RADAR: Learning to Route with Asymmetry-aware Distance Representations

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. INTRODUCTION Vehicle Routing Problem (VRP) represents a classical NP-hard problem in combinatorial optimization with widespread applications such as transportation (Toro O et al., 2016) . It requires finding optimal routes to serve spatially distributed customers under various operational constraints. Traditional solvers (Helsgaun, 2017; Lawler & Wood, 1966) either suffer from exponential computational complexity or depend heavily on handcrafted heuristics, which often limit scalability on largescale VRPs. These challenges have sparked growing interest in neural combinatorial optimization (NCO), which leverages deep learning to develop data-driven solvers that approximate VRP solutions efficiently with reasonable optimality gaps (