Skip to main content

Formalization and Classification of Combinatorial Optimization Problems

Part of the Springer Optimization and Its Applications book series (SOIA,volume 130)

Abstract

Original approach to determination of the concepts “combinatorial object” and “fuzzy combinatorial object” is offered, which allows to strictly formalize both the known and new classes of problems of combinatorial optimization. The offered approach to such formalization which is concept-based a local finiteness of the discrete spaces relies only on properties of the discrete spaces therefore has quite general character and allows to develop the constructive approach to creation of special objects in combinatorial spaces. The obtained results are important both in the theoretical plan and for development of methods of solving the combinatorial optimization problems.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. Alexandrov, P.S.: Introduction to the Theory of Sets General Topology. Nauka, Moscow (1977) (in Russian)

    Google Scholar 

  2. Baudier, F., Lancien, G.: Embeddings of locally finite metric spaces into Banach spaces. Proc. Am. Math Soc. 136, 1029–1033 (2008)

    Google Scholar 

  3. Berge, C.: Principes de Combinatoire. Dunod, Paris (1968) (in French)

    Google Scholar 

  4. Blum, C., Roli, A.: Metaheuristics in combinatorial optimization: overview and conceptual comparison. ACM Comput. Surv. 35(3), 268–308 (2003)

    Google Scholar 

  5. Blum, C., Roli, A., Alba, E.: An introduction to metaheuristic techniques. In: Alba, E. (ed.) Parallel Metaheuristics: A New Class of Algorithms, pp. 3–42. Wiley, Hoboken (2005)

    Google Scholar 

  6. Hulianytskyi, L.F.: On formalization and classification of combinatorial optimization problems. Theory Optimal Decis.-Making 7, 45–49 (2008) (in Ukrainian)

    Google Scholar 

  7. Hulianytskyi, L.F., Mulesa, O.Y.: Applied methods of combinatorial optimization. The University of Kyiv, Kyiv (2016) (in Ukrainian)

    Google Scholar 

  8. Hulianytskyi, L.F., Riasna, I.I.: Automatic classification method based on a fuzzy similarity relation. Cybern. Syst. Anal. 52(1), 30–37 (2016)

    Google Scholar 

  9. Hulianytskyi, L.F., Sirenko, S.I.: Definition and study of combinatorial spaces. Theory Optimal Decis. Making 9, 17–24 (2010) (in Russian)

    Google Scholar 

  10. Kofman, A.: Introduction to Fuzzy Set Theory (Russian translation). Radio i Svyaz’, Moscow (1982)

    Google Scholar 

  11. Korte, B., Vygen, J.: Combinatorial Optimization, vol. 2. Springer, Berlin (2012)

    Google Scholar 

  12. Nakaoka, F., Oda, N.: Some applications of minimal open sets. Int. J. Math. Math. Sci. 27(8), 471–476 (2001)

    Google Scholar 

  13. Papadimitriou, C.H., Stieglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Courier Corporation, North Chelmsford (1982)

    Google Scholar 

  14. Papadimitriou, C.H., Stieglitz, K.: Combinatorial Optimization: Algorithms and Complexity, 2nd edn. Dover Publications, New York (1998)

    Google Scholar 

  15. Parasyuk, I.M., Kaspshitska, M.F.: Fuzzy algorithm for Slump-Vector Method for optimization problems on samples. Comput. Math. (1), 152–163 (2009) (in Russian)

    Google Scholar 

  16. Pospelov, D.A. (ed.): Fuzzy Sets in Models of Control and Artificial Intelligence. Nauka, Moscow (1986) (in Russian)

    Google Scholar 

  17. Rengelking, R.: General Topology. Heldermann, Berlin (1989)

    Google Scholar 

  18. Sachkov, V.N.: Introduction to Combinatorial Methods of Discrete Mathematics. Nauka, Moscow (1982) (in Russian)

    Google Scholar 

  19. Sergienko, I.V., Kaspshytska, M.F.: Models and Methods of Solution of Combinatorial Optimization Problems. Naukova Dumka, Kyiv (1981) (in Russian)

    Google Scholar 

  20. Sergienko, I.V., Shylo, V.P.: Problems of Discrete Optimization. Naukova Dumka, Kyiv (2003) (in Russian)

    Google Scholar 

  21. Talbi, E.G.: Metaheuristics: From Design to Implementation. Wiley, Hoboken (2009)

    Google Scholar 

  22. The Encyclopedia of Mathematics, vol. 2. Sovetskaia Enziklopedia, Moscow (1979) (in Russian)

    Google Scholar 

  23. Yamakami, T.: The world of combinatorial fuzzy problems and the efficiency of fuzzy approximation algorithms. In: Proceedings of the 15th International Symposium on Advanced Intelligent Systems (ISIS 2014), pp. 29–35. IEEE Computer Society Press, New York (2014)

    Google Scholar 

  24. Zaichenko, Y.: Operations Research. PH “Slovo”, Kyiv (2006) (in Ukrainian)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Leonid Hulianytskyi .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Hulianytskyi, L., Riasna, I. (2017). Formalization and Classification of Combinatorial Optimization Problems. In: Butenko, S., Pardalos, P., Shylo, V. (eds) Optimization Methods and Applications . Springer Optimization and Its Applications, vol 130. Springer, Cham. https://doi.org/10.1007/978-3-319-68640-0_11

Download citation