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Steiner Traveling Salesman Problem with Quantum Annealing

Alessia Ciacco, Francesca Guerriero, Eneko Osaba

TL;DR

This work tackles the NP-hard Steiner Traveling Salesman Problem (STSP) by integrating quantum annealing with a classical preprocessing step. It formulates an ILP for STSP, introduces PMRA to prune arcs, and then maps the reduced problem to a QUBO for quantum solvers, evaluating both a QPU and the LeapBQM hybrid on D-Wave hardware alongside Simulated Annealing. The key contributions are (i) the PMRA reduction technique that improves scalability, (ii) a thorough comparison of classical and quantum solution methods, and (iii) evidence that LeapBQM provides higher-quality solutions and better scalability than direct QPU QA for STSP. The findings suggest quantum annealing is a promising direction for STSP, with practical gains arising from problem-size reduction and hybrid quantum-classical strategies.

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.

Steiner Traveling Salesman Problem with Quantum Annealing

TL;DR

This work tackles the NP-hard Steiner Traveling Salesman Problem (STSP) by integrating quantum annealing with a classical preprocessing step. It formulates an ILP for STSP, introduces PMRA to prune arcs, and then maps the reduced problem to a QUBO for quantum solvers, evaluating both a QPU and the LeapBQM hybrid on D-Wave hardware alongside Simulated Annealing. The key contributions are (i) the PMRA reduction technique that improves scalability, (ii) a thorough comparison of classical and quantum solution methods, and (iii) evidence that LeapBQM provides higher-quality solutions and better scalability than direct QPU QA for STSP. The findings suggest quantum annealing is a promising direction for STSP, with practical gains arising from problem-size reduction and hybrid quantum-classical strategies.

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.

Paper Structure

This paper contains 11 sections, 1 equation, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Characteristics of the problem
  4. Mathematical Formulation
  5. Preprocessing method for reducing arcs
  6. Solution approach
  7. Computational study
  8. Scalability of PMRA using Gurobi solver
  9. Results using Simulated Annealing
  10. Results using Quantum Annealing
  11. Conclusions

Figures (1)

  • Figure 1: Representation of the STSP network: the yellow circles represent the $V_R$ nodes, the green circle indicates the depot and the blue circles represent the steiner nodes. The black edges represent the arcs of the graph, while a possible route is highlighted in red, showing one of the potential solutions to the optimization problem.