IDEAS home Printed from https://ideas.repec.org/p/aiz/louvad/2021008.html
   My bibliography  Save this paper

Maxima and near-maxima of a Gaussian random assignment field

Author

Listed:
  • Mordant, Gilles

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

  • Segers, Johan

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

Abstract

The assumption that the elements of the cost matrix in the classical assignment problem are drawn independently from a standard Gaussian distribution motivates the study of a particular Gaussian field indexed by the symmetric permutation group. The correlation structure of the field is determined by the Hamming distance between two permutations. The expectation of the maximum of the field is shown to go to infinity in the same way as if all variables of the field were independent. However, the variance of the maximum is shown to converge to zero at a rate which is slower than under independence, as the variance cannot be smaller than the one of the cost of the average assignment. Still, the convergence to zero of the variance means that the maximum possesses a property known as superconcentration. Finally, the dimension of the set of near-optimal assignments is shown to converge to zero.

Suggested Citation

  • Mordant, Gilles & Segers, Johan, 2021. "Maxima and near-maxima of a Gaussian random assignment field," LIDAM Discussion Papers ISBA 2021008, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    • Handle: RePEc:aiz:louvad:2021008
    as

    Download full text from publisher

    File URL: https://dial.uclouvain.be/pr/boreal/fr/object/boreal%3A243720/datastream/PDF_01/view
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Krokhmal, Pavlo A. & Pardalos, Panos M., 2009. "Random assignment problems," European Journal of Operational Research, Elsevier, vol. 194(1), pages 1-17, April.
    2. Tanguy, Kevin, 2015. "Some superconcentration inequalities for extrema of stationary Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 239-246.
    3. Mittal, Y. & Ylvisaker, D., 1975. "Limit distributions for the maxima of stationary Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 3(1), pages 1-18, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Lifshits, M.A. & Tadevosian, A.A., 2022. "On the maximum of random assignment process," Statistics & Probability Letters, Elsevier, vol. 187(C).

    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. Mordant, Gilles & Segers, Johan, 2021. "Maxima and near-maxima of a Gaussian random assignment field," Statistics & Probability Letters, Elsevier, vol. 173(C).
    2. Panayotis Mertikopoulos & Heinrich H. Nax & Bary S. R. Pradelski, 2019. "Quick or cheap? Breaking points in dynamic markets," ECON - Working Papers 338, Department of Economics - University of Zurich.
    3. Ortega, Josué & Klein, Thilo, 2022. "Improving Efficiency and Equality in School Choice," QBS Working Paper Series 2022/02, Queen's University Belfast, Queen's Business School.
    4. M. Graça Temido, 2000. "Mixture results for extremal behaviour of strongly dependent nonstationary Gaussian sequences," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 9(2), pages 439-453, December.
    5. Kammerdiner, A.R. & Pasiliao, E.L., 2014. "In and out forests on combinatorial landscapes," European Journal of Operational Research, Elsevier, vol. 236(1), pages 78-84.
    6. Kevin Tanguy, 2020. "Talagrand Inequality at Second Order and Application to Boolean Analysis," Journal of Theoretical Probability, Springer, vol. 33(2), pages 692-714, June.
    7. Ortega, Josué & Klein, Thilo, 2023. "The cost of strategy-proofness in school choice," Games and Economic Behavior, Elsevier, vol. 141(C), pages 515-528.
    8. Chrétien, Stéphane & Corset, Franck, 2016. "A lower bound on the expected optimal value of certain random linear programs and application to shortest paths in Directed Acyclic Graphs and reliability," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 221-230.
    9. Michael Z. Spivey, 2011. "Asymptotic Moments of the Bottleneck Assignment Problem," Mathematics of Operations Research, INFORMS, vol. 36(2), pages 205-226, May.
    10. Mădălina M. Drugan, 2015. "Generating QAP instances with known optimum solution and additively decomposable cost function," Journal of Combinatorial Optimization, Springer, vol. 30(4), pages 1138-1172, November.
    11. Sana Bouajaja & Najoua Dridi, 2017. "A survey on human resource allocation problem and its applications," Operational Research, Springer, vol. 17(2), pages 339-369, July.
    12. Panayotis Mertikopoulos & Heinrich H. Nax & Bary S. R. Pradelski, 2019. "Quick or Cheap? Breaking Points in Dynamic Markets," Cowles Foundation Discussion Papers 2217, Cowles Foundation for Research in Economics, Yale University.
    13. Li, Xiaobo & Natarajan, Karthik & Teo, Chung-Piaw & Zheng, Zhichao, 2014. "Distributionally robust mixed integer linear programs: Persistency models with applications," European Journal of Operational Research, Elsevier, vol. 233(3), pages 459-473.
    14. Barbe, Ph. & McCormick, W. P., 2004. "Second-order expansion for the maximum of some stationary Gaussian sequences," Stochastic Processes and their Applications, Elsevier, vol. 110(2), pages 315-342, April.
    15. Paouris, Grigoris & Valettas, Petros & Zinn, Joel, 2017. "Random version of Dvoretzky’s theorem in ℓpn," Stochastic Processes and their Applications, Elsevier, vol. 127(10), pages 3187-3227.
    16. Helena Gaspars-Wieloch, 2021. "The Assignment Problem in Human Resource Project Management under Uncertainty," Risks, MDPI, vol. 9(1), pages 1-17, January.
    17. Mordant, Gilles, 2020. "A Random Assignment Problem: Size of Near Maximal Sets and Correct Order Expectation Bounds," LIDAM Discussion Papers ISBA 2020010, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

    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:aiz:louvad:2021008. 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: Nadja Peiffer (email available below). General contact details of provider: https://edirc.repec.org/data/isuclbe.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.