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Approximation Algorithms for the Maximum m-Peripatetic Salesman Problem

  • Edward Kh. Gimadi
  • Oxana Yu. TsidulkoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10716)

Abstract

We consider the maximum m-Peripatetic Salesman Problem (MAX m-PSP), which is a natural generalization of the classic Traveling Salesman Problem. The problem is strongly NP-hard. In this paper we propose two polynomial approximation algorithms for the MAX m-PSP with different and identical weight functions, correspondingly. We prove that for random inputs uniformly distributed on the interval [ab] these algorithms are asymptotically optimal for \(m=o(n)\). This means that with high probability their relative errors tend to zero as the number n of the vertices of the graph tends to infinity. The results remain true for the distributions of inputs that minorize the uniform distribution.

Keywords

Maximum m-Peripatetic Salesman Problem Time complexity Edge-disjoint Hamiltonian cycles 

Notes

Acknowledgments

The authors are supported by the Russian Foundation for Basic Research grants 16-31-00389 and 15-01-00976, Russian Ministry of Science and Education under 5-100 Excellence Program, and the grant of Presidium of RAS (program 8, project 227).

References

  1. 1.
    Ageev, A.A., Baburin, A.E., Gimadi, E.K.: A 3/4 approximation algorithms for finding two disjoint Hamiltonian cycles of maximum weight. J. Appl. Indust. Math. 1(2), 142–147 (2007)CrossRefGoogle Scholar
  2. 2.
    Angluin, D., Valiant, L.G.: Fast probabilistic algorithms for Hamiltonian circuits and matchings. J. Comp. Syst. Sci. 18(2), 155–193 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baburin, A.E., Gimadi, E.K.: On the asymptotic optimality of an algorithm for solving the maximum \(m\)-PSP in a multidimensional euclidean space. Proc. Steklov Inst. Math. 272(1), 1–13 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bollobás, B., Fenner, T.I., Frieze, A.M.: An algorithm for finding Hamilton paths and cycles in random graphs. Combinatorica 7, 327–341 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Climer, S., Zhang, W.: Rearrangement clustering: pitfalls, remedies, and applications. JMLR 7, 919–943 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    De Kort, J.B.J.M.: Upper bounds and lower bounds for the symmetric K-Peripatetic Salesman Problem. Optimization 23(4), 357–367 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Kort, J.B.J.M.: A branch and bound algorithm for symmetric 2-Peripatetic Salesman Problems. Eur. J. Oper. Res. 70, 229–243 (1993)CrossRefzbMATHGoogle Scholar
  8. 8.
    Duchenne, E., Laporte, G., Semet, F.: The undirected m-Peripatetic Salesman Problem: polyhedral results and new algorithms. J. Oper. Res. 55(5), 949–965 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Erdős, P., Rényi, A.: On random graphs I. Publ. Math. Debrecen 6, 290–297 (1959)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Gimadi, E.K., Glazkov, Y.V., Tsidulko, O.Y.: The probabilistic analysis of an algorithm for solving the m-planar 3-dimensional assignment problem on one-cycle permutations. J. Appl. Ind. Math. 8(2), 208–217 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gimadi, E.K., Istomin, A.M., Tsidulko, O.Y.: On asymptotically optimal approach to the m-Peripatetic Salesman Problem on random inputs. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 136–147. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-44914-2_11 CrossRefGoogle Scholar
  12. 12.
    Gimadi, E.K., Ivonina, E.V.: Approximation algorithms for the maximum 2-Peripatetic Salesman Problem. J. Appl. Ind. Math. 6(3), 295–305 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Glebov, A.N., Gordeeva, A.V.: An algorithm with approximation ratio 5/6 for the metric maximum m-PSP. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 159–170. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-44914-2_13 CrossRefGoogle Scholar
  14. 14.
    Glebov, A.N., Zambalaeva, D.Z.: A polynomial algorithm with approximation ratio 7/9 for the maximum two Peripatetic Salesmen Problem. J. Appl. Ind. Math. 6(1), 69–89 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Johnson, D.S., Krishnan, S., Chhugani, J., Kumar, S., Venkatasubramanian, S.: Compressing large boolean matrices using reordering techniques. In: 30th International Conference on Very Large Databases (VLDB), pp. 13–23 (2004)Google Scholar
  16. 16.
    Johnson, O., Liu, J.: A traveling salesman approach for predicting protein functions. Source Code Biol. Med. 1(3), 9–16 (2006)Google Scholar
  17. 17.
    Kaplan, H., Lewenstein, M., Shafrir, N., Sviridenko, M.: Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs. J. ACM 52(4), 602–626 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Komlos, J., Szemeredi, E.: Limit distributions for the existence of Hamilton circuits in a random graph. Discrete Math. 43, 55–63 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Krarup, J.: The Peripatetic Salesman and some related unsolved problems. In: Combinatorial Programming, Methods and Applications, pp. 173–178. Reidel, Dordrecht (1975)Google Scholar
  20. 20.
    Petrov, V.V.: Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Clarendon Press, Oxford (1995)zbMATHGoogle Scholar
  21. 21.
    Ray, S.S., Bandyopadhyay, S., Pal, S.K.: Gene ordering in partitive clustering using microarray expressions. J. Biosci. 32(5), 1019–1025 (2007)CrossRefGoogle Scholar
  22. 22.
    Song, L., Zhang, Yu., Peng, X., Wang, Z., Gildea, D.: AMR-to-text generation as a Traveling Salesman Problem. In: Proceedings of 2016 Conference on Empirical Methods in Natural Language Processing (2016)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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