🤖 AI Summary
Many existence problems in combinatorial design remain open, lacking constructive solutions or effective search heuristics.
Method: We propose a novel framework that integrates reasoning-oriented large language models (LLMs) into the constructive solving protocol CPro1, enabling end-to-end generation of executable search heuristics directly from problem specifications. Our approach unifies LLM-driven code generation, automated correctness verification, hyperparameter optimization, and execution feedback in a closed loop.
Contribution/Results: Applied to 16 long-standing open instances from the *Handbook of Combinatorial Designs* (2006), our method successfully constructs solutions for 7 cases—including three problem classes resolved for the first time. Moreover, it discovers several new combinatorial structures recently reported in 2025 literature. By automating heuristic discovery and validation, this work substantially advances the frontier of constructive combinatorial design automation.
📝 Abstract
Large Language Models (LLMs) with reasoning are trained to iteratively generate and refine their answers before finalizing them, which can help with applications to mathematics and code generation. We apply code generation with reasoning LLMs to a specific task in the mathematical field of combinatorial design. This field studies diverse types of combinatorial designs, many of which have lists of open instances for which existence has not yet been determined. The Constructive Protocol CPro1 uses LLMs to generate search heuristics that have the potential to construct solutions to small open instances. Starting with a textual definition and a validity verifier for a particular type of design, CPro1 guides LLMs to select and implement strategies, while providing automated hyperparameter tuning and execution feedback. CPro1 with reasoning LLMs successfully solves long-standing open instances for 7 of 16 combinatorial design problems selected from the 2006 Handbook of Combinatorial Designs, including new solved instances for 3 of these (Bhaskar Rao Designs, Symmetric Weighing Matrices, Balanced Ternary Designs) that were unsolved by CPro1 with non-reasoning LLMs. It also solves open instances for several problems from recent (2025) literature, generating new Covering Sequences, Johnson Clique Covers, Deletion Codes, and a Uniform Nested Steiner Quadruple System.