Rethinking Positional Encoding for Neural Vehicle Routing

TL;DR

This paper investigates the importance of specialized positional encodings in transformer models for vehicle routing problems, proposing a geometry-grounded hierarchical encoding that improves performance across variants.

Contribution

It formalizes structural properties for routing-aware positional encoding and introduces a novel hierarchical anisometric encoding that outperforms existing methods.

Findings

Geometry-grounded PE outperforms index-based alternatives.

Gains transfer across different VRP variants and models.

Proposed encoding respects properties like anisometry, cyclic topology, and hierarchy.

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.