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arXiv:2504.02388 · Genetic and Evolutionary Computation Conference Companion (GECCO Companion)

Steiner Traveling Salesman Problem with Quantum Annealing

Alessia Ciacco, Francesca Guerriero, Eneko Osaba

Published Jul 14, 2025• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1145/3712255.3734313

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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.
Quantum computing has paved a new way for faster and more efficient solutions to large-scale, real-world optimization problems that are challenging for classical computing systems. For instance, selective traveling salesman problem (sTSP) that is famous in such fields as logistic optimization and has attracted increasing attention from the research community, however, is known as an NP-Hard problem. Solving the sTSP is, therefore, extremely complex because the optimization function potentially comes with an exponential number of variables that cannot be solved in polynomial time in general. To this end, we propose a quantum annealing framework for time-bounded and near-optimal solutions for the sTSP, overcoming hardware limits of near-term quantum devices. In particular, we put forth an efficient Hamiltonian (QUBO) to encode the complex decision-making for the sTSP on noisy intermediate-scale quantum (NISQ) annealer. Furthermore, experimental results we obtained on the D-Wave 2000QQQquantum hardware demonstrate that the optimal solutions for several instances can be attained.
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.
We propose the Steiner Traveling Salesman Problem with Time Windows and Pickup and Delivery, an advanced and practical extension of classical routing models. This variant integrates the characteristics of the Steiner Traveling Salesman Problem with time-window constraints, pickup and delivery operations and vehicle capacity limitations. These features closely mirror the complexities of contemporary logistics challenges, including last-mile distribution, reverse logistics and on-demand service scenarios. To tackle the inherent computational difficulties of this NP-hard problem, we propose two specialized mathematical formulations: an arc-based model and a node-oriented model, each designed to capture distinct structural aspects of the problem. We further introduce a preprocessing reduction method that eliminates redundant arcs, significantly enhancing computational performance and scalability. We also introduce a benchmark instance generator designed to create diverse and scalable test scenarios, providing a flexible framework for assessing the performance and robustness of both classical and quantum optimization methods. Both formulations are implemented using classical and quantum optimization approaches. In particular, the classical models are solved with Gurobi, whereas the quantum implementation is carried out on D-Wave’s LeapCQMHybrid platform, a hybrid quantum–classical environment that integrates quantum annealing with classical optimization techniques for constrained problem solving. Numerical experiments are conducted to validate the proposed formulations and the preprocessing reduction method. The analyses performed assess the structural properties of the two models, their computational behavior, and the impact of preprocessing on problem size and solution efficiency.
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 (TSP) is a classical NP-hard problem in combinatorial optimization, where determining the shortest route among a set of cities becomes computationally prohibitive as the problem size increases. This work explores quantum computing as an alternative approach to address this complexity. Unlike existing methods that primarily rely on quantum annealing, we propose a quantum iterative framework integrating Quantum Phase Estimation (QPE) and Grover's search algorithm. Route costs are encoded as quantum phases, enabling QPE to efficiently evaluate them, while Amplitude Amplification, implemented via the Grover-Long algorithm, iteratively refines the solution space toward the optimal route. A proof-of-concept case study on a small-scale TSP instance demonstrates the feasibility of this approach and its potential for scaling to larger optimization problems. Furthermore, under an expectation-based analysis, the algorithm exhibits an expected computational complexity of O(m2log⁡2(m)log⁡2(1/ε)ε)O(\frac{m^2\log_2(m)\log_2(1/ε)}{\sqrtε})O(ε​m2log2​(m)log2​(1/ε)​) which depends on the error tolerance parameter εεε. This estimation omits the initialization term, which we expect future refinements to render subdominant to Phase Estimation.
The Steiner Tree Problem (STP) is a well-known NP-hard combinatorial optimization problem, which has wide applications in network design, integrated circuit layout, bioinformatics, and other fields. However, traditional algorithms often struggle to balance efficiency and solution quality when dealing with large-scale STP instances. In this paper, we propose a new quantum annealing-based algorithm for solving the STP: we first model the STP into a quadratic unconstrained binary optimization (QUBO) form suitable for quantum annealing, then design a corresponding encoding strategy, and finally verify the algorithm through experimental tests. The results show that our quantum annealing-based method can obtain high-quality solutions with relatively low computational overhead for moderate-scale STP instances, providing a new feasible path for handling this intractable combinatorial optimization problem.