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On the Complexity of Master Problems

  • Martijn van Ee
  • René Sitters
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9235)

Abstract

A master solution for an instance of a combinatorial problem is a solution with the property that it is optimal for any sub instance. For example, a master tour for an instance of the TSP problem has the property that restricting the solution to any subset S results in an optimal solution for S. The problem of deciding if a TSP instance has a master tour is known to be polynomially solvable. Here, we show that the master tour problem is \(\varDelta _2^p\)-complete in the scenario setting, that means, the subsets S are restricted to some given sets. We also show that the master versions of Steiner tree and maximum weighted satisfiability are also \(\varDelta _2^p\)-complete, as is deciding whether the optimal solution for these problems is unique. Like for the master tour problem, the special case of the master version of Steiner tree where every subset of vertices is a possible scenario turns out to be polynomially solvable. All the results also hold for metric spaces.

Keywords

Computational complexity Polynomial hierarchy Universal optimization Unique optimal solutions 

References

  1. 1.
    Deineko, V.G., Rudolf, R., Woeginger, G.J.: Sometimes travelling is easy: The master tour problem. SIAM J. Discrete Math. 11(1), 81–93 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Gupta, A., Hajiaghayi, M.T., Räcke, H.: Oblivious network design. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 970–979 (2006)Google Scholar
  3. 3.
    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 2010. LNCS, vol. 6302, pp. 178–191. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  4. 4.
    Schalekamp, F., Shmoys, D.B.: Algorithms for the universal and a priori TSP. Oper. Res. Lett. 36(1), 1–3 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Shmoys, D.B., 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) CrossRefGoogle Scholar
  6. 6.
    Stockmeyer, L.J.: The polynomial-time hierarchy. Theor. Comput. Sci. 3(1), 1–22 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Papadimitriou, C.H.: On the complexity of unique solutions. J. ACM 31(2), 392–400 (1984)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Krentel, M.W.: The complexity of optimization problems. J. Comput. Syst. Sci. 36(3), 490–509 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Papadimitriou, C.H., Yannakakis, M.: The complexity of facets (and some facets of complexity). J. Comput. Syst. Sci. 28(2), 244–259 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Wechsung, G.: On the boolean closure of NP. In: Budach, L. (ed.) Fundamentals of Computation Theory. Lecture Notes in Computer Science, vol. 199, pp. 485–493. Springer, Heidelberg (1985) CrossRefGoogle Scholar
  11. 11.
    Blass, A., Gurevich, Y.: On the unique satisfiability problem. Inf. Control 55(1–3), 80–88 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Valiant, L.G., Vazirani, V.V.: NP is as easy as detecting unique solutions. Theor. Comput. Sci. 47(3), 85–93 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hauptmann, M.: Approximation complexity of optimization problems: Structural foundations and Steiner tree problems. Ph.D thesis, Rheinischen Friedrich-Wilhelms-Universität Bonn (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.VU University AmsterdamAmsterdamThe Netherlands
  2. 2.Centrum voor Wiskunde en Informatica (CWI)AmsterdamThe Netherlands

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