# FlexRouter: Learning Complementary Model Sets for Flexible LLM Routing

arXiv: https://arxiv.org/abs/2609.38585 · HTML: https://mlsanity.com/paper/2609.38585/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 36 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 25 of 25)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [Challenging BIG-Bench Tasks and Whether Chain-of-Thought Can Solve Them](https://mlsanity.com/paper/2210.09261/) (2022)
- [Instruction-Following Evaluation for Large Language Models](https://mlsanity.com/paper/2311.07911/) (2023)
- [FrugalGPT: How to Use Large Language Models While Reducing Cost and Improving Performance](https://mlsanity.com/paper/2305.05176/) (2023)
- [LLM-Blender: Ensembling Large Language Models with Pairwise Ranking and Generative Fusion](https://mlsanity.com/paper/2306.02561/) (2023)
- [Mixture-of-Agents Enhances Large Language Model Capabilities](https://mlsanity.com/paper/2406.04692/) (2024)
- [Dynamic Neural Networks: A Survey](https://mlsanity.com/paper/2102.04906/) (2021)
- [RouterBench: A Benchmark for Multi-LLM Routing System](https://mlsanity.com/paper/2403.12031/) (2024)
- [RouterDC: Query-Based Router by Dual Contrastive Learning for Assembling Large Language Models](https://mlsanity.com/paper/2409.19886/) (2024)
- [RouterEval: A Comprehensive Benchmark for Routing LLMs to Explore Model-level Scaling Up in LLMs](https://mlsanity.com/paper/2503.10657/) (2025)
- [EmbedLLM: Learning Compact Representations of Large Language Models](https://mlsanity.com/paper/2410.02223/) (2024)
- [Harnessing the Power of Multiple Minds: Lessons Learned from LLM Routing](https://mlsanity.com/paper/2405.00467/) (2024)
- [Learning the Parameters of Determinantal Point Process Kernels](https://mlsanity.com/paper/1402.4862/) (2014)
- [Multi-Document Summarization with Determinantal Point Processes and Contextualized Representations](https://mlsanity.com/paper/1910.11411/) (2019)
- [Faster Greedy MAP Inference for Determinantal Point Processes](https://mlsanity.com/paper/1703.03389/) (2017)
- [Personalized Bundle Recommendation in Online Games](https://mlsanity.com/paper/2104.05307/) (2021)
- [Learning to Route LLMs from Bandit Feedback: One Policy, Many Trade-offs](https://mlsanity.com/paper/2510.07429/) (2025)

## Built on by (most cited first, 1 of 1)

- [CopRoute: Dependence-Aware Conformal Routing for Language Models](https://mlsanity.com/paper/2610.13276/) (2026)

## Abstract (arXiv)

Existing Large Language Model (LLM) routing methods score LLMs independently to select top-k models. However, this ignores model correlations and enforces a rigid computational budget. Consequently, routers often select redundant models that share failure modes, limiting the overall probability of success. To address this, we propose FlexRouter, a routing framework that explicitly models model complementarity. FlexRouter optimizes for textitanswer coverage, maximizing the probability that at least one selected model yields a correct response. This objective aligns with practical inference pipelines where multiple candidate outputs are generated and a downstream verifier or user selects the final one. We formulate routing as a coverage-oriented subset selection problem and model the routing policy using Determinantal Point Processes (DPPs), which naturally capture both model competence and redundancy. To directly optimize coverage without requiring a ground-truth target subset, we introduce a training objective based on marginalizing over failure sets. During inference, we employ a greedy strategy based on marginal log-determinant gains, enabling the router to adaptively determine subset sizes without a predefined budget. Extensive experiments on the large-scale RouterEval benchmark demonstrate that our proposed FlexRouter achieves higher coverage with lower redundancy across both in-domain and out-of-domain tasks than strong baselines while maintaining flexible inference cost.