# MViewRouter: Internalizing Geometric Equivariance via Multi-view Alternating Attention for Combinatorial Routing

arXiv: https://arxiv.org/abs/2606.01084 · HTML: https://mlsanity.com/paper/2606.01084/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL for games and combinatorial optimization, one of 15 directions in Deep reinforcement learning methods.
- Builds on 31 of its 40 references that are in the map.
- First posted on arXiv 31 May 2026.

## Placed in

- Deep reinforcement learning methods / [RL for games and combinatorial optimization](https://mlsanity.com/deeprl/rl-for-games-and-combinatorial-optimization/)

## Builds on (most cited first, 30 of 31)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [ViViT: A Video Vision Transformer](https://mlsanity.com/paper/2103.15691/) (2021)
- [VGGT: Visual Geometry Grounded Transformer](https://mlsanity.com/paper/2503.11651/) (2025)
- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Pointer Networks](https://mlsanity.com/paper/1506.03134/) (2015)
- [Multi-Task Learning as Multi-Objective Optimization](https://mlsanity.com/paper/1810.04650/) (2018)
- [Attention, Learn to Solve Routing Problems!](https://mlsanity.com/paper/1803.08475/) (2018)
- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018)
- [Neural Combinatorial Optimization with Reinforcement Learning](https://mlsanity.com/paper/1611.09940/) (2016)
- [POMO: Policy Optimization with Multiple Optima for Reinforcement Learning](https://mlsanity.com/paper/2010.16011/) (2020)
- [Reinforcement Learning for Solving the Vehicle Routing Problem](https://mlsanity.com/paper/1802.04240/) (2018)
- [DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2302.08224/) (2023)
- [Learning Improvement Heuristics for Solving Routing Problems](https://mlsanity.com/paper/1912.05784/) (2019)
- [Sym-NCO: Leveraging Symmetricity for Neural Combinatorial Optimization](https://mlsanity.com/paper/2205.13209/) (2022)
- [Neural Large Neighborhood Search for the Capacitated Vehicle Routing Problem](https://mlsanity.com/paper/1911.09539/) (2019)
- [NeuroLKH: Combining Deep Learning Model with Lin-Kernighan-Helsgaun Heuristic for Solving the Traveling Salesman Problem](https://mlsanity.com/paper/2110.07983/) (2021)
- [Multi-Decoder Attention Model with Embedding Glimpse for Solving Vehicle Routing Problems](https://mlsanity.com/paper/2012.10638/) (2020)
- [Efficient Active Search for Combinatorial Optimization Problems](https://mlsanity.com/paper/2106.05126/) (2021)
- [Simulation-guided Beam Search for Neural Combinatorial Optimization](https://mlsanity.com/paper/2207.06190/) (2022)
- [BQ-NCO: Bisimulation Quotienting for Efficient Neural Combinatorial Optimization](https://mlsanity.com/paper/2301.03313/) (2023)
- [Learning to Search Feasible and Infeasible Regions of Routing Problems with Flexible Neural k-Opt](https://mlsanity.com/paper/2310.18264/) (2023)
- [Winner Takes It All: Training Performant RL Populations for Combinatorial Optimization](https://mlsanity.com/paper/2210.03475/) (2022)
- [Pointerformer: Deep Reinforced Multi-Pointer Transformer for the Traveling Salesman Problem](https://mlsanity.com/paper/2304.09407/) (2023)
- [INViT: A Generalizable Routing Problem Solver with Invariant Nested View Transformer](https://mlsanity.com/paper/2402.02317/) (2024)
- [H-TSP: Hierarchically Solving the Large-Scale Travelling Salesman Problem](https://mlsanity.com/paper/2304.09395/) (2023)
- [Combining Reinforcement Learning with Lin-Kernighan-Helsgaun Algorithm for the Traveling Salesman Problem](https://mlsanity.com/paper/2012.04461/) (2020)
- [BOPO: Neural Combinatorial Optimization via Best-anchored and Objective-guided Preference Optimization](https://mlsanity.com/paper/2503.07580/) (2025)

## Abstract (arXiv)

Combinatorial routing problems such as the Traveling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP) are fundamental NP-hard problems with broad real-world applications. While recent deep reinforcement learning methods have shown promising performance, they typically handle geometric symmetries only through data augmentation, resulting in inconsistent decisions and limited generalization. To address this issue, we propose MViewRouter, a multi-view framework that internalizes geometric equivariance as a structural inductive bias to achieve invariant decision-making across routing problem variants. Our approach introduces a Multi-view Alternating Attention (MAA) mechanism that enables parallel processing over the D4 symmetry group, alternating between intra-view relational modeling and inter-view feature alignment. Furthermore, we optimize the policy via Collective Policy Gradient Aggregation (CPGA), leveraging consensus gradients from multiple symmetric views to stabilize training and accelerate convergence. Experiments on TSP and CVRP benchmarks, as well as real-world TSPLIB instances, demonstrate that MViewRouter achieves competitive solution quality and strong zero-shot generalization.