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arXiv:2602.07913

Multi-Agent Route Planning as a QUBO Problem

Renáta Rusnáková, Martin Chovanec, Juraj Gazda

cs.ROquant-phSubmitted Feb 8, 2026• Announced Feb 10, 2026
v1 · Feb 8, 2026v2 · Jun 29, 2026
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Abstract

Multi-Agent Route Planning considers selecting vehicles, each associated with a single predefined route, such that the spatial coverage of a road network is increased while redundant overlaps are limited. This paper gives a formal problem definition, proves NP-hardness by reduction from the Weighted Set Packing problem, and derives a Quadratic Unconstrained Binary Optimization formulation whose coefficients directly encode unique coverage rewards and pairwise overlap penalties. A single penalty parameter controls the coverage-overlap trade-off. We distinguish between a soft regime, which supports multi-objective exploration, and a hard regime, in which the penalty is strong enough to effectively enforce near-disjoint routes. We describe a practical pipeline for generating city instances, constructing candidate routes, building the QUBO matrix, and solving it with an exact mixed-integer solver (Gurobi), simulated annealing, and D-Wave hybrid quantum annealing. Experiments on Barcelona instances with up to 10 000 vehicles reveal a clear coverage-overlap knee and show that Pareto-optimal solutions are mainly obtained under the hard-penalty regime, while D-Wave hybrid solvers and Gurobi achieve essentially identical objective values with only minor differences in runtime as problem size grows.

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Multi-Agent Route Planning considers selecting vehicles, each associated with a single predefined route, such that the spatial coverage of a road network is increased while redundant overlaps are limited. This paper gives a formal problem definition, proves NP-hardness by reduction from the Weighted Set Packing problem, and derives a Quadratic Unconstrained Binary Optimization formulation whose coefficients directly encode unique coverage rewards and pairwise overlap penalties. A single penalty parameter controls the coverage-overlap trade-off. We distinguish between a soft regime, which supports multi-objective exploration, and a hard regime, in which the penalty is strong enough to effectively enforce near-disjoint routes. We describe a practical pipeline for generating city instances, constructing candidate routes, building the QUBO matrix, and solving it with an exact mixed-integer solver (Gurobi), simulated annealing, and D-Wave hybrid quantum annealing. Experiments on Barcelona instances with up to 10 000 vehicles reveal a clear coverage-overlap knee and show that Pareto-optimal solutions are mainly obtained under the hard-penalty regime, while D-Wave hybrid solvers and Gurobi achieve essentially identical objective values with only minor differences in runtime as problem size grows.
This article explores the use of QUBO as a practical and educational framework for solving complex optimization problems. QUBO models are widely applicable across fields such as scheduling, routing, and resource allocation, and are compatible with classical, hybrid, and quantum solvers. We discuss the core principles behind QUBO, its connection to physical systems like the Ising model, and its growing relevance in research and education. To demonstrate the practical side, we apply QUBO to a real-world-inspired Multi-agent Route Planning problem, where the goal is to maximize the network coverage while minimizing the total overlap of vehicle-routes. This use-case helps illustrate how QUBO formulations can model trade-offs, encode constraints, and scale to larger problem sizes. In addition, we evaluate midsized instances of the problem on D-Wave Systems’ Advantage2 quantum annealer. The study aims to bridge theoretical optimization with hands-on experimentation, making it suitable for both teaching and applied research.
Vehicle Routing Problem is a well-known optimization issue, which has many extensions resulting from practical appliances. The problem is NP-hard, therefore it may be impossible to find an optimal solution for big instances, with traditional computation methods. Recent development of quantum technology, may be a response for this challenge, which however requires a different formulation of the model. The problem is then modeled using Quadratic Unconstrained Binary Optimization (QUBO) equations. Such model can be run using quantum annealers (QA), designed specifically for optimization purposes. This paper is a systematic survey of Vehicle Routing Problem models that are formulated using QUBO, to be solved on quantum computers. The main goal of the paper is to check research gaps in this subject, compare and explain different approaches of modeling various problem extensions with QUBO. The results obtained in reviewed papers will be presented and analyzed. Moreover, the theoretical basis for quantum computation will be presented, with focus on D-Wave company solutions.
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.
Quadratic Unconstrained Binary Optimization (QUBO) is a central formulation for combinatorial optimization and has gained increasing attention due to its compatibility with quantum, hybrid quantum-classical, and quantum-inspired solvers. However, translating natural-language problem descriptions into correct QUBO formulations remains difficult, requiring the identification of binary variables, constraints, objective functions, penalty terms, and suitable penalty weights. This process is time-consuming and often demands substantial domain expertise. To address this challenge, we propose an end-to-end multi-agent framework that automatically generates QUBO formulations from natural-language problem descriptions, supported by structured or unstructured test cases. To evaluate its performance, We also introduce QUBOBench, a benchmark containing 100 combinatorial optimization problems across 12 application domains, curated from peer-reviewed literature, competitions, and canonical NP-hard problems. Experimental results show that our framework achieves 68% accuracy on QUBOBench, outperforming a direct single-call baseline by 22%. Further analysis identifies iterative self-repair as the most important component contributing to improved performance. The data and code are open-sourced at https://quitttcat.github.io/QuantumQUBOAgent.
Multi-Agent Path Finding (MAPF) remains a fundamental challenge in robotics, where classical centralized approaches exhibit exponential growth in joint-state complexity as the number of agents increases. This paper investigates Quadratic Unconstrained Binary Optimization (QUBO) as a structurally scalable alternative for simultaneous multi-robot path planning. This approach is a robotics-oriented QUBO formulation incorporating BFS-based logical pre-processing (achieving over 95% variable reduction), adaptive penalty design for collision and constraint enforcement, and a time-windowed decomposition strategy that enables execution within current hardware limitations. An experimental evaluation in grid environments with up to four robots demonstrated near-optimal solutions in dense scenarios and favorable scaling behavior compared to sequential classical planning. These results establish a practical and reproducible baseline for future quantum and quantum-inspired multi-robot coordinations.
Quantum annealing technologies aim to solve computational optimization and sampling problems. QPU (Quantum Processing Unit) machines such as the D-Wave system use the QUBO (Quadratic Unconstrained Binary Optimization) formula to define model optimization problems for quantum annealing. This machine uses quantum effects to speed up computing time better than classical computers. We propose a vehicle routing problem that can be formulated in the QUBO model as a combinatorial problem, which gives the possible route solutions increases exponentially. The solution aims to optimize the vehicle's journey to reach a destination. The study presents a QUBO formulation to solve traffic congestion problems on certain roads. The resulting route selection by optimizing the distribution of the flow of alternative road vehicles based on the weighting of road segments. Constraints formulated as a condition for the level of road density. The road weight parameter influences the cost function for each road choice. The simulations on the D-Wave quantum annealer show optimal results on the route deployment of several vehicles. So that each vehicle will be able to go through different road options and reduce road congestion accurately. This solution provides an opportunity to develop QUBO modeling for more complex vehicle routing problems for road congestion.