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.
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References
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)
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)
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)
Schalekamp, F., Shmoys, D.B.: Algorithms for the universal and a priori TSP. Oper. Res. Lett. 36(1), 1–3 (2008)
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)
Stockmeyer, L.J.: The polynomial-time hierarchy. Theor. Comput. Sci. 3(1), 1–22 (1976)
Papadimitriou, C.H.: On the complexity of unique solutions. J. ACM 31(2), 392–400 (1984)
Krentel, M.W.: The complexity of optimization problems. J. Comput. Syst. Sci. 36(3), 490–509 (1988)
Papadimitriou, C.H., Yannakakis, M.: The complexity of facets (and some facets of complexity). J. Comput. Syst. Sci. 28(2), 244–259 (1984)
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)
Blass, A., Gurevich, Y.: On the unique satisfiability problem. Inf. Control 55(1–3), 80–88 (1982)
Valiant, L.G., Vazirani, V.V.: NP is as easy as detecting unique solutions. Theor. Comput. Sci. 47(3), 85–93 (1986)
Hauptmann, M.: Approximation complexity of optimization problems: Structural foundations and Steiner tree problems. Ph.D thesis, Rheinischen Friedrich-Wilhelms-Universität Bonn (2004)
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van Ee, M., Sitters, R. (2015). On the Complexity of Master Problems. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_47
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DOI: https://doi.org/10.1007/978-3-662-48054-0_47
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