The omnipresence of NP-hard combinatorial optimization problems (COPs) compels domain experts to engage in trial-and-error heuristic design. The long-standing endeavor of design automation has gained new momentum with the rise of large language models (LLMs). This paper introduces Language Hyper-Heuristics (LHHs), an emerging variant of Hyper-Heuristics that leverages LLMs for heuristic generation, featuring minimal manual intervention and open-ended heuristic spaces. To empower LHHs, we present Reflective Evolution (ReEvo), a novel integration of evolutionary search for efficiently exploring the heuristic space, and LLM reflections to provide verbal gradients within the space.

The omnipresence of NP-hard combinatorial optimization problems (COPs) compels domain experts to engage in trial-and-error heuristic design. The long-standing endeavor of design automation has gained new momentum with the rise of large language models (LLMs).

The omnipresence of NP-hard combinatorial optimization problems (COPs) compels domain experts to engage in trial-and-error heuristic design. The long-standing endeavor of design automation has gained new momentum with the rise of large language models (LLMs).

This paper introduces MeLA, a Metacognitive LLM-Driven Architecture that presents a new paradigm for Automatic Heuristic Design (AHD). Traditional evolutionary methods operate directly on heuristic code; in contrast, MeLA evolves the instructional prompts used to guide a Large Language Model (LLM) in generating these heuristics. This process of "prompt evolution" is driven by a novel metacognitive framework where the system analyzes performance feedback to systematically refine its generative strategy. MeLA's architecture integrates a problem analyzer to construct an initial strategic prompt, an error diagnosis system to repair faulty code, and a metacognitive search engine that iteratively optimizes the prompt based on heuristic effectiveness. In comprehensive experiments across both benchmark and real-world problems, MeLA consistently generates more effective and robust heuristics, significantly outperforming state-of-the-art methods. Ultimately, this research demonstrates the profound potential of using cognitive science as a blueprint for AI architecture, revealing that by enabling an LLM to metacogni-tively regulate its problem-solving process, we unlock a more robust and interpretable path to AHD.

Automated Heuristic Design (AHD) using Large Language Models (LLMs) has achieved notable success in recent years. Despite the effectiveness of existing approaches, they only design a single heuristic to serve all problem instances, often inducing poor generalization across different distributions or settings. To address this issue, we propose Automated Heuristic Set Design (AHSD), a new formulation for LLM-driven AHD. The aim of AHSD is to automatically generate a small-sized complementary heuristic set to serve diverse problem instances, such that each problem instance could be optimized by at least one heuristic in this set. We show that the objective function of AHSD is monotone and supermodular. Then, we propose Evolution of Heuristic Set (EoH-S) to apply the AHSD formulation for LLM-driven AHD. With two novel mechanisms of complementary population management and complementary-aware memetic search, EoH-S could effectively generate a set of high-quality and complementary heuristics. Comprehensive experimental results on three AHD tasks with diverse instances spanning various sizes and distributions demonstrate that EoH-S consistently outperforms existing state-of-the-art AHD methods and achieves up to 60\% performance improvements.

The omnipresence of NP-hard combinatorial optimization problems (COPs) compels domain experts to engage in trial-and-error heuristic design. The long-standing endeavor of design automation has gained new momentum with the rise of large language models (LLMs). This paper introduces Language Hyper-Heuristics (LHHs), an emerging variant of Hyper-Heuristics that leverages LLMs for heuristic generation, featuring minimal manual intervention and open-ended heuristic spaces. To empower LHHs, we present Reflective Evolution (ReEvo), a novel integration of evolutionary search for efficiently exploring the heuristic space, and LLM reflections to provide verbal gradients within the space.

Automatic Heuristic Design (AHD) is an active research area due to its utility in solving complex search and NP-hard combinatorial optimization problems in the real world. The recent advancements in Large Language Models (LLMs) introduce new possibilities by coupling LLMs with evolutionary computation to automatically generate heuristics, known as LLM-based Evolutionary Program Search (LLM-EPS). While previous LLM-EPS studies obtained great performance on various tasks, there is still a gap in understanding the properties of heuristic search spaces and achieving a balance between exploration and exploitation, which is a critical factor in large heuristic search spaces. In this study, we address this gap by proposing two diversity measurement metrics and perform an analysis on previous LLM-EPS approaches, including FunSearch, EoH, and ReEvo. Results on black-box AHD problems reveal that while EoH demonstrates higher diversity than FunSearch and ReEvo, its objective score is unstable. Conversely, ReEvo's reflection mechanism yields good objective scores but fails to optimize diversity effectively. With this finding in mind, we introduce HSEvo, an adaptive LLM-EPS framework that maintains a balance between diversity and convergence with a harmony search algorithm. Through experimentation, we find that HSEvo achieved high diversity indices and good objective scores while remaining cost-effective. These results underscore the importance of balancing exploration and exploitation and understanding heuristic search spaces in designing frameworks in LLM-EPS.

The omnipresence of NP-hard combinatorial optimization problems (COPs) compels domain experts to engage in trial-and-error heuristic design process. The long-standing endeavor of design automation has gained new momentum with the rise of large language models (LLMs). This paper introduces Language Hyper-Heuristics (LHHs), an emerging variant of Hyper-Heuristics that leverages LLMs for heuristic generation, featuring minimal manual intervention and open-ended heuristic spaces. To empower LHHs, we present Reflective Evolution (ReEvo), a generic searching framework that emulates the reflective design approach of human experts while far surpassing human capabilities with its scalable LLM inference, Internet-scale domain knowledge, and powerful evolutionary search. Evaluations across 12 COP settings show that 1) verbal reflections for evolution lead to smoother fitness landscapes, explicit inference of black-box COP settings, and better search results; 2) heuristics generated by ReEvo in minutes can outperform state-of-the-art human designs and neural solvers; 3) LHHs enable efficient algorithm design automation even when challenged with black-box COPs, demonstrating its potential for complex and novel real-world applications. Our code is available: https://github.com/ai4co/LLM-as-HH.