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Subclasses of solvable problems from classes of combinatorial optimization problems

  • Systems Analysis
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Cybernetics and Systems Analysis Aims and scope

Well-known subclasses of solvable problems from classes of combinatorial optimization are reviewed. For solvable problems such as the traveling salesman problem, location problem, assignment problem, and clustering problem, the changes in the objective function on a given ordering of combinatorial configurations are analyzed.

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References

  1. C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Englewood Cliffs, Prentice Hall (1982).

    MATH  Google Scholar 

  2. I. V. Sergienko and M. F. Kaspshitskaya, Models and Methods for Computer Solution of Combinatorial Optimization Problems [in Russian], Naukova Dumka, Kyiv (1981).

    Google Scholar 

  3. I. V. Sergienko, Mathematical Models and Methods to Solve Discrete Optimization Problems [in Russian], Naukova Dumka, Kyiv (1988).

    Google Scholar 

  4. A. A. Korbut and Yu. Yu. Finkel’shtein, Discrete Programming [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  5. H. Wagner, Principles of Operations Research, Vol. 1, Prentice-Hall, Englewood Cliffs (1975).

    Google Scholar 

  6. Yu. G. Stoyan and V. Z. Sokolovskii, Solving Some Multiextremum Problems by the Method of Contracting Neighborhoods [in Russian], Naukova Dumka, Kyiv (1980).

    Google Scholar 

  7. V. S. Mikhalevich, “Sequential algorithms of optimization and their application,” I, II, Kibernetika, No. 1, 45–56; No. 2, 85–89 (1965).

  8. V. S. Mikhalevich, V. A. Trubin, and N. Z. Shor, Optimization Problems of Industrial-Transport Scheduling [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  9. V. V. Shkurba, “Mathematical treatment of one class of biochemical elements,” Kibernetika, No. 1, 62–67 (1965).

  10. V. Ya. Burdyuk and V. A. Semenov, “Solvable cases of a new combinatorial problem of optimization,” Cybern. Syst. Analysis, 35, No. 2, 332–334 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  11. D. A Suprunenko, “Values of a linear form on a set of permutations,” Kibernetika, No. 2, 59–63 (1968).

  12. V. Ya. Burdyuk and V. G. Deineko, Optimal Circular Orderings (Traveling Salesman Problem), [in Russian], DGU, Dnepropetrovsk (1983).

    Google Scholar 

  13. V. V. Zaika and S. M. Shvartin, “On the cycles generated by an arbitrary set of elements of a matrix,” Zh. Vych. Mat. Mat. Fiz., 24, No. 8, 1230–1240 (1984).

    MathSciNet  Google Scholar 

  14. V. M. Demidenko, V. S. Gordon, and J.-M. Proth, “Polynomial solvability conditions for the traveling salesman problem and upper-bound estimates of its optimum,” Dokl. NAN Belarusi, 47, No. 1, 36–40 (2003).

    MATH  MathSciNet  Google Scholar 

  15. V. T. Dement’ev and Yu. V. Shamardin, “Two-level assignment problem for the generalized Monge condition,” Diskretn. Analiz Issled. Oper., Ser. 2, 10, No. 2, 19–28 (2003).

  16. J. A. A. van Der Veen, G. Sierksma, and R. van Dal, “Pyramidal tours and the travelling salesman problem,” Eur. J. Oper. Res., 52, No. 1, 90–102 (1991).

    Article  MATH  Google Scholar 

  17. V. G. Deineko, Solvability in Combinatorial Optimization. Solvable Cases of the Traveling Salesman Problem and Heuristic Algorithms [in Ukrainian], Author’s Abstract of PhD Thesis, V. M. Glushkov Institute of Cybernetics, Kyiv (1995).

  18. V. S. Aizenshtat and E. P. Maksimovich, “Certain classes of traveling-salesman problems,” Cybernetics, 14, No. 4, 565–568 (1978).

    Article  Google Scholar 

  19. V. M. Demidenko, “Constructing a relaxation of the polytope of a symmetric traveling salesman problem based on a strongly solvable Kalmanson case,” Diskr. Analiz Issled. Oper., Ser. 2, 11, No. 2, 3–24 (2004).

