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Logical and inequality implications for reducing the size and difficulty of quadratic unconstrained binary optimization problems

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  • Glover, Fred
  • Lewis, Mark
  • Kochenberger, Gary

Abstract

The quadratic unconstrained binary optimization (QUBO) problem arises in diverse optimization applications ranging from Ising spin problems to classical problems in graph theory and binary discrete optimization. The use of preprocessing to transform the graph representing the QUBO problem into a smaller equivalent graph is important for improving solution quality and time for both exact and metaheuristic algorithms and is a step towards mapping large scale QUBO to hardware graphs used in quantum annealing computers. In an earlier paper a set of rules was introduced that achieved significant QUBO reductions as verified through computational testing. Here this work is extended with additional rules that provide further reductions that succeed in exactly solving 10% of the benchmark QUBO problems. An algorithm and associated data structures to efficiently implement the entire set of rules is detailed and computational experiments are reported that demonstrate their efficacy.

Suggested Citation

  • Glover, Fred & Lewis, Mark & Kochenberger, Gary, 2018. "Logical and inequality implications for reducing the size and difficulty of quadratic unconstrained binary optimization problems," European Journal of Operational Research, Elsevier, vol. 265(3), pages 829-842.
    • Handle: RePEc:eee:ejores:v:265:y:2018:i:3:p:829-842
    DOI: 10.1016/j.ejor.2017.08.025
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    References listed on IDEAS

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    1. Mauri, Geraldo Regis & Lorena, Luiz Antonio Nogueira, 2012. "A column generation approach for the unconstrained binary quadratic programming problem," European Journal of Operational Research, Elsevier, vol. 217(1), pages 69-74.
    2. Fred Glover & Gary A. Kochenberger & Bahram Alidaee, 1998. "Adaptive Memory Tabu Search for Binary Quadratic Programs," Management Science, INFORMS, vol. 44(3), pages 336-345, March.
    3. Jeffery L. Kennington & Karen R. Lewis, 2004. "Generalized Networks: The Theory of Preprocessing and an Empirical Analysis," INFORMS Journal on Computing, INFORMS, vol. 16(2), pages 162-173, May.
    4. Kim, Jongkwang & Wilhelm, Thomas, 2008. "What is a complex graph?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(11), pages 2637-2652.
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    Citations

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    Cited by:

    1. Fred Glover & Gary Kochenberger & Rick Hennig & Yu Du, 2022. "Quantum bridge analytics I: a tutorial on formulating and using QUBO models," Annals of Operations Research, Springer, vol. 314(1), pages 141-183, July.
    2. Fred Glover & Gary Kochenberger & Yu Du, 2019. "Quantum Bridge Analytics I: a tutorial on formulating and using QUBO models," 4OR, Springer, vol. 17(4), pages 335-371, December.
    3. Fred Glover & Gary Kochenberger & Moses Ma & Yu Du, 2022. "Quantum Bridge Analytics II: QUBO-Plus, network optimization and combinatorial chaining for asset exchange," Annals of Operations Research, Springer, vol. 314(1), pages 185-212, July.
    4. Mark W. Lewis & Amit Verma & Todd T. Eckdahl, 2021. "Qfold: a new modeling paradigm for the RNA folding problem," Journal of Heuristics, Springer, vol. 27(4), pages 695-717, August.

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