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Dynamic Programming and Strong Bounds for the 0-1 Knapsack Problem

Author Info

Listed author(s):
  • Silvano Martello

    (DEIS, University of Bologna, Viale Risorgimento 2, Bologna, Italy)

  • David Pisinger

    (DIKU, University of Copenhagen, Univ.parken 1, Copenhagen, Denmark)

  • Paolo Toth

    (DEIS, University of Bologna, Viale Risorgimento 2, Bologna, Italy)

Registered author(s):

    Abstract

    Two new algorithms recently proved to outperform all previous methods for the exact solution of the 0-1 Knapsack Problem. This paper presents a combination of such approaches, where, in addition, valid inequalities are generated and surrogate relaxed, and a new initial core problem is adopted. The algorithm is able to solve all classical test instances, with up to 10,000 variables, in less than 0.2 seconds on a HP9000-735/99 computer. The C language implementation of the algorithm is available on the internet.

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    File URL: http://dx.doi.org/10.1287/mnsc.45.3.414
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    Article provided by INFORMS in its journal Management Science.

    Volume (Year): 45 (1999)
    Issue (Month): 3 (March)
    Pages: 414-424

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    Handle: RePEc:inm:ormnsc:v:45:y:1999:i:3:p:414-424
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    Keywords: Knapsack problem; dynamic programming; branch-and-bound; surrogate relaxation;

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    1. Pisinger, David, 1995. "An expanding-core algorithm for the exact 0-1 knapsack problem," European Journal of Operational Research, Elsevier, vol. 87(1), pages 175-187, November.
    2. Freville, Arnaud & Plateau, Gerard, 1993. "An exact search for the solution of the surrogate dual of the 0-1 bidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 68(3), pages 413-421, August.
    3. Silvano Martello & Paolo Toth, 1984. "A Mixture of Dynamic Programming and Branch-and-Bound for the Subset-Sum Problem," Management Science, INFORMS, vol. 30(6), pages 765-771, June.
    4. Silvano Martello & Paolo Toth, 1988. "A New Algorithm for the 0-1 Knapsack Problem," Management Science, INFORMS, vol. 34(5), pages 633-644, May.
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    Cited by:

    1. Tang, Jiafu & Zhiqiao, Wu & Kwong, C.K. & Luo, Xinggang, 2013. "Integrated production strategy and reuse scenario: A CoFAQ model and case study of mail server system development," Omega, Elsevier, vol. 41(3), pages 536-552.
    2. Büther, Marcel & Briskorn, Dirk, 2007. "Reducing the 0-1 knapsack problem with a single continuous variable to the standard 0-1 knapsack problem," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 629, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    3. Klose, Andreas & Gortz, Simon, 2007. "A branch-and-price algorithm for the capacitated facility location problem," European Journal of Operational Research, Elsevier, vol. 179(3), pages 1109-1125, June.
    4. Klose, Andreas & Drexl, Andreas, 2001. "Lower bounds for the capacitated facility location problem based on column generation," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 544, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    5. Reilly, Charles H. & Sapkota, Nabin, 2015. "A family of composite discrete bivariate distributions with uniform marginals for simulating realistic and challenging optimization-problem instances," European Journal of Operational Research, Elsevier, vol. 241(3), pages 642-652.
    6. Elendner, Thomas & Femerling, R., 2003. "Allocation of in-house services: Experimental comparison of allocation mechanisms," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 577, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    7. Sbihi, Abdelkader, 2010. "A cooperative local search-based algorithm for the Multiple-Scenario Max-Min Knapsack Problem," European Journal of Operational Research, Elsevier, vol. 202(2), pages 339-346, April.
    8. Esswein, Carl & Billaut, Jean-Charles & Strusevich, Vitaly A., 2005. "Two-machine shop scheduling: Compromise between flexibility and makespan value," European Journal of Operational Research, Elsevier, vol. 167(3), pages 796-809, December.
    9. Polyakovskiy, S. & Neumann, F., 2017. "The Packing While Traveling Problem," European Journal of Operational Research, Elsevier, vol. 258(2), pages 424-439.
    10. Dahmani, Isma & Hifi, Mhand & Wu, Lei, 2016. "An exact decomposition algorithm for the generalized knapsack sharing problem," European Journal of Operational Research, Elsevier, vol. 252(3), pages 761-774.
    11. Raidl, Günther R., 2015. "Decomposition based hybrid metaheuristics," European Journal of Operational Research, Elsevier, vol. 244(1), pages 66-76.
    12. Büther, Marcel, 2007. "Reducing the elastic generalized assignment problem to the standard generalized assignment problem," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 632, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    13. Andreas Klose & Andreas Drexl, 2005. "Lower Bounds for the Capacitated Facility Location Problem Based on Column Generation," Management Science, INFORMS, vol. 51(11), pages 1689-1705, November.
    14. Cerqueus, Audrey & Przybylski, Anthony & Gandibleux, Xavier, 2015. "Surrogate upper bound sets for bi-objective bi-dimensional binary knapsack problems," European Journal of Operational Research, Elsevier, vol. 244(2), pages 417-433.
    15. Toth, Paolo, 2000. "Optimization engineering techniques for the exact solution of NP-hard combinatorial optimization problems," European Journal of Operational Research, Elsevier, vol. 125(2), pages 222-238, September.
    16. Xiaochuan Shi & Lei Wu & Xiaoliang Meng, 2017. "A New Optimization Model for the Sustainable Development: Quadratic Knapsack Problem with Conflict Graphs," Sustainability, MDPI, Open Access Journal, vol. 9(2), pages 1-10, February.
    17. Ghosh, Diptesh & Bandyopadhyay, Tathagata, 2006. "Spotting Difficult Weakly Correlated Binary Knapsack Problems," IIMA Working Papers WP2006-01-04, Indian Institute of Management Ahmedabad, Research and Publication Department.
    18. 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.
    19. Akinc, Umit, 2006. "Approximate and exact algorithms for the fixed-charge knapsack problem," European Journal of Operational Research, Elsevier, vol. 170(2), pages 363-375, April.
    20. Escudero, Laureano F. & Landete, Mercedes & Rodríguez-Chía, Antonio M., 2011. "Stochastic set packing problem," European Journal of Operational Research, Elsevier, vol. 211(2), pages 232-240, June.
    21. Büther, Marcel, 2008. "Beam search for the elastic generalized assignment problem," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 634, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    22. Alidaee, Bahram, 2014. "Zero duality gap in surrogate constraint optimization: A concise review of models," European Journal of Operational Research, Elsevier, vol. 232(2), pages 241-248.
    23. Pisinger, David & Saidi, Alima, 2017. "Tolerance analysis for 0–1 knapsack problems," European Journal of Operational Research, Elsevier, vol. 258(3), pages 866-876.
    24. Peeters, Marc & Degraeve, Zeger, 2006. "Branch-and-price algorithms for the dual bin packing and maximum cardinality bin packing problem," European Journal of Operational Research, Elsevier, vol. 170(2), pages 416-439, April.

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