NPG-Muse: Scaling Long Chain-of-Thought Reasoning with NP-Hard Graph Problems

Abstract

Reasoning Large Language Models (RLLMs) have recently achieved remarkable progress on complex reasoning tasks, largely enabled by their long chain-of-thought (Long CoT) capabilities. However, developing these Long CoT behaviors relies heavily on post-training with high-quality datasets, which are typically costly and human-curated (e.g., mathematics and code), leaving scalable alternatives unexplored. In this work, we introduce NP-hard (NPH) graph problems as a novel synthetic training corpus, as they inherently require deep reasoning, extensive exploration, and reflective strategies, which are the core characteristics of Long CoT reasoning. Building on this insight, we develop a two-stage post-training framework: (i) Long-CoT Supervised Fine-Tuning (SFT) on rejection-sampled NPH graph instances, which substantially enhances reasoning depth, and (ii) Reinforcement Learning (RL) with a fine-grained reward design, which sharpens reasoning efficiency. The resulting NPG-Muse-series models exhibit substantially enhanced Long CoT reasoning capabilities, achieving consistent gains across mathematics, coding, logical, and graph reasoning benchmarks. NPG-Muse-7B even surpasses QwQ-32B on NPH graph problems in both accuracy and reasoning efficiency. These results position NPH graph problems as an effective and scalable resource for advancing Long CoT reasoning in LLM post-training. Our implementation is available at https://github.com/littlewyy/NPG-Muse.