HiFo-Prompt: Prompting with Hindsight and Foresight for LLM-based Automatic Heuristic Design
Authors: ChentongChen
Mengyuan Zhong
Jialong Shi
Jianyong Sun
Ye Fan
Venue: ICLR 2026 Poster (2026)
Keywords: Large Language Models
Evolutionary Computation
Guided Prompt Synthesis
Knowledge Accumulation
Automated Algorithm Design
Primary area: optimization

Abstract: This paper investigates the application of Large Language Models (LLMs) in Automated Heuristic Design (AHD), where their integration into evolutionary frameworks reveals a significant gap in global control and long-term learning. We propose the Hindsight-Foresight Prompt (HiFo-Prompt), a novel framework for LLM-based AHD designed to overcome these limitations. This is achieved through two synergistic strategies: Foresight and Hindsight. Foresight acts as a high-level meta-controller, monitoring population dynamics(e.g., stagnation and diversity collapse) to switch the global search strategy between exploration and exploitation explicitly. Hindsight builds a persistent knowledge base by distilling successful design principles from past generations, making this knowledge reusable.  This dual mechanism ensures that the LLM is not just a passive operator but an active reasoner, guided by a global plan (Foresight) while continuously improving from its cumulative experience (Hindsight). Empirical results demonstrate that HiFo-Prompt significantly outperforms a comprehensive suite of state-of-the-art AHD methods, discovering higher-quality heuristics with substantially improved convergence speed and query efficiency.  Our code is available at https://github.com/Challenger-XJTU/HiFo-Prompt.

Published as a conference paper at ICLR 2026
HIFO-PROMPT: PROMPTING WITH HINDSIGHT AND FORESIGHT FOR LLM- AUTOMATIC HEURISTIC DESIGN
Chentong Chen'r, Mengyuan Zhong', Ye Fan', Jialong Shi", Jianyong Sun't
Xn'an Jiaotong University
Northwest Polytechnical University
{ [chengt chen, my zhong]@stu. xjtuu.xjuu.ee edu. cn, fanye@nwpu. edu. (jialong.shi, jy.sun)exjtu.
ABSTRACT
This paper investigates the application of Large Language Models (LLMs) in Automated Heuristic Design (AHD), where their integration into evolutionary frameworks reveals significant gap in global control and long-term learning. We propose the Sindsight-Foresight Prompt (HiFo-Prompt), a novel framework for LLM-based AHD designed to overcome these limitations. This is achieved through two synergistic strategies: Foresight and Hindsight. Foresight acts as a high-level meta-controller, monitoring population dynamics(e.g. stagnation and diversity collapse) to switch the global search strategy between exploration and exploitation explicitly. Hindsight builds a persistent knowledge base by distill- ing successful design principles from past generations, making this knowledge reusable. This dual mechanism ensures that the LLM is not just a passive op- erator but active reasoner, guided by global plan (Foresight) while contin- uously improving from its cumulative experience (Hindsight). Empirical results demonstrate that HiFo-Prompt significantly outperforms comprehensive suite of state-of-the-art AHD methods, discovering higher-qualin heuristics with substan- tially improved convergence speed and query efficiency. Our code is available httpghhiuwbboooohahllleeegee WUTU//IiOUUUUUAAAAA Promptt.
INTRODUCTION
Combinatorial Optimization (CO) problems, which involve finding an optimal solution from a dis- crete set of possibilities, are ubiquitous in science and engineering. Because of their NP-hardness, designing effective heuristics for these problems is a complex task, traditionally based on extensive human experience and intuition (Camacho- Villalón al. 2023).
The advent of Large Language Models (LLMs) has catalyzed a paradigm shift toward Automated Heuristic Design (AHD) (Wang & Chen 2023 Liu et al. (024c). A particularly potent approach marries LLMs with Evolutionary Computation (EC), casting the LLM as a high-level semantic mu- tation operator. Foundational works such as FunSearch
Pareeres et al.) 2024] and EoH (Liu et al., 20246) established the viability of this LLM+EC paradigm, demonstrating its capacity to discover novel and effective heuristics.
However, as the field progresses, two fundamental challenges have emerged in AHD: the inability steer the heuristic generation process based on population dynamics and the failure distill and manage the core design principles of high- performance heuristics to guide the subsequent heuristic generation process.
First, many approaches lack a mechanism for global adaptive guidance. They often rely on local or reactive signals; for instance, ReEvo (Ye al. (024) performs reflection on single candi- date, while methods such as MCTS-AHD (Zheng et al. 2025) passively embed the exploration-
"Equal contribution.
I "Correspnnding author.

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Published as a conference paper at ICLR 2026
exploitation trade-off within their search structure. This localized control does not respond to the macroscopical dynamics of the population and cannot proactively intervene when the search en- counters systemic issues such premature convergence a decline in diversity. A more aggressive strategy involves in in-weight adaptation (e.g.. EvoTune (Surina et al. 2025), CALM (Huang et al, ). which uses numerical gradients to fine-tune the LLM. Although powerful, this approach incurs high computational costs from repeated fine-tuning and reduces the LLM to an opaque policy network. Consequently, learning is implicitly encoded in model weights, preventing the extraction of explicit design principles and obscuring the model's symbolic reasoning process.
Second, existing frameworks suffer from poor knowledge persistence, a phenomenon we term knowledge decay. Successful design strategies often remain entangled within specific code imple- mentations; when parent candidates are discarded, the underlying logic is lost. Recent advances, such as evolving the optimizer in MoH (Shi al.) (025)) or automating problem reduction in RedAHD (Thach et al., 2025), operate in orthogonal dimensions. They do not explicitly decouple algorithmic knowledge from executable forms. Consequently, the system fails to achieve cumula- tive learning, perpetually rediscovering similar concepts instead of building on proven algorithmic strategies.
To overcome these fundamental limitations, we propose HiFo-Prompt (Hindsight-Foresigge Prompt), a framework that establishes a hicrarchical control architecture for LLM--based AHD. HiFo-Prompt clevates the LLM from a mere code generator to symbolic meta-optimizer by endow- ing with two synergistic capabilities: First, the Foresight module addresses the control problem by serving as
Navigator) that observes population dynamics. Upon detecting states like performance plateaus, it explicitly modulates the generative process by switch- ing evolutionary regimes via transparent verbal gradients injected at the prompt level, serving symbolic alternative to opaque and expensive numerical gradients. Second, the Hindsight module addresses knowledge decay by implementing the Insight Pool, an evolving repository of abstracted design principles. It distills core algorithmic patterns from specific code implementations, trans- forming them into reusable knowledge. This mechanism allows the system build on validated design principles, effectively seeding subsequent generations with proven algorithmic strategies. In summary, the contributions of our approach are as follows:
We introduce HiFo-Prompt, a novel framework consisting of Hindsight and Foresight mod- ules. It dynamically generates prompts for LLMs by decoupling thoughts from code, thereby enabling independent updates and evaluations. This mechanism leads to sig- nificant reduction in both training time for heuristics and evaluation costs for LLMs.
To improve the Hindsight and Foresight abilities of our method, introduce the Insight Pool and Evolutionary Navigator, respectively. The Insight Pool accumulates knowledge from high-performing codes through iterative updates. The Evolutionary Navigator con- trols population states by monitoring evolution and balancing exploration-exploitaiie dy- namics.
We evaluated the heuristics designed by HiFo-Promp on complex optimization tasks, com- paring them against advanced handcrafted heuristics and existing AHD approaches. Our results achieve state-of-the-art performance in the AHD domain, with substantial improve- ments over prior AHD methods, particularly excelling in the Traveling Salesman Problem (TSP) and Flow Shop Scheduling Problem (FSSP).
RELATED WORK
LLM-driven Automatic Heuristic Design
The integration of Large Language Models (LLMs) into Evolutionary Computation (EC) is vibrant new direction for Automated Heuristic Design (AHD) (Liu et al. 2024al Chauhan et al. 2025). Pioneered by works like FunSearch (Romera
Paredes al. 2024 and BoH (Liu et al.) 2024bj. this paradigm leverages the LLM as powerful
semantic operator to generate heuristics as code. Recent efforts to advance this paradigm can be
categorized along several axes. Some works focus on refining search control through sophisticated prompt engineering and guidance mechanisms (Ye et al.) 2024 Dat et al., (002)), or by redesign- ing the population structure itself (Zheng al. 2025). A distinct, more model-centric approach directly adapts the LLM's parameters via reinforcement learning-based fine-tuning (Surina et al.
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Harritic
Evolution
Foundatiooal Prompt Strulegies
Foundational Prompt
select
Irsight
Insight Pool
Self Evvliviiingg
Composite
Systhesize
Hindsight Prompt
sdact
Prompt
with Regime
Foresight
Prompt
select
Prompt Exploit Regime
Population State Information
Prompt Balonce Regime
The framework ProFop, which comprises two core processes. (1) Prompt Con- struction: A foundational prompt dynamically augmented with design directives from the Fore- sight and insights from the Hindsight to form the final composite prompt. (2) Knowledge Evolu- tion: A self-evolving loop is established where elite heuristics from the evolutionary process distilled into new insights, continuously enriching the Hindsight module's 's knowledge base.
Figure 1:
Huang et al. 2025). At the highest level of abstraction, research has also explored evolving other core components of the optimization process, such as the optimizer (Shi et al. 2025 or the problem representation (Thach et al.) 2025).
2025
Knowledge Management in Generative Search Harnessing historical information stone of efficient search. In classical EC, methods like Cultural Algorithms (Maheri et al. 2021) formalize this via structured Belief Space that stores and evolves collective knowledge. Contem- porary LLM -bbadee approaches often rely in-context reflection mechanisms (Shinn et al. 2023 Bo al. 2024), where the model self-critiques failures inform its next attempt. However, this knowledge is typically transient, unstructured, and instance-specifee. Consequently, these methods lack mechanism for accumulating and generalizing insights over time, preventing the formation of a persistent, structured knowledge base analogous to those in classical EC.
Adaptive Control in Evolutionary Computation Dynamically adapting search strategies a long-standing goal in EC. Historically, this has been addressed through low-level, reactive mecha- nisms like Adaptive Operator Selection (AOS) (Avaro Fialho 2010 Tian et al. 2023) and parameter control (Eiben & Smith 2015 Aleti & Moser 2016). These methods rely on numerical credit as- signment, effectively voting for strategies that recently performed well, but they lack semantic understanding of the search dynamics (e.g., identifying population stagnation). The reasoning capa- bilities of LLMs offer paradigm shift towards higher-level, proactive control (Eiben et al. 1999] Papal 2021). Instead of merely adjusting parameters, LLMs can interpret population-leve statistics escape local optimum. This marks a transition from fine-grained numerical tuning to semantic-based strategic adjustment.
suggest symbolic actions, such increase mutation
3 METHODOLOGY
In this section, we introduce our proposed HiFo- PiFooPrrmpt, a novel framework that equips the evo- lutionary process driven by LLM with mechanisms for learning and adaptation (shown in Figure [].. HiFo-Prompt integrates two synergistic components: a Foresight module for real-time adaptive control, which monitors evolutionary dynamics steer the search strategy, and Hindsight module
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Foundational Prompt Strategles
Hindsight
Insight
Synthesize
conaalering impaci current cectalooo
Prompt for LLM
Ialance
First you need idantify the main componants
(MI
Conskder these successlul design principles Ive obseieed racundl:
evalustion and scoring functions
Design adapthv hybrid masa hourlat es synergisically
Employ machine leoming pottern recegn foon to
(niight 1)
(miight 2)
scduling unnccessary compusationa oveitead and redundarcy
When sinilfying, please pay speriall attention to:
Explons objectio function engi sevring by intraducing
(leighht 3)
Explet
reducting unnecossary computational overteied and radur daney
exploring sokution coretrucioo metho
(Exploit Reginse)
#induuiing tens candomiont on or adaption mecheriioss
Foresight Regime Set
6,04697
5,06035
5,95094
3.000M
5,93836
5,83558
5
Number of generations
Figure 2: Dynamic prompt generation process of HiFo-Prompt.
for long-term knowledge accumulation, which manages a self-evolving repository of successful de- sign principles that we term insights. By uniting foresight-driven strategy with hindsight-informedd knowledge, our framework transforms the generative process into robust, self-regulating system.
3.1 GUIDED PROMPT SYNTHESIS FOR AHD
Our HiFo-Prompt framework applies Guided Prompt Synthesis mechanism that constructs each prompt as a context-aware composite instruction. This mechanism integrates three interlocking modules: Foundational prompt strategies, Hindsight, and Foresight. The resulting composite prompt provides precise, multifaceted guidance to the LLM. A complete example of prompt generation for TSP with step-by-step construction is provided in Figure 2 and the specific templates for all operators (e.g.. I1, E1, MI) are detailed in Appendix
Foundational prompt strategies. Our framework's generative foundation is set of Foundational Prompt Strategies, which function as the I.MM-equvvleent of genetic operators. We first generate heuristics from scratch using the initial prompt strategy 11, then evolve them with five foundational prompts adapted from EoH (Liu et al. 20024b). These prompts are organized into two primary strategies: 1) Reorganization Strategies, which include E1, synthesizing a new algorithm with novel structure from multiple parents, and E2, abstracting shared core ideas to generate con- ceptually distinct variants; and 2) Mutation Strategies, which encompass M1, making structural modifications for functionally equivalent variants; M2, tuning critical parameters; and M3, simpli- fying components prone to overfitting. While this curated set provides the raw generative capability, its effectiveness depends contextual guidance. This is the role of the Hindsight and Foresight modules, which inject insights and a design directive into the prompts to align each action with the search's 's current needs.
Hindsight Module. This module incorporates valid heuristic experience in the form of insights, which are abstract and generalizable design principles distilled from successful heuristics. These insights managed in dynamic Insight Pool, where each is assigned and continuously updated with a credibility score based on its empirical performance. Before generation, high-scoring insights retrieved and embedded into the prompt. They serve validated priors steer the LLM toward
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promising designs. While effective for historical guidance, this module cannot address real-time evolutionary needs.
Foresight Module. The Foresight module implements real-time evolutionary state control, or- chestrated by its core component, the Evolutionary Navigator. The Navigator continuously monitors macroscopic evolutionary indicators, such as performance stagmation and population diversity, to as- sess the state of the search. Based on this analysis, it selects the governing evolutionary regime for the subsequent generation. This regime dictates the evolutionary search direction by choosing one of three explicit modes (Crepinšek et al. 2013): 1) Explore, to foster novelty when diversity is low or progress has stalled; 2) Exploit, to refnne high- perffornnnnn solutions when progress is consistent; 3) Balance, to maintain a synergistic application of all operators. This directive guides the prompt, ensuring the LLM adapts to the current search phase.
3.2 HINDSIGHT: MECHANISMS OF THE SELF-EVOLVING INSIGHT POOT.
Initially, HiFo-Prompt initializes the Insight Pool with seed insights(See Section F33 to bootstrap the generative process. Serving as an initial scaffold, these seeds allow the framework to immedi- ately shift focus to autonomously evolving novel heuristics. This is achieved through continuous lifecycle that systematically transforms transient evolutionary successes into reusable knowledge as- sets, governed by three integrated phases: insight extraction, utility-driven application, and adaptive pruning.
