IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v273y2019i1p102-107.html
   My bibliography  Save this article

Berge-acyclic multilinear 0–1 optimization problems

Author

Listed:
  • Buchheim, Christoph
  • Crama, Yves
  • Rodríguez-Heck, Elisabeth

Abstract

The problem of optimizing a multilinear polynomial f in 0–1 variables arises in applications from many different areas. We are interested in resolution methods based on reformulating the polynomial problem into an equivalent linear one, an approach that attempts to draw benefit from the extensive literature in integer linear programming. More precisely, we characterize instances for which the classical standard linearization procedure guarantees integer optimal solutions. We show that the standard linearization polytope PH is integer if and only if the hypergraph H defined by the higher-degree monomials of f is Berge-acyclic, or equivalently, when the matrix defining PH is balanced. This characterization follows from more general conditions that guarantee integral optimal vertices for a relaxed formulation depending on the sign pattern of the monomials of f.

Suggested Citation

  • Buchheim, Christoph & Crama, Yves & Rodríguez-Heck, Elisabeth, 2019. "Berge-acyclic multilinear 0–1 optimization problems," European Journal of Operational Research, Elsevier, vol. 273(1), pages 102-107.
    • Handle: RePEc:eee:ejores:v:273:y:2019:i:1:p:102-107
    DOI: 10.1016/j.ejor.2018.07.045
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377221718306672
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.ejor.2018.07.045?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Lawrence J. Watters, 1967. "Letter to the Editor—Reduction of Integer Polynomial Programming Problems to Zero-One Linear Programming Problems," Operations Research, INFORMS, vol. 15(6), pages 1171-1174, December.
    2. Goldengorin, Boris & Ghosh, Diptesh & Sierksma, Gerard, 2001. "Branch and peg algorithms for the simple plant location problem," Research Report 01A14, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    3. Yves Crama & Joris van de Klundert, 1999. "Worst‐case performance of approximation algorithms for tool management problems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(5), pages 445-462, August.
    4. Crama, Yves & Oerlemans, Alwin G., 1994. "A column generation approach to job grouping for flexible manufacturing systems," European Journal of Operational Research, Elsevier, vol. 78(1), pages 58-80, October.
    5. Alekseeva, Ekaterina & Kochetov, Yuri & Plyasunov, Alexander, 2008. "Complexity of local search for the p-median problem," European Journal of Operational Research, Elsevier, vol. 191(3), pages 736-752, December.
    6. Fred Glover & Eugene Woolsey, 1973. "Further Reduction of Zero-One Polynomial Programming Problems to Zero-One linear Programming Problems," Operations Research, INFORMS, vol. 21(1), pages 156-161, February.
    7. P. M. Dearing & P. L. Hammer & B. Simeone, 1992. "Boolean and Graph Theoretic Formulations of the Simple Plant Location Problem," Transportation Science, INFORMS, vol. 26(2), pages 138-148, May.
    8. Fred Glover & Eugene Woolsey, 1974. "Technical Note—Converting the 0-1 Polynomial Programming Problem to a 0-1 Linear Program," Operations Research, INFORMS, vol. 22(1), pages 180-182, February.
    9. Crama, Yves, 1997. "Combinatorial optimization models for production scheduling in automated manufacturing systems," European Journal of Operational Research, Elsevier, vol. 99(1), pages 136-153, May.
    10. Zuo-Jun Max Shen & Collette Coullard & Mark S. Daskin, 2003. "A Joint Location-Inventory Model," Transportation Science, INFORMS, vol. 37(1), pages 40-55, February.
    11. Bader F. AlBdaiwi & Diptesh Ghosh & Boris Goldengorin, 2011. "Data aggregation for p-median problems," Journal of Combinatorial Optimization, Springer, vol. 21(3), pages 348-363, April.
    12. ReVelle, C.S. & Eiselt, H.A. & Daskin, M.S., 2008. "A bibliography for some fundamental problem categories in discrete location science," European Journal of Operational Research, Elsevier, vol. 184(3), pages 817-848, February.
    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. Warren Adams & Hanif Sherali, 2005. "A Hierarchy of Relaxations Leading to the Convex Hull Representation for General Discrete Optimization Problems," Annals of Operations Research, Springer, vol. 140(1), pages 21-47, November.
