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Advanced greedy algorithms and surrogate constraint methods for linear and quadratic knapsack and covering problems

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  • Glover, Fred

Abstract

New variants of greedy algorithms, called advanced greedy algorithms, are identified for knapsack and covering problems with linear and quadratic objective functions. Beginning with single-constraint problems, we provide extensions for multiple knapsack and covering problems, in which objects must be allocated to different knapsacks and covers, and also for multi-constraint (multi-dimensional) knapsack and covering problems, in which the constraints are exploited by means of surrogate constraint strategies. In addition, we provide a new graduated-probe strategy for improving the selection of variables to be assigned values. Going beyond the greedy and advanced greedy frameworks, we describe ways to utilize these algorithms with multi-start and strategic oscillation metaheuristics. Finally, we identify how surrogate constraints can be utilized to produce inequalities that dominate those previously proposed and tested utilizing linear programming methods for solving multi-constraint knapsack problems, which are responsible for the current best methods for these problems. While we focus on 0–1 problems, our approaches can readily be adapted to handle variables with general upper bounds.

Suggested Citation

  • Glover, Fred, 2013. "Advanced greedy algorithms and surrogate constraint methods for linear and quadratic knapsack and covering problems," European Journal of Operational Research, Elsevier, vol. 230(2), pages 212-225.
    • Handle: RePEc:eee:ejores:v:230:y:2013:i:2:p:212-225
    DOI: 10.1016/j.ejor.2013.04.010
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    References listed on IDEAS

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    1. Hanafi, Said & Glover, Fred, 2007. "Exploiting nested inequalities and surrogate constraints," European Journal of Operational Research, Elsevier, vol. 179(1), pages 50-63, May.
    2. Fred Glover, 1975. "Surrogate Constraint Duality in Mathematical Programming," Operations Research, INFORMS, vol. 23(3), pages 434-451, June.
    3. Fred Glover, 1965. "A Multiphase-Dual Algorithm for the Zero-One Integer Programming Problem," Operations Research, INFORMS, vol. 13(6), pages 879-919, December.
    4. Yalçın Akçay & Haijun Li & Susan Xu, 2007. "Greedy algorithm for the general multidimensional knapsack problem," Annals of Operations Research, Springer, vol. 150(1), pages 17-29, March.
    5. Fred Glover, 1971. "Flows in Arborescences," Management Science, INFORMS, vol. 17(9), pages 568-586, May.
    6. Vasquez, Michel & Vimont, Yannick, 2005. "Improved results on the 0-1 multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 165(1), pages 70-81, August.
    7. María Osorio & Fred Glover & Peter Hammer, 2002. "Cutting and Surrogate Constraint Analysis for Improved Multidimensional Knapsack Solutions," Annals of Operations Research, Springer, vol. 117(1), pages 71-93, November.
    8. Ablanedo-Rosas, José H. & Rego, César, 2010. "Surrogate constraint normalization for the set covering problem," European Journal of Operational Research, Elsevier, vol. 205(3), pages 540-551, September.
    9. George B. Dantzig, 1957. "Discrete-Variable Extremum Problems," Operations Research, INFORMS, vol. 5(2), pages 266-288, April.
    10. Fred Glover, 1995. "Tabu Thresholding: Improved Search by Nonmonotonic Trajectories," INFORMS Journal on Computing, INFORMS, vol. 7(4), pages 426-442, November.
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    Cited by:

    1. Furini, Fabio & Ljubić, Ivana & Sinnl, Markus, 2017. "An effective dynamic programming algorithm for the minimum-cost maximal knapsack packing problem," European Journal of Operational Research, Elsevier, vol. 262(2), pages 438-448.
    2. Mavrotas, George & Florios, Kostas & Figueira, José Rui, 2015. "An improved version of a core based algorithm for the multi-objective multi-dimensional knapsack problem: A computational study and comparison with meta-heuristics," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 25-43.

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