Initially, HiFo-Prompp initializes the Insight Pool with seed insights(See Section F.3 to bootstrap the generative process. Serving as an initial scaffold, these seeds allow the framework to immediately shift focus to autonomously evolving novel, problem-specific heuristics. This is achieved through a continuous lifecycle that systematically transforms transient evolutionary successes into explicit, reusable knowledge assets, governed by three integrated phases: insight extraction, utility-driven application, and adaptive pruning.
Insight Extraction and Admission. The lifecycle begins by expanding the knowledge base. At the end of each generation, we prompt an LLM distill generalizable design principles (insights) from the elite individuals of the population (See Section F.T.. To preserve informational diversity, a candidate insight knww is admitted to the insight pool Kpeot only if its Jaccard similarity to all existing pool members falls below a novelty threshold Bnovelty- For this comparison, each insight's 's text is preprocessed by converting it to lowercase and tokenizing based on whitespace. The insights themselves become active candidates for the guidance of future generations.
Insight Retrieval and Credit Assignment. To guide heuristic generation, employ utility- based retrieval mechanism. For each new generation attempt, the mechanism selects the top-s in- sights with the highest adaptive utility score tk., This contributing Ke, is then injected into the LLM's prompt. Following the evaluation of the offspring generated, we eredit assignment (Whitacre al. 2006) update the utility scores of all the insights in Kg- The utility function is formulated to balance exploitation and exploration:
of insights, denoted as
U Uki,t)
(t)
W.. (o.((o(()))))))))))))))))(( + 1) (t, &
Effectiveness
Usago Penalty
Recency Bonus
where (t) is the learned effectiveness of insight i. The penalty term, weighted by Wh discourages overuse by penalizing an insight based on its total retrieval count Nt(t). thus promoting exploration of less-used ideas. The recency bonus B, offers temporary reward for insights used recently; specifically, it grants fixed bonus if the insight was used within small generation window T. (that is, ift ttat < Tw), promoting strategic coherence.
The effectiveness score Et) is updated via credit assignment. This process first converts the raw fitness g(hnew) of an offssping into normalized, problem-agnostic agnostic score b. scaling its performance relative to the best (g( (hrea) )) and worst (g(hwors) solutions of the current population:
g(hwars)) - g ((eew)
p g (hwara)) ) (Aihestt +
(2)
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where is small constant prevent division by zero. Drawing upon techniques for handling sparse rewards in Hierarchical Reinforcement Learning, we posit that an offspring's evolutionary contribution is non-linear. Therefore, map a final credit signal, gen. using tiered. piecewise function. This design creates distinct reward regimes for qualitatively different outcomes:
0.8+000 0.2
if g(hnew) 2 92 (hhes))
Jeff
0.6
if g(hbest) > ((hnew)
2 g(wag)
-0.3 0.5 p if g(hnew) < glaag)
This structure provides strong positive signals for -hanning improvements (p 2 1), mod- erate rewards for incremental progress (0 S 1), and penalties for below-average performance. The allowance of negative credit is a deliberate design choice to accelerate the pruning of detrimental ideas.
This piecewise reward structure incorporates techniques for handling sparse rewards in Hierarchical Reinforcement Learning. It provides discrete and explicit signals for distinct performance levels to enhance interpretability. The coefficients follow specific asymmetric philosophy where the reward for elite solutions is significantly larger than the penalty magnitude, which turn larger than the reward for incremental improvements. This prioritization provides strong signal for paradigm- shifting breakthroughs while effectively pruning unproductive directions. Furthermore, the parame- ter sensitivity analysis detailed in Appendix C.7 demonstrates that the framework maintains robust performance across a range of coefficient settings. This confirms that the structural logic of the reward mechanism is more critical than precise tuning.
To ensure stable updates, the raw credit is then clipped to a final value Jeff max(-1.0. min(1.0, seft") )). Finally, for each insight in the contributing set Ke, its effectiveness score is updated through an Exponential Moving Average (EMA):
(t + 1) = (1 (((() + a 9elf
where the learning rate a E [0, 1J smooths the stochastic credit signals from individual evaluations. This allows an insight's long-term utility to emerge statistically, preventing its score from being skewed by outlier performances.
Adaptive Pruning and Pool Maintenance. To maintain quality within its finite capacity Cpoot. the pool employs an adaptive pruning mechanism triggered when |pooal > Cpeot This process removes the insight with the minimum eviction score, Seviets which balances the proven performance of an insight against the risk of its obsolescence:
S,vie t) = Et(t) - Rossay - (t used
(5)
where (t) is the current effectiveness of the insight, fasttused is the generation of its last use, and the time decay rate Raway targets not only low-effectiveness insights, but also those that have become inactive. The decay rate set conservatively to prevent the premature removal of valuable but temporarily dormant knowledge. Additionally, to prevent the premature elimination of under- explored principles, insights with usage counts below a threshold Tusage are exempt from pruning, allowing sufficient iterations to establish their true utility.
3.3 FORESIGHT: THE EVOLUTIONARY NAVIGATOR FOR STATE- GUIDANCE
While the Hindsight module grounds the search in historically validated principles, the Foresight module acts as an Evolutionary Navigator, providing real-time, state-aware guidance. Its primary role is to modulate the pnploraaion-exploiattti
trade-off via a control policy, : S; - 9r This pol- icy maps the current evolutionary state S, to a high-level search orientation, denoted as the Evolu- tionary Regime 0,. Unlike traditional adaptive EAs that solely tune numerical parameters, Foresight establishes a semantic feedback loop that directly steers the conceptual strategy of the LLMs. We define rule-based system that selects the appropriate strategic modes based evolutionary
state
Oxxplore Oexploit
if Ostag (t) 2 Tstag (p 8p
0,-
if Ppoo Tprog otherwise
Obalance

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To implement this policy, the Navigator continuously monitors two primary aspects of the evolu- tionary search: its performance trajectory and its population diversity. The performance trajectory is tracked via two mutually exclusive counters, Gpoy and Catgg based on the improvement in the best raw fitness Ag. If Ag > 10-A, it is generation of progress (increment Cprog. reset Csnng); otherwise, it is stagnation (increment Csug. reset Cgrog. Simultaneously. the Navigator assesses population health through novel measure of phenotypic diversity, Ap(t). This metric is crucial because relying on fitness alone can be misleading; a population of high-fitness but structurally similar individuals often indicates entrapment in a local optimum. To counteract this, our metric quantifies the semantic variety within the population by measuring the dissimilarity of the generated algorithms'' textual forms, rather than their fitness values. is computed as the normalized fraction of unique pairs of algorithms in the population P whose textual descriptions are non-identical:
IPI
IPI
(p(t) = [P((P] 1)/2
Σ I(alg, * alg,)
where I(-) is the indicator function and alg, denotes the textual description of the i-th algorithm. Unlike embedding- -based metrics, which risk obscuring critical logical alterations (e.g., dynamic vs. static) through semantic smoothing (Agarwal et al. (025)) this metric relies on exact string matching of the generated algorithmic descriptions. By enforcing a standardized prompt template to eliminate trivial syntactic noise, any remaining lexical difference indicates deliberate structural modification by the LLM. Consequently, this approach provides robust and computationally efficient proxy for phenotypic diversity, ensuring that even subtle yet functional changes are preserved.
This hybrid state representation, which marries quantitative performance trends with qualitative measure of semantic diversity, provides the Navigator with a much more holistic understanding of the search landscape than fitness-based metrics alone can. These state indicators evaluated against empirically determined thresholds, with their specific values detailed in our experimental setup. The progress and stagnation counters designed to track immediate performance trends. This focus on recent history is crucial for capturing the rapid dynamics inherent in LLM-bbsed evo- lution, enabling swift strategic adjustments in response even short periods of stagnation or consis- tent improvement. Similarly, the diversity threshold is calibrated to detect significant collapse in semantic variety. acts as an early warning mechanism against premature convergence. triggering stronger exploratory pressure when a substantial portion of the population becomes homogeneous. Once the regime 0, is determined, Section F2 that is injected into the LLM'' prompt. For instance, Desplore might yield directive to try a significantly different approach from conventional solutions, whereas fesplon would instruct the model focus on refining and optimizing the most effective patterns. This mechanism ensures the LLM's generative focus is explicitly aligned with the high-level strategy dictated by the current evolutionary state.
is translated into a natural language Design Directive(Sec (See
EXPERIMENTS
In this section, we present the results of heuristics designed by our proposed HiFo-Prompt on dif- ferent complex tasks. including Traveling Salesman Problem (TSP) (Matai et al. 2010). Online Bin Packing Problem (Online BPP) (Seiden) 2002), Flow Shop Scheduling Problem (FSSP) (Emmons) & Vairaktarakis 2012). and Bayesian Optimization (BO) (Shahriari et al.) 2016). Task definitions and details are given in Appendix B Results on TSP, online BPP, and FSSP are presented in this section, while the results on BO are provided in Appendix C.4. where HiFo-Prompt demonstrates competitive and reliable performance.
Experimental Settings. Experiments were conducted on a workstation equipped with an Intel Core 17-12700 CPU, employing Qwen2.5-Mav 5-Max as the core LLM. The evolutionary process main- tains a population size 8f running for 8 generations on Combinatorial Optimization (CO) tasks and 4 generations Bayesian Optimization (BO) tasks. Regarding specific module hyperparameters, the Hindsight Module maintains an Insight Pool of capacity Opeol = 30 (Jaccard threshold 0.7), with retrieval parameters set to selection count = 3. usage penalty w. = 0.1, recency bonus mag- nitude T, = 0.2, EMA rate a = 0.3, decay Ratecay = 0.01, and probation Taage = 3. The Foresight Module utilizes Navigator thresholds of Taag = (stagnation), Tprog (progress), and 8,. = 0.3
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Table 1: Results TSP with step-by-step construction. Gap(%) denotes the performance gap compared to advanced heuristic algorithms. Time(s) represents the running time of the designed heuristics. This result of LLM-based AHD method is the average three runs. The best-performing LLM--based AHD results are shown in bold.
TSP50
TSP100
TSP200
Method
Gap
Time(s)
Gap
Time(s)
Gap
Time(s)
LKH3
0.000%
323.3
0.000%
1450
0.000%
6312
POMO
0.163%
1.636%
13.961%
LEHD
0.117%
0.452%
0.367%
EoH
12.820%
1.4
15.361 %
16. 16.658%
78
ReEvo
10.239%
21.5
12.577%
224
14.890%
3013
HSEvo 10.467%
89.5
12.008%
1286
13.578%
24835
MCTS-AHD
10.642%
91.5
12.521% %
1084
13.510%
14521
Ours 6.625
244.7
8.582%
1843
8.877%
16099
Table 2: Results on TSP with GLS. Comparison relative to the results of advanced heuristic on TSP100, TSP200 and TSP500. The result of the LLM-based AHD method the average of three runs. The best results are shown in bold.
TSP100
TSP200
TSP500
Method
Gap
Time(s)
Gap
Time(s)
Gap
Time(s)
LKH3
0.000%
0.000%
0.000%
EoH
0.026%
210.3
0.453%
368.1
2.037%
1100.7
ReEvo
0.049%
357.1
0.424%
775.8
2.090%
1103.4
HSEvo
0.087%
543.4
0.886%
792.9
2.507%
1105.1
Ours
0.015%
217.0
0.382
392.4
1.520%
1100.8
(diversity). To ensure fair comparison, all LLM-based AHD baselines utilize the same underlying LLM, while their hyperparameters follow the settings reported in their original publications.
Baselines. To demonstrate the effectiveness of our proposed method designing heuristies, we in- troduce several approaches for solving these complex optimization tasks. (1) handcrafted heuristics, e.g., LKH3(Lin & Kernighan]
1973) for TSP, First and Best (Romera- Paredes et al. 2024)
for Online BPP, NEH (Nawaz et al. 1983) and NEHFF (Fernandez- Viagas & Framinan 20 for
FSSP. (2) Neural Combinatorial Optimization (NCO) methods, c.g., POMO (Kwon et al. 2020)
and LEHD (Luo et al.) 2023) for TSP, NES (Pan et al. (022) for FSSP. (3) M-bsedd
AHD methods, c.g., Funsearch (Romera-Pareder et al. 2024). EoH (Liu et al. 2024b). ReEvo (Ye
al. 2024), HSEvo (Dat et al. (025) and MCTS-AIE AHD (Zenng al. 2025) Most of our test
datasets follow EoH. Notably, Funsearch, ReEvo, and HSEvo require a seed function to initialize
their populations, while EoH, MCTS-AHD, and our method can run without it.
Traveling Salesman Problem. Step-by-Step Construction (Asani al. 2023) and Guided Lo- cal Search (GLS) (Alsheddy et al. 2016) are two different strategies for solving TSP. The detailed be found in the Appendix B.I We design key heuristics for
procedure of the two strategies these two strategies. Table compares the results of our method with other baselines in step- step-by-step construction. We evaluated the performance strate performance on out-of-distribution instances, we also conducted experiments on the TSPLib dataset (Reineli) 119T), and the results can be found in the Appendix C.T. We further evaluated method on 100 instances of TSP100 TSP200, and TSP500 in GLS. The results are shown in Table The heuristic designed by our method achieves significant performance improvement compared to LLM- based AHD methods.
100 instances each of three sizes. To demon-
2
Online Bin Packing Problem. The key function of Online BPP is a scoring function that outputs score for each bin, based the current item's size and the remaining capacities of the bins. We
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Table 3: Results Online BPP. Gap(%) denotes the ratio of excess bins compared to the lower bound on Weibull instances. Obj. represents the value of the objective function. Results with are from EoH (Liu et al, 2024b). This result of LLM-based AHD method the average of three runs. The best results are highlighted in bold.
5k.C100
5k_C300
5k-C500
Method
Gap
Obj
Gap
Obj
Gap
Obj
lower bound
0.00%
2006.2
0.00%
1740.2
0.00%
1687.0
First Fit
4.40%
4.18%
4.27%
Best Fit*
4.08%
3.83%
3.91 %
Funsearch
0.80%
2022.2
1.07%
1758.8
1.47%
1711.8
EoH
1.02%
2026.6
1.00%
1757.6
1.00%
1703.9
ReEvo
0.78%
2021.8
4.47%
1818.0
3.24%
1741.6
HSEvo
1.91% %
2044.6
5.47%
1835.4
4.39%
1761.1
MCTS-AHD
0.99%
2026.0
0.95%
1756.8
0.95%
1703.0
Ours
0.69%
2020.1
0.66%
1751.7
0.66%
1698.1
Table 4: Results FSSP. The results show the comparison of makespan relative to the baseline Taillard instances. n denotes the number of jobs and m denotes the number of machines. Results with # from EoH (Liu et 2024b). This result of LLM-based AHD method is the average of three runs. The best results are highlighted in bold.
Method
n20,m10
n50,m10
n100,m10
n20,m20
n50,m20
n100,m20
NEH
4.05%
3,47%
2.07%
3.06%
5.48%
3,58%
NEHFF
4.15%
3.62%
1.88%
2.72%
5.10%
3.73%
PFSPNeL
4.04%
3.48%
1.72%
2.96%
5.05%
3.56%
EoH
0.31%
0.29%
0.23%
0.20%
0.84%
0.94%
Ours
0.17%
0.17%
0.13%
0.10%
0.58%
0.51 %
evaluate our method on Weibull BPP instances Koomera-Paredes et al., 2024) following the EoH setting. The results with three scales are shown Table 3 Our method not only significantly outperforms handcrafted heuristics but also surpasses LLM-base based AHD approaches. More results are provided Appendix C.2
Flow Shop Scheduling Problem. FSSP involves scheduling jobs m machines to minimize the makespan, where each job comprises operations processed in fixed order. We evaluate the heuristic designed by our method on Taillard instances (Taillard 1993). The dataset includes stances with 20 to 100 johs (n) and 10 to 20 machines (m). Table presents some of the results, with additional results provided in Appendix C.3 Our method performs well on all datasets, consistently delivering strong and reliable results across different scenarios.
