A priori TSP in the Scenario Model

  • Martijn van Ee
  • Leo van Iersel
  • Teun Janssen
  • René Sitters
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10138)

Abstract

In this paper, we consider the a priori traveling salesman problem (TSP) in the scenario model. In this problem, we are given a list of subsets of the vertices, called scenarios, along with a probability for each scenario. Given a tour on all vertices, the resulting tour for a given scenario is obtained by restricting the solution to the vertices of the scenario. The goal is to find a tour on all vertices that minimizes the expected length of the resulting restricted tour. We show that this problem is already NP-hard and APX-hard when all scenarios have size four. On the positive side, we show that there exists a constant-factor approximation algorithm in three restricted cases: if the number of scenarios is fixed, if the number of missing vertices per scenario is bounded by a constant, and if the scenarios are nested. Finally, we discuss an elegant relation with an a priori minimum spanning tree problem.

Keywords

Traveling salesman problem A priori optimization Master tour Optimization under scenarios 

Notes

Acknowledgments

We would like to thank Karen Aardal, Jan Driessen and Neil Olver for useful discussions. A part of the work by Teun Janssen has been performed in the project INTEGRATE “Integrated Solutions for Agile Manufacturing in High-mix Semiconductor Fabs”, co-funded by grants from France, Italy, Ireland, The Netherlands and the ECSEL Joint Undertaking. Martijn van Ee and René Sitters are supported by the NWO Grant 612.001.215. Leo van Iersel was partially supported by NWO and partially by the 4TU Applied Mathematics Institute.

References

  1. 1.
    Driessen, J., Janssen, T.: Minimizing the blading in lithography machines: an application of the a priori TSP problem. Unpublished manuscript (2016)Google Scholar
  2. 2.
    Shmoys, D., Talwar, K.: A constant approximation algorithm for the a priori traveling salesman problem. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 331–343. Springer, Heidelberg (2008). doi:10.1007/978-3-540-68891-4_23 CrossRefGoogle Scholar
  3. 3.
    Zuylen, A.: Deterministic sampling algorithms for network design. Algorithmica 60, 110–151 (2011)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Schalekamp, F., Shmoys, D.B.: Algorithms for the universal and a priori TSP. Oper. Res. Lett. 36, 1–3 (2008)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Hajiaghayi, M.T., Kleinberg, R., Leighton, T.: Improved lower and upper bounds for universal TSP in planar metrics. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 649–658 (2006)Google Scholar
  6. 6.
    Gorodezky, I., Kleinberg, R.D., Shmoys, D.B., Spencer, G.: Improved lower bounds for the universal and a priori TSP. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX/RANDOM 2010. LNCS, vol. 6302, pp. 178–191. Springer, Heidelberg (2010). doi:10.1007/978-3-642-15369-3_14 CrossRefGoogle Scholar
  7. 7.
    Immorlica, N., Karger, D.R., Minkoff, M., Mirrokni, V.S.: On the costs and benefits of procrastination: approximation algorithms for stochastic combinatorial optimization problems. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 691–700 (2004)Google Scholar
  8. 8.
    Ravi, R., Sinha, A.: Hedging uncertainty: approximation algorithms for stochastic optimization. Math. Program. 108, 97–114 (2006)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Chen, L., Megow, N., Rischke, R., Stougie, L.: Stochastic and robust scheduling in the cloud. In: Proceedings of the 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pp. 175–186 (2015)Google Scholar
  10. 10.
    Feuerstein, E., Marchetti-Spaccamela, A., Schalekamp, F., Sitters, R., Ster, S., Stougie, L., Zuylen, A.: Scheduling over scenarios on two machines. In: Cai, Z., Zelikovsky, A., Bourgeois, A. (eds.) COCOON 2014. LNCS, vol. 8591, pp. 559–571. Springer, Heidelberg (2014). doi:10.1007/978-3-319-08783-2_48 Google Scholar
  11. 11.
    Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical report, DTIC Document (1976)Google Scholar
  12. 12.
    Deineko, V.G., Rudolf, R., Woeginger, G.J.: Sometimes travelling is easy: the master tour problem. SIAM J. Discrete Math. 11, 81–93 (1998)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Ee, M., Sitters, R.: On the complexity of master problems. In: Italiano, G.F., Pighizzini, G., Sannella, D.T. (eds.) MFCS 2015. LNCS, vol. 9235, pp. 567–576. Springer, Heidelberg (2015). doi:10.1007/978-3-662-48054-0_47 CrossRefGoogle Scholar
  14. 14.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations, pp. 85–103. Springer, Heidelberg (1972)CrossRefGoogle Scholar
  15. 15.
    Lovász, L.: Coverings and colorings of hypergraphs. In: Proceedings of the 4th Southeastern Conference on Combinatorics, Graph Theory and Computing, pp. 3–12 (1973)Google Scholar
  16. 16.
    Arora, S., Grigni, M., Karger, D.R., Klein, P.N., Woloszyn, A.: A polynomial-time approximation scheme for weighted planar graph TSP. In: Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 33–41 (1998)Google Scholar
  17. 17.
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM (JACM) 42, 1115–1145 (1995)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Khot, S., Kindler, G., Mossel, E., O’Donnell, R.: Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM J. Comput. 37, 319–357 (2007)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Håstad, J.: Some optimal inapproximability results. J. ACM 48, 798–859 (2001)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Guruswami, V., Håstad, J., Manokaran, R., Raghavendra, P., Charikar, M.: Beating the random ordering is hard: every ordering CSP is approximation resistant. SIAM J. Comput. 40, 878–914 (2011)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Galil, Z., Megiddo, N.: Cyclic ordering is NP-complete. Theor. Comput. Sci. 5, 179–182 (1977)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Austrin, P., Manokaran, R., Wenner, C.: On the NP-hardness of approximating ordering-constraint satisfaction problems. Theor. Comput. 11, 257–283 (2015)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Bertsimas, D.: Probabilistic combinatorial optimization problems. Ph.D. thesis, Massachusetts Institute of Technology (1988)Google Scholar
  24. 24.
    Boria, N., Murat, C., Paschos, V.: On the probabilistic min spanning tree problem. J. Math. Model. Algorithms 11, 45–76 (2012)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM (JACM) 45, 753–782 (1998)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Martijn van Ee
    • 1
  • Leo van Iersel
    • 2
  • Teun Janssen
    • 2
  • René Sitters
    • 1
    • 3
  1. 1.Vrije Universiteit AmsterdamAmsterdamThe Netherlands
  2. 2.Delft University of TechnologyDelftThe Netherlands
  3. 3.Centrum voor Wiskunde en Informatica (CWI)AmsterdamThe Netherlands

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