Graph-Coarsening Approach for the Capacitated Vehicle Routing Problem with Time Windows

TL;DR

This work addresses the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW) by introducing a multilevel, spatio-temporal graph coarsening approach that aggregates customers into meta-nodes using a distance that combines travel time and schedule compatibility. The reduced problem is solved with classical heuristics and optionally QUBO-based solvers, followed by an inflation step that reconstructs feasible timed routes in the original graph. Experiments on Solomon benchmark instances demonstrate significant reductions in computation time while preserving or improving key metrics such as total distance, number of vehicles, and total route duration, with feasibility maintained after inflation. The study also highlights potential benefits from integrating quantum-inspired optimization to accelerate large-scale CVRPTW solving.

Abstract

The Capacitated Vehicle Routing Problem with Time Windows (CVRPTW) is a fundamental NP-hard optimization problem in logistics. Solving large-scale instances remains computationally challenging for exact solvers. This work introduces a multilevel graph coarsening and refinement framework that aggregates customers into meta-nodes using a spatio-temporal distance metric. The reduced problem is solved with classical heuristics and subsequently expanded back into the original space with feasibility corrections. Preliminary experiments on Solomon benchmark instances show that the proposed method reduces computation time while preserving or improving solution quality, particularly with respect to capacity and time window constraints. The paper also explores the integration of quantum-inspired optimization techniques, highlighting their potential to further accelerate large-scale vehicle routing tasks.

Figures (19)

• Figure 1: Illustration of the graph coarsening procedure. Nodes $C$ and $D$ are merged into super-node $CD$, and nodes $F$ and $G$ into super-node $FG$.
• Figure 2: Overall workflow of the proposed spatio--temporal graph coarsening framework for CVRPTW. The pipeline proceeds top--down: from graph augmentation and coarsening, through reduced-graph optimization, to route inflation and feasibility restoration.
• Figure 3: A subset of instances of the Solomon solomon1987 dataset used for the experiments. For clarity, edges are omitted.
• Figure 4: Comparison of the number of vehicles required across coarsened and inflated solutions for both Greedy and Savings heuristics. uncoarsened solutions show a larger spread and generally higher vehicle counts due to local infeasibilities. After inflation, the number of vehicles decreases significantly, indicating that the coarsening process successfully aggregates demand and leverages the reduced search space to find more efficient routes that better utilize vehicle capacity.
• Figure 5: Distribution of time window violations for coarsened and inflated solutions under Greedy and Savings heuristics. Coarsened runs, especially with the Savings heuristic, exhibit higher violation counts due to the relaxed temporal merging criteria. Inflation substantially reduces violations, confirming that the uncoarsening stage effectively restores temporal feasibility.