Ablation Study. To evaluate the contribution of each component in our method, we conducted a series of ablation studies, as shown in Table 5 We separately removed the Hindsight obtained by the insight pool and the Foresight provided the navigator, and then removed both Hindsight and Foresight entirely. These experiments are conducted on TSP and Online BPP. The design and parameters of the experiments are aligned with those used in the main experiments.
Discussion. We provide more experiments and discussion in the appendix. In Appendix we present comparative studies between our method and EoH in terms convergence speed. We also evaluate our method across different LLMs to further demonstrate its effectiveness and generality. In Appendix C.6 we conduct a thorough sensitivity analysis of the key hyperparameters involved in method, providing detailed explanations of their impact performance. In Appendix C7 we conduct additional ablation studies and statistical analyses to further investigate the necessity, underlying mechanisms, and stability of the key design components of HiFo-Prompt.

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Published as a conference paper at ICLR 2026
Table 5: Ablations Insight Pool and Navigator in TSP and Online BPP. The best results highlighted in bold.
TSP
TSP20
TSP50
TSPI00
Gap
Obj.
Gap
Obj.
Gap
Obj.
w/o Insight Pool w/o Navigator
11.07%
4.29
13. 76%
6.50
16.26%
9.06
5,82%
4.09
10.31%
6.30
11.48%
8.69
w/o Insight Pool and Navigator
11.49%
4.31
14.36%
6.53
18. 88800
9.26
HiFo-Prompt
3.62%
4.00
6.63%
6.09
8.58%
8,46
Online BPP
1k, 100
5k, 100
1k, 300
5k, 300
1k, 500
5k, 500
w/o Insight Pool
4.33%
1.27%
4.07%
1.22%
4.14%
1.29%
w/o Navigator
2.83%
1.26%
2.64%
1.24%
2.60%
1.24%
w/o Insight Pool and Navigator
4.53%
2.19%
4.30%
2.01%
4.26%
2.05%
HiFo-Prompt
2.19%
0.69 %
2.08%
0.66%
2.07%
0.66 %
5 CONCLUSION
We introduced HiFo-Prompt, a novel evolutionary framework that advances LLM-driven heuristic design through hierarchical foresight-hindsight prompting mechanism. By synergizing an Evolu- tionary Navigator for adaptive control with a self-evolving Insight Pool for knowledge reuse, our framework transforms the search process into closed-looy -oop, self-regulating system. Across diverse optimization benchmarks, HiFo-Prompt consistently outperforms state-of-the the-art methods with re- markable sample efficiency, often finding superior solutions using only 200 LLM requests. This work provides not only powerful method for heuristic automated algorithm design but also con- crete step towards agents that can learn to invent their mroblem-solving methodologies.
ACKNOWLEDGMENTS
This work was partially supported by National Natural Science Foundation of China (grants 12571590, 12426305, 62401473), National Key Research and Development Program of China (grant 2022YFA 1004201), Tianyuan Fund for Mathematics of the National Natural Science Foundation of China (grant 12426105), Fundamental Research Funds for the Central Universi- ties (grant xzd01 2024049), National Key Laboratory Fund Project for Space Microwave Com- munication (grant HTK/2024KL504010) Shenzhen Science and Technology Program (grant ICY120240081355073500 and Key Research and Development Project Shaanxi Province (grant 2025CY-YBXM-055)
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A PRELIMINARY
A.1 PROBLEM FORMULATION OF AUTOMATIC HEURISTIC DESIGN
Automatic Heuristic Design (AHD) (Burke et al. 2009 Voudouris et al. 2016 aims to identify an optimal heuristic h² from a vast search space H for a given computationa task P. This process can be formally expressed as the following optimization problem (Zheng et al. 2025);
h" arg g()),
hEw
where g(h) is a fitness function (maximized) that maps a heuristic to a real number, estimating its quality based its expected performance representative set of problem instances D. For task with a minimization objective of (minimized, e.g., minimizing cost or error), this performance metric is defined
g(h)= = Rinse d [ (h(ins))
This formulation reframes the original task maximization problem the heuristic space H. thereby enabling the search for robust heuristics that yield high-quality solutions across diverse instances.
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A.2 LLM-DRIVEN EVOLUTIONARY COMPUTATION
The LLM-driven Evolutionary Computation (LLM+EC) framewor(Liu et al. 2024b Romera-
Paredes al. 2024 Yao al. 2025 Chauhan al. 2025) casts the AHD problem as iterative,
population-based search process. Let P(t) = {h]- h M be the population of M heuristics at
generation i, where each heuristic 1, H is represented by its thought and code. The transition from p(() p((+1) is governed by stochastic evolutionary kernel F, which is parameterized by control vector ((0) E O:
p(t+1) ~ F(P(t) ((()).
(10)
Here, the control vector ((0) encapsulates contextual information that guides heuristic generation, such prompting strategies performance feedback. In the absence of adaptive control, g(+) can be considered constant or null, i.e., g(t)
const or g = Ø. The kernel F comprises two phases:
Generation Phase: An LLM,, acting as conditional generator L. creates new heuristics through prompted crossover and mutation. set of parents hp S p(t) is selected, and their symbolic representations p(hp), which include the thought aadi--nnd-coon code, are used. A prompt function IT constructs a conditional prompt from p(hpp and ((t) (e.g., exploration or modification strategies). The LLM then generates offspring:
he = ((TT()()p); (()), O(t) (he,,, ha,, ..., Ne,N);
(11)
where N is the number of offspring (e.g., N = AM, where À is the reproduction rate).
2. Selection Phase: The parent and offspring populations are merged into a candidate pool,
(+t)) p(t) U O(1), selection operator S then chooses the top M heuristics from this pool based on the fitness metric to form the next generation:
p(++1) S(U :g).
(12)
This framework, introduced within the heuristic evolution paradigm, leverages LLM-driven genera- tion to explore the heuristic space and a selection mechanism to exploit high-performing solutions.
A.3 BHOWLEDGE-AUGMENTTE EVOLUTIONARY COMPUTATION
To extend the capabilities of evolutionary algorithms beyond simple adaptive control, prominent research direction involves incorporating an explicit knowledge component. Cultural Algorithms (Coello & Becerra 2010 Yan et al.) (017)) provide a classic framework for this idea, decoupling the evolutionary system into two interacting spaces: a population space containing candidate solutions p(t) and a belief space K, serving as repository of experiential knowledge (Maheri et al. 2021 These two spaces co-evolve through dual-inheritance communication protocol.
Crucially, knowledge k, extracted from the belief space K, becomes part of the high-level control vector g(t) This knowledge then guides the generation of offspring via an influence mechanism:
he ~ ((I(p))))) B())), where k, S g())
(13)
Currently, the successes of the population are fed back into the belief space. An acceptance function A identifies and extracts potentially valuable experiences, A(P())), from the current population. These are then used by knowledge update function UR to update the belief space:
K++1 = Uk (K,, A(p()))))
(14)
Khowledgg-augmentee evolution is thus a coupled dynamical system where the population and knowledge base co-evolve interdependently. This explicit mechanism for knowledge management allows the framework to achieve a more sophisticated form of learning based on abstracted experi- ence.
B OPTIMIZATION PROBLEM DETAILS
B. TRAVELING SALESMAN PROBLEM
The Traveling Salesman Problem (TSP) Matai et al. 2010 Voudouris & Tsang/ 1999) is a canonical NP-hard combinatorial optimization problem. Given of N cities and the distances between
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each pair, the objective is to find the shortest possible tour that visits each city exactly once before returning to the starting city.
Formally, let V (2 ..., UN be the set of cities. We model the problem complete, undirected graph G E), where a non-negative distance di is associated with each edge (v4, E. A tour is permutation of the indices {1,2..... ....... N). The goal is to find per- mutation n that minimizes the total tour length (Zheng et al. 2024):
min Σ
dx + dux
where Un, denotes the i-th city in the tour.
Step-by-step Construction. Construction heuristics build feasible solution from empty by making a sequence of decisions (Glover et al., 2001). At each step, a component is added to the partial solution based on specific greedy criterion until complete tour is formed (Martí & Reinelt 2011. A fundamental example is the Nearest Neighbor (NN) heuristic. Starting from a node Uni constructs a tour by iteratively selecting the closest unvisited node. At step t, with partial tour S. = (vx, ..... Vr..)
and the set of unvisited nodes U, = V (Um....., J, the next node U7++1 is
chosen according to the rule (Marinakis) 2024):
UF arg min d DyEU,
Regarding the integration of our method with the framework, we will discuss this more detail. While fast, the myopic nature of such simple rules often leads to suboptimal solutions. Our ap- proach, HiFo-Prompt, addresses this limitation by automating the discovery of more sophisticated construction heuristies. We retain the foundational step-by-step by-p framework where a solution is built incrementally.
During training phase, replace the fixed, handcrafted decision logic with heuristic function synthesized by an LLM within our evolutionary framework. Crucially, rather than employing the LLM for online, per-step inference during the solving process, our method's core contribution lies in the offline synthesis of complete, executable heuristic function.
During inference phase, this generated function-not the L.M tisefftthatio is invoked at each step This function receives the full problem state, including the current partial tour Sg, the set of unvisited candidate nodes Ur, and the global distance matrix C' (Liu et al. 2024b 2023), and computes the next node to add to the tour locally and efficiently.
Guided Local Search Improvement heuristics, such as local search, start with a complete tour and iteratively refine it. Guided Local Search (GLS) is an advanced metaheuristic that enhances local search by introducing a guidance mechanism to escape local optima (Voudouris & Tsang 1999. Voudouris et al. 2016). GLS achieves this by modifying the objective function with penalties on certain solution features that appear in locally optimal solutions (Wu et al., 2015). For the Traveling Salesperson Problem (TSP), the most natural features to penalize are the edges of the tour (Tairan & Zhang 2010).
The standard GLS cost function augmented with penalty term:
Trang 8) = I(s) +
Σ (u,o)Es
Poe
(15)
where L(s) is the original tour length, the sum over all edges (u, v) in tour 8, Puv is a penalty counter for using the edge between cities and v, and À is regularization parameter. An efficient implementation involves creating a penalized distance matrix D' for the local search:
D' we = dap + Puv
(16)
Minimizing the tour length using is equivalent to minimizing Lang (s). The critical challenge lies in designing the rule for updating the penalty matrix P = {pun). When the local search becomes
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trapped in a local optimum s", state-dependent update heuristic, Hupdates is invoked to determine which edges s" should be penalized:
Pne
Hupdate (oldd s" DD,N)
(17)
where is a matrix of edge usage frequencies. Typically, this heuristic is a static, handcrafted rule that increments the penalties for subset of edges in s*. This update reshapes the search landscape to guide the search away from the current basin of attraction.
HiFo-Prompp automates the discovery of the update heuristic Hupdate- Instead of relying on static, human-designed rule, we task an LLM with synthesizing a complete, executable Python function to serve Hupdate-
At each GLS iteration, after converging to a local optimum 8", this L.LM-generated function is invoked. It receives the full state necessary for intelligent penalization- -the current penalty matrix (Poi),, the local optimum tour (s*), the original distance matrix (D), and the edge frequency matrix (N)-and and outputs new penalty matrix, P Feew. This updated matrix then modifies the search costs via Eq. [6 for the next major iteration. The local search itself is driven by fundamental prompt strategies like Relocate [Tuononen] (022) and 2-opt (Sengupta et al., 2019). Our contribution lies not in these prompt strategies but in the automated design of the sophisticated Hupdate function through HiFo-Prompt, which provides the intelligence to guide these prompt strategies effectively.
We further elevate this concept by embedding heuristic generation within an evolutionary frame- work. We treat the distinct Hupdate functions as a population of individuals. The LLM itself serves as the primary genetic operator. Through structured prompts for crossover and mutation, the LLM in- telligently combines or modifies existing et al. 2024b).
strategies to produce novel offspring
(Liu
B.2 ONLINE BIN PACKING PROBLEM
The Online Bin Packing Problem (OBP) (Seiden 2002 is classic sequential docision-making problem. We are presented with sequence of items, A (a1, 02,.... ur), arriving one at a time. Each item o, has a size 8, E (0, 1]. We have unlimited supply of bins, each with unit capacity of 1. The core constraints of the OBP are:
Online Constraint: When item ag arrives, a decision must be made to place it into a bin without any knowledge of future items (a++1, ...,,,....000 ar).
Irrevocable Placement: Once an item is placed in bin, it cannot be moved.
Capacity Constraint: For any bin By, the sum of the sizes of all items placed within must not exceed 1.
The objective is to minimize the total number of bins used after placing all T items (Epstein et al.] 2012 Yarimcam et al. 2014). If we let y if bin is used and y 0 otherwise, the goal is to minimize Σ3 V/ (Sgal 2014).
Given the online nature of the problem, optimal solutions are generally not achievable. Instead, high- performance algorithms rely sophisticated greedy placement policies heuristics. Following the approach of Kkomera-Paredee et al.) 2024). we frame the task as learning a superior placement heuristic. Specifically, we use the HiFo-Prompt framework design scoring function, Hscore that determines the most suitable bin for an incoming item.
When item a, with size 8t arrives, the HiFo Prifo--rompt-eeeerttedd heuristic is invoked. It takes as input the item's 's size and the state of all currently open bins. The state of bin B, is captured by its residual capacity, Cj candidate bin:
EogEB, Sk. The scoring function produces a scalar value for each
oj Hsoore (St. cj)
(18)
where oj represents the desirability of placing item at into bin By. The placement policy then to assign the item to the valid bin with the highest score:
5 arg max Øj
(19)
61o22)
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If existing bin accommodate the item (i.e., the set of valid bins is empty), new bin is opened. The intelligence of our method lies in MiiFo-promp''s abilitt to discover mon-trivial scoring function Home that implicitly balances competing objectives, such as leaving space for potentially larger future items versus consolidating small items efficiently. This contrasts with classic heuristics like Best Fit (which is equivalent to Hoore Cg) -c) First Fit, by allowing for much richer and more adaptive decision-making process.
B.3 FLOW SHOP SCHEDULING PROBLEM
The Flow Shop Scheduling Problem (FSSP) [Emmons & Vairaktarakis/ 2012 Komaki et al. 2019) a canonical NP-hard scheduling challenge. The task is schedule a set of n jobs, J [I.... .......... on series of m machines, M = {1.... m). Each job i E J requires operations, with the j-th operation occurring on machine j. The processing time for job on machine j is given by T; from processing time matrix T (Vaessens et al., 1996 Ribas et al., 2010). A solution (Juan et al. 2014) is a permutation of jobs, æ = (I.,,.... ),), which dictates the processing order on machines. Key constraints include that machine can process multiple jobs at once, and job can be on multiple machines simultancously.
