide the same direction. Therefore, by abstracting core components, the unspecificity (entropy) can decrease within the (0,log(k+1)] interval."

    IG is defined as H(Ω_t)−Σ p_j H(Ω_j), and the 'with abstraction' entropy is H(direction | subset membership), where subsets are defined post hoc by which core component an output direction is associated with. The proof reduces IG to −Σ p_j log p_j; the advertised (0, log(k+1)] range requires this entropy of p_j to be positive. The paper obtains positivity by asserting 'the LLM will not consistently provide the same direction.' Under the paper's formalization, 'CAP reduces unspecificity' is exactly IG>0, so this assertion is the conclusion assumed as a property of the output distribution. No mechanism of CAP is used to derive it; the proof is a conditional identity, not an independent derivation.

full rationale

The empirical pipeline is self-contained and benchmarked against external baselines (Random, EoH, ReEvo; KGLS, ACO, POMO, LEHD seeds), so the SOTA claims are not circular. The CAP ablation, rank-based selection, and PPP/EXEMPLAR/ConS ablations are evaluated with true fitness on test sets. The only load-bearing derivation that reduces by definition is the information-gain proof of CAP: its conclusion is built into the assumption that the post-abstraction direction distribution is non-degenerate. Self-citations in the paper (e.g., [43]-[45], [48], [49]) are background references and are not load-bearing for any central claim. The paper's own tables also contain cells where Hercules does not beat baselines and few significance tests, but that is a correctness/robustness concern rather than circularity. Overall, there is one definitional circularity in a theoretical sub-claim, while the central empirical contribution has independent content.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The main algorithm depends on several hand-chosen hyperparameters (lambda, delta, alpha, beta, k) and on assumptions about LLM behavior that are asserted rather than derived. The information-gain proof relies on a partition of the direction space and non-degeneracy of the induced distribution, both assumed. The performance predictor relies on unverified calibration of LLM confidence. No new physical or conceptual entities are introduced.

free parameters (5)
  • lambda (CAP elite ratio) = 0.7
    Tuned via ablation on the TSP GLS task (Table 7) and applied unchanged to all other tasks.
  • delta (ConS confidence interval width) = 0.1
    Selected in the ablation study (Table 7) on the TSP GLS task and used everywhere.
  • alpha (ConS acceptance decay base) = 0.5
    Set heuristically in Eq. (7) with no sensitivity analysis.
  • beta (ConS acceptance decay rate) = 0.8
    Set heuristically in Eq. (7) with no sensitivity analysis.
  • k (number of elite heuristics for CAP) = 5
    Chosen without a tuning analysis for the number of core components.
assumptions (4)
  • ad hoc to paper The space of possible search directions can be partitioned into k+1 subsets associated with the k core components plus one independent subset, with p_j times p(omega_i | Omega_j) equal to p(omega_i | Omega_t).
    Used in Proposition 1 (Appendix A) to derive the information gain formula. The one-to-one correspondence between core components and heuristics is an assumption the paper itself later relaxes.
  • domain assumption The LLM will not always output the same direction, so p_j is strictly less than 1 for every subset, which makes the information gain positive.
    Stated in Appendix A after Eq. (9) as due to the "diverse nature of LLM training corpora". It is not measured, and it is exactly what the theorem needs to conclude that CAP reduces unspecificity.
  • domain assumption Gain computed relative to seed heuristics on training and test datasets from the prior study [53] is a valid measure of heuristic quality.
    Used throughout Section 4. The paper relies on [53] for task definitions and datasets, and does not independently calibrate the metric.
  • domain assumption LLM-predicted fitness values and confidence scores correlate with true fitness well enough for ConS to filter unreliable predictions.
    Underlies Eq. (7) and the Hercules-P time-savings claims. The paper reports a Pearson correlation of 0.39, so the assumption is only partially supported.

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Pith. "Pith review of Efficient Heuristics Generation for Solving Combinatorial Optimization Problems Using Large Language Models." pith.science (2026). https://pith.science/paper/HSI7Z3T6

@misc{pith2026250512627,
  author       = {Pith},
  title        = {Pith review of: Efficient Heuristics Generation for Solving Combinatorial Optimization Problems Using Large Language Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HSI7Z3T6}},
  note         = {Machine review of arXiv:2505.12627}
}
read the original abstract

Recent studies exploited Large Language Models (LLMs) to autonomously generate heuristics for solving Combinatorial Optimization Problems (COPs), by prompting LLMs to first provide search directions and then derive heuristics accordingly. However, the absence of task-specific knowledge in prompts often leads LLMs to provide unspecific search directions, obstructing the derivation of well-performing heuristics. Moreover, evaluating the derived heuristics remains resource-intensive, especially for those semantically equivalent ones, often requiring omissible resource expenditure. To enable LLMs to provide specific search directions, we propose the Hercules algorithm, which leverages our designed Core Abstraction Prompting (CAP) method to abstract the core components from elite heuristics and incorporate them as prior knowledge in prompts. We theoretically prove the effectiveness of CAP in reducing unspecificity and provide empirical results in this work. To reduce computing resources required for evaluating the derived heuristics, we propose few-shot Performance Prediction Prompting (PPP), a first-of-its-kind method for the Heuristic Generation (HG) task. PPP leverages LLMs to predict the fitness values of newly derived heuristics by analyzing their semantic similarity to previously evaluated ones. We further develop two tailored mechanisms for PPP to enhance predictive accuracy and determine unreliable predictions, respectively. The use of PPP makes Hercules more resource-efficient and we name this variant Hercules-P. Extensive experiments across four HG tasks, five COPs, and eight LLMs demonstrate that Hercules outperforms the state-of-the-art LLM-based HG algorithms, while Hercules-P excels at minimizing required computing resources. In addition, we illustrate the effectiveness of CAP, PPP, and the other proposed mechanisms by conducting relevant ablation studies.

Figures

Figures reproduced from arXiv: 2505.12627 by the authors.

Figure 2
Figure 2. Illustration of two heuristics with identical seman [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Overview of the proposed Hercules and Hercules-P algorithms. Hercules exploits CAP to provide specific search [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Illustration of the prediction process using the pro [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: Convergence curves of different HG algorithms. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Ablation study on dif￾ferent EXEMPLAR variants. We further present the predictive accuracy of PPP with and with￾out EXEMPLAR, both of which are executed ten times, aiming to per￾form meaningful statisti￾cal tests. In addition, we include w/ EXEMPLAR￾U as an additional …

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.