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A Horizontal Method of Localizing Values of a Linear Function in Permutation-Based Optimization

  • Liudmyla Koliechkina
  • Oksana PichuginaEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)

Abstract

This paper is dedicated to linear constrained optimization on permutation configurations’ set, namely, to permutation-based subset sum problem (PB-SSP). To this problem, a directed structural graph is associated connected with a skeleton graph of the permutohedron and allowing to perform a directed search to solve this linear program. To solve PB-SSP, a horizontal method for localizing values of a linear objective function is offered combining Graph Theory tools, geometric and structural properties of a permutation set mapped into Euclidean space, the behavior of linear functions on the set, and Branch and Bound techniques.

Keywords

Discrete optimization Linear constrained optimization Combinatorial configuration Permutation Skeleton graph Grid graph  Search tree 

References

  1. 1.
    Donec, G.A., Kolechkina, L.M.: Construction of Hamiltonian paths in graphs of permutation polyhedra. Cybern. Syst. Anal. 46(1), 7–13 (2010).  https://doi.org/10.1007/s10559-010-9178-1MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Donec, G.A., Kolechkina, L.M.: Extremal Problems on Combinatorial Configurations. RVV PUET, Poltava (2011)Google Scholar
  3. 3.
    Donets, G.A., Kolechkina, L.N.: Method of ordering the values of a linear function on a set of permutations. Cybern. Syst. Anal. 45(2), 204–213 (2009).  https://doi.org/10.1007/s10559-009-9092-6MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gimadi, E., Khachay, M.: Extremal Problems on Sets of Permutations. Ural Federal University, Yekaterinburg (2016). [in Russian]Google Scholar
  5. 5.
    Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Berlin, New York (2010)Google Scholar
  6. 6.
    Koliechkina, L.M., Dvirna, O.A.: Solving extremum problems with linear fractional objective functions on the combinatorial configuration of permutations under multicriteriality. Cybern. Syst. Anal. 53(4), 590–599 (2017).  https://doi.org/10.1007/s10559-017-9961-3CrossRefzbMATHGoogle Scholar
  7. 7.
    Koliechkina, L.N., Dvernaya, O.A., Nagornaya, A.N.: Modified coordinate method to solve multicriteria optimization problems on combinatorial configurations. Cybern. Syst. Anal. 50(4), 620–626 (2014).  https://doi.org/10.1007/s10559-014-9650-4CrossRefzbMATHGoogle Scholar
  8. 8.
    Koliechkina, L., Pichugina, O.: Multiobjective Optimization on Permutations with Applications. DEStech Trans. Comput. Sci. Eng. Supplementary Volume OPTIMA 2018, 61–75 (2018). https://doi.org/10.12783/dtcse/optim2018/27922
  9. 9.
    Kozin, I.V., Maksyshko, N.K., Perepelitsa, V.A.: Fragmentary structures in discrete optimization problems. Cybern. Syst. Anal. 53(6), 931–936 (2017).  https://doi.org/10.1007/s10559-017-9995-6MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms. Springer, New York (2018)Google Scholar
  11. 11.
    Lengauer, T.: Combinatorial Algorithms for Integrated Circuit Layout. Vieweg+Teubner Verlag (1990)Google Scholar
  12. 12.
    Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implementations. Wiley, Chichester, New York (1990)Google Scholar
  13. 13.
    Mehdi, M.: Parallel Hybrid Optimization Methods for permutation based problems (2011). https://tel.archives-ouvertes.fr/tel-00841962/document
  14. 14.
    Pichugina, O.: Placement problems in chip design: Modeling and optimization. In: 2017 4th International Scientific-Practical Conference Problems of Infocommunications. Science and Technology (PIC S&T). pp. 465–473 (2017). https://doi.org/10.1109/INFOCOMMST.2017.8246440
  15. 15.
    Pichugina, O., Farzad, B.: A human communication network model. In: CEUR Workshop Proceedings, pp. 33–40. KNU, Kyiv (2016)Google Scholar
  16. 16.
    Pichugina, O., Yakovlev, S.: Convex extensions and continuous functional representations in optimization, with their applications. J. Coupled Syst. Multiscale Dyn. 4(2), 129–152 (2016).  https://doi.org/10.1166/jcsmd.2016.1103CrossRefGoogle Scholar
  17. 17.
