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IEEE Transactions on Intelligent Transportation Systems

Solving Vehicle Routing Problem Using Grover Adaptive Search Algorithm

Lu Liu, Ling Qian, Xing-Yu Wu, Chen‐Rui Fan, Lu‐Fan Zhang, Dun-Bo Cai, Hui-Jun Lu, Tie-Jun Wang, Chuan Xi Wang

Published May 1, 2025• 11 citations
Publisher

Abstract

Intelligent Transportation Systems (ITS) occupy a pivotal position in the contemporary societal landscape, with one of their widespread applications being the vehicle routing problem, renowned for its classification as an NP-hard challenge. Traditional algorithms often require considerable time to address such problems or can only provide approximate solutions. Recent progress in quantum computation, however, offer a promising alternative by utilizing the parallelism inherent in quantum mechanics. In this study, we propose to solve the vehicle routing problem using the Grover adaptive algorithm. Compared to classical algorithms and existing quantum methods, our innovative algorithm achieves a quadratic speedup, with a time complexity of$O(K\sqrt{\frac{N}{M}})$. Additionally, it is applicable to a broader range of real-world scenarios, as it can directly handle linear inequality constraints without introducing slack variables. Notably, the algorithm operates independently of classical optimizers and gradient-based methods, thereby circumventing potential challenges such as the “barren plateaus” phenomenon. Experimental results from four instances demonstrate the feasibility and correctness of the proposed approach, highlighting its capability to handle inequality constraints easily.

