# RADAR: Learning to Route with Asymmetry-aware DistAnce Representations

arXiv: https://arxiv.org/abs/2603.03388 · HTML: https://mlsanity.com/paper/2603.03388/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 42 references that are in the map.
- First posted on arXiv 3 Mar 2026.

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

- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [DeepWalk: Online Learning of Social Representations](https://mlsanity.com/paper/1403.6652/) (2014)
- [Attention, Learn to Solve Routing Problems!](https://mlsanity.com/paper/1803.08475/) (2018)
- [POMO: Policy Optimization with Multiple Optima for Reinforcement Learning](https://mlsanity.com/paper/2010.16011/) (2020)
- [An Efficient Graph Convolutional Network Technique for the Travelling Salesman Problem](https://mlsanity.com/paper/1906.01227/) (2019)
- [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)
- [Neural Combinatorial Optimization with Heavy Decoder: Toward Large Scale Generalization](https://mlsanity.com/paper/2310.07985/) (2023)
- [Hybrid Genetic Search for the CVRP: Open-Source Implementation and SWAP* Neighborhood](https://mlsanity.com/paper/2012.10384/) (2020)
- [Learning to Iteratively Solve Routing Problems with Dual-Aspect Collaborative Transformer](https://mlsanity.com/paper/2110.02544/) (2021)
- [Deep Policy Dynamic Programming for Vehicle Routing Problems](https://mlsanity.com/paper/2102.11756/) (2021)
- [Matrix Encoding Networks for Neural Combinatorial Optimization](https://mlsanity.com/paper/2106.11113/) (2021)
- [GLOP: Learning Global Partition and Local Construction for Solving Large-scale Routing Problems in Real-time](https://mlsanity.com/paper/2312.08224/) (2023)
- [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)
- [Graph Neural Network Guided Local Search for the Traveling Salesperson Problem](https://mlsanity.com/paper/2110.05291/) (2021)
- [MVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts](https://mlsanity.com/paper/2405.01029/) (2024)
- [Towards Generalizable Neural Solvers for Vehicle Routing Problems via Ensemble with Transferrable Local Policy](https://mlsanity.com/paper/2308.14104/) (2023)
- [Unsupervised Learning for Solving the Travelling Salesman Problem](https://mlsanity.com/paper/2303.10538/) (2023)
- [RouteFinder: Towards Foundation Models for Vehicle Routing Problems](https://mlsanity.com/paper/2406.15007/) (2024)
- [UDC: A Unified Neural Divide-and-Conquer Framework for Large-Scale Combinatorial Optimization Problems](https://mlsanity.com/paper/2407.00312/) (2024)
- [GOAL: A Generalist Combinatorial Optimization Agent Learner](https://mlsanity.com/paper/2406.15079/) (2024)
- [Rethinking Light Decoder-based Solvers for Vehicle Routing Problems](https://mlsanity.com/paper/2503.00753/) (2025)
- [RRNCO: Towards Real-World Routing with Neural Combinatorial Optimization](https://mlsanity.com/paper/2503.16159/) (2025)

## Built on by (most cited first, 8 of 8)

- [Rethinking Constraint Awareness for Efficient State Embedding of Neural Routing Solver](https://mlsanity.com/paper/2605.10122/) (2026)
- [Two-Stage Learned Decomposition for Scalable Routing on Multigraphs](https://mlsanity.com/paper/2605.05389/) (2026)
- [Edge-aware Decoding for Neural Asymmetric Routing](https://mlsanity.com/paper/2606.02136/) (2026)
- [AGDN: Learning to Solve Traveling Salesman Problem with Anisotropic Graph Diffusion Network](https://mlsanity.com/paper/2606.19185/) (2026)
- [GeoPAR: Large-Scale Multi-Agent Combinatorial Optimization with Geometry-Guided Parallel Autoregressive Learning](https://mlsanity.com/paper/2609.00577/) (2026)
- [Hyper Algorithm Design Agent: Evolving Learnable Optimizer from Zero](https://mlsanity.com/paper/2609.35328/) (2026)
- [MiLoop: Selective Memory Propagation for Neural Combinatorial Optimization](https://mlsanity.com/paper/2610.01685/) (2026)
- [GREAT-ESDD: Graph Edge Attention Network for Effectively Steered Delivery Decisions](https://mlsanity.com/paper/2610.14792/) (2026)

## Abstract (arXiv)

Recent neural solvers have achieved strong performance on vehicle routing problems (VRPs), yet they mainly assume symmetric Euclidean distances, restricting applicability to real-world scenarios. A core challenge is encoding the relational features in asymmetric distance matrices of VRPs. Early attempts directly encoded these matrices but often failed to produce compact embeddings and generalized poorly at scale. In this paper, we propose RADAR, a scalable neural framework that augments existing neural VRP solvers with the ability to handle asymmetric inputs. RADAR addresses asymmetry from both static and dynamic perspectives. It leverages Singular Value Decomposition (SVD) on the asymmetric distance matrix to initialize compact and generalizable embeddings that inherently encode the static asymmetry in the inbound and outbound costs of each node. To further model dynamic asymmetry in embedding interactions during encoding, it replaces the standard softmax with Sinkhorn normalization that imposes joint row and column distance awareness in attention weights. Extensive experiments on synthetic and real-world benchmarks across various VRPs show that RADAR outperforms strong baselines on both in-distribution and out-of-distribution instances, demonstrating robust generalization and superior performance in solving asymmetric VRPs.