用大模型生成代码，解决6类组合设计的开放难题。
Using Code Generation to Solve Open Instances of Combinatorial Design Problems
- 用大模型自动生成构造组合设计的代码，结合验证器反馈迭代优化。
- 在16类设计中成功解决6个长期未解的开放实例。
- 适合对组合数学与自动化求解感兴趣的算法研究者。
《组合设计手册》列出了多种组合设计类型及尚未确定存在的开放实例。本文提出一种构造性协议CPro1，利用大语言模型（LLMs）生成代码以构建组合设计，并解决其中部分开放实例。该协议从特定设计定义和可靠验证器出发，由大模型选择策略并编码实现，辅以自动化超参数调优与执行反馈。尽管多数生成代码失败，但通过大量候选方案的探索，自动尝试了多种标准方法（如模拟退火、遗传算法）及其变体（如不同代价函数），最终在16类设计中成功构造出6类的解决方案：对称与斜称加权矩阵、等距置换数组、打包数组、平衡三元设计以及佛罗伦萨矩形。
原文摘要 · Abstract (English)
The Handbook of Combinatorial Designs catalogs many types of combinatorial designs, together with lists of open instances for which existence has not yet been determined. We develop a constructive protocol CPro1, which uses Large Language Models (LLMs) to generate code that constructs combinatorial designs and resolves some of these open instances. The protocol starts from a definition of a particular type of design, and a verifier that reliably confirms whether a proposed design is valid. The LLM selects strategies and implements them in code, and scaffolding provides automated hyperparameter tuning and execution feedback using the verifier. Most generated code fails, but by generating many candidates, the protocol automates exploration of a variety of standard methods (e.g. simulated annealing, genetic algorithms) and experimentation with variations (e.g. cost functions) to find successful approaches. Testing on 16 different types of designs, CPro1 constructs solutions to open instances for 6 of them: Symmetric and Skew Weighing Matrices, Equidistant Permutation Arrays, Packing Arrays, Balanced Ternary Designs, and Florentine Rectangles.
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