Counting Cycles with Deepseek
This work addresses a difficult open problem in mathematics and AI, specifically for researchers in graph theory and computational methods, though it appears incremental as it relies on AI guidance rather than a fully autonomous solution.
The paper tackled the problem of deriving a Computationally Efficient Equivalent Form (CEEF) for the cycle count statistic, which lacks general solutions and involves complex combinatorics, by combining a novel approach with AI coding skills, resulting in new formulas for general cases that were previously undiscovered.
Despite recent progress, AI still struggles on advanced mathematics. We consider a difficult open problem: How to derive a Computationally Efficient Equivalent Form (CEEF) for the cycle count statistic? The CEEF problem does not have known general solutions, and requires delicate combinatorics and tedious calculations. Such a task is hard to accomplish by humans but is an ideal example where AI can be very helpful. We solve the problem by combining a novel approach we propose and the powerful coding skills of AI. Our results use delicate graph theory and contain new formulas for general cases that have not been discovered before. We find that, while AI is unable to solve the problem all by itself, it is able to solve it if we provide it with a clear strategy, a step-by-step guidance and carefully written prompts. For simplicity, we focus our study on DeepSeek-R1 but we also investigate other AI approaches.