    Pichugina, O.S., Yakovlev, S.V.: Functional and analytic representations of the general permutation. East. Eur. J. Enterp. Technol. 79(4), 27–38 (2016).  https://doi.org/10.15587/1729-4061.2016.58550CrossRefGoogle Scholar
  18. 18.
    Pichugina, O.S., Yakovlev, S.V.: Continuous representations and functional extensions in combinatorial optimization. Cybern. Syst. Anal. 52(6), 921–930 (2016).  https://doi.org/10.1007/s10559-016-9894-2MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pichugina, O., Yakovlev, S.: Optimization on polyhedral-spherical sets: Theory and applications. In: 2017 IEEE 1st Ukraine Conference on Electrical and Computer Engineering, UKRCON 2017-Proceedings, pp. 1167–1174. KPI, Kiev (2017). https://doi.org/10.1109/UKRCON.2017.8100436
  20. 20.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin, New York (2003)Google Scholar
  21. 21.
    Semenova, N.V., Kolechkina, L.M., Nagirna, A.M.: Multicriteria lexicographic optimization problems on a fuzzy set of alternatives. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki. (6), 42–51 (2010)Google Scholar
  22. 22.
    Semenova, N.V., Kolechkina, L.N., Nagornaya, A.N.: On an approach to the solution of vector problems with linear-fractional criterion functions on a combinatorial set of arrangements. Problemy Upravlen. Inform. 1, 131–144 (2010)MathSciNetGoogle Scholar
  23. 23.
    Sergienko, I.V., Kaspshitskaya, M.F.: Models and Methods for Computer Solution of Combinatorial Optimization Problems. Naukova Dumka, Kyiv (1981). [in Russian]Google Scholar
  24. 24.
    Sergienko, I.V., Shilo, V.P.: Discrete Optimization Problems: Challenges. Methods of Solution and Analysis. Naukova Dumka, Kyiv (2003). [in Russian]Google Scholar
  25. 25.
    Stoyan, Y.G., Yakovlev, S.V.: Mathematical Models and Optimization Methods of Geometrical Design. Naukova Dumka, Kyiv (1986). [in Russian]Google Scholar
  26. 26.
    Stoyan, Y.G., Yakovlev, S.V., Pichugina O.S.: The Euclidean Combinatorial Configurations: A Monograph. Constanta (2017). [in Russian]Google Scholar
  27. 27.
    Stoyan, Y.G., Yemets, O.O.: Theory and Methods of Euclidean Combinatorial Optimization. ISSE, Kyiv (1993). [in Ukrainian]Google Scholar
  28. 28.
    Yakovlev, S.: Convex Extensions in Combinatorial Optimization and Their Applications. Optim. Methods Appl. 567–584. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-68640-0_27CrossRefGoogle Scholar
  29. 29.
    Yakovlev, S.V., Grebennik, I.V.: Localization of solutions of some problems of nonlinear integer optimization. Cybern. Syst. Anal. 29(5), 727–734 (1993).  https://doi.org/10.1007/BF01125802CrossRefzbMATHGoogle Scholar
  30. 30.
    Yakovlev, S.V., Pichugina, O.S.: Properties of combinatorial optimization problems over polyhedral-spherical sets. Cybern. Syst. Anal. 54(1), 99–109 (2018).  https://doi.org/10.1007/s10559-018-0011-6CrossRefzbMATHGoogle Scholar
  31. 31.
    Yakovlev, S., Pichugina, O., Yarovaya, O.: On optimization problems on the polyhedral-spherical configurations with their properties. In: 2018 IEEE First International Conference on System Analysis Intelligent Computing (SAIC), pp. 94–100 (2018).  https://doi.org/10.1109/SAIC.2018.8516801
  32. 32.
    Yakovlev, S.V., Pichugina, O.S., Yarovaya, O.V.: Polyhedral spherical configuration in discrete optimization. J. of Autom. Inf. Sci. 51, 38–50 (2019)CrossRefGoogle Scholar
  33. 33.
    Yakovlev, S., Pichugina, O., Yarovaya, O.: Polyhedral spherical configuration in discrete optimization. J. of Autom. Inf. Sci. 51(1), 38–50 (2019)CrossRefGoogle Scholar
  34. 34.
    Yakovlev, S.V., Valuiskaya, O.A.: Optimization of linear functions at the vertices of a permutation polyhedron with additional linear constraints. Ukr. Math. J. 53(9), 1535–1545 (2001).  https://doi.org/10.1023/A:1014374926840CrossRefGoogle Scholar
  35. 35.
    Yemelichev, V.A., Kovalev, M.M., Kravtsov, M.K.: Polytopes. Graphs and Optimisation. Cambridge University Press, Cambridge (1984)Google Scholar
  36. 36.
    Ziegler, G.M.: Lectures on Polytopes. Springer, New York (1995)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.University of LodzLodzPoland
  2. 2.National Aerospace University Kharkiv Aviation InstituteKharkivUkraine

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