The objective is to find permutation x" that minimizes the makespan, Cmax. This is the total time elapsed until the last job completes its final operation. The completion time C(T,,)) for job x; on machine j is calculated recursively:
C(T,.)) max (C( C(T.j 1)) Tn.
(20)
with base cases () 0 and C(T,, 0) = 0. The makespan for sequence a therefore Cinaxe (T)) = C(T,, m..
Due to the problem's complexity, we employ local search metaheuristic (Liu et al., 2024b). To prevent the search from being trapped in local optima, we use the HiFo-Prompt framework to design a sophisticated guidance strategy automatically. When the search converges to a locally optimal sequence x², PerFo-Promt-geeee original processing time matrix, and problem dimensions to produce both new time matrix T and designated list of jobs to perturb, perturh:
heuristic, guide. is invoked. It takes the current sequence, the
(T', Jpenum) guide (TT ,T, m)
(21)
This dual output provides a powerful guidance mechanism. The new matrix T' reshapes the search landscape by penalizing attributes of the local optimum, effectively steering the search toward un- explored regions. Concurrently, the list Jpenrrb directs subsequent local search operators, such as insertions swaps, to focus their computational effort a specific subset of critical jobs. This combined strategy of altering the problem's ' cost structure while focusing on the search operators constitutes a complete and intelligent guidance component, designed automatically by our frame- work.
B.4 BAYESIAN OPTIMIZATION
Bayesian Optimization (BO) (Shahriari et al. 2016) and its ongoing development are of paramount
importance, as it provides the leading framework for sample-efficiently navigating the complex, high-cost search spaces prevalent in modern science and engineering. Bayesian Optimization (BO) has emerged as principal framework for this task, excelling applications like hyperparameter tuning and automated scientific discovery. The power of BO lies in its sample efficiency. It builds a probabilistic surrogate model of the objective function. It then uses an acquisition function (Lam) et al. 2016) to intelligently decide where to sample next, thereby minimizing the number of costly evaluations.
Our work addresses particularly demanding variant: cost-aware BO (Snoek et al. 2012 Yao et al.) 2024 Zheng et al.) 2025). In this setting, each function evaluation f(x) has a heterogeneous and unknown cost, denoted by c(x). The goal is to find the global maximum of If fxx within a fixed total
budget B This requires sequentially choosing evaluation points [x.....xx, } to maximize the
outcome, subject to the constraint that Σ o(x;) S B (Lee et al. 2020). To manage this, the BO
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agent maintains two surrogate models, typically Gaussian Processes (GPs) (Rasmussen & Williams 2006). One GP models the objective function, predicting its posterior mean y and standard deviation of (x). A second GP models the evaluation cost, providing cost prediction (x(x)..
The core intelligence of the agent is encoded in its acquisition function, o(x). This function must navigate a complex trade-off between seeking high rewards (exploitation), reducing model uncer- tainty (exploration), and managing evaluation costs. At each iteration t, the next evaluation point X selected by maximizing this utility:
X arg max ax (x),,, Brem) XEA
(22)
where D, {(x,, yo, 4=) is the set of previously evaluated points and Brem is the remaining budget. The design of a(x) is the single most critical factor for achieving high performance.
We propose to automate the discovery of superior acquisition funetions using the HiFo-Prompp framework. Rather than relying on static, human-designed heuristics, HiFo-Prompt generates a novel utility function, Huniliy, from scratch. This generated function highly context-aware, syn- thesizing all critical information available at each decision step. It explicitly considers the surrogate models' predictions, the best-found solution so far (yī max, Wc), and the dynamic state of the budget:
o(x) = Huntay (15(()), ( (x()),,((x)) VE. Bused, Bustal)
(23)
By generating a holistic function that reasons about the interplay between potential gain, uncertainty, cost, and remaining resources, Prompt creates powerful and adaptive sampling strategies. This approach moves beyond hand-crafted designs, enabling superior performance in complex, budget- constrained optimization scenarios.
C
MORE RESULTS
C.I TRAVELING SALESMAN PROBLEM
To demonstrate the generality and robustness of the heuristic designed by our proposed method, conduct comprehensive evaluation using the real-world benchmark dataset TSPLib (Reinelt, 1991]. TSPLib is a well-established collection of TSP instances, widely used in the research com- munity for benchmarking optimization algorithms.
For our experiments, we select a diverse subset of TSP instances from TSPLib, specifically focusing on those containing more than 500 nodes. This selection criterion ensures manageable problem scale while still retaining the complexity necessary to assess algorithmic performance effectively.
In our evaluation framework, adopt step-by-step construction approach to solve the TSP, which incrementally builds tour by selecting the next node based on a learned heuristic. This paradigm allows evaluate the quality of decisions made each step and better observe the contribution of the designed heuristic to the final solution quality.
To rigorously assess the effectiveness and competitiveness of our proposed heuristic, compare its performance against state-of-the-art Lba-d AHD baselines. These methods represent recent advances in leveraging large language models for combinatorial optimization tasks and serve as strong comparative basclines in our study. The experimental results, which include performance metries, are summarized Table These results provide empirical evidence that our method not only performs competitively but also generalizes well across a variety of TSP instances in real-world scenarios. addition, we further conduct experiments on small-scale instances (TSP10 and TSP20) under the step-by-ster construction framework for TSP. The results are presented in Table 7
C.2 ONLINE BIN PACKING PROBLEM
To provide more comprehensive empirical validation and to assess the robustness of our frame- work, we conducted further evaluations of the online Bin Packing Problem (BPP) on broader and more challenging set of real-world scenarios than uniform distributions, were generated following the protocol established in prior work on M--M-based heuristic discovery (Romera-Paredes et al. 2024). Table presents
instances. These instances, which more closely mimic
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Table 6: Results TSPLib. The best results highlighted in bold.
name
EoH
MCTS-AHD
ReEvo
HSEvo
Ours
eil51
7.665%
17.133%
4.409%
4.295% %
6.691%
berlin52
16.080%
17.481%
15.678%
19.814%
10.973%
ei176
10.566%
15.193%
8.450%
10.034%
8.439 %
pr76
25.222%
29.371%
21.089%
21.718%
20.505% %
kroA100
24.347%
24.315%
17.509%
18.758%
16.616 %
troB100
30.274%
22.593%
9.059%
13.266%
20.502%
kroC100
25.542%
10.212%
27.343%
36.176%
10.000 %
kroD100
31.497%
25.765%
25.797%
13.258% %
32.315%
kroE100
24.618%
21.299%
23.905%
22.218%
14.925%
rd100
12.654%
11,470%
8.660%
12.995%
7.921 %
eil101
14.421%
22.947%
12.163%
12.372%
7.229 %
lin
40.297%
16.797%
27.774%
42.519%
13.974% %
pr107
9.900%
5.004%
7.894%
7.537%
3.811 %
ch130
21.406%
6.930%
7.352%
13.107%
7.090%
ch150
19.604%
9.874%
12.644%
10.185%
4.275%
kroA150
27.891%
20.517%
25.887%
21.869%
20.070%
kroB
22.403%
27.716%
29.499%
32.729%
13.677% %
pr152
12.302%
10.457%
12.473%
11.960%
6.545 %
u159
13.444%
7.074%
7.951%
13.644%
7.828%
kroA200
26.395%
26.219%
27.366%
28.537%
22.702%
kroB200
20.980%
20.083%
23.861%
21.590%
14.314%
tsp225
32.938%
20.193%
21.703%
22.503%
18.683%
a280
34.081%
24.095%
27.921%
31.736%
15.936 %
rd400
14.612%
14.745%
13.865%
15.213%
8.075
d493
18.833%
13.248%
14.776%
21.179%
7.976 %
avg.
21.519%
17.629%
17.401%
19.168%
12.843% %
Table 7: Results on TSP with step-by-step construction in 10 and 20.
TSP10
TSP20
Method
Gap
Time(s)
Gap
Time(s)
LKH3
0.000%
6.492
0.000%
24.948
POMO LEHD
0.246%
0.248%
0. 0.183%
0.010%
EoH
7.148%
0.042
10.064% %
0.168
ReEvo
5.227%
0.228
6.811%
1.215
HSEvo
5.461%
0.689
7.950%
3.273
MCTS-AHD
4.829%
0.440
8.045%
4.087
Ours
1.654%
0.709
3.619%
12.961
detailed comparative analysis of our method against wide spectrum of competitors, including both classical, widely-adopted handcrafted heuristics (First Fit, Best Fit) and suite of contemporary approaches based on Large Language Models.
For each combination of bin capacity and problem size, the dataset includes five unique instances. Performance quantified by the average percentage gap to the known theoretical lower bound across these instances, where a smaller value signifies more efficient and effective packing solution.
The empirical results demonstrate the superior performance of the heuristic have discovered. As shown in the table, our method consistently outperforms all baseline methods in nearly every setting, securing the smallest average gap in 8 out of the distinct configurations tested. This high- lights a remarkable level of consistency and dominance. The only exception the (Capacity=100, Size=10k) case, where Funsearch achieves marginally lower gap.
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Table 8: Results Online BPP in Weibull instances with varied capacities and problem sizes.
Results marked with denote values taken from (Liu et al. 2024b).
Capacity
Size
First Fit
Best Fit
EoH
ReEvo
HSEvo
MCTS-AHD
Ours
5.32%
4.87%
3.10%
3.63%
3.51%
3.38%
2.19%
100
5k
4.40%
4.08%
1.02%
0.78%
1.91%
0.99%
0.69 %
10k
4.44%
4.09%
0.80%
0.35% %
1.68%
0.84%
0.42%
1k 4.93%
4.48%
3.04%
7.34%
6.88%
3.21%
2.08 %
300
5k
4.18%
3.83%
1.00%
4.47%
5.47%
0.95%
0.66 %
10k
4.20%
3.87%
0.78%
4.05%
5.26%
0.85%
0.39% %
1k 4.97%
4.50%
3.04%
5.92%
5.80%
3.20%
2.07 %
500
4.27%
3.91%
1.00%
3.24%
4.39%
0.95%
0.66 %
10k
4.28%
3.95%
0.78%
2.79%
4.14%
0.85%
0.40 %
However, our method's strong performance across the entire parameter space underscores its greater reliability and generalizability compared to methods that may excel only in specific, narrow sce- narios. Notably, our approach scales gracefully, achieving extremely low gap values (e.g.. 0.41 %, 0.42%) the largest and most complex problems, significantly surpassing both traditional algo- rithms and other state-of-the-art LLM-driven frameworks. This comprehensive evaluation on chal- lenging instances further validates the efficacy of evolutionary framework, showcasing its capac- ity to discover sophisticated and high-performance heuristics that are robust across varied problem characteristics.
C.3 FLOW SHOP SCHEDULING PROBLEM
Table presents comparative analysis of the performance of EoH and proposed method on FSSP of varying scales, defined by the number of jobs and machines m. Performance evaluated using two key metrics: the objective function value and the relative gap, where the gap is measured with respect to the solution quality of an advanced handcrafted heuristic. smaller gap reflects a solution that closer the heuristic baseline and, therefore, indicates higher solution quality. The results demonstrate that our method consistently achieves smaller gaps across all tested problem configurations, regardless of scale. Moreover, many cases, it outperforms the advanced heuristic itself in terms of the raw objective value. This consistent superiority suggests that our approach not only generalizes well across diverse FSSP instances (Liu al. (024b) but also offers a competitive alternative to domain-specife heuristics, exhibiting both strong effectiveness and robustness.
Table 9: Results on FSSP. The best results are highlighted in bold.
EoH
Ours
Gap
Time(s)
Gap
Time(s)
5
0.25%
7.5
-0.01 %
5.5
20
10
0.31% %
13.0
0.17% %
8.8
20
0.20%
23.8
0.10%
18.5
5
0.01%
45.1
0.00 %
29.7
50
10
0.29%
83.9
0.17% %
50.6
20
0.84%
168.4
0.58 %
101.1
5
-0.02%
230.2
-0.04 %
146.7
100
10
0.23%
299.6
0.13% %
243.5
20
0.94%
305.2
0.51 %
303.0
10
0.37%
337.1
0.12%
317.2
200
20
1.23%
334.6
0.71 %
321.4
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Table 10: Results on BO. The results denote the absolute relative the optimal solution. Results marked with are from
Yao et al. 2024). The best results are highlighted in bold.
Three
Ackley
Rastrigin
Griewank
Rosenbrock
Levy
HumpCamel
E
2.66
4.74
0.49
1.26
0.01
0.05
Elpu*
2.33
5.62
0.34
2.36
0.01
0.12
El-cool
2.74
5.78
0.34
2.29
0.01
0.07
EoH
3.11
3,48
0.72
2.57
0.04
0.18
MCTS-AHD
3.23
0.87
0,43
1.30
0.01
0.05
Ours
1.78
0.45
0.41
1.50
0.00
0.01
Styblinski Tang
Hartmann
Powell
Shekel
Hartmann
Cosine8
EI*
0.03
0.00
18.89
7.91
0.03
0.47
Elpu
0.02
0.00
19.83
7.92
0.03
0.47
El-cool*
0.03
0.00
14.95
8.21
0.03
0.54
EoH
2.89
0.01
13.71
8.71
0.47
1.04
MCTS-AHD
0.02
0.00
1.91
5.14
0.08
0.29
Ours
0.02
0.01
2.65
4.08
0.57
0.28
C.4 BAYESIAN OPTIMIZATION
Table 10 presents the experimental results the design of cost-aware acquisition functions (CAFs) Yao al. 2024) Bayesian Optimization. We compare our method against both man- ually crafted CAFs and those generated by LLM-bbssed AHD methods. To ensure fair comparison, all LLM-based AHD approaches utilize the same underlying language model. Our method achieves superior performance the majority of benchmark functions, outperforming traditional CAFs (e.g.. EI, Elpu, EI-cool) (aoo et al., 2024) as well as recent LLM-based AHD baselines (e.g.. EoH (Liu al. 2024b), MCTS (Zheng et al.) 2025)). The performance advantage particularly pronounced on challenging functions such as Rastrigin, ThreeHumpCame and Shekel. These results underscore the robustness and strong generalization ability of our approach across wide range of optimization landscapes.
C.5 COMPARATIVE RESULTS
Convergence Analysis. We compared the progression of objective function values during the evo- lutionary process of our method and EoH on both the TSP and Online BPP. Figure 3 presents the convergence analysis curves. The figure demonstrates that method converges requiring fewer individuals in the population, indicating higher efficiency. For a detailed breakdown of computational costs, including runtime and token consumption comparisons, please refer to Ap- pendix C.8
rapidly while
Results with Different LLMs. Furthermore, we evaluated it across diverse set of LLMs. To ensure fairesss and consistency. we adopted the evaluation protocol established in MCTS-AHD (Zheng et al., 2025). Specifically, we selected two representative problem scales for each task to validate scalability, and each experiment was conducted across three independent runs, with average values reported to mitigate stochasticity. The detailed comparative results are presented Table []
The results indicate that our framework exhibits strong adaptability, consistently discovering effec- tive heuristics across diverse LLMs. While Qwen-2.5-max yields the leading metrics we observe that performance does not follow the generic reasoning benchmarks of these models. Instead, suc- cess appears to rely on the synergy between the framework's guidance mechanisms and the specific model's 's characteristics (c.g., instruction following). Even smaller models like GPT--o-mimi show competitive results, highlighting that optimizing the coordination between the LLM-based AHD and the underlying model remains promising and open direction for future research.
