Neural Decomposition Strategies for Multigraph Vehicle Routing Problems

Licentiate thesis, 2026

Vehicle Routing Problems (VRPs) are a canonical class of combinatorial optimization problems. In recent years, end-to-end machine learning has shown great promise for this class. Besides often achieving competitive solution quality, learning-based methods offer advantages such as amortized inference, scalability and adaptation across problem variants without extensive tuning.

Yet, progress has been confined to VRPs defined on simple graphs, with a single edge between each node pair. In many practically relevant scenarios, routing takes place on multigraphs, where many parallel edges with distinct attributes connect the same node pair. Multigraphs capture effects that are important in practice, but they also pose substantial challenges for existing neural methods. Beyond the node order, the routing policy must also decide which parallel edge to traverse, creating a substantially larger action space as instance size and edge multiplicity increase.

This thesis proposes three Graph Neural Network (GNN)-based methods for routing on multigraphs, each trained with reinforcement learning. The methods span a spectrum from operating directly on the dense multigraph to decomposing the policy by separating the choice of node permutation from the choice of connecting edges, improving computational scalability. First, GMS-EB (GNN-based Multigraph Solver, Edge-Based) operates end-to-end on the graph edges, building routes autoregressively one edge at a time. Second, GMS-DH (Dual-Head) prunes the multigraph into a simple graph by retaining one edge per node pair and then selects nodes autoregressively. Third, NEPF (Node-Edge Policy Factorization) reverses the stage order by first fixing the node permutation and then selecting the connecting edges.

We evaluate the three methods on a wide suite of single- and multi-objective routing problems against classical (non-learned) and neural baselines. Across these settings we show that the proposed methods are efficient and effective, remaining competitive with strong classical solvers while retaining the amortization benefits of end-to-end learning. By exploring the design space of neural combinatorial optimization for multigraph routing, this work bridges the gap between learned routing methods and the realistic settings in which they typically must operate