IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v73y2019i4d10.1007_s10898-018-0729-8.html
   My bibliography  Save this article

On the complexity of quasiconvex integer minimization problem

Author

Listed:
  • A. Yu. Chirkov

    (Lobachevsky State University of Nizhny Novgorod)

  • D. V. Gribanov

    (Lobachevsky State University of Nizhny Novgorod
    National Research University Higher School of Economics)

  • D. S. Malyshev

    (National Research University Higher School of Economics)

  • P. M. Pardalos

    (National Research University Higher School of Economics
    University of Florida)

  • S. I. Veselov

    (Lobachevsky State University of Nizhny Novgorod)

  • N. Yu. Zolotykh

    (Lobachevsky State University of Nizhny Novgorod)

Abstract

In this paper, we consider the class of quasiconvex functions and its proper subclass of conic functions. The integer minimization problem of these functions is considered, assuming that the optimized function is defined by the comparison oracle. We will show that there is no a polynomial algorithm on $$\log R$$ log R to optimize quasiconvex functions in the ball of radius R using only the comparison oracle. On the other hand, if the optimized function is conic, then we show that there is a polynomial on $$\log R$$ log R algorithm (the dimension is fixed). We also present an exponential on the dimension lower bound for the oracle complexity of the conic function integer optimization problem. Additionally, we give examples of known problems that can be polynomially reduced to the minimization problem of functions in our classes.

Suggested Citation

  • A. Yu. Chirkov & D. V. Gribanov & D. S. Malyshev & P. M. Pardalos & S. I. Veselov & N. Yu. Zolotykh, 2019. "On the complexity of quasiconvex integer minimization problem," Journal of Global Optimization, Springer, vol. 73(4), pages 761-788, April.
    • Handle: RePEc:spr:jglopt:v:73:y:2019:i:4:d:10.1007_s10898-018-0729-8
    DOI: 10.1007/s10898-018-0729-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-018-0729-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10898-018-0729-8?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. Ravi Kannan, 1987. "Minkowski's Convex Body Theorem and Integer Programming," Mathematics of Operations Research, INFORMS, vol. 12(3), pages 415-440, August.
    2. Wojciech Banaszczyk & Alexander E. Litvak & Alain Pajor & Stanislaw J. Szarek, 1999. "The Flatness Theorem for Nonsymmetric Convex Bodies via the Local Theory of Banach Spaces," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 728-750, August.
    3. H. W. Lenstra, 1983. "Integer Programming with a Fixed Number of Variables," Mathematics of Operations Research, INFORMS, vol. 8(4), pages 538-548, November.
    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. Klaus Jansen & Roberto Solis-Oba, 2011. "A Polynomial Time OPT + 1 Algorithm for the Cutting Stock Problem with a Constant Number of Object Lengths," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 743-753, November.
    2. Friedrich Eisenbrand & Gennady Shmonin, 2008. "Parametric Integer Programming in Fixed Dimension," Mathematics of Operations Research, INFORMS, vol. 33(4), pages 839-850, November.
    3. Matthias Bentert & Robert Bredereck & Péter Györgyi & Andrzej Kaczmarczyk & Rolf Niedermeier, 2023. "A multivariate complexity analysis of the material consumption scheduling problem," Journal of Scheduling, Springer, vol. 26(4), pages 369-382, August.
    4. William Cook & Thomas Rutherford & Herbert E. Scarf & David F. Shallcross, 1991. "An Implementation of the Generalized Basis Reduction Algorithm for Integer Programming," Cowles Foundation Discussion Papers 990, Cowles Foundation for Research in Economics, Yale University.
    5. D. V. Gribanov & D. S. Malyshev & P. M. Pardalos & S. I. Veselov, 2018. "FPT-algorithms for some problems related to integer programming," Journal of Combinatorial Optimization, Springer, vol. 35(4), pages 1128-1146, May.
    6. Li, Weidong & Ou, Jinwen, 2024. "Machine scheduling with restricted rejection: An Application to task offloading in cloud–edge collaborative computing," European Journal of Operational Research, Elsevier, vol. 314(3), pages 912-919.
    7. Niclas Boehmer & Edith Elkind, 2020. "Stable Roommate Problem with Diversity Preferences," Papers 2004.14640, arXiv.org.
    8. Klaus Jansen & Kim-Manuel Klein & José Verschae, 2020. "Closing the Gap for Makespan Scheduling via Sparsification Techniques," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1371-1392, November.
    9. Phablo F. S. Moura & Matheus J. Ota & Yoshiko Wakabayashi, 2023. "Balanced connected partitions of graphs: approximation, parameterization and lower bounds," Journal of Combinatorial Optimization, Springer, vol. 45(5), pages 1-27, July.
    10. M. Köppe & M. Queyranne & C. T. Ryan, 2010. "Parametric Integer Programming Algorithm for Bilevel Mixed Integer Programs," Journal of Optimization Theory and Applications, Springer, vol. 146(1), pages 137-150, July.
    11. K. Aardal & R. E. Bixby & C. A. J. Hurkens & A. K. Lenstra & J. W. Smeltink, 2000. "Market Split and Basis Reduction: Towards a Solution of the Cornuéjols-Dawande Instances," INFORMS Journal on Computing, INFORMS, vol. 12(3), pages 192-202, August.
    12. Alberto Del Pia & Robert Hildebrand & Robert Weismantel & Kevin Zemmer, 2016. "Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 511-530, May.
    13. Elizabeth Baldwin & Paul Klemperer, 2019. "Understanding Preferences: “Demand Types”, and the Existence of Equilibrium With Indivisibilities," Econometrica, Econometric Society, vol. 87(3), pages 867-932, May.
    14. Jaykrishnan, G. & Levin, Asaf, 2024. "Scheduling with cardinality dependent unavailability periods," European Journal of Operational Research, Elsevier, vol. 316(2), pages 443-458.
    15. Masing, Berenike & Lindner, Niels & Borndörfer, Ralf, 2022. "The price of symmetric line plans in the Parametric City," Transportation Research Part B: Methodological, Elsevier, vol. 166(C), pages 419-443.
    16. Chen, Lin & Ye, Deshi & Zhang, Guochuan, 2018. "Parallel machine scheduling with speed-up resources," European Journal of Operational Research, Elsevier, vol. 268(1), pages 101-112.
    17. Sanchari Deb & Kari Tammi & Karuna Kalita & Pinakeswar Mahanta, 2018. "Review of recent trends in charging infrastructure planning for electric vehicles," Wiley Interdisciplinary Reviews: Energy and Environment, Wiley Blackwell, vol. 7(6), November.
    18. Kenneth J. Arrow & Timothy J. Kehoe, 1994. "Distinguished Fellow: Herbert Scarf's Contributions to Economics," Journal of Economic Perspectives, American Economic Association, vol. 8(4), pages 161-181, Fall.
    19. Kubale, Marek, 1996. "Preemptive versus nonpreemptive scheduling of biprocessor tasks on dedicated processors," European Journal of Operational Research, Elsevier, vol. 94(2), pages 242-251, October.
    20. Danny Nguyen & Igor Pak, 2020. "The Computational Complexity of Integer Programming with Alternations," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 191-204, February.

    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:spr:jglopt:v:73:y:2019:i:4:d:10.1007_s10898-018-0729-8. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.