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Convergence Salesman
Ouline
Sal
B4R29
802
Obpecier
60
Figure 3: Key Example of Convergence Analysis for TSP and Online BPP.
Table 1I: Results with different LLMs.
Problems
TSP-construction
TSP-GLS
Scale
N=50
N=100
N=100
N=200
Qwen-2.5-max
6.625%
8.582%
0.015%
0.382%
GPT-4o-mini
7.058%
9.579%
0.021%
0.433%
DeepSeek-v3
7.717%
10.455%
0.041%
0,480%
DeepSeek-rl
10.412%
12.638%
0.074%
0.589%
Claude3. 1ilee3.5--one
8.276%
11.087%
0.061%
0.454%
Problems
Online BPP
FSSP-GLS
Scale
N=lk, C=100
N=10k, C=300
n= 100, m=10
n=200, m=20
Qwen-2.5-max
2.188%
0.391
0.128%
0.710%
GPT-4o-mini
2.287%
0.492%
0.137%
0.787%
DeepSeek-v3
2.238%
0.978%
0.185%
0.916%
DeepSeek-rl
2.984%
1.487%
0.313%
1.485%
Claude3.5 5-nnnet
2.089%
1.487%
0.179%
1.256%
C.6 PARAMETER SENSITIVITY ANALYSIS
To rigorously quantify the impact of key hyperparameters the performance of HiFooPrompt, conducted a comprehensive ablation study involving eight core parameters. All sensitivity experi- ments were conducted on the TSP-50 construction task using the training dataset. To ensure statis- tical reliability, each configuration was executed across 3 independent runs, with results reported as the average objective function value. Unless otherwise specified, the framework operated under default configuration of population size Nypp 8 and maximum generations Gmax 8. The results confirm that the framework consistently converges to sugh-tuality solutions across broad spectrum of configurations, demonstrating that our settings are both effective and robust without relying on ietance-specifie fine-tuning. The results, visualized Figure and detailed Table T2. not only justify our default configurations but also reveal the underlying mechanics of the framework. Col- lectively, these experiments demonstrate that PaFo-Pr maintains consistent performance across reasonable range of parameter settings, confirming the robustness and reproducibility of the pro- posed method.
Recency Bonus Magnitude (7,) The role of this parameter is grounded in the formulation of the recency bonus function B,, designed to capture the temporal dynamics of the search process. Formally defined as:
B, clust = - Tr if e T
(24)
otherwise
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Table 12:
Ablation analysis of recency bonus magnitude (T,).
usage penalty weight (w,, and
decay rate (Raceay).
Wa
run
run2
run3
avg.
0 5.958
5.989
5.978
5.975
0.05
5.894
5,901
5.907
5.901
0.1 (default)
5.885
5.838
5.873
5,866
0.3
5,921
5.907
5.922
5,917
Tr
run
run2
run3
avg.
0
5.954
5.972
5.979
5,968
0.1
5.920
5.925
5,901
5.915
0.2 (default)
5.885
5.838
5.873
5,866
0.4
5.937
5.945
5.933
5.938
Riseaa
run
run2
run3
avg.
0
5,998
6.097
6.075
6.057
0.005
5,899
5.908
5,906
5,904
0.01 (default)
5.885
5.838
5.873
5,866
0.04
5.985
5.956
5.967
5.969
where denotes the current generation index, eaat records the last generation in which insight i was successfully applied, and Tw represents the sliding window size (set to Tu = 2). Here, controls the magnitude of the reward assigned to recently validated insights. Theoretically, this mechanism addresses the non-stationary nature of evolutionary search- an insight highly effecttve during early exploration may become obsolete during later exploitation. This design aligns with the principles of Sliding- Sliding-Window Upper Confidence Bound (SW-UCB) (Garivier & Moulines 2011) 1) algorithms used in non-stationary Multi- Armed Bandits. By restricting the bonus to narrow temporal window Tw. the framework implicitly discounts outdated information, ensuring that the selection process prioritizes insights that are not just historically successful, but contextually relevant to the current state of the population.
The sensitivity analysis presented in Table 12 validates this design choice. Specifically. when this short-term memory is deactivated (r, 0). the algorithm inherently lacks the incentive to maintain search continuity. Consequently, the search explores widely but inefficiently, resulting in degraded performance. However, it is crucial to note that simply increasing the reward is not monotonically beneficial. As observed in the table, setting the reward magnitude too high (T, 0.4) leads to a deterioration in algorithm quality. This outcome suggests that excessive recency bias induces myopic behavior, wherein the algorithm becomes overly focused the specific traits of the most recent iteration. Such short-sighted focus ultimately overshadows the robust, long-term effectiveness signals (E.) accumulated over time. In contrast, introducing moderate reward (7, 0.2) achieves consistently strong performance. This finding confirms that explicitly rewarding recent successes allows the L.M effectively build upon immediate gains, thus guiding the population through the shifting fitness landscape without becoming distracted by transient fluctuations.
Usage Penalty Weight (w.). This parameter regulates the exploration pressure within the Insight Pool by modulating the usage cost term, defined as ww, log( (!) + +1,, where N2(() represents the retrieval count of insight The choice of logarithmic growth function is deliberate: ensures that the penalty accumulates rapidly for the first few uses to prevent immediate overfitting, but saturates quickly thereafter. This design enables the score of a frequently used insight stabilize rather than suffer precipitous decline, thereby avoiding sharp priority reversals or oscillatory behavior while granting other high-potential candidates a fair chance. The sensitivity analysis Table 12 confirms the necessity of this balanced regulation. When the penalty is removed (w., = 0), the framework exhibits a significant performance drop. This indicates a phenomenon of knowledge collapse, where the algorithm over-exploits a few early dominant insights, effectively depriving the search of diversity. Conversely, an aggressive penalty (w, = 0.3) yields suboptimal results by discarding valid insights prematurely before their full potential is realized. The default value of
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Wa = 0.1 demonstrates robust efficacy, showing that gentle, saturating penalty is sufficient to maintain a healthy turnover of valid ideas without suppressing fundamental design principles.
Decay Rate (Raeaa). This parameter governs the temporal lifecycle of insights, controlling the rate which older knowledge becomes obsolete. By applying linear decay to the eviction score, Rasea enforces a mechanism of knowledge turnover, preventing the Insight Pool from becoming static archive of legacy strategies that, while successful in early generations, may longer be competitive. The sensitivity analysis in Table 12 reveals that this dynamic maintenance is critical for sustained performance. When the decay mechanism is disabled (Rasay = =0), the algorithm yields the worst results among all tested configurations. This confirms that a static pool eventually becomes saturated with stale information, effectively locking out superior novel insights due to capacity constraints. Conversely, setting the rate too high (Rueaay 0.04) causes the system to rapidly forget valid principles, disrupting the accumulation of algorithmic knowledge. The default rate of Raecay- = 0.01 demonstrates robust performance, continuously purging outdated or inactive insights to ensure the pool remains relevant and valid, thereby staying aligned with the evolving needs of the population.
Diversity Threshold (6p). This parameter establishes the minimum acceptable level of pheno- typic diversity, acting as a sensitivity trigger for the Evolutionary Navigator. As defined in Eq. 7, diversity is computed via exact string matching of the generated algorithmic descriptions. This rig- orous design choice is motivated by the fact that our standardized prompt template eliminates trivial syntactic noise (e.g., variable renaming), ensuring that any remaining lexical difference reflects deliberate structural modification by the LLM. Moreover, exact matching captures subtle but criti- cal logical alterations-succ as dynamic vs. static-that embedding-basse The threshold 6, dictates when the system perceives the population as stagnated; dropping below it triggers shift to an Exploration regime. The sensitivity analysis in Figure reveals the delicate balance required here. A low threshold (6p- = 0.1) proves overly tolerant of redundancy, allowing the LLM to repeatedly output identical text without intervention. This failure to trigger exploration traps the search in local optima, leading to premature convergence. Conversely, an excessively high threshold (8p = 0.5) imposes a hypersensitive constraint that misinterprets healthy convergence as stagnation. As evidenced by the significant instability in the green curve, this causes frequent, un- necessary disruptions that abort the fine-tuning of high-potential heuristics. Therefore, the default setting (6p- 0.3) represents a robust operating point. By triggering intervention only when over 70% of the population consists of exact duplicates, it functions as an indicator that effectively antic- ipates local-optimum collapse, enforcing proactive exploration before the search fully stalls while still permitting sufficient iterations for the local refinement of elite patterns.
metrics might obscure.
Population Size pop). This parameter determines the breadth of the heuristic search space cov- ered per generation. The sensitivity analysis in Figure 4 reveals distinct trade-off: excessively small populations (e.g.. Npop = 4) lead to premature convergence, as the limited gene pool lacks sufficient diversity to sustain effective recombination. Conversely, overly large populations (c.g., Npop = 12) significantly increase the computational burden (token consumption) without yielding proportional performance gains. The results confirm that size of 8 serves as an effective balance point, providing sufficient genetic material for the LLM to synthesize high-quality heuristics while maintaining manageable query budget.
Maximum Generations (Gmux). This parameter defines the temporal budget of the evolutionary process. The convergence trajectories in Figure near-optimal solutions within the first 6-8 generations. While extending the search to 12 or more generations might offer slight refinements, it yields diminishing returns. The marginal performance improvements observed in these later stages do not not justify the significant additional inference costs. Consequently, we selected Gmax 8 as the standard configuration, representing trade-off that secures high solution quality while maximizing computational efficiency.
that the algorithm typically achieves
Insight Pool Size (Cpwot). The capacity of the Insight Pool regulates the system's 's long-term mem- ory. As indicated by the variance and trends in Figure 4 small pool (e.g., < 10) leads to catas- trophic forgetting, where valuable high-level principles are evicted before they can be effectively reused. On the other hand, an unconstrained or overly large pool acts as noise accumulator, dilut-
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ing high-quality insights with mediocre ones and confusing the LLM with irrelevant context. Our experiments indicate that a capacity of 30 effectively balances the retention of diverse, high-utility insights with the need to filter out noise, maintaining both stability and agility.
Stagnation Threshold (Tstag). This parameter determines the sensitivity of the Evolutionary Nav- igator in detecting performance plateaus. It sets the number of consecutive non- non-improving genera- tions required trigger a shift from Exploitation to Exploration. The sensitivity analysis in Figure highlights the critical balance involved. A low threshold (e.g., Tsaag 1) makes the system hy- persensitive, frequently interrupting the refinement of promising solutions due to minor stochastic fluctuations. This prevents the algorithm from fully exploiting local basins of attraction. Conversely, a high threshold (e.g., Tstag = 5) renders the framework sluggish. Given the constrained evolutionary window (typically 12 generations) and the high cost of LLM inference, waiting for 5 generations represents a significant waste of the computational budget on unproductive search. The default value of Tsug 3 provides a trade-off, allowing for rapid detection of diminishing returns while filtering out transient noise in the optimization trajectory.
4
Population Size
Max Generations
Insight
soart
aed, Jlaw
6.2
6.1
4.
0.4
5.3
Swerton
Oonundor
Stagnation Thrushold
Diversity Threshold
-O- Fander
-O- Dennooni
-O: Bwankss
Figure 4: Parameter sensitivity analysis of population size, maximum generations, insight pool size, stagnation threshold and diversity threshold.
C.7 ADDITIONAL ABLATION STUDIES AND ANALYSIS
To provide comprehensive understanding of the internal mechanisms of HiFo- and to dis- sociate the contribution of specific components from the overall framework, we conducted a series of additional ablation studies. These experiments investigate the architectural necessity of the In- sight Pool, the efficacy of the adaptive Evolutionary Navigator, the behavioral dynamics of regime switchnng, and the fine-grained design of the utility function.
Architectural Necessity of the Insight Pool. To determine whether the performance gains of HiFo-Prompt stem principally from the initialization with high-quality seed insights or from the framework's 's capability to manage them, we injected the identical set of seed insights into the initial populations or prompts of four state-of-the-art baselines: EoH, ReEvo, HSEvo, and MCTS-AHD. As detailed in Table [3] the incorporation of seed insights into these baselines yields negligible performance improvements, and in certain cases (e.g., MCTS-AHD, results in slight performance degradation due to interference with their native search logic. This stands in sharp contrast to the significant gains achieved by HiFo-Prompt. These results substantiate that standard L--e-b
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evolutionary methods lack the necessary architectural mechanisms ceaaaiisssssppsssilllll.l the Insight Pool and the Hindsight reflection process to effectively retain, retrieve, and utilize abstract knowledge. Consequently, the superior performance of HiFo- ProFopproppttt attributed to its unique ability to leverage high-level guidance rather than merely the quality of initialization.
Table 13: Impact of seed insights baselines, using TSP construction as an example. The results show the objective function values. Lower is better.
Method
TSPI0
TSP20
TSP50
TSPI00
TSP200
EoH (w/ seed insight)
3.024
4.233
6.512
8.998
12.470
EoH (w/o seed insight)
3.049
4.252
6.441
8.990
12.514
ReEvo (w/ seed insight)
2.993
4.132
6.344
8.759
12.198
ReEvo (w/o seed insight)
2.994
4.126
6.294
8.773
12.324
HSEvo (w/ seed insight)
3.064
4.324
6.452
8.842
12.270
HSEvo (w/o seed insight)
3.001
4.170
6.307
8.729
12.183
MCTS-AHD (w/ seed insight)
2.938
4.142
6.294
8.724
12.076
MCTS-AHD (w/o seed insight)
2.983
4.174
6.317
8.769
12.176
Efficacy of Adaptive Regime Switching, The Evolutionary Navigator is designed to dynamically switch between Explore. Exploit, and Balance regimes based on real-time population states. To validate the necessity of this adaptive control, we compared the full tiFo-Prompt against a variant fixed permanently in the Balance mode, which attempts to combine exploration every prompt without explicit directional guidance. Table 14 demonstrates that the adaptive mecha- nism significantly outperforms the fixed strategy. This performance gap widens as the problem scale increases (c.g., reducing the gap TSP200 from 22. 15% to 8.877%). This indicates that a static prompt strategy is insufficient for navigating complex search landscapes. The Navigator's ' ability to explicitly enforce exploration during stagnation and exploitation during progress is critical for escaping local optima and refining solutions efficiently.
exploitation in
Table 14: Ablation study of the Evolutionary Navigator's adaptive mechanism fixed strategy.
Method
TSP10
TSP20
TSP50
TSP100
TSP200
HiFooPromptt (adaptive) 1.654 %
3.619% %
6.625 %
8.582 %
8.877 %
HiFoo-Prompt (fixed Balance)
5.812%
9.676%
15.599%
18.924%
22.115%
Behavioral Analysis of the Evolutionary Navigator. To further elucidate the operational behav- ior of the Navigator, we analyzed the frequency of each regime triggered during the evolutionary process across three independent runs. As shown in Table the system exhibits a balanced distri- bution of states. While Balance is the most frequent state (providing stability), the system frequently triggers Explore (avg. 102.3 times) and Exploit (avg. 83.7 times). This confirms that the diversity (6,) and stagnation (Tstag) thresholds actively functioning, ensuring the search dynamically cillates between structural diversification and local refinement rather than converging prematurely.
Table 15: Frequency statisties of Evolutionary Regimes triggered by the Navigator.
Regime
run 1
run
run 3
avg.
