Directing to Diversify: Multi-Directional Search for Automatic Heuristic Design

Large language models (LLMs) have demonstrated strong potential to automate heuristic design for NP-hard combinatorial optimization (CO) tasks without human intervention. Recent methods predominantly rely on evolutionary computation frameworks, where heuristic diversification is mainly induced through EC-like operators or downstream selection, while the direction of heuristic transformation remains largely unharnessed. In this paper, we propose PRS, a multi-directional **P**erturbation–**R**ewrite **S**earch (**PRS**) framework for automatic heuristic design that factorizes heuristic transformation through directional search primitives at the generation level. Specifically, PRS operates on heuristic ideas through three complementary directions: expansion toward richer and more expressive designs, compression toward simpler and less specialized designs, and steering for fine-grained refinement, together enabling broader and more sustained exploration of diverse regions in the heuristic design space. PRS further organizes candidate exploration within a breadth-first search-based process with adaptive operator control, ensuring thorough candidate utilization and sustained exploration, thereby maintaining strong search dynamics while preventing premature convergence. Extensive empirical results across diverse CO tasks show that PRS consistently discovers high-quality heuristics, demonstrating the effectiveness and generality of explicitly directional exploration design.