DOI
10.1109/tits.2025.3562860

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Intelligent Transportation Systems (ITS) occupy a pivotal position in the contemporary societal landscape, with one of their widespread applications being the vehicle routing problem, renowned for its classification as an NP-hard challenge. Traditional algorithms often require considerable time to address such problems or can only provide approximate solutions. Recent progress in quantum computation, however, offer a promising alternative by utilizing the parallelism inherent in quantum mechanics. In this study, we propose to solve the vehicle routing problem using the Grover adaptive algorithm. Compared to classical algorithms and existing quantum methods, our innovative algorithm achieves a quadratic speedup, with a time complexity ofO(KNM)O(K\sqrt{\frac{N}{M}})O(KMN​​). Additionally, it is applicable to a broader range of real-world scenarios, as it can directly handle linear inequality constraints without introducing slack variables. Notably, the algorithm operates independently of classical optimizers and gradient-based methods, thereby circumventing potential challenges such as the “barren plateaus” phenomenon. Experimental results from four instances demonstrate the feasibility and correctness of the proposed approach, highlighting its capability to handle inequality constraints easily.
Vehicle Routing Problem (VRP) is considered one of the most challenging problems with application in several domains, including transportation and logistics distribution. VRP is known to be NP-Hard problem. Several algorithms have been proposed to solve VRP in polynomial time. However, these algorithms are inefficient if the VRP instance increases. Recently, researchers investigated how quantum computing can be used to solve VRP. In particular, two promising variational quantum algorithms have been studied, i.e., Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA). In this paper, we implement both algorithms on IBM Qiskit and compare their evaluation in solving several instances of VRP. We observe that current Noisy-Intermediate Scale Quantum (NISQ) devices cannot solve VRP instances beyond 5 nodes and 2 vehicles. Furthermore, we observe that classical optimizers still provide better results. However, we believe that with the rapid advancement of quantum computing manufacturing, VQE and QAOA can provide better performance as compared to classical computing,
With the expansion of complex transport infrastructure, new paths for linking different destinations have emerged, allowing the definition of multiple routes. The introduction of new roads may also lead to disruptions in traffic flow. This problem can be generalized into a special case of VRP, which aims to find the optimal route for a fleet of vehicles within a network of destinations. Finding solutions to this problem requires an exponential number of operations, hindering the ability of exact solvers to deal with large instances, which must therefore be tackled using heuristics, often leading to suboptimal results. Quantum-based algorithms may allow us to find exact solutions in polynomial time with high probability, offering a potential future alternative to classical solvers and in this work, we focus on the QAOA. Our novel contributions are a parameter-initialization strategy that improves the convergence probability and the integration of the subtour constraints directly into the optimization without using additional qubits by employing an approximation of exponential equality constraints. We believe that our results pave the way for effectively solving instances of VRP using quantum-based solutions, leveraging the space-efficient problem encoding and our initialization strategy, which limits the effect of barren-plateau under random parameter initialization.
This research addresses the Vehicle Routing Problem (VRP) with uncertain data and proposes a novel approach using quantum computing techniques. The problem involves optimizing vehicle routes considering uncertain customer demands, time windows, and vehicle capacities. We formulate the problem mathematically and develop an algorithmic framework to tackle it. The approach incorporates multiple scenarios based on the uncertainty distribution and selects the one with the minimum cost to optimize the vehicle routes. Through a numerical example, we demonstrate the effectiveness of the proposed approach in generating optimal routes that minimize the total distance traveled by the vehicles. The results highlight the solution quality, adaptability to uncertainty, and potential benefits in terms of cost reduction and resource utilization. While the computational efficiency of quantum computing approaches is a consideration, this research provides a promising direction for addressing uncertain optimization problems in logistics and transportation. Future research should focus on scalability and refinement of the algorithm to further enhance its applicability in real-world scenarios.
There has never been a more exciting time for the future of quantum computing than now. Real-world quantum computing usage is now the next XPRIZE. With that challenge in mind we have explored a new approach as a hybrid quantum-classical algorithm for solving NP-Hard optimization problems. We have focused on the classic problem of the Capacitated Vehicle Routing Problem (CVRP) because of its real-world industry applications. Heuristics are often employed to solve this problem because it is difficult. In addition, meta-heuristic algorithms have proven to be capable of finding reasonable solutions to optimization problems like the CVRP. Recent research has shown that quantum-only and hybrid quantum/classical approaches to solving the CVRP are possible. Where quantum approaches are usually limited to minimal optimization problems, hybrid approaches have been able to solve more significant issues. Still, the hybrid approaches often need help finding solutions as good as their classical counterparts. In our proposed approach, we created a hybrid quantum/classical meta-heuristic algorithm capable of finding the best-known solution to a classic CVRP problem. Our experimental results show that our proposed algorithm often outperforms other hybrid approaches.
Intelligent transportation systems (ITS) are a critical component of Industry 4.0 and 5.0, particularly having applications in logistic management. One of their crucial utilization is in supply-chain management and scheduling for optimally routing transportation of goods by vehicles at a given set of locations. This paper discusses the broader problem of vehicle traffic management, more popularly known as the Vehicle Routing Problem (VRP), and investigates the possible use of near-term quantum devices for solving it. For this purpose, we give the Ising formulation for VRP and some of its constrained variants. Then, we present a detailed procedure to solve VRP by minimizing its corresponding Ising Hamiltonian using a hybrid quantum-classical heuristic called Quantum Approximate Optimization Algorithm (QAOA), implemented on the IBM Qiskit platform. We compare the performance of QAOA with classical solvers such as CPLEX on problem instances of up to 15 qubits. We find that performance of QAOA has a multifaceted dependence on the classical optimization routine used, the depth of the ansatz parameterized byppp, initialization of variational parameters, and problem instance itself.
Route determination and optimization is a persistent challenge in transportation networks due to combinatorial explosion of options, dynamically changing conditions, competing optimization criteria, and complexity of real-world systems. Classical algorithms struggle to efficiently solve route optimization at scale due to exponential growth of possibilities, NP-hardness, limitations of dynamic programming approaches, and inability to leverage quantum parallelism. This paper explores the potential application of Grover’s algorithm in optimizing road transportation. Grover’s quantum algorithm offers a potential solution by using an oracle function to recognize optimal routes based on constraints and objectives encoded in a quantum state representing all possible paths. Constructive interference and amplitude amplification drive the system toward optimal solutions. By utilizing Grover’s algorithm, road transportation networks can benefit from improved route planning, traffic management, and resource allocation, ultimately enhancing efficiency and reducing congestion. The transportation network must be efficiently encoded into a quantum representation. An optimization oracle tailored to routing objectives then recognizes optimal solutions, which are amplified via a diffusion operator utilizing interference. Repeated application concentrates amplitude in optimal routes, which can be measured. Key challenges include encoding the network effectively within qubit resource constraints, designing an oracle to evaluate routes based on real-world conditions and optimization criteria, and translating the optimized quantum state into actionable routing guidance. If these challenges can be met, Grover’s algorithm may enable route optimization at a scale intractable for classical techniques.