Explore
110
95
102
102.3
Exploit
81
82
83.7
Balance
129
137
136
134.0
Synergistic Effect of Utility Function Components. sum of terms but establishes dynamic interplay among three constituents to ensure stable, conver- gent gent selection. The learned effectiveness Et()) serves the dominant, low-variance signal reflecting
The utility function Uk; t)) is not merely
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long-term value via Exponential Moving Average (EMA). The usage penalty and recency bonus act as critical modulating factors. Specifically, the usage penalty employs a logarithmic growth design ()og(-N,(t) + 1)); this ensures the penalty saturates quickly, meaning insight's s priority is gently down-weighted rather than suffering a precipitous decline. This prevents sharp priority reversals and oscillatory behavior while preserving diversity. Conversely, the recency bonus injects fixed, time-dependent reward to encourage search continuity along recently validated trajectories.
The ablation study in Table 16 empirically confirms this synergy. Removing the penalty (w,, 0) leads to inferior results, indicating that without exploration pressure, the system over-exploits early dominant insights, leading to premature knowledge collapse. Removing the reward (T, 0) sim- ilarly harms performance, as the search loses its momentum and explores widely but inefficiently. The poorest performance occurs when both modulators are removed, validating that the simultane- ous application of penalizing overuse and rewarding recency is essential for robust credit assignment.
Table 16: Ablation analysis of usage penalty (w,) and recency bonus (T,) in the Utility Function.
Setting
run
run 2
run 3
avg.
Wy > 0, Tr > 0 (default)
5.885
5.838
5,873
5,866
Wy 0, T. (w/o penalty)
5.958
5.989
5.978
5.975
Vy > O, Tr 0 (w/o reward)
5.954
5.972
5,979
5.968
Wa 0, Tr (w/o both)
5.984
6.026
5.982
5.997
Calibration of Base Credit Assignment. Finally, we analyzed the sensitivity of the base credit assignment coefficients defined in Eq. 3 (sel"). ). In Table 17 each configuration is presented as triplet [Sbest: mm::: 3pel); representing the base intercept values for three distinct performance tiers: the first value (Boet) rewards offspring that surpass the current population best serraddi--ssiftting the second (8me) rewards those improving upon the average but not the best (incremental); and the third (Spen) penalizes individuals performing below the average (sub-optimal).
Table 17: Sensitivity analysis of base reward configurations.
Configuration
run 1
run 2
run 3
avg.
(0.8, 0.2, 0.0) 1.0, -055
6.046
5.976
5.998
6.007
5.921
5,932
5,902
5,918
0.6, 0.3, -0.1
5,916
5,917
5.915
5,916
0.8, 0.2, -0.3] (default)
5.885
5.838
5.873
5,866
Our default configuration [0.8, -0.3) is not arbitrary but follows specific asymmetric design philosophy: Elite Reward > |Penalty| 2 Incremental Reward. The substantial elite reward (0.8) provides a strong signal for breakthroughs, prioritizing paradigm discovery. The incremental reward (0.2) acknowledges smaller gains, while the penalty tal reward effectively prunes unproductive directions. This asymmetry improves search efficiency by aggressively filtering noise while amplifying true innovations. Additionally, the coefficients fol- low a A-shaped sensitivity design, where the intermediate tier receives the largest gradient to encour- age rapid upward transitions, while the elite tier is assigned lower sensitivity to maintain stability. The results in Table [7 confirm that this theoretically grounded configuration yields the lowest ob- jective value. Deviating towards more aggressive rewards (c.g.. [1.0, 0.1, -.55) or different penalty structures disrupts this balance, resulting in inferior performance.
in magnitude than the incremen-
Standard Deviation Analysis. To comprehensively evaluate the stability and consistency of the models, we conducted three independent runs of all LLM-based AHD methods on several repre- sentative scales. We then summarized the objective function values of each run and reported their average (avg.) and standard deviation (std.), as shown in Table [8] These statistics provide clear indication of the variability across runs, offering a more reliable basis for subsequent performance comparison and robustness analysis.
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Table 18: Objective function values of three and their average(avg.) and standard deviation(std.))
Task
Method
runl
run2
run3
avg.
std.
TSPSO-construction
EoH
6.369
6.554
6.401
6.441
0.09887534
ReEvo
6.264
6.298
6.320
6,294
0.02821347
HSEvo
6.348
6,259
6.314
6.307
0.04491102
MCTS-AHD
6.359
6.288
6.303
6.317
0.03742103
Ours
6.135
6.087
6.041
6.088
0.04700355
TSP100-GLS
EoH
7.738
7.738
7.739
7.738
0.00034451
ReEvo
7.742
7.741
7.738
7.740
0,00173957
HSEvo
7.739
7.737
7.753
7.743
0.00894281
Ours
7.737
7.737
7.738
7.737
0,00007013
Online BPP 5k C100
EoH
2028
2027
2025
2027
1.11355287
ReEvo
2021
2023
2022
2022
0.60880932
HSEvo
2022
2024
2088
2045
37.59627641
MCTS-AHD
2027
2026
2026
2026
0.66901381
Ours
2019
2020
2021
2020
0.64291005
FSSP-GLS n100 ml0
EoH
5640
5645
5644
5643
2.60832002
Ours
5635
5638
5638
5637
1.67470225
C.8 CONSUMPTION OF TIME AND TOKEN
Compared to other LLM-based AHD methods, our approach demonstrates advantages in both time and token consumption, with particularly significant reductions in computational time. Based on qwen2. 5-max model, we calculated the runtime and token consumption of various methods across three problem domains: CSP-construction, Online BPP, and BO-CAF in Table 19
Table 19:
Time and token consumption with different methods.
Methods
Consumption
TSP-construction
Online BPP
BO-CAF
Time
1.2h
lh
2h
EoH
Input Token
0.8.M
0.5M
1.2M
Output Token
0.2M
0.2M
0.5M
Time
2h
ReEvo
Input Token
1.IM
Output Token
0.4M
Time
4h
3h
14h
MCTS-AHD
Input Token
1M
1M
1.3M
Output Token
0.3M
0.2M
0.6M
Time
40 min
36min
1h 0.8M
Ours
Input Token
0.5M
0.28M
Output Token
0.2M
0.12M
0.2M
D LIMITATION AND FUTURE WORK
D.1 LIMITATION
While HiFo-Prompt demonstrates significant advancements in automated heuristic design, its cur- rent architecture possesses inherent limitations that define the boundaries of its present capabilities.
First, the decision-mall making mechanism of the Evolutionary Navigator is predicated on static, handcrafted control logic. This rule-based system, while interpretable, lacks the capacity for self- adaptation. Its fixed thresholds for stagnation, progress, and diversity are calibrated for the evaluated
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problems but may not be universally optimal. potentially constraining its performance novel problem landscapes or over different evolutionary timescales. The Navigator can react to predefined states but cannot learn refine its control strategy from experience.
Second, the knowledge evolution within the Insight Pool is fundamentally intra-task. The framework excels at capturing, refining. and reusing design principles within the context of single optimization problem. However, the true generalizability of these learned insights across different problem do- mains remains an unevaluated and open question. We have not yet systematically validated whether insights distilled from solving one problem class can effectively bootstrap the learning process on a structurally different, unseen one.
D.2 FUTURE WORK
The limitations mentioned above naturally chart course for several high-impact avenues for future research, aimed at enhancing the framework's ' autonomy, generality, and strategic depth.
A primary research thrust will be to transcend the Evolutionary Navigator's current heuristic-driven design by developing it into a learned metacontroller. We propose parameterizing its control func- tion and leveraging techniques from the domain of Rent-eeeinforeementt Learning (Meta-RL). In this paradigm, the Navigator would operate as a high-level agent, learning a control policy that dynamically maps observational data from the evolutionary process- such as population diversity metries, fitness landscape topology, and rates of convergence to nuanced control vector. This vector would modulate critical evolutionary parameters in real-time, including operator probabili- ties, selection pressure, and even strategic alterations to the LLM prompts themselves. The reward signal for this meta-agent would be carefully engineered to reflect overarching goals of evolutionary efficiency, such maximizing the rate of fitness improvement or minimizing the computational cost to reach a performance threshold. Successfully implementing this would elevate the framework from system guided by static rules to one capable of learning and deploying its own adaptive control strategies online, thereby achieving superior order of autonomy and problem-specificity
Another critical, complementary direction is the systematic investigation of inter-task knowledge transfer, intending to evolve the framework from a single-task solver into a more general algo- rithmic discovery platform. This research will proceed along two parallel vectors. First, will conduct rigorous empirical evaluation of the framework's ' zero-shot and few-shot transfer capa- bilities. By applying mature Insight Pool-developed for a source task-to novel target domains, we can precisely quantify the generality of the learned principles and measure the extent to which discovery costs can be amortized across problems. Second, we will pioneer the exploration of more abstract and powerful knowledge representations that transcend the inherent ambiguities of natural language strings. This includes investigating structured canonical forms, such as predicate logic or probabilistic program sketches, and rich semantic representations like knowledge graphs. The cen- tral hypothesis is that such formalisms can more effectively decouple a core algorithmic invariant from its domain-speciff, instantiation, thereby creating more robust foundation for seamless and compositional cross-domain knowledge transfer.
E PROMPTS WITH FORESIGHT AND HINDSIGHT
This section provides the specific, concrete templates of the prompts used to guide the LLM, of- fering a transparent and reproducible view of the core interaction engine the heart of our HiFo- Prompt framework. The HiFo Hhindsight-Foresigee core methodology: the strategic deployment of distinct, context-aware prompts different stages of the evolutionary process. Foresight prompts are strategically employed during the initial generation and creative mutation stages, encouraging the I.I. to explore a diverse space of novel heuristic possibilities. In stark contrast, Hindsight prompts play critical role in reflection and data-driven refinement, leveraging empirical performance feedback from past evaluations to methodically prune the search space and systematically improve promising solutions.
name is not arbitrary; it directly reflects our
To ground these abstract concepts and make our prompts tangible, we anchor our examples in the canonical Online Bin Packing (OBP) problem. The objective in OBP is to assign incoming sequence of items of varying sizes into a minimum number of fixed-capacity bins, with the crucial constraint that each item must be placed without knowledge of future items. Within this problem
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context, our framework's specific task is to evolve high-performance Python scoring function, score(item, bin), which acts as the core decision-making logic. This function evaluates the suitability of specific candidate bin for incoming item. A higher returned score signifies placement, and the overarching control logic of the framework places the item in the bin that yields the maximum score. The following subsections provide verbatim templates for each distinct phase of the evolutionary process, clearly demonstrating how the system's 's guidance dynamically shifts its focus-from broad exploration focused exploitation-aa the search progresses.
desirable
E.1 INITIAL PROMPT STRATEGY II
The il operator uses the following prompt template to generate the initial population of heuristics. The guidance provided to the LLM at this stage comes from the initial state of the framework's components, including set of high-qualit, seed insights.
Prompt for Operator i1
Given a sequence items and set identical bins with fixed capacity, you need to assign each item to a bin to minimize the total number of bins used. The task can solved step-by-step by taking the next item and deciding which bin to place it in based on a score.
First, deseribe your algorithm and main steps in sentence. The description must be inside brace. Next, implement in Python as function named score. This function should accept 2 imput(s): fitem'. "bins'.
The function should return output(s): scores'...
The score function is designed to evaluate the placement options for a given item. It takes the item be placed and the current list of bins input. It returns list of numerical scores, with each score corresponding to a bin in the input list. This list of scores guides the heuristic in selecting the most suitable bin Consider these successful design principles I've observed recently:
item according to the generated logic.
< successful design principle from Insights pool>
<Another successful ful design principle...>
For the evolutionary regime, please pay special attention to: <A specific instruction from Design Directiven
Depending the regime, try significantly different parameter values exploseex)))oation)) or fine-tune existing ones (focus exploossxpliaaiinn) or combine both strategies (balanced_search), not give additional explanations.
E.2 RECOMBINATION PROMPT STRATEGY E1
The following template for the el operator a representative example of prompt used during the main evolutionary loop. It demonstrates how parent heuristies and the full, dynamic guidance from the meta-cognitive components are integrated.
Prompt for Operator el
Given sequence of items and of identical bins with fixed capacity, you need to assign each item a bin to minimize the total number bins used. The task can be solved step-by-step by taking the next item and deciding which bin to place it in based on a score. I have k existing algorithms with their codes as follows: No.1 algorithm and the corresponding code are: <leecrr
iessrrippii ip on the Fi rst algorithm:
<The Python code implemente ementat inn of the first al algorithm>
No.k algorithm and the corresponding code are: <Description the last algorithis thm>
<The Python code implementation
last algorithm>
Please help me create a new algorithm that has a totally different form from the given ones.
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First, describe your algorithm and main steps in sentence, enclosed in braces (). Next, implement in Python function named score. This function should accept 2 input(s): item', 'bins'>. The function should return output(s); <' scores' >. <Additional info on inputs & outputs> <0ther constraints or requirentn rementss: Consider these successful design principles I've observed recently:
<A successful design principle from Insights pool>
<Another success ful design principle...>
For the evolutionary regime, please pay special attention to: <A specific Design Directive for promoting structural noveity, preserving diversity, and aveiding premature convergence>
Depending on the regime, try significantly different parameter values (focus exploration), or fine-tune existing ones (focus_es) explootato),,i)) or combine both strategies (balanced search). Do not give additional explanations.
E.3 RECOMBINATION PROMPT STRATEGY E2
The e2 operator focuses on "motivated recombination." It prompts the LLM to first identify a com- mon "backhone" core principle shared by the parent heuristics and then to create new, improved algorithm based on that shared foundation.
Prompt for Operator e2
Given a sequence of items and set of identical bins with fixed capacity, you need to assign each item to a bin to minimize the total number of bins used. The task can be solved step-by-step by taking the next item and deciding which bin to place it based on score. I have k existing algorithms with their codes as follows:
No.1 algorithm and the corresponding code are:
<Description of the first algorithm>
<The Python code implemetaatio of the first algorithm>
No.k algorithm and corresponding code are: <Description of the Last algorithm>
<The Python code implementation of the Last algorithm>
Please help me create new algorithm that has totally different form from the given ones but can be motivated from them.
Firstly, identify the common backbone idea in the provided algorithms. Secondly, based on the back- bone idea, describe your new algorithm and main steps in one sentence, enclosed in braces (). Thirdly,
implement it in Python a function named score. This function should accept 2 imput(s): <" item'' "bins'>, The function should return output(s): scores'' <Additional info inputs outputs> <0ther constraints or requirements> Consider these successful design principles I've observed recently:
<A successful design principle from Insights pool>
<Another successful design principle>
For this recombination, please pay special attention to: <A speci fic Design Di rect vv for simplis poollification cation prune low- impact features>
Depending the regime, try significuntly different parameter values (focus exploration), or fine-tune existing ones exploitatio). combine both stralegies search), Do give additional explanations.
E.4 MUTATION PROMPT STRATEGY MI
The MI operator implements a form of targeted mutation, generating single-parent descendant by refining its most critical components. It performs surgical modifications to elements like scor- ing rules parameter weights, while meticulously safeguarding the parent's established, high- performing logic. This process is not random; it is guided by a synthesis recent, high-utility insights and high-level strategic directive. By injecting controlled, purposeful diversity, M1 en-
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ables the search to escape local optima without dismantling the parent's effective architecture. This deliberate balance between exploitation (refining what works) and exploration (seeking novelty) is erucial for accelerating convergence towards superior heuristics.
