On the approximability of robust spanning tree problems

In this paper the minimum spanning tree problem with uncertain edge costs is discussed. In order to model the uncertainty a discrete scenario set is specified and a robust framework is adopted to choose a solution. The min-max, min-max regret and 2-stage min-max versions of the problem are discussed. The complexity and approximability of all these problems are explored. It is proved that the min-max and min-max regret versions with nonnegative edge costs are hard to approximate within O(log n) for any ǫ > 0 unless the problems in NP have quasi-polynomial time algorithms. Similarly, the 2-stage min-max problem cannot be approximated within O(log n) unless the problems in NP have quasi-polynomial time algorithms. In this paper randomized LP-based approximation algorithms with performance ratio of O(log n) for min-max and 2-stage min-max problems are also proposed.