Counting Cycles with Deepseek

2025-05-23

AI Summary

The long-standing open problem of deriving computationally efficient equivalent forms (CEEFs) for cycle counting statistics in graph theory lacks a general, scalable solution. Method: We propose a novel “human-strategy-guided + AI-precise-execution” paradigm, leveraging the advanced programming capabilities of the DeepSeek-R1 large language model, integrated with graph-theoretic modeling, symbolic computation, and structured prompt engineering. Contribution/Results: We derive, for the first time, a closed-form CEEF for k-cycle counting on arbitrary graphs. The resulting analytic expression unifies and generalizes known formulas across multiple standard graph families (e.g., complete, bipartite, regular graphs) and has been rigorously verified by human experts. It substantially surpasses manual derivation in both computational efficiency and combinatorial generality. This work overcomes key limitations of end-to-end AI systems in higher-order combinatorial reasoning and establishes a reproducible, extensible methodology for AI-assisted mathematical discovery.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Humans and AI: Other Foundations of Human Computation & AI
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation

Abstract

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.

Problem

• Derive Computationally Efficient Equivalent Form for cycle count statistic
• Solve CEEF problem using AI and novel combinatorial approach
• Develop new graph theory formulas for general cycle counting cases

Innovation

• Combines novel approach with AI coding
• Uses delicate graph theory formulas
• Provides step-by-step guidance for AI