Prompt for Operator ml
Given a sequence of items and set of identical bins with fixed capacity, you need to assign each item bin to minimize the total number of bins used. The task can be solved step-by-step by taking the next item and deciding which bin place it based on have one algorithm with code as follows:
Algorithm description: <itescript i of the parent algorithms
Code:
<The Python code implementation the parent algorithm> Please assist me in creating new algorithm that diffferent but can modified version of the algorithm provided.
First, deseribe your new algorithm and main steps in one sentence, enclosed in braces ( Next, implement it in Python function named score. This function should accept 2 input(s): < iiemm, 'bins'>. The function should return output(s): <'scores' <Additional info inputs outputs> <0ther constraints or requirements> Consider these successful design principles I've observed recently:
< successful design principle from Insights pool>>
<Another successful design principle> For this mutation, please pay special attention to: <A speci fíc nesign pirectin ive for mutation ion
Depending the regime, try significantly different parameter values (focus exploration), fine-tuee existing ones (focus exploitation). combine both stralegies (balanced search). not give additional explanations.
E.5 MUTATION PROMPT STRATEGY M2
The M2 operator implements parameter mutation, targeted process to modify heuristic's numer- ical behavior. It instructs the LLM to first deconstruct the parent's score function to identify its key hyperparameters. Following this analysis, the model prompted to generate a new set parameter values. This generation can either explore novel configurations through significant changes fine- tune existing ones via subtle adjustments. This surgical approach methodically injects parametric diversity into the population while preserving the integrity of the core algorithmic logic.
Prompt for Operator m2
Given a sequence of items and set of identical hins with fixed capacity, you need to assign each item to a bin to minimize the total number of bins used. The task can be solved step-by-step by taking the next item and deciding which bin place it based on score.
have one algorithm with code as follows:
Algorithm description: <Description of the parent algorithm>
Code:
<The Python code implementation of the parent algorithm>>
Please identify the main algorithm parameters and
creating a new algorithm that has
different parameter settings of the score function provided. First, describe your new algorithm and main steps in one sentence, enclosed in braces ()- Next, implement it in Python
function named score. This function should accept 2 input(s): item', "bins'>, >. The function should return output(s): scocress'S <Additional info inputs outputs>> <0ther constraints or requirements> Consider these successful design principles I've observed recently:
<A successfui design principle from Insights pool>
<Ancther successful design principle>
When adjusting parameters, please pay special attention to: <A specific Design Directive for parameter tuning>
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Depending the regime, try significuntly different parameter values (focus exploration), or fine-tune existing ones (focus exploitation), combine both strategies (balanced search). not give additional explanations.
E.6 MUTATION PROMPT STRATEGY M3
The M3 operator is designed for structural simplification to enhance heuristic robustness and com- overfitting. It instructs the LLM to perform a critical analysis of the parent score function, specifically targeting components suspected of being over-specialized to in-distribution data. These potentially brittle or overly complex segments are then strategically pruned or streamlined. The out- come is more parsimonious and computationally lean implementation that is theorized to exhibit superior generalization to out-of-distribution scenarios, all while preserving the original function signature ensure architectural compatibility.
Prompt for Operator m3
Given sequence of items and of identical bins with fixed capacity, you need to assign each item a bin to minimize the total number bins used. The task can be solved step-by-step by taking the next item and deciding which bin to place it in based on a score.
I have one algorithm with its code as follows:
Code:
<The Python code implementar ementat. First, identify the main components in the function above. Next, analyze which these components may be overfitting to the in-distribution instances. Then, simplify those components to enhance gen- eralization to aut-of-distribution cases. Finally, provide the revised code, keeping the function name, inputs, and outputs unchanged.
the parent al algorithm>
Provide complete revised function implementation, preserving original signature. <hhe function name, inputs & outputs iicetiiicaioon from your prompt> Consider these successful design principles I've observed recently:
< successful design principle from Insights poo1>
<Another successful design principle>
When simplifying, please pay special attention to: <A specific Design Directive for s simplif wwllllfffiiiiin con>
Depending the regime, try significantly different parameter values (focus exploration), or fine-tune existing ones expliseex)l)iaatioo)) or combine both strategies (balch)ed_search)) Do not give additional explanations.
F MANAGING FORESIGHT AND HINDSIGHT KNOWLEDGE
F.1 HINDSIGHT EVOLUTION VIA INSIGHT DISTILLATION
To continually enrich our system's understanding of effective optimization strategies, we employ a Large Language Model (LLM) to distill high-level design principles from high-performing al- gorithms. This process is guided by a structured prompt, shown below, which provides the LLM with the descriptions and/or code of elite solutions discovered during an evolutionary run. The core instruction tasks the model with synthesizing concise, generalizable, and performance-positive pat- terns, drawing insights from both the conceptual descriptions and the code implementations. This automated extraction mechanism allows our system learn from own successes and progres- sively build more sophisticated knowledge base.
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Prompt Template for Insight Extraction
The following are core descriptions and/or code of high-performanee optimization algorithms evolved recently:
Algorithm 1:
<Ntural language description and/or code of elite
indivi vuua 1>
Algorithm 2:
<Natural Tanguage description and/or code of elite (and so on for the top 308 of the population)
individ dual
Algorithm n:
Please extract 1-2 concise, generic, and terns] from the above algorithms. These principles should be applicable to various combinatorial optimization problems, not just the specific problem domain. When formulating these principles, is essential draw insights from both the conceptual natural language descriptions and their cor- responding code implementations. Focus on identifying the underlying strategic design choices and algorithmic methodologies rather than superficial characteristics or specific implementation minutiae.
coarmmancc-positiive [design principles] or [effective pat-
Each principle/pattern must be expressed independent sentence in the following format:
Balance local optimization with global solution structure when making decisions.
Prioritize choices that muintain flexibility for future decision dleisioon-making making steps.
Implement adaptive mechanisms that respond to problem instance characteristics.
Provide only the list of principles, without any preamble or other explanatory text.
F.2 THE DIRECTIVE POOL FOR FORESIGHT
To operationalize the Evolutionary Navigator's high-level strategy, we map the chosen Exploitation, Balance-to specific Design Directive. Each directive is a fine-grained textual instruction, uniformly sampled from predefined pool corresponding to the active regime. This sampled directive then integrated into the generation prompt. This two-tiered mechanism facilitates fine fine-graid une-ggrined control over the generation process, ensuring model outputs are precisely aligned with the overarching evolutionary objective.
Design Directive
Balance:
Optimizing objective function evaluation criteria.
Considering the long-term impact of current decisions.
Balancing local optimality with global search strategies.
Improving algorithu robustness across different problem instances.
Managing computational complexity and time efficiency. Exploitation:
Refining core evaluation and scoring functions. Fine-ture Fine-tuning critical algorithm parameters and thresholds. Improving precision of existing heuristics and rules.
Reducing unnecessary computational overhead.
Exploration:
Exploring novel solution construction methodologies.
Investigating alternative problem decomposition approaches. Introducing new randomization or adaptive mechanisms. Experimenting with hybrid strategy combinations.
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F.3 INSIGHT SEED POOL
At the inception of the framework's 's execution, the LLM bootstraps the heuristic generation process by drawing from curated repository of Seed Insights, mechanism designed to mitigate the classic cold start problem and channel the model's creativity. These insights, which encapsulate established principles and canonical rules-of-thumb distilled from decades of human expertise in heuristic de- sign-such as the Shortest Processing Time principle in scheduling the Nearest Neighbor concept routing-arr are not rigid constraints but rather high-level conceptual guidelines. They are strate- gically injected into the foundational prompts, often framed within dedicated human knowledge block, to act as an intellectual scaffold. This initial infusion well-vetted knowledge serves to ground the search, preventing the generation of naive logically flawed heuristics and providing potent directional bias that immediately steers the early stages of algorithmic evolution away from vast, unproductive regions of the design space. By ensuring the process begins not from tabula rasa but from a high-quality, well-founded starting point, these seed insights exert persistent influence that accelerates convergence and significantly enhances the quality and novelty of all subsequent automated discovery.
Seed Insights
Design adaptive hybrid meta meta-heuristics synergistically fusing multiple search paradigms and dynamically operator parameters based search stage problem features.
Employ machine learning to mine problem structures and use learned insights to intelligently bias towards promising search regions.
Explore objective function engineering by introducing auxiliary objectives or dynamically adjusting weights to reshape the search landscape.
Construct problem-specializg solution representations and co-design dedicated operators fully leverage the representation's structure.
Implement intelligent diversification based on solution feature space analysis to systemati- cally target uncovered regions and escape local optima.
G THE USE OF LARGE LANGUAGE MODELS (LLMs)
Throughout the drafting and revision process of this manuscript, we employed Google's Gemini large language model as advanced writing and editing tool. Its the language and presenting our existing ideas and research. The model's contributions include: (1) correcting grammatical, spelling, and punctuation errors; (2) rephrasing complex sentences to improve readability and precision; and (3) suggesting adjustments to tone and style to ensure con- sistency with academic standards.
strictly limited to refining
Critically, the LLM was not used for any substantive intellectual contribution. All aspects of the research, including the formulation of research questions. literature review, methodology, data anal- ysis. and the drawing of conclusions, were conducted exclusively by the human authors. The authors critically evaluated each suggestion provided by the L.M for accuracy and appropriateness, and we retain full responsibility for all claims, arguments, and the final articulation of the work.
H GENERATED HEURISTICS
In this section, we compile the most successful heuristics produced by HiFo-Prompt, spanning the entire suite of experimental settings.
H.1 THE BEST TSP-CONSTRUCTIVE HEURISTIC
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Listing 1: Python Implementation of the Node Selection Strategy
import numpy as np
def scleet_ncxt_node (ourrent_nddd destination_node; unvisited
distance_matrix)
Selects the next node to visit based on hybrid scoring mechanisn involving
MST lookahead and cluster-aware simulation.
def calculate pruned mst_ccstttt_ttctt cost (nodes): """aacculates the MST cost for subset nodes, """
if len (nodes)
return
Create edges only between relevant nodes
edges
(distance ['distance_atrri([i]] matrix(i]] i, i) for i in nodes for i in nodes if i 51
edges sort sort (key=lambda x: x(0])
parent (node: node for nede in nodes)
def find
while parent [u] !- u:
parent [u] parent (parent [u]]
u parent [u]
return
=
for cost, u, v in edges: root_u, root_v fu), fvnd(v)
if root_u != root_v: parent [root_v] reet_u
mst_cost += cost return mst_cost
def veriorm_clusted aware snier__smmlaiiin (node)
"""Simulates a greedy path from the candidate node. ...
future_nodes [n for in unvisited_nodes if n !- node]
total_cost distame_natrixcuurrenntt current node
[node]
while future future_nodes: nodes:
# Greedy selection based on normalized distance ncxt_candidate = min(
future_nodes,
key=lambda x: distance_matrix (curreett] [x]
(np. sitiiinndddtttnnne_attiitit/ 1e-9)
tetal_cost +- distance dwstttnceemarriiieeeeeeeo [ctrrentt [next_cand] coxt__aandddtee
current next_ cixt__canidate
nuture_nodes remove (current))
total_cost +- di distance_matrix [current] [destination_nee return total_cost
def evaluate_candidaan (node)
"""Computes the weighted score for a candidate node.
future_nodes = [n for in nnvisited_nodes if n != node]
1. Cost (Lookahead)
pruneo_nst_cost = caloulltte_rrneedmet.oooo (future_nodes destination_node))
Normal 1 ized Immed ate Cost
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normalized cost distance natrix current mode] [node]
semaaaddddsaacccmacccoooooonnnod 1e-9)
# 3. Simulation Cost simulation_cost = httpmm_ccsssteeeeaeessmmmmiimiie (node)
Dynamic Weighting based en search progress arogress_ratio len (unvisited / len
weight_mst 0,4 - 0.15 progress_ratee
weight_normmaizee 0.3 0.1 progress_ratee
weight_simulation 0.3
return (weight_mst pruned_mst_cost Beight_normaltte Slight_simlation simulation_cost)
normalized cost
Beam search filtering to reduce computation beam_width max (2, (6, len (unvisited_nodes) // 3))
Pre-filter candidates using simple heuristic candidates sorted(
unvisited_nodes,,
key=lambda x: [nttane_matrixxcorreentt
[x] /
inp.mean (distance matrix [current_ndee] + 1e-9)
| :beam :beam_width 31
acored_candidates []
for candidate in candidates:
eval_score evaluate_candidee (candidate)
scored_candidates append
candidate)
Select the node with the minimum weighted score
next_node min (scored__anddddates, kcy=lambda x: x(0)) ) [1]
return next_node
H.2 THE BEST ONLINE BIN PACKING HEURISTIC
Listing 2: Evolved Scoring Function for Online BPP
import numpy as np
def score (item, bins):
***
Calculates the placement scores for candidate bins given an iten. Higher scores indicate better placement suitability.
Compute residuals (remaining space) after placing the item resicuals bins - item
Stochastic perturoation for enhanced exploration Helps break ties and avoid deterministic local optima perturbation = np.