    MathSciNet  Google Scholar 

  20. K. Kalmanson, “Edgeconvex circuits and the traveling salesman problem,” Canad. J. Math., 27, No. 5, 1000–1010 (1975).

    MathSciNet  Google Scholar 

  21. F. Supnick, “Extreme Íamiltonian lines,” Annals of Math., 66, 179–201 (1957).

    Article  MathSciNet  Google Scholar 

  22. V. M. Demidenko, “A special case of the traveling salesman problem,” Izv. AN BSSR, Ser. Fiz.-Mat. Nauk, No. 5, 28–32 (1976).

  23. Ya. M. Volodarskii, E. Ya. Gabovich, and A. Ya. Zakharin, “A solvable case in discrete programming,” Izv. AN SSSR, Tekhn. Kibern., No. 1, 34–44 (1976).

  24. M. I. Rubinshtein, “Symmetric traveling salesman problem,” Avtom. Telemekh., No. 9, 126–133 (1971).

  25. V. M. Demidenko, “Optimization of v (A, τ) on a set of cycles,” Vestsi AN BSSR, Ser. Fiz.-Mat. Navuk, No. 1, 21–26 (1978).

  26. V. M. Demidenko, Extremum Problems on Permutations [in Russian], Author’s Abstract of PhD Thesis: 01.01.09, Institute of Mathematics AS BSSR, Minsk (1980).

  27. V. S. Aizenshtat and D. N. Kravchuk, “Extremum of a linear form on a set of all cyclic permutations,” Kibernetika, No. 6, 76–81 (1968).

  28. V. S. Aizenshtat and D. N. Kravchuk, “Minimum linear form on the set of all complete cycles of a symmetric group,” Kibernetika, No. 2, 64–66 (1968).

  29. V. M. Demidenko, “Generalizing strong solvability conditions for a quadratic assignment problem with Anti-Monge and Toeplitz matrices,” Dokl. NAN Belarusi, 47, No. 2, 15–18 (2003).

    MATH  MathSciNet  Google Scholar 

  30. G. Monge “Mémoire sur la théorie des déblais et des remblais,” in: Histoire de l’Académie Royale des Sciences, Année M. DCCLXXXI, avec les Mémoires de Mathématique et de Physique, pour la mé eme Année, Tirés des Registres de cette Académie, Paris (1781), pp. 666–704.

  31. R. E. Burkard, B. Klinz, and R. Rudolf, “Perspectives of Monge properties in optimization,” Discrete Appl. Math., 70, 95–161 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  32. V. M. Demidenko, “Conic characterization of Monge matrices,” Cybern. Syst. Analysis, 40, No. 4, 87–98 (2004).

    Article  MathSciNet  Google Scholar 

  33. V. G. Deineko and V. N. Trofimov, “Possible values of the functional in special traveling salesman problems,” Kibernetika, No. 6, 135–136 (1981).

  34. V. Ya. Burdyuk and V. N. Trofimov, “Generalizing Gilmore’s and Gomory’s solutions of the traveling salesman problem,” Izv. AN SSSR, Tekhn. Kibern., No. 3, 16–22 (1976).

  35. V. M. Demidenko and J.-M. Proth, “Quadratic assignment problem: Attaining optimum on given permutations,” Vestsi NAN Belarusi, Ser. Fiz.-Mat. Navuk, No. 2, 60–64 (2005).

  36. V. M. Demidenko and A. Dolgui, “Efficiently solvable cases of quadratic assignment problem with generalized monotonic and incomplete Anti-Monge matrices,” Cybern. Syst. Analysis, 43, No. 1, 112–125 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  37. M. Parnas, D. Ron, and R. Rubinfeld, “On testing convexity and submodularity,” SIAM J. Comput., 32, No. 5, 1158–1184 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  38. N. K. Timofeeva, “Some features of constructing mathematical models for combinatorial optimization problems,” USiM, No. 5, 38–45 (2004).

  39. A. F. Turbin and N. V. Pratsevityi, Fractal Sets, Functions, Distributions [in Russian], Naukova Dumka, Kyiv (1992).

    MATH  Google Scholar 

  40. S. I. Salpagarov, Assignment Problem on Fractal and Prefractal Graphs (Multicriterion Formulation) [in Russian], Dep. VINITI Dec. 31 2003, No. 2323–B2003, Karachaevo-Cherkes. Gos. Tekhn. Akad., Cherkessk (2003).

