We demonstrate the power of human-LLM collaboration in tackling open problems in theoretical computer science. Focusing on combinatorial optimization, we refine outputs from the FunSearch algorithm [Romera-Paredes et al., Nature 2023] to derive state-of-the-art lower bounds for standard heuristics. Specifically, we target the generation of adversarial instances where these heuristics perform poorly. By iterating on FunSearch's outputs, we identify improved constructions for hierarchical $k$-median clustering, bin packing, the knapsack problem, and a generalization of Lovász's gasoline problem - some of these have not seen much improvement for over a decade, despite intermittent attention. These results illustrate how expert oversight can effectively extrapolate algorithmic insights from LLM-based evolutionary methods to break long-standing barriers. Our findings demonstrate that while LLMs provide critical initial patterns, human expertise is essential for transforming these patterns into mathematically rigorous and insightful constructions. This work highlights that LLMs are a strong collaborative tool in mathematics and computer science research.

翻译：我们展示了人类与大语言模型协作在解决理论计算机科学开放性问题方面的强大能力。聚焦于组合优化领域，我们通过改进FunSearch算法[Romera-Paredes等人，《自然》2023]的输出，为标准启发式算法推导出最先进的性能下界。具体而言，我们致力于生成使这些启发式算法表现不佳的对抗性实例。通过对FunSearch输出结果进行迭代优化，我们为层次化$k$-中值聚类、装箱问题、背包问题以及Lovász汽油问题的推广形式提出了改进的构造方案——其中某些问题尽管持续受到关注，但其下界在过去十余年间鲜有实质突破。这些成果表明，专家监督能够有效从基于大语言模型的进化方法中提炼算法洞见，从而突破长期存在的理论障碍。我们的研究证明，虽然大语言模型提供了关键性的初始模式，但人类专业知识对于将这些模式转化为数学严谨且富有洞察力的构造至关重要。本工作凸显了大语言模型在数学与计算机科学研究中作为强大协作工具的价值。