IDEAS home Printed from https://ideas.repec.org/a/wly/navres/v58y2011i2p153-164.html
   My bibliography  Save this article

Quadratic bottleneck problems

Author

Listed:
  • Abraham P. Punnen
  • Ruonan Zhang

Abstract

We study the quadratic bottleneck problem (QBP) which generalizes several well‐studied optimization problems. A weak duality theorem is introduced along with a general purpose algorithm to solve QBP. An example is given which illustrates duality gap in the weak duality theorem. It is shown that the special case of QBP where feasible solutions are subsets of a finite set having the same cardinality is NP‐hard. Likewise the quadratic bottleneck spanning tree problem (QBST) is shown to be NP‐hard on a bipartite graph even if the cost function takes 0–1 values only. Two lower bounds for QBST are derived and compared. Efficient heuristic algorithms are presented for QBST along with computational results. When the cost function is decomposable, we show that QBP is solvable in polynomial time whenever an associated linear bottleneck problem can be solved in polynomial time. As a consequence, QBP with feasible solutions form spanning trees, s‐t paths, matchings, etc., of a graph are solvable in polynomial time with a decomposable cost function. We also show that QBP can be formulated as a quadratic minsum problem and establish some asymptotic results. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011

Suggested Citation

  • Abraham P. Punnen & Ruonan Zhang, 2011. "Quadratic bottleneck problems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 58(2), pages 153-164, March.
    • Handle: RePEc:wly:navres:v:58:y:2011:i:2:p:153-164
    DOI: 10.1002/nav.20446
    as