  41. V. A. Perepelitsa, I. V. Sergienko, and A. M. Kochkarov, “Recognition of fractal graphs,” Cybern. Syst. Analysis, 35, No. 4, 572–586 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  42. E. V. Bobyleva, “Special case of the recognition problem for a complete noncanonical prefractal graph,” Matem. Mash. Sistemy, No. 2, 3–14 (2005).

  43. G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge Univ. Press, Cambridge (1934).

    Google Scholar 

  44. M. R. Garey and D. S. Johnson, Computers and Intractability. A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., New York (1979).

    MATH  Google Scholar 

  45. E. Corrizosa, H. W. Hamacher, R. Klein, and S. Nickel, “Solving nonconvex planar location problems by finite dominating sets,” J. Glob. Optimiz., 18, No. 2, 195–210 (2000).

    Article  Google Scholar 

  46. G. Cornuejols, D. Naddef, and W. Pulleyblank, “The traveling salesman problem in graphs with 3-edge cutsets,” J. Assoc. Comput. Mach., 32, No. 2, 383–410 (1985).

    MATH  MathSciNet  Google Scholar 

  47. R. E. Burkard and U. Fincke, “The asymptotic probabilistic behavior of quadratic sum assignment problems,” J. Oper. Res., A27, No. 3, 73–81 (1983).

    Article  MathSciNet  Google Scholar 

  48. N. K. Timofeeva, “Dependence of objective functions in combinatorial optimization problems on the ordering of combinatorial configurations,” Komp. Matem., No. 2, 135–146 (2005).

  49. N. K. Timofeeva, “Approach to finding the optimal solution in the assignment problem,” Cybern. Syst. Analysis, 32, No. 1, 104–112 (1996).

    Article  MathSciNet  Google Scholar 

  50. N. K. Timofeeva, “Combinatorial functions in the assignment problem,” Komp. Matem., No. 1, 47–56 (2004).

  51. N. K. Timofeeva, “Analyzing the objective function in the location problem,” Dep. VINITI March 01 1991, No. 930–B91, V. M. Glushkov Institute of Cybernetics NAS Ukraine, Kyiv (1991).

  52. N. K. Timofeeva, “Special case of the traveling salesman problem,” in: Developing Mathematics and Software for an Application Package and Solving Discrete Optimization Problems [in Russian], V. M. Glushkov Institute of Cybernetics NAS Ukraine, Kyiv (1992), pp. 95–101.

    Google Scholar 

  53. N. K. Timofeeva, “Finding optimal solution in the traveling salesman problem,” in: Software and Algorithms to Solve Problems of Applied Mathematics [in Russian], V. M. Glushkov Institute of Cybernetics NAS Ukraine, Kyiv (1993), pp. 21–26.

    Google Scholar 

  54. N. K. Timofeeva, “Solvable case of the arrangement problem,” Visn. Kyiv. Univ., Ser. Fiz.-Mat. Nauky, Taras Shevchenko Kyiv University, No. 4, 227–231 (2006).

  55. G. P. Donets and N. K. Timofeeva, “Method to model the structure of input data and subclasses of solvable problems,” in: Mathematics and Software for Intelligent Systems (MPZIS—2007) [in Ukrainian], 5th Intern. Sci.-Pract. Conf., Dnipropetrovsk, November 14–16, 2007, Dnipropetrovsk (2007), pp. 52–53.

  56. N. K. Timofeeva, “Some properties of splitting sets into subsets,” USiM, No. 5, 6–23 (2002).

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  1. International Research and Training Center for Information Technologies and Systems, National Academy of Sciences of Ukraine and Ministry of Education and Science of Ukraine, Kyiv, Ukraine

    N. K. Timofeeva

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Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 97–105, March–April 2009.

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Timofeeva, N.K. Subclasses of solvable problems from classes of combinatorial optimization problems. Cybern Syst Anal 45, 245–252 (2009). https://doi.org/10.1007/s10559-009-9088-2

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