In this paper we consider probabilistic models for steiner tree. Under these models, the problem is defined in a two-stage setting over a complete weighted graph whose vertices are associated with a probability of presence in the second stage. A first-stage feasible solution on the input graph might become infeasible in the second stage, when certain vertices of the graph fail (with the specified probability). Therefore, a well defined modification strategy is devised which transforms a partial solution to a feasible second-stage solution. The objective is to devise an algorithm for the first-stage solution (sometimes called the a priori or anticipatory solution) so that the expected second-stage solution cost is minimized. An important feature of this framework is that the modification strategy is essentially a part of the problem, while algorithmic treatment is required in the construction of the anticipatory solution. We essentially provide approximation results regarding two modification strategies. Furthermore, for the first of them we also formulate the associated objective function and prove complexity results for the computation of the anticipatory solution optimizing it.
