SoS1: O1 and R1-Like Reasoning LLMs are Sum-of-Square Solvers

TL;DR

This paper demonstrates that large language models can be guided to solve complex polynomial nonnegativity problems, achieving high accuracy with minimal fine-tuning and structured reasoning instructions.

Contribution

The introduction of the SoS-1K dataset and expert-designed reasoning instructions enables LLMs to effectively address a computationally intractable problem, advancing mathematical reasoning capabilities.

Findings

Structured instructions significantly improve LLM accuracy.

Fine-tuned 7B model outperforms larger models in polynomial nonnegativity tasks.

LLMs can be trained efficiently to solve NP-hard problems with minimal resources.

Abstract

Large Language Models (LLMs) have achieved human-level proficiency across diverse tasks, but their ability to perform rigorous mathematical problem solving remains an open challenge. In this work, we investigate a fundamental yet computationally intractable problem: determining whether a given multivariate polynomial is nonnegative. This problem, closely related to Hilbert's Seventeenth Problem, plays a crucial role in global polynomial optimization and has applications in various fields. First, we introduce SoS-1K, a meticulously curated dataset of approximately 1,000 polynomials, along with expert-designed reasoning instructions based on five progressively challenging criteria. Evaluating multiple state-of-the-art LLMs, we find that without structured guidance, all models perform only slightly above the random guess baseline 50%. However, high-quality reasoning instructions significantly improve accuracy, boosting performance up to 81%. Furthermore, our 7B model, SoS-7B, fine-tuned on SoS-1K for just 4 hours, outperforms the 671B DeepSeek-V3 and GPT-4o-mini in accuracy while only requiring 1.8% and 5% of the computation time needed for letters, respectively. Our findings highlight the potential of LLMs to push the boundaries of mathematical reasoning and tackle NP-hard problems.