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arXiv:2609.04593

GNN-Guided Graph Coarsening and Adaptive QUBO Penalties for the Capacitated Vehicle Routing Problem with Time Windows on a Quantum Annealer

Youssef Kamel Rezk, Paweł Góra

Published Sep 4, 2026• 0 citations
arXivHTMLPDF

Abstract

Graph coarsening reduces the large Quadratic Unconstrained Binary Optimization (QUBO) formulations arising when vehicle-routing problems are solved by quantum annealing. Nearby customers with compatible time windows are merged into super-nodes, the reduced problem is solved, and the solution is expanded to the original graph. For the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW), existing coarsening heuristics require family-specific tuning and remain unreliable on random instances. We address these limitations on the Solomon benchmark using simulated annealing and a D-Wave Advantage2 processor. We first introduce adaptive penalty calibration. Uniform penalty scaling has little effect, whereas controlling the internal coefficient range substantially improves raw samples. Removing non-binding constraints, normalising binding ones, and scaling the remaining penalties reduces mean raw constraint violations from 33.0 to 0.06 at the same solver budget (p=3.7e-11, n=56). A variable-count-preserving control attributes this gain to conditioning rather than problem size. Second, we replace the hand-tuned merge score with a graph neural network (GNN) using one configuration across all families. At N=10, it achieves 100% feasibility across all Solomon families, including R-type (100% vs. 80% for the tuned heuristic). Across N=10,...,100, feasibility is 83% vs. 69%, with the GNN better or tied on 85/90 instance-size pairs. At N=80,100, the difference is significant (p=0.002; 25/25 pairs), while the QUBO remains approximately 5-6 times smaller. Finally, hardware experiments reproduce the conditioning effect at fixed logical variable count: feasible samples increase from 0.02% to 39% across 13 instances. Classical repair with local search remains a reference bound for end-to-end solution cost.

