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Cybernetics and Computer Technologies

An Overview of Algorithms for Solving Vehicle Routing Problems in the Quantum-Classical Cloud

Leonid Hulianytskyi, Vyacheslav Yu. Korolyov, Oleksandr Khodzinskyi

Published Jul 28, 2023• 1 citation
Publisher

Abstract

Introduction. The hope of solving the problem of the avalanche-like growth of requirements for computing power, essential for solving complex routing problems and other problems of combinatorial optimization, relies on the latest quantum computers, in the development of which governments and corporations invest multi-billion investments. The article examines modern routing algorithms and performs their analysis and verification, if the authors of the algorithm provided appropriate test programs. The purpose of the article is to review the current state of development in the field of development of routing algorithms for hybrid quantum-classical clouds, analyze them and propose a classification of algorithms. Results. Modern quantum computers (QCs) make it possible to find approximate solutions to some of mathematical problems faster than classical computers. The inaccuracy of the solutions obtained by the QC is a consequence of physical and technological limitations: calculation errors are caused by thermal noise, a small number of computational elements - qubits and connections between them, which requires the decomposition of the problem and the use of heuristic algorithms. The analysis of approaches to the solution of optimization problems on QC allows us to single out: quantum response and variational search of eigenvalues based on quantum logic gates as the general directions of development of the vast majority of algorithms for solving routing problems. The considered algorithms reduce the vehicle routing problem to a quadratic unconstrained binary optimization problem, which is isomorphic to the Hamilton-Ising model. In this form, the problem is suitable for embedding in QC, which finds an approximate solution that has the best statistical reliability or corresponds to the quantum state with the lowest energy. As a separate class, vehicle routing algorithms for classical computers that use quantum computing to accelerate problem solving can be distinguished. For example, neural networks that calculate weighting factors using QC or an ant algorithm that calculates a pheromone trail in a hybrid cloud. It should be mentioned the quantum-inspired algorithms, which are based on software tools for the simulation of QC and the corresponding libraries and allow creating an effective class of algorithms for solving problems of vehicle routing. Conclusions. Combining hardware quantum annealing with a number of software tools for calculating optimization problems for classical computers in a hybrid quantum-classical cloud service allows to obtain advantages in speed and accuracy of some types of complex optimization problems of a commercial scale, in particular, routing vehicles, which is already bringing substantial profits to a number of corporations. Keywords: vehicle routing problem, quantum computer, annealing, combinatorial optimization, traveling salesman problem, clustering, qubit.

