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Improved Dynamic Programming in Connection with an FPTAS for the Knapsack Problem

Author

Listed:
  • Hans Kellerer

    (University of Graz)

  • Ulrich Pferschy

    (University of Graz)

Abstract

A vector merging problem is introduced where two vectors of length n are merged such that the k-th entry of the new vector is the minimum over ℓ of the ℓ-th entry of the first vector plus the sum of the first k − ℓ + 1 entries of the second vector. For this problem a new algorithm with O(n log n) running time is presented thus improving upon the straightforward O(n 2) time bound. The vector merging problem can appear in different settings of dynamic programming. In particular, it is applied for a recent fully polynomial time approximation scheme (FPTAS) for the classical 0–1 knapsack problem by the same authors.

Suggested Citation

  • Hans Kellerer & Ulrich Pferschy, 2004. "Improved Dynamic Programming in Connection with an FPTAS for the Knapsack Problem," Journal of Combinatorial Optimization, Springer, vol. 8(1), pages 5-11, March.
    • Handle: RePEc:spr:jcomop:v:8:y:2004:i:1:d:10.1023_b:joco.0000021934.29833.6b
    DOI: 10.1023/B:JOCO.0000021934.29833.6b
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    Citations

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    Cited by:

    1. Halman, Nir & Kellerer, Hans & Strusevich, Vitaly A., 2018. "Approximation schemes for non-separable non-linear boolean programming problems under nested knapsack constraints," European Journal of Operational Research, Elsevier, vol. 270(2), pages 435-447.
    2. Jooken, Jorik & Leyman, Pieter & De Causmaecker, Patrick, 2022. "A new class of hard problem instances for the 0–1 knapsack problem," European Journal of Operational Research, Elsevier, vol. 301(3), pages 841-854.
    3. Zhong, Xueling & Ou, Jinwen & Wang, Guoqing, 2014. "Order acceptance and scheduling with machine availability constraints," European Journal of Operational Research, Elsevier, vol. 232(3), pages 435-441.
    4. Luca Bertazzi, 2012. "Minimum and Worst-Case Performance Ratios of Rollout Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 152(2), pages 378-393, February.
    5. Ranka Gojković & Goran Đurić & Danijela Tadić & Snežana Nestić & Aleksandar Aleksić, 2021. "Evaluation and Selection of the Quality Methods for Manufacturing Process Reliability Improvement—Intuitionistic Fuzzy Sets and Genetic Algorithm Approach," Mathematics, MDPI, vol. 9(13), pages 1-17, June.
    6. Shang-Chia Liu & Chin-Chia Wu, 2016. "A Faster FPTAS for a Supply Chain Scheduling Problem to Minimize Holding Costs with Outsourcing," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 33(05), pages 1-11, October.
    7. Luo, Wenchang & Gu, Boyuan & Lin, Guohui, 2018. "Communication scheduling in data gathering networks of heterogeneous sensors with data compression: Algorithms and empirical experiments," European Journal of Operational Research, Elsevier, vol. 271(2), pages 462-473.
    8. Stephan Helfrich & Arne Herzel & Stefan Ruzika & Clemens Thielen, 2022. "An approximation algorithm for a general class of multi-parametric optimization problems," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1459-1494, October.
    9. Zhou Xu, 2013. "The knapsack problem with a minimum filling constraint," Naval Research Logistics (NRL), John Wiley & Sons, vol. 60(1), pages 56-63, February.

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