Can AI based methods help us make advances in complexity theory? We provide evidence towards answering this in the affirmative, using AlphaEvolve (an LLM code mutation agent) to obtain new results in three settings: a) We improve a recent result of Kunisky and Yu to obtain near-optimal upper and (conditional) lower bounds on certification algorithms for MAX-CUT and MAX-Independent Set on random 3- and 4-regular graphs. Our improved lower bounds are obtained by constructing nearly extremal Ramanujan graphs on as many as $163$ vertices, and our upper bounds are obtained via analytical arguments. b) We obtain new inapproximability results for MAX-4-CUT and MAX-3-CUT, proving that it is NP-hard to approximate them within factors of $0.987$ and $0.9649$ respectively, using AlphaEvolve to discover new gadget reductions. Our MAX-4-CUT result improves upon the SOTA of $0.9883$, and our MAX-3-CUT result improves on the current best gadget-based inapproximability result of $0.9853$, but falls short of the SOTA of $16/17$ that relies on a custom PCP (rather than a reduction from ``standard'' Håstad-style PCPs). c) Inapproximability for the metric Traveling Salesman Problem (TSP): We show that it is NP-hard to approximate the minimum cost tour within a factor of $111/110$ using AlphaEvolve to discover a new gadget, thus improving the SOTA of $117/116$. Along the way, we provide new modular soundness and completeness arguments that can be of independent interest. A key technical challenge we faced: verifying a candidate construction produced by AlphaEvolve is costly (sometimes requiring time exponential in the size of the construction). We used AlphaEvolve itself to evolve the verification procedure to be faster (sometimes by $10,000\times$ for our gadgets). Our results suggest that gadget based proofs would benefit from a pass through AI-based tools to obtain stronger results.

翻译：基于人工智能的方法能否帮助我们在复杂性理论方面取得进展？我们提供了肯定答案的证据，使用AlphaEvolve（一种大语言模型代码变异智能体）在三种场景中获得新结果：a) 我们改进了Kunisky和Yu最近的结果，在随机3-正则图和4-正则图上，为MAX-CUT和MAX-Independent Set的认证算法获得了接近最优的上界和（条件）下界。改进的下界通过构造多达163个顶点的近极值Ramanujan图获得，上界则通过解析论证获得。b) 我们获得了MAX-4-CUT和MAX-3-CUT新的不可近似性结果，证明在因子0.987和0.9649内近似它们分别是NP难的，这利用了AlphaEvolve发现新的小工具归约。我们的MAX-4-CUT结果改进了现有最优结果（0.9883），MAX-3-CUT结果改进了当前基于小工具的最佳不可近似性结果（0.9853），但未达到依赖自定义PCP（而非来自“标准”Håstad风格PCP的归约）的现有最优结果16/17。c) 度量旅行商问题的不可近似性：我们证明使用AlphaEvolve发现的新小工具，在因子111/110内近似最小成本环路是NP难的，从而改进了现有最优结果117/116。在此过程中，我们提供了新的模块化可靠性与完备性论证，这些论证本身可能具有独立价值。我们面临的关键技术挑战：验证AlphaEvolve生成的候选构造代价高昂（有时需要指数级时间于构造规模）。我们使用AlphaEvolve自身演化出更快的验证程序（对某些小工具提速达10,000倍）。我们的结果表明，基于小工具的证明可通过人工智能工具辅助获得更强结果。