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Decomposition Algorithms for Solving NP-hard Problems on a Quantum Annealer

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Abstract

NP-hard problems such as the maximum clique or minimum vertex cover problems, two of Karp’s 21 NP-hard problems, have several applications in computational chemistry, biochemistry and computer network security. Adiabatic quantum annealers can search for the optimum value of such NP-hard optimization problems, given the problem can be embedded on their hardware. However, this is often not possible due to certain limitations of the hardware connectivity structure of the annealer. This paper studies a general framework for a decomposition algorithm for NP-hard graph problems aiming to identify an optimal set of vertices. Our generic algorithm allows us to recursively divide an instance until the generated subproblems can be embedded on the quantum annealer hardware and subsequently solved. The framework is applied to the maximum clique and minimum vertex cover problems, and we propose several pruning and reduction techniques to speed up the recursive decomposition. The performance of both algorithms is assessed in a detailed simulation study.

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Acknowledgments

Research presented in this article was supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory under project numbers 20180267ER and 20190065DR.

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Authors and Affiliations

  1. Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Elijah Pelofske & Hristo Djidjev

  2. T.H. Chan School of Public Health, Harvard University, Boston, MA, 02115, USA

    Georg Hahn

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  1. Elijah Pelofske
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Pelofske, E., Hahn, G. & Djidjev, H. Decomposition Algorithms for Solving NP-hard Problems on a Quantum Annealer. J Sign Process Syst 93, 405–420 (2021). https://doi.org/10.1007/s11265-020-01550-1

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