    Download full text from publisher

    File URL: https://doi.org/10.1002/nav.20446
    Download Restriction: no
    File URL: https://libkey.io/10.1002/nav.20446?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. R. S. Garfinkel & M. R. Rao, 1971. "The bottleneck transportation problem," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 18(4), pages 465-472, December.
    2. Arjang Assad & Weixuan Xu, 1992. "The quadratic minimum spanning tree problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 39(3), pages 399-417, April.
    3. Ulrich Derigs, 1982. "On three basic methods for solving bottleneck transportation problems," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 29(3), pages 505-515, September.
    4. Katagiri, Hideki & Sakawa, Masatoshi & Ishii, Hiroaki, 2004. "Fuzzy random bottleneck spanning tree problems using possibility and necessity measures," European Journal of Operational Research, Elsevier, vol. 152(1), pages 88-95, January.
    5. Punnen, Abraham P. & Nair, K. P. K., 1996. "An O(m log n) algorithm for the max + sum spanning tree problem," European Journal of Operational Research, Elsevier, vol. 89(2), pages 423-426, March.
    6. Abraham P. Punnen & K. P. K. Nair, 1996. "An Improved Algorithm for the Constrained Bottleneck Spanning Tree Problem," INFORMS Journal on Computing, INFORMS, vol. 8(1), pages 41-44, February.
    7. Elena Fernandez & Robert Garfinkel & Roman Arbiol, 1998. "Mosaicking of Aerial Photographic Maps Via Seams Defined by Bottleneck Shortest Paths," Operations Research, INFORMS, vol. 46(3), pages 293-304, June.
    8. Zhou, Gengui & Gen, Mitsuo, 1999. "Genetic algorithm approach on multi-criteria minimum spanning tree problem," European Journal of Operational Research, Elsevier, vol. 114(1), pages 141-152, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lacour, Renaud, 2014. "Approches de résolution exacte et approchée en optimisation combinatoire multi-objectif, application au problème de l'arbre couvrant de poids minimal," Economics Thesis from University Paris Dauphine, Paris Dauphine University, number 123456789/14806 edited by Vanderpooten, Daniel.
    2. de Meijer, Frank, 2023. "Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization," Other publications TiSEM b1f1088c-95fe-4b8a-9e15-c, Tilburg University, School of Economics and Management.
    3. Delorme, Xavier & Gandibleux, Xavier & Degoutin, Fabien, 2010. "Evolutionary, constructive and hybrid procedures for the bi-objective set packing problem," European Journal of Operational Research, Elsevier, vol. 204(2), pages 206-217, July.
    4. Juan Villegas & Fernando Palacios & Andrés Medaglia, 2006. "Solution methods for the bi-objective (cost-coverage) unconstrained facility location problem with an illustrative example," Annals of Operations Research, Springer, vol. 147(1), pages 109-141, October.
    5. Perny, Patrice & Spanjaard, Olivier, 2005. "A preference-based approach to spanning trees and shortest paths problems***," European Journal of Operational Research, Elsevier, vol. 162(3), pages 584-601, May.
    6. Zhou, Gengui & Min, Hokey & Gen, Mitsuo, 2003. "A genetic algorithm approach to the bi-criteria allocation of customers to warehouses," International Journal of Production Economics, Elsevier, vol. 86(1), pages 35-45, October.
    7. Wen, Hao & Sang, Song & Qiu, Chenhui & Du, Xiangrui & Zhu, Xiao & Shi, Qian, 2019. "A new optimization method of wind turbine airfoil performance based on Bessel equation and GABP artificial neural network," Energy, Elsevier, vol. 187(C).
    8. Francis Sourd & Olivier Spanjaard, 2008. "A Multiobjective Branch-and-Bound Framework: Application to the Biobjective Spanning Tree Problem," INFORMS Journal on Computing, INFORMS, vol. 20(3), pages 472-484, August.
    9. I. Stancu-Minasian & R. Caballero & E. Cerdá & M. Muñoz, 1999. "The stochastic bottleneck linear programming problem," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 7(1), pages 123-143, June.
    10. Xiucui Guan & Xinyan He & Panos M. Pardalos & Binwu Zhang, 2017. "Inverse max $$+$$ + sum spanning tree problem under Hamming distance by modifying the sum-cost vector," Journal of Global Optimization, Springer, vol. 69(4), pages 911-925, December.
    11. Prabuddha De & Jay B. Ghosh & Charles E. Wells, 1992. "On the solution of a stochastic bottleneck assignment problem and its variations," Naval Research Logistics (NRL), John Wiley & Sons, vol. 39(3), pages 389-397, April.
    12. Fanrong Xie & Zuoan Li, 2022. "An iterative solution technique for capacitated two-stage time minimization transportation problem," 4OR, Springer, vol. 20(4), pages 637-684, December.
    13. Knowles, Joshua D. & Corne, David W., 2002. "Enumeration of Pareto optimal multi-criteria spanning trees - a proof of the incorrectness of Zhou and Gen's proposed algorithm," European Journal of Operational Research, Elsevier, vol. 143(3), pages 543-547, December.
    14. Diabat, Ali & Kannan, Devika & Kaliyan, Mathiyazhagan & Svetinovic, Davor, 2013. "An optimization model for product returns using genetic algorithms and artificial immune system," Resources, Conservation & Recycling, Elsevier, vol. 74(C), pages 156-169.
    15. Dali Jiang & Haitao Li & Tinghong Yang & De Li, 2016. "Genetic algorithm for inventory positioning problem with general acyclic supply chain networks," European Journal of Industrial Engineering, Inderscience Enterprises Ltd, vol. 10(3), pages 367-384.
    16. Mak, Brenda & Blanning, Robert & Ho, Susanna, 2006. "Genetic algorithms in logic tree decision modeling," European Journal of Operational Research, Elsevier, vol. 170(2), pages 597-612, April.
    17. Masatoshi Sakawa & Hideki Katagiri, 2012. "Stackelberg solutions for fuzzy random two-level linear programming through level sets and fractile criterion optimization," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 20(1), pages 101-117, March.
    18. Andréa Santos & Diego Lima & Dario Aloise, 2014. "Modeling and solving the bi-objective minimum diameter-cost spanning tree problem," Journal of Global Optimization, Springer, vol. 60(2), pages 195-216, October.
    19. Selcen (Pamuk) Phelps & Murat Köksalan, 2003. "An Interactive Evolutionary Metaheuristic for Multiobjective Combinatorial Optimization," Management Science, INFORMS, vol. 49(12), pages 1726-1738, December.
    20. Altannar Chinchuluun & Panos Pardalos, 2007. "A survey of recent developments in multiobjective optimization," Annals of Operations Research, Springer, vol. 154(1), pages 29-50, October.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wly:navres:v:58:y:2011:i:2:p:153-164. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://doi.org/10.1002/(ISSN)1520-6750 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.