RADAR: Learning to Route with Asymmetry-aware DistAnce Representations

TL;DR

RADAR is a scalable neural framework that augments existing neural VRP solvers with the ability to handle asymmetric inputs and outperforms strong baselines on both in-distribution and out-of-distribution instances, demonstrating robust generalization and superior performance in solving asymmetric VRPs.

Abstract

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.

Figures (8)

• Figure 1: Framework of RADAR, which features two key designs for asymmetric VRPs: (1) an SVD-based embedding method that captures static asymmetry in the distance matrix; and (2) Sinkhorn-normalized attention in the encoder layers, replacing standard Softmax to enforce balanced incoming and outgoing attention, modeling dynamic asymmetry in representation learning.
• Figure 2: Different initialization performance under varying size of ATSP.
• Figure 3: Efficiency Score of informed initialization across different top k on ATSP.
• Figure 4: The left panel shows wall‐clock time versus ATSP size (log scale) for the total pipeline, SVD, and Sinkhorn. The right panel reports the percentage share of the total runtime attributable to SVD and Sinkhorn at each size. Each measurement averages 1,000 test instances.
• Figure 5: ATSP-100: Softmax vs. Sinkhorn. Left: Early training (epochs 1–10) showing faster convergence with Sinkhorn. Right: Late training (epochs 2091–2100) zoo m-in, where Sinkhorn stabilizes at a lower objective than Softmax.