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Strengthening a linear reformulation of the 0-1 cubic knapsack problem via variable reordering

Author

Listed:
  • Richard J. Forrester

    (Dickinson College)

  • Lucas A. Waddell

    (Bucknell University)

Abstract

The 0-1 cubic knapsack problem (CKP), a generalization of the classical 0-1 quadratic knapsack problem, is an extremely challenging NP-hard combinatorial optimization problem. An effective exact solution strategy for the CKP is to reformulate the nonlinear problem into an equivalent linear form that can then be solved using a standard mixed-integer programming solver. We consider a classical linearization method and propose a variant of a more recent technique for linearizing 0-1 cubic programs applied to the CKP. Using a variable reordering strategy, we show how to improve the strength of the linear programming relaxation of our proposed reformulation, which ultimately leads to reduced overall solution times. In addition, we develop a simple heuristic method for obtaining good-quality CKP solutions that can be used to provide a warm start to the solver. Computational tests demonstrate the effectiveness of both our variable reordering strategy and heuristic method.

Suggested Citation

  • Richard J. Forrester & Lucas A. Waddell, 2022. "Strengthening a linear reformulation of the 0-1 cubic knapsack problem via variable reordering," Journal of Combinatorial Optimization, Springer, vol. 44(1), pages 498-517, August.
    • Handle: RePEc:spr:jcomop:v:44:y:2022:i:1:d:10.1007_s10878-021-00840-z
    DOI: 10.1007/s10878-021-00840-z
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    References listed on IDEAS

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    1. Adams, Warren P. & Guignard, Monique & Hahn, Peter M. & Hightower, William L., 2007. "A level-2 reformulation-linearization technique bound for the quadratic assignment problem," European Journal of Operational Research, Elsevier, vol. 180(3), pages 983-996, August.
    2. Fred Glover, 1975. "Improved Linear Integer Programming Formulations of Nonlinear Integer Problems," Management Science, INFORMS, vol. 22(4), pages 455-460, December.
    3. Alberto Caprara & David Pisinger & Paolo Toth, 1999. "Exact Solution of the Quadratic Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 11(2), pages 125-137, May.
    4. H. Martin Weingartner, 1966. "Capital Budgeting of Interrelated Projects: Survey and Synthesis," Management Science, INFORMS, vol. 12(7), pages 485-516, March.
    5. repec:grz:wpsses:2014-04 is not listed on IDEAS
    6. G. Edward Fox & Norman R. Baker & John L. Bryant, 1984. "Economic Models for R and D Project Selection in the Presence of Project Interactions," Management Science, INFORMS, vol. 30(7), pages 890-902, July.
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