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Approximation Algorithms for the Multiple Knapsack Problem with Assignment Restrictions

Author

Listed:
  • M. Dawande

    (T.J. Watson Research Center)

  • J. Kalagnanam

    (T.J. Watson Research Center)

  • P. Keskinocak

    (Georgia Institute of Technology)

  • F.S. Salman

    (Georgia Institute of Technology)

  • R. Ravi

    (Carnegie Mellon University)

Abstract

Motivated by a real world application, we study the multiple knapsack problem with assignment restrictions (MKAR). We are given a set of items, each with a positive real weight, and a set of knapsacks, each with a positive real capacity. In addition, for each item a set of knapsacks that can hold that item is specified. In a feasible assignment of items to knapsacks, each item is assigned to at most one knapsack, assignment restrictions are satisfied, and knapsack capacities are not exceeded. We consider the objectives of maximizing assigned weight and minimizing utilized capacity. We focus on obtaining approximate solutions in polynomial computational time. We show that simple greedy approaches yield 1/3-approximation algorithms for the objective of maximizing assigned weight. We give two different 1/2-approximation algorithms: the first one solves single knapsack problems successively and the second one is based on rounding the LP relaxation solution. For the bicriteria problem of minimizing utilized capacity subject to a minimum requirement on assigned weight, we give an (1/3,2)-approximation algorithm.

Suggested Citation

  • M. Dawande & J. Kalagnanam & P. Keskinocak & F.S. Salman & R. Ravi, 2000. "Approximation Algorithms for the Multiple Knapsack Problem with Assignment Restrictions," Journal of Combinatorial Optimization, Springer, vol. 4(2), pages 171-186, June.
    • Handle: RePEc:spr:jcomop:v:4:y:2000:i:2:d:10.1023_a:1009894503716
    DOI: 10.1023/A:1009894503716
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    References listed on IDEAS

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    1. Martello, Silvano & Toth, Paolo, 1980. "Solution of the zero-one multiple knapsack problem," European Journal of Operational Research, Elsevier, vol. 4(4), pages 276-283, April.
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    Cited by:

    1. Hans Kellerer & Joseph Y.‐T. Leung & Chung‐Lun Li, 2011. "Multiple subset sum with inclusive assignment set restrictions," Naval Research Logistics (NRL), John Wiley & Sons, vol. 58(6), pages 546-563, September.
    2. Lixin Tang & Ying Meng & Zhi-Long Chen & Jiyin Liu, 2016. "Coil Batching to Improve Productivity and Energy Utilization in Steel Production," Manufacturing & Service Operations Management, INFORMS, vol. 18(2), pages 262-279, May.
    3. Fontaine, Pirmin & Crainic, Teodor Gabriel & Jabali, Ola & Rei, Walter, 2021. "Scheduled service network design with resource management for two-tier multimodal city logistics," European Journal of Operational Research, Elsevier, vol. 294(2), pages 558-570.
    4. Kataoka, Seiji & Yamada, Takeo, 2014. "Upper and lower bounding procedures for the multiple knapsack assignment problem," European Journal of Operational Research, Elsevier, vol. 237(2), pages 440-447.
    5. Jinwen Ou & Xueling Zhong & Xiangtong Qi, 2016. "Scheduling parallel machines with inclusive processing set restrictions and job rejection," Naval Research Logistics (NRL), John Wiley & Sons, vol. 63(8), pages 667-681, December.
    6. J. Álvaro Gómez-Pantoja & M. Angélica Salazar-Aguilar & José Luis González-Velarde, 2021. "The food bank resource allocation problem," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(1), pages 266-286, April.
    7. Francisco Castillo-Zunino & Pinar Keskinocak, 2021. "Bi-criteria multiple knapsack problem with grouped items," Journal of Heuristics, Springer, vol. 27(5), pages 747-789, October.
    8. Elif Akçalı & Alper Üngör & Reha Uzsoy, 2005. "Short‐term capacity allocation problem with tool and setup constraints," Naval Research Logistics (NRL), John Wiley & Sons, vol. 52(8), pages 754-764, December.
    9. David Bergman, 2019. "An Exact Algorithm for the Quadratic Multiknapsack Problem with an Application to Event Seating," INFORMS Journal on Computing, INFORMS, vol. 31(3), pages 477-492, July.
    10. Homsi, Gabriel & Jordan, Jeremy & Martello, Silvano & Monaci, Michele, 2021. "The assignment and loading transportation problem," European Journal of Operational Research, Elsevier, vol. 289(3), pages 999-1007.
    11. Fleszar, Krzysztof, 2022. "A branch-and-bound algorithm for the quadratic multiple knapsack problem," European Journal of Operational Research, Elsevier, vol. 298(1), pages 89-98.
    12. Li, Shuguang, 2017. "Approximation algorithms for scheduling jobs with release times and arbitrary sizes on batch machines with non-identical capacities," European Journal of Operational Research, Elsevier, vol. 263(3), pages 815-826.
    13. Stefka Fidanova & Krassimir Todorov Atanassov, 2021. "ACO with Intuitionistic Fuzzy Pheromone Updating Applied on Multiple-Constraint Knapsack Problem," Mathematics, MDPI, vol. 9(13), pages 1-7, June.
    14. Burke, E.K. & Landa Silva, J.D., 2006. "The influence of the fitness evaluation method on the performance of multiobjective search algorithms," European Journal of Operational Research, Elsevier, vol. 169(3), pages 875-897, March.
    15. Xueqi Wu & Zhi‐Long Chen, 2022. "Fulfillment scheduling for buy‐online‐pickup‐in‐store orders," Production and Operations Management, Production and Operations Management Society, vol. 31(7), pages 2982-3003, July.

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