Email this article Probabilistic models for the analysis of inverse extremal problems in combinatorics
- Authors: Enatskaya N.Y.1
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Affiliations:
- National Research University “Higher School of Economics”, Moscow Institute of Electronics and Mathematics
- Issue: Vol 26, No 3 (2022)
- Pages: 571-591
- Section: Mathematical Modeling, Numerical Methods and Software Complexes
- URL: https://journals.eco-vector.com/1991-8615/article/view/108999
- DOI: https://doi.org/10.14498/vsgtu1947
- ID: 108999
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Abstract
In an inverse extremal problem for a combinatorial scheme with a given value of the objective function of the form of a certain extreme value of its characteristic, a probabilistic model is developed that ensures that this value is obtained in its outcomes. Two types of such characteristics are considered, relating to each of the schemes or to a set of outcomes.
The pre-asymptotic analysis of such a model is carried out by the author's enumerative method. It is based on the construction of an iterative random process with iterations of successive stages of a numbered non-repetitive enumeration and the formation of outcomes of the scheme. The iterative development of the process is represented by a probabilistic graph.
The study of the outcomes of the scheme according to the model in the enumerative method is carried out in the following areas: visual numbering of the outcomes of the scheme, finding their number, establishing a one-to-one correspondence between the types and numbers of outcomes of the scheme, obtaining their probabilistic distribution (controlled by a random process of listing the outcomes of the scheme), and modeling them with this distribution.
Along with the direct study of circuits in these areas, algorithms are proposed to obtain results for them by partially recalculating them from the results of a similar analysis of more general, previously studied circuits without restrictions or with less restrictions on the values of the characteristics under consideration.
About the authors
Nataliya Yu. Enatskaya
National Research University “Higher School of Economics”,Moscow Institute of Electronics and Mathematics
Author for correspondence.
Email: nat1943@mail.ru
ORCID iD: 0000-0003-1241-7543
SPIN-code: 9706-9900
Scopus Author ID: 6504731611
ResearcherId: L-6102-2015
http://www.mathnet.ru/person28100
Cand. Phys. & Math. Sci.; Associate Professor; Dept. of Applied Mathematics
Russian Federation, 34, Tallinskay st, Moscow, 123458, Russian FederationReferences
- Afanas’ev M. Yu., Suvorov B. P. Issledovanie operatsii v ekonomike. Modeli, zadachi, resheniia [Operations Research in Economics. Models, Tasks, Solutions]. Moscow, INFRA-M, 2003, 202 pp. (In Russian)
- Baranov V. I., Stechkin B. S. Ekstremal’nye kombinatornye zadachi i ikh prilozheniia [Extremal Combinatorial Problems and Their Applications]. Moscow, Fizmatlit, 2004, 240 pp. (In Russian). EDN: RXGTQD.
- Kolchin V. F. On the limiting behavior of extreme order statistics in a polynomial scheme, Theory Probab. Appl., 1969, vol. 14, no. 3, pp. 458–469. DOI: https://doi.org/10.1137/1114058.
- Khakimullin E.R. Asymptotic behavior of the maximum occupancy in an equiprobable scheme of allocation of particles by complexes, Math. Notes, 1981, vol. 30, no. 2, pp. 626–633. DOI: https://doi.org/10.1007/BF01708846.
- Viktorova I. I. Asymptotic behavior of maximum of an equiprobable polynomial scheme, Math. Notes, 1969, vol. 5, no. 3, pp. 184–191. EDN: ZZYIEX. DOI: https://doi.org/10.1007/BF01388624.
- Enatskaya N. Yu. Probabilistic models of combinatorial schemes, Bulletin of the South Ural State University, Ser. Mathematical Modelling, Programming and Computer Software, 2020, vol. 13, no. 3, pp. 103–111 (In Russian). EDN: DKMIUC. DOI: https://doi.org/10.14529/mmp200312.
- Enatskaya N. Yu. Analysis of combinatorial schemes in the pre-asymptotic region of parameter change, Proceedings of the Karelian Research Centre of the Russian Academy of Sciences, 2018, no. 7, pp. 117–133 (In Russian). EDN: XRQZZR. DOI: https://doi.org/10.17076/mat750.
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