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arXiv:2604.20321

Cutting-plane methodology via quantum optimization for solving the Traveling Salesman Problem

Alessia Ciacco, Luigi Di Puglia Pugliese, Francesca Guerriero

quant-phSubmitted Apr 22, 2026• Announced Apr 23, 2026• 0 citations
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Abstract

The Traveling Salesman Problem is a classical NP-hard combinatorial optimization problem that has been extensively studied in operations research. A major challenge in Traveling Salesman Problem formulations is the large number of subtour elimination constraints required to ensure a valid tour. To address this issue, we adopt an iterative approach grounded in well-established operations research techniques, in which subtour elimination constraints are generated dynamically. In addition, we integrate a preprocessing phase to reduce the number of candidate arcs. In this work, we investigate both classical and quantum optimization approaches for solving the problem using the proposed framework. In particular, for quantum optimization we analyze quantum annealing techniques within the D-Wave framework, considering both direct quantum execution on the QPU and hybrid quantum classical solvers. Computational experiments show that the proposed strategies significantly reduce the model size and lead to positive improvements in computational performance across classical, direct quantum, and hybrid optimization approaches.

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The Traveling Salesman Problem is a classical NP-hard combinatorial optimization problem that has been extensively studied in operations research. A major challenge in Traveling Salesman Problem formulations is the large number of subtour elimination constraints required to ensure a valid tour. To address this issue, we adopt an iterative approach grounded in well-established operations research techniques, in which subtour elimination constraints are generated dynamically. In addition, we integrate a preprocessing phase to reduce the number of candidate arcs. In this work, we investigate both classical and quantum optimization approaches for solving the problem using the proposed framework. In particular, for quantum optimization we analyze quantum annealing techniques within the D-Wave framework, considering both direct quantum execution on the QPU and hybrid quantum classical solvers. Computational experiments show that the proposed strategies significantly reduce the model size and lead to positive improvements in computational performance across classical, direct quantum, and hybrid optimization approaches.
The Traveling Salesman Problem is a fundamental combinatorial optimization problem widely studied in operations research. Despite its simple formulation, it remains computationally challenging due to the exponential growth of the search space and the large number of constraints required to eliminate subtours. This paper introduces a preprocessing strategy that significantly reduces the size of the optimization model by restricting the set of candidate arcs and retaining only the lowest-cost neighbors for each vertex. Computational experiments on TSPLIB benchmark instances demonstrate that the proposed approach substantially reduces the number of decision variables. The method is evaluated using both classical and quantum optimization techniques, showing improvements in computational time and reductions in optimality gaps. Overall, the results indicate that the proposed preprocessing enhances the scalability of the formulations and makes them more suitable for both classical solvers and emerging quantum optimization frameworks.
The traveling salesman problem is a well-known NP-hard problem in combinatorial optimization. This paper shows how to solve it on an Ising Hamiltonian based quantum annealer by casting it as a quadratic unconstrained binary optimization (QUBO) problem. Results of practical experiments are also presented using D-Wave’s 5,000 qubit Advantage 1.1 quantum annealer and the performance is compared to a classical solver. It is found the quantum annealer can only handle a problem size of 8 or less nodes and its performance is subpar compared to the classical solver both in terms of time and accuracy.
The Steiner Traveling Salesman Problem (STSP) is a variant of the classical Traveling Salesman Problem. The STSP involves incorporating steiner nodes, which are extra nodes not originally part of the required visit set but that can be added to the route to enhance the overall solution and minimize the total travel cost. Given the NP-hard nature of the STSP, we propose a quantum approach to address it. Specifically, we employ quantum annealing using D-Wave's hardware to explore its potential for solving this problem. To enhance computational feasibility, we develop a preprocessing method that effectively reduces the network size. Our experimental results demonstrate that this reduction technique significantly decreases the problem complexity, making the Quadratic Unconstrained Binary Optimization formulation, the standard input for quantum annealers, better suited for existing quantum hardware. Furthermore, the results highlight the potential of quantum annealing as a promising and innovative approach for solving the STSP.
The Traveling Salesman Problem is a classical NP-hard problem with significant implications in logistics, circuit design, and operations research. This paper presents a comparative study of four approaches to solving the Traveling Salesman Problem: brute-force enumeration, a 2-approximation algorithm using minimum spanning trees, simulated annealing, and the Quantum Approximate Optimization Algorithm. We implement each technique and evaluate them on graphs of varying sizes to analyze performance, solution quality, and scalability. In doing so, we have also developed an open-source framework that allows researchers and practitioners to explore, test and extend these methods.
The Traveling Salesman Problem (TSP) is a widely studied NP-complete optimization challenge with significant theoretical and practical implications. This study proposes a hybrid quantum-classical framework using a Quantum-Inspired Evolutionary Algorithm (QEA) with Sort Gray Binary Encoding to solve the TSP. The proposed method guarantees the generation of valid TSP tours by eliminating invalid solutions. It employs quantum superposition with intrinsic randomness to enhance computational efficiency and scalability. The framework was implemented on cloud-based NISQ platforms, including IBM Quantum and AWS Braket, demonstrating its practicality and effectiveness. Experimental evaluations revealed that the proposed framework successfully solved TSP instances with up to 15 cities, achieving superior performance compared to classical methods and showcasing its ability to scale under NISQ constraints. The results also highlight the potential of hybrid quantum-classical approaches to overcome hardware limitations in current quantum systems. This study establishes a robust hybrid methodology for solving combinatorial optimization problems. It also sets a benchmark for leveraging the capabilities of NISQ-era quantum devices in real-world applications, thereby providing a foundation for future research in hybrid quantum-classical optimization techniques.
NP-hard problems are one of the most computationally intensive challenges in computer science, appearing in many domains such as logistics, cybersecurity, drug design, and financial modelling. Classical algorithms become impractical as problem size increases due to combinatorial explosion. This paper presents a structured framework for analysing quantum algorithms targeting NP-hard optimization problems, like the Traveling Salesman Problem (TSP), Max-Cut, and Boolean Satisfiability (SAT). We have developed a taxonomy which categorizes quantum approaches—quantum annealing, variational algorithms (QAOA/VQE), and Grover-based search—based on problem encoding strategies, resource requirements, and theoretical guarantees. From the comparative analysis of existing literature, we: 1) Map algorithm classes to problem domains (e.g., QAOA for Max-Cut, annealing for TSP) 2) Identify performance trade-offs between solution quality, qubit counts, and circuit depth 3) Analyse hardware compatibility constraints across NISQ platforms 4) Reveal fundamental scalability barriers including noise susceptibility and embedding overhead The framework establishes implementation guidelines for near- term quantum hardware and proposes hybrid quantum-classical pathways to bridge theoretical potential and practical deployment. This systematic analysis prioritizes research directions for achieving quantum advantage in combinatorial optimization.