DOI
10.34229/2707-451x.23.2.3

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Introduction. The hope of solving the problem of the avalanche-like growth of requirements for computing power, essential for solving complex routing problems and other problems of combinatorial optimization, relies on the latest quantum computers, in the development of which governments and corporations invest multi-billion investments. The article examines modern routing algorithms and performs their analysis and verification, if the authors of the algorithm provided appropriate test programs. The purpose of the article is to review the current state of development in the field of development of routing algorithms for hybrid quantum-classical clouds, analyze them and propose a classification of algorithms. Results. Modern quantum computers (QCs) make it possible to find approximate solutions to some of mathematical problems faster than classical computers. The inaccuracy of the solutions obtained by the QC is a consequence of physical and technological limitations: calculation errors are caused by thermal noise, a small number of computational elements - qubits and connections between them, which requires the decomposition of the problem and the use of heuristic algorithms. The analysis of approaches to the solution of optimization problems on QC allows us to single out: quantum response and variational search of eigenvalues based on quantum logic gates as the general directions of development of the vast majority of algorithms for solving routing problems. The considered algorithms reduce the vehicle routing problem to a quadratic unconstrained binary optimization problem, which is isomorphic to the Hamilton-Ising model. In this form, the problem is suitable for embedding in QC, which finds an approximate solution that has the best statistical reliability or corresponds to the quantum state with the lowest energy. As a separate class, vehicle routing algorithms for classical computers that use quantum computing to accelerate problem solving can be distinguished. For example, neural networks that calculate weighting factors using QC or an ant algorithm that calculates a pheromone trail in a hybrid cloud. It should be mentioned the quantum-inspired algorithms, which are based on software tools for the simulation of QC and the corresponding libraries and allow creating an effective class of algorithms for solving problems of vehicle routing. Conclusions. Combining hardware quantum annealing with a number of software tools for calculating optimization problems for classical computers in a hybrid quantum-classical cloud service allows to obtain advantages in speed and accuracy of some types of complex optimization problems of a commercial scale, in particular, routing vehicles, which is already bringing substantial profits to a number of corporations. Keywords: vehicle routing problem, quantum computer, annealing, combinatorial optimization, traveling salesman problem, clustering, qubit.
Quantum computing (QC) is expected to solve incredibly difficult problems, including finding optimal solutions to combinatorial optimization problems. However, to date, QC alone has still been unable to demonstrate this capability except for small-sized problems. Hybrid approaches, where QC and classical computing work together, have shown the most potential for solving real-world-scale problems. This work aims to show that we can enhance a classical optimization algorithm with QC so that it can overcome this limitation. We present a new Hybrid Quantum-classical Tabu Search (HQTS) algorithm to solve the Capacitated Vehicle Routing Problem (CVRP). Based on our prior work, HQTS leverages QC for routing within a classical tabu search framework. The quantum component formulates the Traveling Salesman Problem (TSP) for each route as a Quadratic Unconstrained Binary Optimization (QUBO) and solves it using D-Wave’s Advantage system. Experiments investigate the impact of quantum routing frequency and starting solution methods. Across different starting solution methods, including quantum-based and classical heuristics, it shows minimal overall impact. HQTS achieved optimal or near-optimal solutions for several CVRP problems, outperforming other hybrid CVRP algorithms and significantly reducing the optimality gap relative to preliminary research. The experimental results demonstrate that more frequent quantum routing improves solution quality and runtime. The findings highlight the potential of integrating QC within meta-heuristic frameworks for complex optimization in vehicle routing problems.
The vehicle routing problem (VRP) is an NP-hard optimization problem that has been an interest of research for decades in science and industry. The objective is to plan routes of vehicles to deliver goods to a fixed number of customers with optimal efficiency. Classical tools and methods provide good approximations to reach the optimal global solution. Quantum computing and quantum machine learning provide a new approach to solving combinatorial optimization of problems faster due to inherent speedups of quantum effects. Many solutions of VRP are offered across different quantum computing platforms using hybrid algorithms such as quantum approximate optimization algorithm and quadratic unconstrained binary optimization. In this work, we build a basic VRP solver for 3 and 4 cities using the variational quantum eigensolver on a fixed ansatz. The work is further extended to evaluate the robustness of the solution in several examples of noisy quantum channels. We find that the performance of the quantum algorithm depends heavily on what noise model is used. In general, noise is detrimental, but not equally so among different noise sources.
Quantum computing (QC) is expected to solve incredibly difficult problems, including finding optimal solutions to combinatorial optimization problems. However, to date, QC alone is still far to demonstrate this capability except on small-sized problems. Hybrid approaches where QC and classical computing work together have shown the most potential for solving real-world scale problems. This work aims to show that we can enhance a classical optimization algorithm with QC so that it can overcome this limitation. We present a new hybrid quantum-classical tabu search (HQTS) algorithm to solve the capacitated vehicle routing problem (CVRP). Based on our prior work, HQTS leverages QC for routing within a classical tabu search framework. The quantum component formulates the traveling salesman problem (TSP) for each route as a QUBO, solved using D-Wave's Advantage system. Experiments investigate the impact of quantum routing frequency and starting solution methods. While different starting solution methods, including quantum-based and classical heuristics methods, it shows minimal overall impact. HQTS achieved optimal or near-optimal solutions for several CVRP problems, outperforming other hybrid CVRP algorithms and significantly reducing the optimality gap compared to preliminary research. The experimental results demonstrate that more frequent quantum routing improves solution quality and runtime. The findings highlight the potential of integrating QC within meta-heuristic frameworks for complex optimization in vehicle routing problems.
The determination of vehicle routes fulfilling connectivity, time, and operational constraints is a well-studied combinatorial optimization problem. The NP-hard complexity of vehicle routing problems has fostered the adoption of tailored exact approaches, matheuristics, and metaheuristics on classical computing devices. The ongoing evolution of quantum computing hardware and the recent advances of quantum algorithms (i.e., VQE, QAOA, and ADMM) for mathematical programming make decision-making for routing problems an avenue of research worthwhile to be explored on quantum devices. In this article, we propose several mathematical formulations for inventory routing cast as vehicle routing with time windows and comment on their strengths and weaknesses. The optimization models are compared from a quantum computing perspective, specifically with metrics to evaluate the difficulty in solving the underlying quadratic unconstrained binary optimization problems. Finally, the solutions obtained on simulated quantum devices demonstrate the relative benefits of different algorithms and their robustness when put into practice.
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.
Vehicle routing problem (VRP) is an NP-hard optimization problem that has been an interest of research for decades in science and industry. The objective is to plan routes of vehicles to deliver a fixed number of customers with optimal efficiency. Classical tools and methods provide good approximations to reach the optimal global solution. Quantum computing and quantum machine learning provide a new approach to solving combinatorial optimization of problems faster due to inherent speedups of quantum effects. Many solutions of VRP are offered across different quantum computing platforms using hybrid algorithms such as quantum approximate optimization algorithm and quadratic unconstrained binary optimization. Quantum computers such as IBM-Q experience along with Qiskit framework offer tools to solve combinatorial optimization problems. This work proposed here builds a basic VRP solution for 3 and 4 cities using variational quantum eigensolver on a variable ANSATZ. The work is further extended to evaluate the robustness of the solution in noisy channels available within the ambit of Qiskit framework.