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On the rectangular knapsack problem

Author

Listed:
  • Fritz Bökler

    (Osnabrück University)

  • Markus Chimani

    (Osnabrück University)

  • Mirko H. Wagner

    (Osnabrück University)

Abstract

A recent paper by Schulze et al. (Math Methods Oper Res 92(1):107–132, 2020) presented the Rectangular Knapsack Problem (Rkp) as a crucial subproblem in the study on the Cardinality-constrained Bi-objective Knapsack Problem (Cbkp). To this end, they started an investigation into its complexity and approximability. The key results are an NP -hardness proof for a more general scenario than Rkp, and a 4.5-approximation for Rkp, raising the question of improvements for either result. In this note we settle both questions conclusively: we show that (a) Rkp is indeed NP -hard in the considered setting (and even in more restricted settings), and (b) there exists both a pseudopolynomial algorithm and a fully-polynomial time approximation scheme (i.e., efficient approximability within any desired ratio $$\alpha >1$$ α > 1 ) for Rkp.

Suggested Citation

  • Fritz Bökler & Markus Chimani & Mirko H. Wagner, 2022. "On the rectangular knapsack problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(1), pages 149-160, August.
    • Handle: RePEc:spr:mathme:v:96:y:2022:i:1:d:10.1007_s00186-022-00788-8
    DOI: 10.1007/s00186-022-00788-8
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    References listed on IDEAS

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    1. Thomas Erlebach & Hans Kellerer & Ulrich Pferschy, 2002. "Approximating Multiobjective Knapsack Problems," Management Science, INFORMS, vol. 48(12), pages 1603-1612, December.
    2. Britta Schulze & Michael Stiglmayr & Luís Paquete & Carlos M. Fonseca & David Willems & Stefan Ruzika, 2020. "On the rectangular knapsack problem: approximation of a specific quadratic knapsack problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(1), pages 107-132, August.
    3. Xu, Zhou, 2012. "A strongly polynomial FPTAS for the symmetric quadratic knapsack problem," European Journal of Operational Research, Elsevier, vol. 218(2), pages 377-381.
    4. Ulrich Pferschy & Joachim Schauer, 2016. "Approximation of the Quadratic Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 28(2), pages 308-318, May.
    5. G. L. Nemhauser & Z. Ullmann, 1969. "Discrete Dynamic Programming and Capital Allocation," Management Science, INFORMS, vol. 15(9), pages 494-505, May.
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