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Traveling Salesman Problem with a preprocessing method for classical and quantum optimization

Alessia Ciacco, Luigi Di Puglia Pugliese, Francesca Guerriero

Abstract

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.

Traveling Salesman Problem with a preprocessing method for classical and quantum optimization

Abstract

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.
Paper Structure (12 sections, 1 theorem, 3 equations, 1 figure, 4 tables, 1 algorithm)

This paper contains 12 sections, 1 theorem, 3 equations, 1 figure, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem definition
  3. Mathematical Formulation
  4. Cost-based Arc Filtering (CAF)
  5. Feasibility Preservation
  6. Optimality Considerations
  7. Solution approach
  8. Computational study
  9. Impact of CAF on the model size
  10. Classical optimization experiments
  11. Quantum hybrid solver experiments
  12. Conclusions

Key Result

Theorem 1

Let $G=(V,A)$ be the complete directed graph and let $\tilde{G}=(V,\tilde{A})$ be the directed graph obtained by applying the CAF procedure with $k=\lceil n/2\rceil$. Then the undirected graph induced by $\tilde{A}$ is Hamiltonian.

Figures (1)

  • Figure 1: Number of decision variables in the TSP as the number of vertices $V$ increases. The red curve represents the number of variables in the standard formulation (n.var without CAF), while the green curve shows the number of variables after applying the CAF preprocessing procedure (n.var with CAF).

Theorems & Definitions (2)

  • Theorem 1: Feasibility of the Reduced Graph
  • proof