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An asymptotically optimal algorithm for the m-Peripatetic Salesman Problem on random inputs with discrete distribution

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Abstract

We consider the m-Peripatetic Salesman Problem (m-PSP) on random inputs with discrete distribution function. In this paper we present a polynomial approximation algorithm which, under certain conditions, with high probability (w.h.p.) gives optimal solution for both the m-PSP on random inputs with identical weight functions and the m-PSP with different weight functions.

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References

  1. A. A. Ageev, A. E. Baburin, and E. Kh. Gimadi, “A 3/4-Approximation Algorithm for Finding Two Disjoint Hamiltonian Cycles of Maximum Weight,” Diskretn. Anal. Issled. Oper. Ser. 1, 13 (2), 11–20 (2006) [J. Appl. Indust. Math. 1 (2), 142–147 (2007)].

    MATH  Google Scholar 

  2. A. A. Ageev and A. V. Pyatkin, “A 2-Approximation Algorithm for the Metric 2-Peripatetic Salesman Problem,” Diskretn. Anal. Issled. Oper. 16 (4), 3–20 (2009).

    MathSciNet  MATH  Google Scholar 

  3. A. E. Baburin and E. Kh. Gimadi, “On the Asymptotic Optimality of an Algorithm for Solving the Maximum m-PSP in a Multidimensional Euclidean Space,” Trudy Inst. Mat. Mekh. 16 (3), 12–24 (2010) [Proc. Steklov Inst. Math. 272, Suppl. 1, S1–S13 (2011)]

    MATH  Google Scholar 

  4. A. E. Baburin and E. Kh. Gimadi, and N. M. Korkishko, “Approximation Algorithms for Finding Two Edge-Disjoint Hamiltonian Cycles of Minimal Total Weight,” Diskretn. Anal. Issled. Oper. Ser. 2, 11 (1), 11–25 (2004).

    MathSciNet  MATH  Google Scholar 

  5. E. Kh. Gimadi, Yu. V. Glazkov, and A. N. Glebov, “Approximation Algorithms for Solving the 2-Peripatetic Salesman Problem on a Complete Graph with Edge Weights 1 and 2,” Diskretn. Anal. Issled. Oper. Ser. 2, 14 (2), 41–61 (2007) [J. Appl. Indust. Math. 3 (1), 46–60 (2009)].

    MATH  Google Scholar 

  6. E. Kh. Gimadi, Yu. V. Glazkov, and O. Yu. Tsidulko, “The Probabilistic Analysis of an Algorithm for Solving the m-Planar 3-Dimensional Assignment Problem on One-Cycle Permutations,” Diskretn. Anal. Issled. Oper. 21 (1), 15–29 (2014) [J. Appl. Indust. Math. 8 (2), 208–217 (2014)].

    MATH  Google Scholar 

  7. E. Kh. Gimadi, A. M. Istomin, I. A. Rykov, and O. Yu. Tsidulko, “Probabilistic Analysis of an Approximation Algorithm for the m-Peripatetic Salesman Problem on Random Instances Unbounded from Above,” Trudy Inst. Mat. Mekh. 20 (2), 88–98 (2014) [Proc. Steklov Inst. Math. 289, Suppl. 1, S77–S87 (2015)].

    MATH  Google Scholar 

  8. E. Kh. Gimadi and E. V. Ivonina, “Approximation Algorithms for the Maximum 2-Peripatetic Salesman Problem,” Diskretn. Anal. Issled. Oper. 19 (1), 17–32 (2012) [J. Appl. Indust. Math. 6 (3), 295–305 (2012)].

    MATH  Google Scholar 

  9. E. Kh. Gimadi and V. A. Perepelitsa, “A Statistically Effective Algorithm for Selection of a Hamiltonian Contour or Cycle,” in Discrete Analysis, Vol. 22 (Inst. Mat. SO AN SSSR, Novosibirsk, 1973), pp. 15–28.

    Google Scholar 

  10. A. N. Glebov and D. Zh. Zambalaeva, “A Polynomial Algorithm with Approximation Ratio 7/9 for the Maximum Two Peripatetic Salesmen Problem,” Diskretn. Anal. Issled. Oper. 18 (4), 17–48 (2011) [J. Appl. Indust. Math. 6 (1), 69–89 (2012)].

    MathSciNet  MATH  Google Scholar 

  11. A. N. Glebov and D. Zh. Zambalaeva, “An Approximation Algorithm for the Minimum Two Peripatetic Salesmen Problem with Different Weight Functions,” Diskretn. Anal. Issled. Oper. 18 (5), 11–37 (2011) [J. Appl. Indust. Math. 6 (2), 167–183 (2012)].