    2. Yves Crama & Joris van de Klundert, 1999. "Worst‐case performance of approximation algorithms for tool management problems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(5), pages 445-462, August.
    3. Matzliach, Barouch & Tzur, Michal, 2000. "Storage management of items in two levels of availability," European Journal of Operational Research, Elsevier, vol. 121(2), pages 363-379, March.
    4. Konak, Abdullah & Kulturel-Konak, Sadan & Azizoglu, Meral, 2008. "Minimizing the number of tool switching instants in Flexible Manufacturing Systems," International Journal of Production Economics, Elsevier, vol. 116(2), pages 298-307, December.
    5. Juan S. Borrero & Colin Gillen & Oleg A. Prokopyev, 2017. "Fractional 0–1 programming: applications and algorithms," Journal of Global Optimization, Springer, vol. 69(1), pages 255-282, September.
    6. Renato de Matta & Vernon Ning Hsu & Timothy J. Lowe, 1999. "Capacitated selection problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(1), pages 19-37, February.
    7. Gebennini, Elisa & Gamberini, Rita & Manzini, Riccardo, 2009. "An integrated production-distribution model for the dynamic location and allocation problem with safety stock optimization," International Journal of Production Economics, Elsevier, vol. 122(1), pages 286-304, November.
    8. Melo, M.T. & Nickel, S. & Saldanha-da-Gama, F., 2009. "Facility location and supply chain management - A review," European Journal of Operational Research, Elsevier, vol. 196(2), pages 401-412, July.
    9. Sven Mallach, 2018. "Compact linearization for binary quadratic problems subject to assignment constraints," 4OR, Springer, vol. 16(3), pages 295-309, September.
    10. M. Hosein Zare & Juan S. Borrero & Bo Zeng & Oleg A. Prokopyev, 2019. "A note on linearized reformulations for a class of bilevel linear integer problems," Annals of Operations Research, Springer, vol. 272(1), pages 99-117, January.
    11. Karthik Natarajan & Dongjian Shi & Kim-Chuan Toh, 2014. "A Probabilistic Model for Minmax Regret in Combinatorial Optimization," Operations Research, INFORMS, vol. 62(1), pages 160-181, February.
    12. Zheng, Xiaojin & Yin, Meixia & Zhang, Yanxia, 2019. "Integrated optimization of location, inventory and routing in supply chain network design," Transportation Research Part B: Methodological, Elsevier, vol. 121(C), pages 1-20.
    13. Sergio García & Martine Labbé & Alfredo Marín, 2011. "Solving Large p -Median Problems with a Radius Formulation," INFORMS Journal on Computing, INFORMS, vol. 23(4), pages 546-556, November.
    14. M. Fattahi & M. Mahootchi & S. M. Moattar Husseini, 2016. "Integrated strategic and tactical supply chain planning with price-sensitive demands," Annals of Operations Research, Springer, vol. 242(2), pages 423-456, July.
    15. Kim, Nayeon & Montreuil, Benoit & Klibi, Walid & Zied Babai, M., 2023. "Network inventory deployment for responsive fulfillment," International Journal of Production Economics, Elsevier, vol. 255(C).
    16. Congdong Li & Hao Guo & Ying Zhang & Shuai Deng & Yu Wang, 2018. "An Improved Differential Evolution Algorithm for a Multicommodity Location-Inventory Problem with False Failure Returns," Complexity, Hindawi, vol. 2018, pages 1-13, October.
    17. Cavory, G. & Dupas, R. & Goncalves, G., 2005. "A genetic approach to solving the problem of cyclic job shop scheduling with linear constraints," European Journal of Operational Research, Elsevier, vol. 161(1), pages 73-85, February.
    18. Paul, Henrik J. & Bierwirth, Christian & Kopfer, Herbert, 2007. "A heuristic scheduling procedure for multi-item hoist production lines," International Journal of Production Economics, Elsevier, vol. 105(1), pages 54-69, January.
    19. Sven Mallach, 2021. "Inductive linearization for binary quadratic programs with linear constraints," 4OR, Springer, vol. 19(4), pages 549-570, December.
    20. Spencer Leitch & Zhiyuan Wei, 2024. "Improving spatial access to healthcare facilities: an integrated approach with spatial analysis and optimization modeling," Annals of Operations Research, Springer, vol. 341(2), pages 1057-1074, October.

    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:eee:ejores:v:273:y:2019:i:1:p:102-107. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/eor .

    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.