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Upper Bounds and Algorithms for Hard 0-1 Knapsack Problems

Author

Listed:
  • Silvano Martello

    (University of Bologna, Italy)

  • Paolo Toth

    (University of Bologna, Italy)

Abstract

It is well-known that many instances of the 0-1 knapsack problem can be effectively solved to optimality also for very large values of n (the number of binary variables), while other instances cannot be solved for n equal to only a few hundreds. We propose upper bounds obtained from the mathematical model of the problem by adding valid inequalities on the cardinality of an optimal solution, and relaxing it in a Lagrangian fashion. We then introduce a specialized iterative technique for determining the optimal Lagrangian multipliers in polynomial time. A branch-and-bound algorithm is finally developed. Computational experiments prove that several classes of hard instances are effectively solved even for large values of n .

Suggested Citation

  • Silvano Martello & Paolo Toth, 1997. "Upper Bounds and Algorithms for Hard 0-1 Knapsack Problems," Operations Research, INFORMS, vol. 45(5), pages 768-778, October.
    • Handle: RePEc:inm:oropre:v:45:y:1997:i:5:p:768-778
    DOI: 10.1287/opre.45.5.768
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    Citations

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    Cited by:

    1. Silvano Martello & Paolo Toth, 2003. "An Exact Algorithm for the Two-Constraint 0--1 Knapsack Problem," Operations Research, INFORMS, vol. 51(5), pages 826-835, October.
    2. Silvano Martello & David Pisinger & Paolo Toth, 1999. "Dynamic Programming and Strong Bounds for the 0-1 Knapsack Problem," Management Science, INFORMS, vol. 45(3), pages 414-424, March.
    3. Charles H. Reilly, 2009. "Synthetic Optimization Problem Generation: Show Us the Correlations!," INFORMS Journal on Computing, INFORMS, vol. 21(3), pages 458-467, August.
    4. Mhand Hifi & Rym M'Hallah, 2005. "An Exact Algorithm for Constrained Two-Dimensional Two-Staged Cutting Problems," Operations Research, INFORMS, vol. 53(1), pages 140-150, February.
    5. Mervat Chouman & Teodor Gabriel Crainic & Bernard Gendron, 2018. "The impact of filtering in a branch-and-cut algorithm for multicommodity capacitated fixed charge network design," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 6(2), pages 143-184, June.
    6. Wishon, Christopher & Villalobos, J. Rene, 2016. "Robust efficiency measures for linear knapsack problem variants," European Journal of Operational Research, Elsevier, vol. 254(2), pages 398-409.
    7. Mervat Chouman & Teodor Gabriel Crainic & Bernard Gendron, 2017. "Commodity Representations and Cut-Set-Based Inequalities for Multicommodity Capacitated Fixed-Charge Network Design," Transportation Science, INFORMS, vol. 51(2), pages 650-667, May.
    8. Arnaud Fréville & SaÏd Hanafi, 2005. "The Multidimensional 0-1 Knapsack Problem—Bounds and Computational Aspects," Annals of Operations Research, Springer, vol. 139(1), pages 195-227, October.
    9. Nicholas G. Hall & Marc E. Posner, 2007. "Performance Prediction and Preselection for Optimization and Heuristic Solution Procedures," Operations Research, INFORMS, vol. 55(4), pages 703-716, August.
    10. Markus Leitner & Andrea Lodi & Roberto Roberti & Claudio Sole, 2024. "An Exact Method for (Constrained) Assortment Optimization Problems with Product Costs," INFORMS Journal on Computing, INFORMS, vol. 36(2), pages 479-494, March.

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