(1.0, 0.2, size=len (bins))
Cubic residual scoring emphasize finer space usage precision Smaller residuals yield significantly higher base scores residual_scores
(1 + residuals ** 3) perturbation
Delayed penalty adjustment based on long-term impact of underutilization
Adaptive threshold with increased sensitivity threshold = mp.mean (bins) 0.4
delayed_penalty = np.where (residuals threshold,
1 1 (1 residuals ** 2),
0)
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Reward function bins maintaining balanced utilization over time Gaussian-like reward centered around the threshold inicoddutilizattionrooooa = oxpe -np.abs (residuals threshcld) 2 / (threshold ** 2)
Dynamic scaling factor based en cumulative fit ratio and trends fit rat_ratios item Dins
factor 0.6 0.4 np.tanh (fit_ration 5) Sharper scaling
Combine components:
Emphasizes long-term efficiency via scaling and delayed penalties (scaling_factor balanced sttncceddtiiizzatin..o reward) dolayod_ponalty)
factor aesidual_scores
return scores
H.3 THE BEST TSP-GLS HEURISTIC
Listing 3: Hybrid Edge Update Function for TSP
import numpy as np
def update edge distance (edge distance, local opp__tutt tour, edge used)
=**
Updates the edge distance matrix to penalize edges used in local
optima, facilitating escape from local basins using hybrid penalty strategy ***
Farameters for the novel hybrid strategy
learning_rate 0.3
siversity_factor 0.4
penalty_decay 0.85
min_penalty = 0.01
smoothing_term 1e-6
Compute normalized edge usage probabilities potal_usage = np. sum (edge_n_used) + smoothing term edge_prooabilitiis edge_n_used / total_usage
Measure global asymmetry to gauge search bias asymmetry np. cattee__rbbbbiitieee Normal i ze asymmetry to [0, 1] range
edge_probabilittiss T) mean ()
normal zed asymmetry asymmetry (1 smoothing_term
Icentity edges the Local optimal tour tour tour_edges =
(Iocal_opt_tourr] [i], local_opt_tourrit] [i+1]) for i in range (len (Iocal__ptt_toorr-11
Your_edges.appee (1ocalloppttcurr [-1], local_opt_tour(01)> F Close loop
@eaated_edge_disttee
sige__distance.copy
for (1, 1) in tour_edgest
if > 1: # Ensure symmetry handling
i, = j,
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Phase 1: Adaptive learning penalty
learning_penaltyy learning_rate 1 expexp heapp/dgg__oobaalttitellleele j1)
Phase 2: Diversity-driven modulation
civersity_penalty udiversity_facton aeemaliied_ssymmette np. (1 adge_probabilitiesll. il))
Dynamic weight modulation based edge usage frequency modulation_ penalty_decay ** edge_n_used[i, jl smoothing_term
penally diversity_penalty combined c modulation_factto
Add stochastic noise to encourage exploration
noise np, random. uniform(0, 1.5)
exploration_noise inniiiieeeneeeeeee min_penalty noise modulation_factor
# Combine penalties and update edge distance (symretric)
jotal_penaltyy combined_pendeeeeeeeeeeeeeeeeeee updated eege_ distance [i, Lotal_penale
exploration_noise
updated eediddegee______ distance il cotal_penall
return updated__ede__distaace
H.4 THE BEST ESSP-GLS HRURISTIC
Listing 4: Chaotic Matrix Adjustment and Job Selection
import numpy as np from clluser import KMeans
def eet_ (ourrent__ sequence, time matrix, m,
Generates a perturbed time matrix using chaotic dynamics and identifies
critical jobs to disturb based on clustering dispersion patterns.
def wawculate_makeeteeeeaaaaa (sequence, matrix, m):
"""Helper to compute the makospan of sequence,
machine_times [0]
for job in sequence:
machine_times [0] matrix 0]
for i in range (1, m):
(max(machine_tines) machire_tines Bechine_timeslle
-1])
matrixljob, il)
return max (aw.ciiine.ttiiai times)
Step 1: Compute current makespan baseline current_makespan calculate_makespa. (current_sequenee time_matrix, m)
Step 2: Chaotic adjustment using Logistic Map dynamics Introduces non-linear variance escape local optima new_matrix cime_matrix.copy
chaos_factor = 3.99 # Parameter near the edge of chaos chactic_values np. random. rand (n, m)
for range (10) Iterative chaotic updates
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chactic values chaos factor Shaotic_valuer chaotic_values)
Scalc chactic values to perturbation range [0.9, 1.1]
llaatic_perturationn = 0.9 0.2 chaotic_valaes nex_matrix pertuiignett new_matria np.clip (nex_matrix, 1, None) + Ensure valid processing times
Step 3: Cluster jops based execution time patterns KMeans on the perturbed matrix regarding the current sequence kmeans KMeans
n_clusters-max (2, // 10),
randem_stade Stnden_state=42,
autiit='uuto
) ). fit nnw__mtrrix currentt_sequence]]
job_labels kmeans, labels_
Calculate dispersion: average std deviation within each cluster "'uster_dispersion np.array
np. (new_matris [cprreett_seeee_eeeeea job_labels -- c], axis=0).
mean ()
for in range (kmeans clusters)
1)
Assign scores: jobs in highly dispersed clusters get higher priority
job _ro_sccres_ ar.array [Icluster_disperssiieee [label] for label in job_labels])
Sclect top diverse jobs for structural perturbation num_perturb = max (1, // 6) perturb_jobs np. argsort (-jeb_scores) [snun_perturb]
return new_matrix, perturb_jebs
42


Reviews:
[{"review_id": "XFHCXBdWLL", "reviewer": "Reviewer_44Vb", "summary": "The paper presents HiFo-Prompt, a framework for LLM-based automated heuristic design that combines Hindsight, which builds an evolving Insight Pool of distilled design principles, and Foresight, an Evolutionary Navigator that adaptively balances exploration and exploitation. The method is applied to several optimization tasks (TSP, Online BPP, FSSP, and BO), and the authors report improvements over prior AHD methods in both solution quality and sample efficiency.", "questions": "See the weakness.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 1, "presentation": 3, "contribution": 2, "strengths": "1. The motivation of this paper is clear and reasonable. The design ideas of global guidance and the insight pool are interesting and inspiring.  \n2. The similarity-based diversity discussion for the insight pool is conceptually stimulating.  \n3. The paper is clearly written and well organized, making it easy to follow.", "weaknesses": "1. I have concerns about the novelty threshold. The Insight Pool’s novelty filtering relies on Jaccard similarity over token sets. While this removes near-duplicate sentences, such a pure text-based comparison cannot capture semantic overlap. For example, one insight might be expressed in different ways. Since this novelty threshold is crucial for ensuring diversity, I worry this design may harm the actual effectiveness of the diversity mechanism.  \n\n2. The combination of a usage penalty and a recency bonus in $U(k, t)$ aims to balance exploration and exploitation, but the dynamics between these opposing terms are not analyzed. This could be sensitive in practice, and it would be helpful to justify or empirically demonstrate that this interaction leads to stable selection rather than oscillatory behavior. In particular, $w_u$ is a hyperparameter without ablation or sensitivity analysis, and the calculation of $B_r$ is not clearly presented. This reduces the soundness and reproducibility of the method.  \n\n3. The mapping from normalized performance $\\tilde{\\rho}$ to the effective credit $g_{\\text{eff}}$ uses manually chosen piecewise constants (0.8, 0.6, 0.5, -0.3, etc.) with no theoretical justification or ablation. While the idea of tiered reward regimes is understandable, the specific scaling choices seem ad hoc and may not generalize across tasks. It would strengthen the work to at least provide hints or guidelines on how to select these values.  \n\n4. The definition of phenotypic diversity as the fraction of non-identical algorithm text strings feels coarse and potentially misleading. The measure is a bit lexical that two code snippets are treated as completely different even if they differ only by refactoring or variable renaming, ignoring actual semantic or functional similarity (similar with my commen in 2.). As a result, the system may overestimate diversity and trigger unnecessary exploration. Moreover, this approach scales as $O(|P|^2)$ comparisons per generation, which may become inefficient for larger populations and increase token consumption for LLM-based evaluations. The diversity threshold is also arbitrary and not justified or ablated. Overall, the lack of semantic grounding and unclear efficiency raises concerns about the robustness and practicality of the Navigator’s diversity control.  \n\n5. The experimental section raises several concerns about fairness, reproducibility, and efficiency. Although the paper states that all LLM-based baselines were evaluated under the same Qwen 2.5-Max model, implementation details and prompt adaptations are not provided, so fairness remains unclear (like baselines might use different LLMs, thus they can not be comparied directly). HiFo-Prompt’s runtime on small TSP instances (Table 1) is about an order of magnitude slower than competing methods, with no explanation, contradicting the claim of improved convergence speed. Token-usage statistics are summarized only coarsely (Appendix C.7) without breakdown or cost analysis, leaving uncertainty about true computational overhead. The brief multi-LLM comparison (Table 9) covers only two tasks and lacks analysis, providing little evidence of model generality. Finally, runtime behavior is inconsistent across tables (slower in TSP 10–50 but faster in TSP 100–500) with no explanation. Together, these issues make it difficult to assess the practical efficiency and generalizability of the proposed framework.  \n\n6. The code does not seem to be provided. Even though the authors share the core prompts, several computational details remain unclear, as mentioned in earlier points. This makes it hard to guarantee reproducibility and verify the soundness of the proposed method.  \n\n**Minors**\n\n1. Very minor: for LaTeX quotation marks, please use the proper “…” format instead of plain double quotes. For example, in L055 the quotation marks are incorrectly formatted.  \n2. There are a few missing or incomplete citations in the appendix, such as at L811 and L1017. These should be corrected for completeness and consistency.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Q9sWuMG6CZ", "reviewer": "Reviewer_GjDu", "summary": "The paper proposes HiFo-Prompt with (i) a Hindsight module that distills reusable principles from successful candidates, and (ii) a Foresight module that adaptively switches explore/exploit/balance based on population state to guide LLM-based AHD. The proposed method is evaluated on TSP, Online BPP, FSSP, and BO.", "questions": "1. Line 387 says “100 instances at each of five sizes,” but Table 1 shows three, please fix the mismatch. Also, there are several misplaced “?” characters around lines 811, 854, 946, 967 that need cleanup.\n2. Can you present some of the actual heuristics generated and used to produce the reported results?\n3. In Table 5, removing the Insight Pool would make the method perform worse than EoH, which is surprising to me since the setup still retains the Foundational Prompts adapted from EoH and the Navigator module. Can you analyze the concrete differences between EoH and HiFo-Prompt w/o Insight Pool & Navigator that can explain this gap? Will the Navigator module improve baselines like EoH as a drop-in controller?\n4. How frequently does the Navigator select explore or exploit across runs? Have you tried an ablation that fixes the state to “balance” throughout to isolate the benefit of adaptive switching?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. The proposed method is well motivated and outperforms recent LLM-based AHD baselines across several tasks.\n2. The design details are well presented.\n3. The limitations and future directions are clearly analyzed.", "weaknesses": "1. For TSP step-by-step construction (i.e., Table 1), Appendix B.1 states that HiFo-Prompt involves LLM calls at inference time, however, it is unclear to me that whether such strategy also applies to the baselines. Please disambiguate: (a) If baselines also call the LLM at inference, please explain why HiFo-Prompt’s runtime is longer; (b) If they do not, please also report HiFo-Prompt under the same inference protocol for fair comparisons.\n2. The main text claims TSPLIB results are in Appendix C.1, but C.1 contains only descriptive text and a placeholder “Table ??”, with no actual results. Please add the promised table/metrics or revise the pointer.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "j37WZ6TvZN", "reviewer": "Reviewer_xien", "summary": "HiFo-Prompt tackles two common gaps in LLM-based Automatic Heuristic Design (AHD): lack of global search control and poor knowledge persistence. It adds  a rule-based Foresight meta-controller that watches population progress/diversity and switches prompts among explore/exploit/balance regimes, and a Hindsight Insight Pool that distills reusable design principles from elites with utility-based credit assignment, then injects top-scoring insights into subsequent prompts. The method obtains the best results among various LLM-AHD baselines.", "questions": "1. About dissimilarity (Eq 7): how are the textual descriptions calculated, and how do you ensure these are the same? (e.g.: will changing a single word make two descriptions different?)\n    \n2. It appears that there is a massive degradation if Qwen 2.5 max is not used in Table 9. How do you explain this?\n    \n3. What would happen if baseline methods also have the seed insights as part of the generator prompt?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The idea of tracking both local and global evolution dynamics via specialized modules is interesting and well executed\n    \n- Useful ablation studies\n    \n- Strong performance with few function evaluations", "weaknesses": "1. Seed insights are required by the method. Importantly, these insights could significantly improve generation quality: “Design adaptive hybrid meta-heuristics synergistically fusing multiple search paradigms and dynamically tune operator parameters based on search stage or problem features.” particularly is a high-quality handcrafted prompt that can have a substantial effect on the generation. For fairness of comparison, one should provide the same information in the prompt of other baselines, say EoH.\n    \n2. The novelty regarding global control and historical information aggregation is overstated, e.g., ReEvo already implements a short and long-term reflection that could be seen as a simpler version of hindsight. Discussions would be appreciated.\n    \n3. I am not convinced about the population size being chosen as 4. How can diversity be maintained in such as small population and avoid inbreeding?\n    \n4. I found the methodology section quite confusing, with many quite complicated implementations. For example, a decay rate is introduced, but there is no ablation or sensitivity analysis on it.  Eq. 3, which describes the evolutionary contribution, is full of hardcoded parameters, which are hard to parse, and the rationale for choosing them is not explained.\n    \n    1. On this point, please clarify whether $g$ is a minimization or maximization objective. In EoH, this is maximization, but Fig. 2 and equations suggest otherwise. However, section A.1 again takes $g$ as argmax. This is confusing.\n        \n5. No code is provided", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5XZJO2ogHY", "reviewer": "Reviewer_99Cj", "summary": "The paper proposes HiFo-Prompt, a prompting framework for LLM-based automated heuristic design that marries two modules: Foresight (an Evolutionary Navigator that steers exploration vs. exploitation from population signals) and Hindsight (an Insight Pool that distills and reuses design principles from successful code across generations). By decoupling “thoughts” from code, HiFo-Prompt supplies state-aware guidance and persistent knowledge. Experiments on TSP, FSSP, online bin packing, and black-box functions show state-of-the-art quality, faster convergence, and lower token/time cost than prior AHD systems; ablations confirm both modules matter.", "questions": "See weaknesses.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- Dual Foresight/Hindsight design elevates the LLM from code generator to meta-optimizer. \n- Evaluation sees evident performance gain.", "weaknesses": "- It’s unclear how you ensured a fair comparison under “the same query budget.” Does distilling insights consume additional queries? How many times did you run your method and the baselines? Did you use the same number of heuristic evaluations? Standard deviations are not reported, so the performance gains are not fully convincing.\n\n- The approach involves many hyperparameters. It’s unclear how they were chosen and how robust the method is to their settings.\n\n- The method relies heavily on pre-engineered prompts.\n\n- Similar ideas appear in EoH and ReEvo, where thoughts and reflections are distilled (both) and accumulated (the latter). Please clarify the novelty relative to these.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9MFzar3zyC", "reviewer": "Reviewer_9fSr", "summary": "This paper introduces HiFo-Prompt, a new method for LLM-based AHD. It integrates a Foresight module, featuring an evolutionary navigator that monitors population dynamics and steers the search using interpretable \"verbal gradients,\" and a Hindsight module, which maintains a self-evolving insight pool that distills successful design principles from high-performing code into knowledge. Evaluated on different heuristic design tasks like TSP, BPP, and FSSP, HiFo-Prompt demonstrates competetive performance, achieving superior results with greater computational efficiency than existing LLM-based AHD methods.", "questions": "The stagnation is measured by raw fitness, delta g, which is a fixed value and may suffer from poor generalization to different heuristic design tasks.\n\nThe semantic variety is calculated based on the textual descriptions of algorithms (eq. 7). Is it the thought or the code text? It seems that the indicator only counts when the two algorithms are exactly the same. Will it be too greedy?\n\nFor the Foresight module, how was the specific set of Design Directives in the pool (Appendix G.3) designed? Was there an ablation study on the impact of different directive wordings on the LLM's output quality?\n\nThe framework's knowledge management is confined to a single task; can the learned insights be generalized or transferred to new, unseen problem domains?\n\nWhat does the final algorithm look like? and how the insights and foresight prompts contribute to the generation of better heuristics, could you provide example illustrations?\n\nA discussion and comparison with related works on prompt evolution and hierarchical search is suggested [1-3].\n\n[1] MeLA: A Metacognitive LLM-Driven Architecture for Automatic Heuristic Design, arXiv\n\n[2] Large Language Model-driven Large Neighborhood Search for Large-Scale MILP Problems, ICML\n\n[3] Experience-guided reflective co-evolution of prompts and heuristics for automatic algorithm design, arXiv\n\n\nThere are typos and inadequate descriptions: e.g., \n\nline 811 ?\n\nline 854, 788\n\nFigure 1, The Left and Right can be misleading", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The self-evolving insight pool (Hindsight) and foresight instructions effectively prevent knowledge decay while enabling more strategic exploration of the heuristic space.\n\nIt achieves superior performance with fewer LLM calls and lower runtime compared to other state-of-the-art AHD methods.", "weaknesses": "The evolutionary navigator uses a fixed, rule-based policy with hand-tuned thresholds, which may lack generalization.\n\nThe paper would benefit from additional illustrations and a more extensive set of results to further support its claims", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)