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Approximation schemes for r-weighted Minimization Knapsack problems

Author

Listed:
  • Khaled Elbassioni

    (Masdar Institute, Khalifa University of Science and Technology)

  • Areg Karapetyan

    (Masdar Institute, Khalifa University of Science and Technology)

  • Trung Thanh Nguyen

    (Hai Phong University)

Abstract

Stimulated by salient applications arising from power systems, this paper studies a class of non-linear Knapsack problems with non-separable quadratic constrains, formulated in either binary or integer form. These problems resemble the duals of the corresponding variants of 2-weighted Knapsack problem (a.k.a., complex-demand Knapsack problem) which has been studied in the extant literature under the paradigm of smart grids. Nevertheless, the employed techniques resulting in a polynomial-time approximation scheme (PTAS) for the 2-weighted Knapsack problem are not amenable to its minimization version. We instead propose a greedy geometry-based approach that arrives at a quasi PTAS (QPTAS) for the minimization variant with boolean variables. As for the integer formulation, a linear programming-based method is developed that obtains a PTAS. In view of the curse of dimensionality, fast greedy heuristic algorithms are presented, additionally to QPTAS. Their performance is corroborated extensively by empirical simulations under diverse settings and scenarios.

Suggested Citation

  • Khaled Elbassioni & Areg Karapetyan & Trung Thanh Nguyen, 2019. "Approximation schemes for r-weighted Minimization Knapsack problems," Annals of Operations Research, Springer, vol. 279(1), pages 367-386, August.
    • Handle: RePEc:spr:annopr:v:279:y:2019:i:1:d:10.1007_s10479-018-3111-9
    DOI: 10.1007/s10479-018-3111-9
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    References listed on IDEAS

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    1. Bretthauer, Kurt M. & Shetty, Bala, 2002. "The nonlinear knapsack problem - algorithms and applications," European Journal of Operational Research, Elsevier, vol. 138(3), pages 459-472, May.
    2. Satoru Fujishige & Naoki Katoh & Tetsuo Ichimori, 1988. "The Fair Resource Allocation Problem with Submodular Constraints," Mathematics of Operations Research, INFORMS, vol. 13(1), pages 164-173, February.
    3. Gerhard J. Woeginger, 2000. "When Does a Dynamic Programming Formulation Guarantee the Existence of a Fully Polynomial Time Approximation Scheme (FPTAS)?," INFORMS Journal on Computing, INFORMS, vol. 12(1), pages 57-74, February.
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    Cited by:

    1. Areg Karapetyan & Khaled Elbassioni & Majid Khonji & Sid Chi-Kin Chau, 2023. "Approximations for generalized unsplittable flow on paths with application to power systems optimization," Annals of Operations Research, Springer, vol. 320(1), pages 173-204, January.

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