Rethinking Positional Encoding for Neural Vehicle Routing

2026-05-12

AI Summary

Standard position encodings in natural language processing exhibit limited effectiveness for the Vehicle Routing Problem (VRP) due to their neglect of the problem’s geometric and topological structure. This work proposes a geometry-guided hierarchical anisotropic position encoding that explicitly models VRP’s cyclic directionality, anisotropic distances, and depot-centered hierarchical organization by integrating intra-route distance-based cyclic consistency with inter-route angular information anchored at the depot. The proposed encoding unifies insights from natural language processing, graph Transformers, and route planning, significantly outperforming conventional index-based encodings across multiple VRP variants. Moreover, it demonstrates strong generalization capabilities across diverse model architectures, problem types, and distribution shifts.

Abstract

Transformer-based models have become the dominant paradigm for neural combinatorial optimization (NCO) of vehicle routing problems (VRPs), yet the role of positional encoding (PE) in these architectures remains largely unexplored. Unlike natural language, where tokens are uniformly spaced on a line, routing solutions exhibit several properties that render standard NLP positional encodings inadequate. In this work, we formalize three such structural properties that a routing-aware PE should respect, namely anisometric node distances, cyclic and direction-aware topology, and hierarchical depot-anchored global multi-route structure, combining them with a unifying design principle of geometric grounding. Guided by these criteria, we analyze and compare PE methods spanning NLP, graph-transformer, and routing-specific families, and propose a hierarchical anisometric PE that combines a distance-indexed, circularly consistent in-route encoding with a depot-anchored angular cross-route encoding. Extensive experiments across diverse VRP variants demonstrate that geometry-grounded PE consistently outperforms index-based alternatives, with gains that transfer across problem variants, model architectures, and distribution shifts.

Problem

• positional encoding
• vehicle routing problem
• neural combinatorial optimization
• Transformer
• geometric grounding

Innovation

• positional encoding
• vehicle routing problem
• geometric grounding
• anisometric distances
• hierarchical structure