IDEAS home Printed from https://ideas.repec.org/a/inm/orijoc/v32y3i2020p747-762.html
   My bibliography  Save this article

A Lagrangian Bound on the Clique Number and an Exact Algorithm for the Maximum Edge Weight Clique Problem

Author

Listed:
  • Seyedmohammadhossein Hosseinian

    (Industrial & Systems Engineering, Texas A&M University, College Station, Texas 77843)

  • Dalila B. M. M. Fontes

    (Institute for Systems and Computer Engineering, Technology and Science and Faculdade de Economia, Universidade do Porto, 4200-465 Porto, Portugal)

  • Sergiy Butenko

    (Industrial & Systems Engineering, Texas A&M University, College Station, Texas 77843)

Abstract

This paper explores the connections between the classical maximum clique problem and its edge-weighted generalization, the maximum edge weight clique (MEWC) problem. As a result, a new analytic upper bound on the clique number of a graph is obtained and an exact algorithm for solving the MEWC problem is developed. The bound on the clique number is derived using a Lagrangian relaxation of an integer (linear) programming formulation of the MEWC problem. Furthermore, coloring-based bounds on the clique number are used in a novel upper-bounding scheme for the MEWC problem. This scheme is employed within a combinatorial branch-and-bound framework, yielding an exact algorithm for the MEWC problem. Results of computational experiments demonstrate a superior performance of the proposed algorithm compared with existing approaches.

Suggested Citation

  • Seyedmohammadhossein Hosseinian & Dalila B. M. M. Fontes & Sergiy Butenko, 2020. "A Lagrangian Bound on the Clique Number and an Exact Algorithm for the Maximum Edge Weight Clique Problem," INFORMS Journal on Computing, INFORMS, vol. 32(3), pages 747-762, July.
    • Handle: RePEc:inm:orijoc:v:32:y:3:i:2020:p:747-762
    DOI: 10.1287/ijoc.2019.0898
    as

