$A_2$DEPT: Large Language Model–Driven Automated Algorithm Design via Evolutionary Program Trees

Designing heuristics for combinatorial optimization problems (COPs) is a fundamental yet challenging task that traditionally requires extensive domain expertise. Recently, Large Language Model (LLM)-based Automated Heuristic Design (AHD) has shown promise in autonomously generating heuristic components with minimal human intervention. However, most existing LLM-based AHD methods enforce fixed algorithmic templates to ensure executability, which confines the search to component-level tuning and limits system-level algorithmic expressiveness. To enable open-ended solver synthesis beyond rigid templates, we propose Automated Algorithm Design via Evolutionary Program Trees (A$_2$DEPT), which treats LLMs as system-level algorithm architects. A$_2$DEPT explores the vast program space via a tree-structured evolutionary search with hybrid selection and hierarchical operators, enabling iterative refinement of complete algorithms. To make open-ended generation practical, we enforce executability with a lightweight program-maintenance loop that performs feedback-driven repair. In experiments, A$_2$DEPT consistently outperforms state-of-the-art baselines across standard and highly constrained benchmarks, reducing the optimality gap by an average of 9.8%. Our work implies that system-level algorithm synthesis is a viable and scalable paradigm for LLM-driven optimization.