Approximation and hardness results for Robust Network design with Exponential Scenarios

In a classical optimization problem, we are usually given a s ystem with some known parameters and constraints and the goal is to find a feasible solution of minimum cost (or m axi um profit) with respect to the constraints. Often these parameters and constraints, which heavily influ e ce the optimum solution, are assumed to be precisely known. For example, we might be given a graph G(V,E) with weights on the edges and a set T ⊆ V of terminals and are asked to find a minimum cost Steiner tree. Cl arly changing the value of even one edge or changing the set of terminals can have a significant impact on the optimum solution. However, in reality, often it is very costly (or maybe imposs ible) to have an accurate picture about the values of the parameters or even the constraints of the optimizatio n pr blem at hand at the time of planning. Two of the common approaches studied in the literature to address t his uncertainty about future are referred to as robust optimization and stochastic optimization. Traditional robust optimization, which has been studied in both decision theory [15] and Mathematical Programming [9], deal with the uncertainty in data. In a typical d ta-robust model, we have a finite set of scenarios that can materialize and each scenario contains one possibl e et of data values. The goal is to find a solution that is good with respect to all or most scenarios. One example in t his category is absolute robustness or min-max, where the goal is to find a solution such that the maximum cost o ver all possible scenarios is minimized. Another example is min-max regret, in which the goal is to minimize th maximum regret over all possible scenarios, where the regret of a scenario is defined as the difference bet ween the cost of the solution in that scenario with respect to optimal solution for that scenario. In stochastic optimization, we assume we are provided with s ome probability distribution about the possible scenarios. The goal is then to find a solution that minimizes t h expected cost, over all possible scenarios. This approach is useful in situations that (i) we have a good idea a bout the probability distribution (which may be a strong requirement), and (ii) we have a repeated decision ma king framework. One particular version of stochastic optimization, that has attracted much attention in the last decade, is two-stage (or multi-stage) stochastic optimization, where the solution is built in two stages: in the fir st stage we have to decide to build a partial solution based on the probability distribution of possible scenario s. In stage two, once the actual scenario is revealed we have to complete our partial solution to a feasible solution for the given scenario. There has been considerable amount of research focused on two-stage (or multi-stage) st ochastic version of classical optimization problems such as set-cover, Steiner tree, vertex cover, facility loc ati n, cut problems, and other network design problems [16, 18, 12, 13] and efficient approximation algorithms have be n developed for many of these problem. In some cases, the set of possible scenarios and the corresponding p robabilities are given explicitly [16, 13], and some