Skip to main content

Approximation Algorithms for the Maximum m-Peripatetic Salesman Problem

  • Conference paper
  • First Online:
Analysis of Images, Social Networks and Texts (AIST 2017)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 10716))

  • 2300 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  Google Scholar 

  2. Angluin, D., Valiant, L.G.: Fast probabilistic algorithms for Hamiltonian circuits and matchings. J. Comp. Syst. Sci. 18(2), 155–193 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Climer, S., Zhang, W.: Rearrangement clustering: pitfalls, remedies, and applications. JMLR 7, 919–943 (2006)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Erdős, P., Rényi, A.: On random graphs I. Publ. Math. Debrecen 6, 290–297 (1959)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. Johnson, O., Liu, J.: A traveling salesman approach for predicting protein functions. Source Code Biol. Med. 1(3), 9–16 (2006)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Komlos, J., Szemeredi, E.: Limit distributions for the existence of Hamilton circuits in a random graph. Discrete Math. 43, 55–63 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  21. Ray, S.S., Bandyopadhyay, S., Pal, S.K.: Gene ordering in partitive clustering using microarray expressions. J. Biosci. 32(5), 1019–1025 (2007)

    Article  Google Scholar 

  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 

Download references

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Oxana Yu. Tsidulko .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Gimadi, E.K., Tsidulko, O.Y. (2018). Approximation Algorithms for the Maximum m-Peripatetic Salesman Problem. In: van der Aalst, W., et al. Analysis of Images, Social Networks and Texts. AIST 2017. Lecture Notes in Computer Science(), vol 10716. Springer, Cham. https://doi.org/10.1007/978-3-319-73013-4_28

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-73013-4_28

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-73012-7

  • Online ISBN: 978-3-319-73013-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics