# RADAR: Reasoning-Ability and Difficulty-Aware Routing for Reasoning LLMs

arXiv: https://arxiv.org/abs/2509.25426 · HTML: https://mlsanity.com/paper/2509.25426/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 35 of its 71 references that are in the map.
- First posted on arXiv 29 Sep 2025.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 30 of 35)

- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [s1: Simple test-time scaling](https://mlsanity.com/paper/2501.19393/) (2025)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [DROP: A Reading Comprehension Benchmark Requiring Discrete Reasoning Over Paragraphs](https://mlsanity.com/paper/1903.00161/) (2019)
- [FrugalGPT: How to Use Large Language Models While Reducing Cost and Improving Performance](https://mlsanity.com/paper/2305.05176/) (2023)
- [RouteLLM: Learning to Route LLMs with Preference Data](https://mlsanity.com/paper/2406.18665/) (2024)
- [L1: Controlling How Long A Reasoning Model Thinks With Reinforcement Learning](https://mlsanity.com/paper/2503.04697/) (2025)
- [Hybrid LLM: Cost-Efficient and Quality-Aware Query Routing](https://mlsanity.com/paper/2404.14618/) (2024)
- [The Illusion of Thinking: Understanding the Strengths and Limitations of Reasoning Models via the Lens of Problem Complexity](https://mlsanity.com/paper/2506.06941/) (2025)
- [RouterBench: A Benchmark for Multi-LLM Routing System](https://mlsanity.com/paper/2403.12031/) (2024)
- [Are We Done with MMLU?](https://mlsanity.com/paper/2406.04127/) (2024)
- [ThinkPrune: Pruning Long Chain-of-Thought of LLMs via Reinforcement Learning](https://mlsanity.com/paper/2504.01296/) (2025)
- [Fact, Fetch, and Reason: A Unified Evaluation of Retrieval-Augmented Generation](https://mlsanity.com/paper/2409.12941/) (2024)
- [Between Underthinking and Overthinking: An Empirical Study of Reasoning Length and correctness in LLMs](https://mlsanity.com/paper/2505.00127/) (2025)
- [Don't Overthink it. Preferring Shorter Thinking Chains for Improved LLM Reasoning](https://mlsanity.com/paper/2505.17813/) (2025)
- [Does Thinking More always Help? Mirage of Test-Time Scaling in Reasoning Models](https://mlsanity.com/paper/2506.04210/) (2025)
- [AdaCtrl: Towards Adaptive and Controllable Reasoning via Difficulty-Aware Budgeting](https://mlsanity.com/paper/2505.18822/) (2025)
- [IRT-Router: Effective and Interpretable Multi-LLM Routing via Item Response Theory](https://mlsanity.com/paper/2506.01048/) (2025)
- [Don't Overthink It: A Survey of Efficient R1-style Large Reasoning Models](https://mlsanity.com/paper/2508.02120/) (2025)
- [Scalable Chain of Thoughts via Elastic Reasoning](https://mlsanity.com/paper/2505.05315/) (2025)
- [Beyond GPT-5: Making LLMs Cheaper and Better via Performance-Efficiency Optimized Routing](https://mlsanity.com/paper/2508.12631/) (2025)
- [Think Smarter not Harder: Adaptive Reasoning with Inference Aware Optimization](https://mlsanity.com/paper/2501.17974/) (2025)
- [How to Select Datapoints for Efficient Human Evaluation of NLG Models?](https://mlsanity.com/paper/2501.18251/) (2025)
- [SMART: Simulated Students Aligned with Item Response Theory for Question Difficulty Prediction](https://mlsanity.com/paper/2507.05129/) (2025)
- [Thinking Fast and Right: Balancing Accuracy and Reasoning Length with Adaptive Rewards](https://mlsanity.com/paper/2505.18298/) (2025)
- [AdaReasoner: Adaptive Reasoning Enables More Flexible Thinking in Large Language Models](https://mlsanity.com/paper/2505.17312/) (2025)

## Built on by (most cited first, 6 of 6)

- [When Do LLMs Reason? A Dynamical Systems View via Entropy Phase Transitions](https://mlsanity.com/paper/2605.22873/) (2026)
- [VDAR-Router: Adaptive LLMs Routing via Verbalized Query Difficulty Analysis Retrieval](https://mlsanity.com/paper/2607.18098/) (2026)
- [RSI-Router: Evolving Subtask-Level LLM Routing and Skills for Cost-Efficient Agents](https://mlsanity.com/paper/2609.34712/) (2026)
- [When Do We Need LLMs? A Diagnostic for Language-Driven Bandits](https://mlsanity.com/paper/2604.05859/) (2026)
- [SeLMRoute: Probabilistic Semantic Evidence for Large Language Model Routing](https://mlsanity.com/paper/2609.34736/) (2026)
- [Rethinking Quality in Large Language Model Routing: Security-Conditioned Model-and-Budget Selection](https://mlsanity.com/paper/2610.18566/) (2026)

## Abstract (arXiv)

Reasoning language models have demonstrated remarkable performance on many challenging tasks in math, science, and coding. Choosing the right reasoning model for practical deployment involves a performance and cost tradeoff at two key levels: model size and reasoning budget, where larger models and higher reasoning budget lead to better performance but with increased cost and latency. In this work, we tackle this tradeoff from the angle of model configuration routing for different queries, and present RADAR (Reasoning-Ability and Difficulty-Aware Routing), a lightweight, interpretable, and scalable routing framework. Inspired by psychometrics, RADAR learns an item response model from model responses with different budgets to different queries, with interpretable parameters including query difficulties and model-budget abilities. RADAR then routes queries with higher difficulty to model-budget pairs with higher ability, and vice versa. We conduct extensive experiments on 8 widely used challenging reasoning benchmarks, demonstrating the superior performance of RADAR compared to state-of-the-art model routing methods. RADAR also exhibits query generalization capabilities, showing strong performance on out-of-distribution queries in all benchmarks. RADAR is also scalable and can efficiently integrate additional models by dynamically selecting a small set of evaluation queries to estimate their abilities.