DOI
10.48550/arxiv.2609.04593

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Graph coarsening reduces the large Quadratic Unconstrained Binary Optimization (QUBO) formulations arising when vehicle-routing problems are solved by quantum annealing. Nearby customers with compatible time windows are merged into super-nodes, the reduced problem is solved, and the solution is expanded to the original graph. For the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW), existing coarsening heuristics require family-specific tuning and remain unreliable on random instances. We address these limitations on the Solomon benchmark using simulated annealing and a D-Wave Advantage2 processor. We first introduce adaptive penalty calibration. Uniform penalty scaling has little effect, whereas controlling the internal coefficient range substantially improves raw samples. Removing non-binding constraints, normalising binding ones, and scaling the remaining penalties reduces mean raw constraint violations from 33.0 to 0.06 at the same solver budget (p=3.7e-11, n=56). A variable-count-preserving control attributes this gain to conditioning rather than problem size. Second, we replace the hand-tuned merge score with a graph neural network (GNN) using one configuration across all families. At N=10, it achieves 100% feasibility across all Solomon families, including R-type (100% vs. 80% for the tuned heuristic). Across N=10,...,100, feasibility is 83% vs. 69%, with the GNN better or tied on 85/90 instance-size pairs. At N=80,100, the difference is significant (p=0.002; 25/25 pairs), while the QUBO remains approximately 5-6 times smaller. Finally, hardware experiments reproduce the conditioning effect at fixed logical variable count: feasible samples increase from 0.02% to 39% across 13 instances. Classical repair with local search remains a reference bound for end-to-end solution cost.
In this paper, we explore the potential for quantum annealing to solve realistic routing problems. We focus on two NP-Hard problems, including the Traveling Salesman Problem with Time Windows and the Capacitated Vehicle Routing Problem with Time Windows. We utilize D-Wave's Quantum Annealer and Constrained Quadratic Model (CQM) solver within a hybrid framework to solve these problems. We demonstrate that while the CQM solver effectively minimizes route costs, it struggles to maintain time window feasibility as the problem size increases. To address this limitation, we implement a heuristic method that fixes infeasible solutions through a series of swapping operations. Testing on benchmark instances shows our method achieves promising results with an average optimality gap of 3.86%.
The Capacitated Vehicle Routing Problem (CVRP) involves finding the shortest possible routes for a fleet of vehicles, each with a fixed capacity limit, to serve a set of customers where every customer is visited exactly once. When CVRP is encoded as a Quadratic Unconstrained Binary Optimization (QUBO) model, one of the biggest practical hurdles is setting the penalty coefficients correctly — a process that is typically done by hand and has to be repeated for every new problem instance. This paper introduces a new technique called BOSA (Bayesian Optimization for Simulated Annealing), a framework that removes this manual step by using Gaussian Process-based Bayesian Optimization to find good penalty values automatically, while the underlying QUBO is solved using Simulated Annealing. Our hybrid approach integrates multi-start greedy construction and 2-opt local search to ensure solution feasibility when SA produces invalid routes. Experiments on 11 benchmark instances demonstrate solution distances of 12.04%- 33.26% relative to the best-known optima. On the two instances where comparison is possible, our proposed technique, BOSA achieves shorter routes than D-Wave (949 vs 1085 for A-n32-k5, and 2240 vs 2660 for A-n80-k10), suggesting that automated classical penalty tuning can match or exceed commercial quantum annealing solutions on these problem sizes without requiring quantum resources. Although not competitive with state-of-the-art CVRP solvers such as HGS and branch and cut algorithm, our automated tuning framework provides a foundation for future quantum-classical hybrid algorithms. It demonstrates the viability of BO-driven parameter optimization for QUBO-based combinatorial problems.
The Vehicle Routing Problem (VRP) is an NP-Hard optimization problem that has been widely studied over the last decades due to its wide practical applications. The goal of the VRP is to design efficient routes for vehicles performing service functions such as the distribution of goods from one or more central locations. There exist many variants to the VRP. These variations arise due to application-specific constraints such as vehicle capacities, delivery time windows, split deliveries, and many more. Due to the complexity of the problem, solvers are usually designed for specific variants and do not generally scale well to larger problem instances. As we approach the physical scaling limits of Moore's law, different novel special-purpose hardware are being designed for solving combinatorial optimization problems such as quantum and quantum-inspired devices. However, these devices are able to solve problems up to fixed variable sizes. In this work, we carry out experiments on the Fujitsu Digital Annealer (DA), quantum-inspired hardware, and show that we are able to get near-optimal results for instances of the Capacitated VRP (CVRP) that can be directly solved on the DA. We further develop a hybrid method for solving large-scale VRP instances. In order to achieve this, we develop a novel candidate route generation method, whose core solves a multi-objective clustering problem on the DA. Once a set of candidate routes has been selected, we then solve a Set Partitioning Problem on the DA to get a final solution. We apply our method to the CVRP and the CVRP with Time-Windows and show that we are able to get near-optimal solutions for problem instances significantly larger the base solvers could solve directly, within the same amount of computation time.
The Capacitated Vehicle Routing Problem (CVRP) is a combinatorial optimization problem that can be formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem, making it a potential application for near-term quantum optimization. This paper benchmarks the Quantum Approximate Optimization Algorithm (QAOA, p = 2) against Google OR-Tools Guided Local Search on a seeded instance with three customers and two vehicles. The study examines the QUBO formulation, energy spectrum, optimization behavior, penalty weights, Hamiltonian scaling, and circuit depths. The experiments show that the tested QAOA configurations do not reliably concentrate probability on optimal solutions. The classical solver identifies the optimal solution for the selected instance. The study provides a reproducible baseline and discusses limitations and future research directions. The findings are limited to the tested instance, formulation, and optimization configurations.
Capacitated vehicle routing with time windows (CVRPTW) is a natural target for coherent Ising machines (CIMs), but a direct multi-vehicle arc encoding scales as O(mN2) and exceeds the variable budget of current CIM-compatible systems. We argue the bottleneck is encoding density, not expressiveness, and present LSQ, a hardware-aware sparse Quadratic Unconstrained Binary Optimization (QUBO) framework that decouples CVRPTW into a compact customer-to-route assignment QUBO and a classical intra-route ordering step under a soft no-wait service convention. LKH candidate edges compress the per-route edge space from O(N2) to O(KN), and a per-route dynamic-penalty subroutine encodes time-window sensitivities as binary variables in a round-wise outer loop. On a six-vehicle, 51-node reference instance curated from long-term operational data, LSQ shrinks the maximum single-submission QUBO from 15,300 arc variables to 342 logicalQUBOvariables (∼45× compression), cuts travel time by 22.9% (74 vs. 96), and cuts route duration by 11.2% (174 vs. 196) against an OR-Tools soft-window baseline at the same fleet size. OR-Tools retains an advantage on raw time-window penalty (1600 vs. 3540) and runtime; under the scalar operational cost ∑kT(πk)+∑iℓi(τi), OR-Tools is therefore the better single-objective solver, and the comparison is a multi-objective trade-off rather than a scalar dominance claim. Ablations confirm that the LKH prior recovers Held–Karp on a 15-customer TSP at 53 vs. 120 variables and that the dynamic-penalty encoding reduces compressed time-window loss by 15.25% at constant travel. All hardware claims refer to QUBO sizing on a Kaiwu/CIM-compatible backend, not physical CIM execution.
This study develops a hybrid quantum-classical framework for constrained vehicle routing problems, focusing on the pickup-and-delivery problem with time windows. Instead of casting the full routing problem as a stand-alone quantum optimization task, we embed shallow quantum samplers inside the repair phase of an Adaptive Large Neighbourhood Search (ALNS) heuristic. A Deep Q-Network controller decides whether each reduced repair subproblem should be handled by a classical repair heuristic or by a quantum sampler, using features that describe the local repair structure and predicted hardware reliability. IBM Heron experiments are used to calibrate an empirical noise-aware model for local quantum repair circuits. Across the tested instances, quantum repair is admissible in only about 16% of reduced repair states and is not superior on average. However, under selected matched repair budgets, quantum-enabled repair reduces the final gap relative to standard ALNS in 29 of 36 tested settings. These results suggest that near-term quantum sampling is most useful as a selective local repair mechanism rather than as a replacement for classical routing heuristics.