    MathSciNet  MATH  Google Scholar 

  12. A. D. Korshunov, “On the Power of Some Classes of Graphs,” Dokl. Akad. Nauk SSSR 193 (6), 1230–1233 (1970) [Soviet Math. Dokl. 11 (6), 1100–1104 (1970)].

    MathSciNet  Google Scholar 

  13. V. V. Petrov, Limit Theorems of Probability Theory: Sequences of Independent Random Variables (Nauka, Moscow, 1987; Clarendon Press, Oxford, 1995).

    Google Scholar 

  14. D. Angluin and L. G. Valiant, “Fast Probabilistic Algorithms for Hamiltonian Circuits and Matchings,” J. Comput. Syst. Sci. 18 (2), 155–193 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  15. B. Bollobás, T. I. Fenner, and A. M. Frieze, “An Algorithm for Finding Hamilton Paths and Cycles in Random Graphs,” Combinatorica 7 (4), 327–341 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  16. M. J. D. De Brey and A. Volgenant, “Well-Solved Cases of the 2-Peripatetic Salesman Problem,” Optimization 39 (3), 275–293 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  17. J. B. J. M. De Kort, “Lower Bounds for Symmetric K-Peripatetic Salesman Problems,” Optimization 22 (1), 113–122 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  18. J. B. J. M. De Kort, “Bounds for the Symmetric K-Peripatetic Salesman Problem,” Optimization 23 (4), 357–367 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  19. J. B. J. M. De Kort, “A Branch and Bound Algorithm for Symmetric 2-Peripatetic Salesman Problems,” European J. Oper. Res. 70 (2), 229–243 (1993).

    Article  MATH  Google Scholar 

  20. P. Erdös and A. Rényi, “On Random Graphs I,” Publ. Math. 6, 290–297 (1959).

    MathSciNet  MATH  Google Scholar 

  21. A. M. Frieze, “An Algorithm for Finding Hamiltonian cycles in Random Directed Graphs,” J. Algorithms 9 (2), 181–204 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  22. A. M. Frieze and S. Haber, “An Almost Linear Time Algorithm for Finding Hamiltonian cycles in Sparse Random Graphs with Minimum Degree at Least Three,” Random Struct. Algorithms 47 (1), 73–98 (2015).

    Article  MATH  Google Scholar 

  23. E. Kh. Gimadi, A. N. Glebov, A. A. Skretneva, O. Yu. Tsidulko, and D. Zh. Zambalaeva, “Combinatorial Algorithms with Performance Guarantees for Finding Several Hamiltonian Circuits in a Complete Directed Weighted Graph,” Discrete Appl. Math. 196 (11), 54–61 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  24. A. N. Glebov and A. V. Gordeeva, “An Algorithm with Approximation Ratio 5/6 for the Metric Maximumm-PSP,” in Discrete Optimization and Operations Research: Proceedings of 9th International Conference DOOR, Vladivostok, Russia, Sept. 19–23, 2016 (Springer, Cham, Switzerland, 2016), pp. 159–170.

    Chapter  Google Scholar 

  25. J. Komlós and E. Szemerédi, “Limit Distributions for the Existence of Hamilton Circuits in a Random Graph,” Discrete Math. 43 (1), 55–63 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Krarup, “The Peripatetic Salesman and Some Related Unsolved Problems,” in Combinatorial Programming: Methods and Applications, (Proceedings of NATO Advance Study Institute, Versailles, France, September 2–13, 1974) (D. Reidel, Dordrecht, 1975), pp. 173–178.

    Google Scholar 

  27. L. Posa, “Hamiltonian Circuits in Random Graphs,” Discrete Math. 14 (4), 359–364 (1976).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia

    E. Kh. Gimadi & O. Yu. Tsidulko

  2. Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

    E. Kh. Gimadi & O. Yu. Tsidulko

Authors
  1. E. Kh. Gimadi
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  2. O. Yu. Tsidulko
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Correspondence to E. Kh. Gimadi.

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Original Russian Text © E.Kh. Gimadi, O.Yu. Tsidulko, 2017, published in Diskretnyi Analiz i Issledovanie Operatsii, 2017, Vol. 24, No. 3, pp. 5–19.

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Gimadi, E.K., Tsidulko, O.Y. An asymptotically optimal algorithm for the m-Peripatetic Salesman Problem on random inputs with discrete distribution. J. Appl. Ind. Math. 11, 354–361 (2017). https://doi.org/10.1134/S1990478917030061

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  • DOI: https://doi.org/10.1134/S1990478917030061

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