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A Dynamic Subgradient-Based Branch-and-Bound Procedure for Set Covering

Author

Listed:
  • Egon Balas

    (Carnegie Mellon University, Pittsburgh, Pennsylvania)

  • Maria C. Carrera

    (Queues Enforth Development, Inc., Cambridge, Massachusetts)

Abstract

We discuss a branch and bound algorithm for set covering, whose centerpiece is a new integrated upper bounding/lower bounding procedure called dynamic subgradient optimization (DYNSGRAD). This new procedure, applied to a Lagrangean dual at every node of the search tree, combines the standard subgradient method with primal and dual heuristics that interact to change the Lagrange multipliers and tighten the upper and lower bounds, fix variables, and periodically restate the Lagrangean itself. Extensive computational testing is reported. As a stand-alone heuristic, DYNSGRAD performs significantly better than other procedures in terms of the quality of solutions obtainable with a certain computational effort. When incorporated into a branch-and-bound algorithm, DYNSGRAD considerably advances the state of the art in solving set covering problems.

Suggested Citation

  • Egon Balas & Maria C. Carrera, 1996. "A Dynamic Subgradient-Based Branch-and-Bound Procedure for Set Covering," Operations Research, INFORMS, vol. 44(6), pages 875-890, December.
    • Handle: RePEc:inm:oropre:v:44:y:1996:i:6:p:875-890
    DOI: 10.1287/opre.44.6.875
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    Cited by:

    1. Patrizia Beraldi & Andrzej Ruszczyński, 2002. "The Probabilistic Set-Covering Problem," Operations Research, INFORMS, vol. 50(6), pages 956-967, December.
    2. Victor Reyes & Ignacio Araya, 2021. "A GRASP-based scheme for the set covering problem," Operational Research, Springer, vol. 21(4), pages 2391-2408, December.
    3. Sven de Vries & Rakesh V. Vohra, 2003. "Combinatorial Auctions: A Survey," INFORMS Journal on Computing, INFORMS, vol. 15(3), pages 284-309, August.
    4. Wang, Yiyuan & Pan, Shiwei & Al-Shihabi, Sameh & Zhou, Junping & Yang, Nan & Yin, Minghao, 2021. "An improved configuration checking-based algorithm for the unicost set covering problem," European Journal of Operational Research, Elsevier, vol. 294(2), pages 476-491.
    5. 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.
    6. Alberto Caprara & Matteo Fischetti & Paolo Toth, 1999. "A Heuristic Method for the Set Covering Problem," Operations Research, INFORMS, vol. 47(5), pages 730-743, October.
    7. Ran Wei & Alan Murray & Rajan Batta, 2014. "A bounding-based solution approach for the continuous arc covering problem," Journal of Geographical Systems, Springer, vol. 16(2), pages 161-182, April.
    8. Siddhartha Syam & Bala Shetty, 1998. "Coordinated replenishments from multiple suppliers with price discounts," Naval Research Logistics (NRL), John Wiley & Sons, vol. 45(6), pages 579-598, September.
    9. Gao, Chao & Yao, Xin & Weise, Thomas & Li, Jinlong, 2015. "An efficient local search heuristic with row weighting for the unicost set covering problem," European Journal of Operational Research, Elsevier, vol. 246(3), pages 750-761.
    10. Boschetti, Marco A. & Golfarelli, Matteo & Graziani, Simone, 2020. "An exact method for shrinking pivot tables," Omega, Elsevier, vol. 93(C).
    11. José García & Gino Astorga & Víctor Yepes, 2021. "An Analysis of a KNN Perturbation Operator: An Application to the Binarization of Continuous Metaheuristics," Mathematics, MDPI, vol. 9(3), pages 1-20, January.
    12. Jihong Yan & Wenliang Cheng & Chengyu Wang & Jun Liu & Ming Gao & Aoying Zhou, 2015. "Optimizing word set coverage for multi-event summarization," Journal of Combinatorial Optimization, Springer, vol. 30(4), pages 996-1015, November.
    13. Krzysztof C. Kiwiel & Torbjörn Larsson & P. O. Lindberg, 2007. "Lagrangian Relaxation via Ballstep Subgradient Methods," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 669-686, August.
    14. Lan, Guanghui & DePuy, Gail W. & Whitehouse, Gary E., 2007. "An effective and simple heuristic for the set covering problem," European Journal of Operational Research, Elsevier, vol. 176(3), pages 1387-1403, February.
    15. Hernández-Leandro, Noberto A. & Boyer, Vincent & Salazar-Aguilar, M. Angélica & Rousseau, Louis-Martin, 2019. "A matheuristic based on Lagrangian relaxation for the multi-activity shift scheduling problem," European Journal of Operational Research, Elsevier, vol. 272(3), pages 859-867.
    16. Kedong Yan & Dongjing Miao & Cui Guo & Chanying Huang, 2021. "Efficient feature selection for logical analysis of large-scale multi-class datasets," Journal of Combinatorial Optimization, Springer, vol. 42(1), pages 1-23, July.
    17. Peeters, Marc & Degraeve, Zeger, 2006. "An linear programming based lower bound for the simple assembly line balancing problem," European Journal of Operational Research, Elsevier, vol. 168(3), pages 716-731, February.
    18. Yagiura, Mutsunori & Kishida, Masahiro & Ibaraki, Toshihide, 2006. "A 3-flip neighborhood local search for the set covering problem," European Journal of Operational Research, Elsevier, vol. 172(2), pages 472-499, July.
    19. Noriyoshi Sukegawa & Yoshitsugu Yamamoto & Liyuan Zhang, 2013. "Lagrangian relaxation and pegging test for the clique partitioning problem," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 7(4), pages 363-391, December.
    20. Ibrahim, Walid & El-Sayed, Hesham & El-Chouemie, Amr & Amer, Hoda, 2009. "An adaptive heuristic algorithm for VLSI test vectors selection," European Journal of Operational Research, Elsevier, vol. 199(3), pages 630-639, December.
    21. Masoud Yaghini & Mohammad Karimi & Mohadeseh Rahbar, 2015. "A set covering approach for multi-depot train driver scheduling," Journal of Combinatorial Optimization, Springer, vol. 29(3), pages 636-654, April.
    22. Youngho Lee & Hanif D. Sherali & Ikhyun Kwon & Seongin Kim, 2006. "A new reformulation approach for the generalized partial covering problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 53(2), pages 170-179, March.
    23. Torbjörn Larsson & Michael Patriksson, 2006. "Global Optimality Conditions for Discrete and Nonconvex Optimization---With Applications to Lagrangian Heuristics and Column Generation," Operations Research, INFORMS, vol. 54(3), pages 436-453, June.

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