Given a nominal combinatorial optimization problem, we consider a robust two-stages variant with cost uncertainty, called DDID. In the first stage, DDID selects a subset of uncertain cost coefficients to be observed, and in the second-stage, DDID selects a solution to the nominal problem, where the remaining cost coefficients are still uncertain. We further assume that the uncertainty set is a polytope defined by upper bounds and a constant number of knapsack constraints. Given a compact linear programming formulation for the nominal problem, we provide a compact mixed-integer linear programming formulation for DDID. The formulation leads to polynomial-time algorithms for DDID when the number of possible observations is polynomially bounded. We extend this formulation to more general nominal problems through column generation and constraint generation algorithms. We illustrate our reformulations and algorithms numerically on the selection problem, the orienteering problem, and the spanning tree problem.