On the Complexity of Master Problems
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.
KeywordsComputational complexity Polynomial hierarchy Universal optimization Unique optimal solutions
- 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
- 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