    Download full text from publisher

    File URL: https://doi.org/10.1287/ijoc.2019.0898
    Download Restriction: no
    File URL: https://libkey.io/10.1287/ijoc.2019.0898?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. Wu, Qinghua & Hao, Jin-Kao, 2015. "A review on algorithms for maximum clique problems," European Journal of Operational Research, Elsevier, vol. 242(3), pages 693-709.
    2. Wayne Pullan, 2006. "Phased local search for the maximum clique problem," Journal of Combinatorial Optimization, Springer, vol. 12(3), pages 303-323, November.
    3. Mikhail Batsyn & Boris Goldengorin & Evgeny Maslov & Panos M. Pardalos, 2014. "Improvements to MCS algorithm for the maximum clique problem," Journal of Combinatorial Optimization, Springer, vol. 27(2), pages 397-416, February.
    4. Seyedmohammadhossein Hosseinian & Dalila B. M. M. Fontes & Sergiy Butenko, 2018. "A nonconvex quadratic optimization approach to the maximum edge weight clique problem," Journal of Global Optimization, Springer, vol. 72(2), pages 219-240, October.
    5. Martí, Rafael & Gallego, Micael & Duarte, Abraham, 2010. "A branch and bound algorithm for the maximum diversity problem," European Journal of Operational Research, Elsevier, vol. 200(1), pages 36-44, January.
    6. Seyedmohammadhossein Hosseinian & Dalila B. M. M. Fontes & Sergiy Butenko & Marco Buongiorno Nardelli & Marco Fornari & Stefano Curtarolo, 2017. "The Maximum Edge Weight Clique Problem: Formulations and Solution Approaches," Springer Optimization and Its Applications, in: Sergiy Butenko & Panos M. Pardalos & Volodymyr Shylo (ed.), Optimization Methods and Applications, pages 217-237, Springer.
    7. Macambira, Elder Magalhaes & de Souza, Cid Carvalho, 2000. "The edge-weighted clique problem: Valid inequalities, facets and polyhedral computations," European Journal of Operational Research, Elsevier, vol. 123(2), pages 346-371, June.
    8. Hunting, Marcel & Faigle, Ulrich & Kern, Walter, 2001. "A Lagrangian relaxation approach to the edge-weighted clique problem," European Journal of Operational Research, Elsevier, vol. 131(1), pages 119-131, May.
    9. Alidaee, Bahram & Glover, Fred & Kochenberger, Gary & Wang, Haibo, 2007. "Solving the maximum edge weight clique problem via unconstrained quadratic programming," European Journal of Operational Research, Elsevier, vol. 181(2), pages 592-597, September.
    10. Dijkhuizen, G. & Faigle, U., 1993. "A cutting-plane approach to the edge-weighted maximal clique problem," European Journal of Operational Research, Elsevier, vol. 69(1), pages 121-130, August.
    11. R Aringhieri & R Cordone, 2011. "Comparing local search metaheuristics for the maximum diversity problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 62(2), pages 266-280, February.
    12. Park, Kyungchul & Lee, Kyungsik & Park, Sungsoo, 1996. "An extended formulation approach to the edge-weighted maximal clique problem," European Journal of Operational Research, Elsevier, vol. 95(3), pages 671-682, December.
    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. Seyedmohammadhossein Hosseinian & Dalila B. M. M. Fontes & Sergiy Butenko, 2018. "A nonconvex quadratic optimization approach to the maximum edge weight clique problem," Journal of Global Optimization, Springer, vol. 72(2), pages 219-240, October.
    2. San Segundo, Pablo & Coniglio, Stefano & Furini, Fabio & Ljubić, Ivana, 2019. "A new branch-and-bound algorithm for the maximum edge-weighted clique problem," European Journal of Operational Research, Elsevier, vol. 278(1), pages 76-90.
    3. Wu, Qinghua & Hao, Jin-Kao, 2015. "A review on algorithms for maximum clique problems," European Journal of Operational Research, Elsevier, vol. 242(3), pages 693-709.
    4. Sorensen, Michael M., 2004. "New facets and a branch-and-cut algorithm for the weighted clique problem," European Journal of Operational Research, Elsevier, vol. 154(1), pages 57-70, April.
    5. Wu, Qinghua & Hao, Jin-Kao, 2013. "A hybrid metaheuristic method for the Maximum Diversity Problem," European Journal of Operational Research, Elsevier, vol. 231(2), pages 452-464.
    6. Lourenco, Lidia Lampreia & Pato, Margarida Vaz, 2007. "The effect of strengthened linear formulations on improving the lower bounds for the part families with precedence constraints problem," European Journal of Operational Research, Elsevier, vol. 183(1), pages 181-196, November.
    7. Yi Chu & Boxiao Liu & Shaowei Cai & Chuan Luo & Haihang You, 2020. "An efficient local search algorithm for solving maximum edge weight clique problem in large graphs," Journal of Combinatorial Optimization, Springer, vol. 39(4), pages 933-954, May.
    8. Michael M. Sørensen, 2004. "b-Tree Facets for the Simple Graph Partitioning Polytope," Journal of Combinatorial Optimization, Springer, vol. 8(2), pages 151-170, June.
    9. Carvalho, Filipa D. & Almeida, M. Teresa, 2011. "Upper bounds and heuristics for the 2-club problem," European Journal of Operational Research, Elsevier, vol. 210(3), pages 489-494, May.
    10. Zhou, Yi & Hao, Jin-Kao & Goëffon, Adrien, 2017. "PUSH: A generalized operator for the Maximum Vertex Weight Clique Problem," European Journal of Operational Research, Elsevier, vol. 257(1), pages 41-54.
    11. Hunting, Marcel & Faigle, Ulrich & Kern, Walter, 2001. "A Lagrangian relaxation approach to the edge-weighted clique problem," European Journal of Operational Research, Elsevier, vol. 131(1), pages 119-131, May.
    12. Macambira, Elder Magalhaes & de Souza, Cid Carvalho, 2000. "The edge-weighted clique problem: Valid inequalities, facets and polyhedral computations," European Journal of Operational Research, Elsevier, vol. 123(2), pages 346-371, June.
    13. Ricardo M. Lima & Ignacio E. Grossmann, 2017. "On the solution of nonconvex cardinality Boolean quadratic programming problems: a computational study," Computational Optimization and Applications, Springer, vol. 66(1), pages 1-37, January.
    14. Borgwardt, S. & Schmiedl, F., 2014. "Threshold-based preprocessing for approximating the weighted dense k-subgraph problem," European Journal of Operational Research, Elsevier, vol. 234(3), pages 631-640.
    15. Bahram Alidaee & Haibo Wang, 2017. "A note on heuristic approach based on UBQP formulation of the maximum diversity problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 68(1), pages 102-110, January.
    16. Sinha, Ankur & Das, Arka & Anand, Guneshwar & Jayaswal, Sachin, 2023. "A general purpose exact solution method for mixed integer concave minimization problems," European Journal of Operational Research, Elsevier, vol. 309(3), pages 977-992.
    17. Martí, Rafael & Martínez-Gavara, Anna & Pérez-Peló, Sergio & Sánchez-Oro, Jesús, 2022. "A review on discrete diversity and dispersion maximization from an OR perspective," European Journal of Operational Research, Elsevier, vol. 299(3), pages 795-813.
    18. Sebastian Lamm & Peter Sanders & Christian Schulz & Darren Strash & Renato F. Werneck, 2017. "Finding near-optimal independent sets at scale," Journal of Heuristics, Springer, vol. 23(4), pages 207-229, August.
    19. Alidaee, Bahram & Glover, Fred & Kochenberger, Gary & Wang, Haibo, 2007. "Solving the maximum edge weight clique problem via unconstrained quadratic programming," European Journal of Operational Research, Elsevier, vol. 181(2), pages 592-597, September.
    20. Sinha, Ankur & Das, Arka & Anand, Guneshwar & Jayaswal, Sachin, 2021. "A General Purpose Exact Solution Method for Mixed Integer Concave Minimization Problems," IIMA Working Papers WP 2021-03-01, Indian Institute of Management Ahmedabad, Research and Publication Department.

    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:inm:orijoc:v:32:y:3:i:2020:p:747-762. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    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.