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The set partitioning problem in a quantum context

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Abstract

The set partitioning problem and its decision variant (i.e., the exact cover problem) are combinatorial optimization problems that were historically crucial in the quantum optimization community. This problem is also employed in the main problem of the branch-and-price approach in many real-world optimization problems, including, but not limited to, redistricting and scheduling. Motivated by recent claims on the capability of quantum computers in “solving” hard combinatorial optimization problems, we propose a quadratic unconstrained binary optimization (\({\text {QUBO}}\)) formulation for the set partitioning problem with penalty coefficients that are at least as tight as the existing naive ones. We also employ five reduction techniques of Garfinkel and Nemhauser (Operations Research 17(5):848–856, 1969) to reduce the size of an existing set of benchmark instances. We finally use variational quantum eigensolver (\({\text {VQE}}\)) as a heuristic to find feasible solutions for the problem. Our computational experiments show the efficacy of employing the proposed penalty coefficients and the existing classical reduction techniques in the quantum context. Our codes and data are available on GitHub.

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Notes

  1. The goal of demonstrating that a quantum computer with a tailored quantum algorithm can efficiently perform a task that its classical counterparts cannot.

  2. Here, index gs stands for ground state.

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Acknowledgements

The authors acknowledge the High Performance Computing Center (HPCC) at Texas Tech University for providing computational resources that have contributed to the research results reported within this paper. The authors thank the anonymous reviewer for helpful comments.

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Authors and Affiliations

  1. Industrial, Manufacturing and Systems Engineering, Texas Tech University, Lubbock, USA

    Rafael Cacao, Lucas R. C. T. Cortez, Hamidreza Validi & Ismael R. de Farias

  2. Operations Research and Engineering Management, Southern Methodist University, Dallas, USA

    Jackson Forner

  3. Computational Applied Mathematics and Operations Research, Rice University, Houston, USA

    Illya V. Hicks

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  1. Rafael Cacao
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  2. Lucas R. C. T. Cortez
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Correspondence to Hamidreza Validi.

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This paper is dedicated to the memory of our coauthor, Ismael R. de Farias Jr., who sadly passed away on Wednesday, August 3rd, 2022.

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Cacao, R., Cortez, L.R.C.T., Forner, J. et al. The set partitioning problem in a quantum context. Optim Lett 18, 1–17 (2024). https://doi.org/10.1007/s11590-023-02029-1

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